STRING ALGEBRAS IN REPRESENTATION THEORY · over a domestic string algebra, this is also the value of the Krull-Gabriel dimension of the category of nitely presented functors from

STRING ALGEBRAS IN

REPRESENTATION THEORY

A thesis submitted to the University of Manchester

for the degree of Doctor of Philosophy

in the Faculty of Engineering and Physical Sciences

2016

Rosanna Laking

School of Mathematics

Contents

Abstract 6

Declaration 7

Copyright Statement 8

Acknowledgements 9

I The module category 19

1 The functor category and Ziegler spectrum 20

1.1 The functor category . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.1.1 The category of finitely presented functors . . . . . . . . . . . . 22

1.1.2 Extensions of functors along direct limits . . . . . . . . . . . . . 24

1.2 The Ziegler spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.1 Cantor-Bendixson rank . . . . . . . . . . . . . . . . . . . . . . . 28

1.3.2 Krull-Gabriel dimension . . . . . . . . . . . . . . . . . . . . . . 29

1.3.3 m-dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.3.4 Fundamental bijections . . . . . . . . . . . . . . . . . . . . . . . 30

1.3.5 Localisation with respect to a hereditary torsion pair . . . . . . 34

1.3.6 Artin algebras and fp idempotent ideals . . . . . . . . . . . . . 36

1.4 Representation type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.5 Elementary duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2

2 String algebras 43

2.1 Definition of a string algebra . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2 Strings and bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2.1 Finite strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.2.2 Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.2.3 Infinite strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.2.4 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . 47

2.3 Some indecomposable modules . . . . . . . . . . . . . . . . . . . . . . . 48

2.3.1 Finite-dimensional string modules . . . . . . . . . . . . . . . . . 48

2.3.2 Finite-dimensional band modules . . . . . . . . . . . . . . . . . 49

2.3.3 Infinite-dimensional string modules . . . . . . . . . . . . . . . . 50

2.3.4 Infinite-dimensional band modules . . . . . . . . . . . . . . . . . 52

2.4 The module category of a string algebra . . . . . . . . . . . . . . . . . 54

2.4.1 Auslander-Reiten sequences for string algebras . . . . . . . . . . 54

2.4.2 Morphisms between string modules . . . . . . . . . . . . . . . . 58

2.4.3 Morphisms between string and band modules . . . . . . . . . . 59

2.4.4 Morphisms between band modules . . . . . . . . . . . . . . . . 60

2.4.5 Hammock posets and factorisations of graph maps . . . . . . . . 62

2.4.6 Hammock posets and morphisms between pure-injective modules 65

2.5 The Ziegler spectrum of a domestic string algebra . . . . . . . . . . . . 67

2.5.1 The transfinite radical and a lower bound for the Krull-Gabriel

dimension of a string algebra . . . . . . . . . . . . . . . . . . . . 69

3 The Cantor-Bendixson rank and Krull-Gabriel dimension of a do-

mestic string algebra 71

3.1 The Cantor-Bendixson rank of points . . . . . . . . . . . . . . . . . . . 72

3.1.1 The Cantor-Bendixson rank of N-string modules . . . . . . . . . 72

3.1.2 The Cantor-Bendixson rank of Z-string modules . . . . . . . . . 77

3.1.3 The Cantor-Bendixson rank of Prufer modules . . . . . . . . . . 79

3.1.4 The Cantor-Bendixson rank of adic modules . . . . . . . . . . . 81

3.1.5 The Cantor-Bendixson rank of generic modules . . . . . . . . . 84

3.2 The CB rank and KG dimension of a string algebra . . . . . . . . . . . 84

3

II The derived category 86

4 The bounded derived category of a gentle algebra 87

4.1 Homotopy strings and homotopy bands . . . . . . . . . . . . . . . . . . 88

4.1.1 Homotopy strings . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.1.2 Homotopy bands . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.1.3 Resolvable homotopy strings . . . . . . . . . . . . . . . . . . . . 93

4.1.4 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . 94

4.2 Indecomposable complexes . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2.1 String complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2.2 Band complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5 Morphisms between indecomposable objects in the bounded derived

category of a gentle algebra 100

5.1 Morphisms between string and band complexes . . . . . . . . . . . . . 100

5.1.1 The setup and some preliminary lemmas . . . . . . . . . . . . . 101

5.1.2 A basis in Cb,−(A-proj) . . . . . . . . . . . . . . . . . . . . . . . 103

5.1.3 A basis in Kb,−(A-proj) . . . . . . . . . . . . . . . . . . . . . . 107

5.2 Morphisms between r-dimensional band complexes . . . . . . . . . . . 115

5.2.1 A basis in Cb,−(A-proj) . . . . . . . . . . . . . . . . . . . . . . . 116

5.2.2 A basis in Kb,−(A-proj) . . . . . . . . . . . . . . . . . . . . . . 118

5.3 Homotopy category of a derived-discrete algebra . . . . . . . . . . . . . 124

5.3.1 A bound on the dimension of Hom(PV , PW ) . . . . . . . . . . . 127

5.3.2 The category when gldim Λ <∞ . . . . . . . . . . . . . . . . . 130

5.3.3 The category when gldim Λ =∞ . . . . . . . . . . . . . . . . . 134

6 The Ziegler spectrum of a compactly generated triangulated category138

6.1 Compactly generated triangulated categories . . . . . . . . . . . . . . . 138

6.2 The Ziegler spectrum of T . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.3 Homeomorphism between Zg(T) and Zg(Abs-Tc) . . . . . . . . . . . . . 141

6.4 Σ-pure-injective objects . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.5 Localisation with respect to closed subsets . . . . . . . . . . . . . . . . 144

4

7 The Ziegler spectrum of the homotopy category of a derived-discrete

algebra 145

7.1 Krull-Gabriel analysis of Coh(K) . . . . . . . . . . . . . . . . . . . . . 146

7.1.1 Simple functors in Coh(K) . . . . . . . . . . . . . . . . . . . . . 146

7.1.2 Simple functors in Coh(K)/ann(X0) . . . . . . . . . . . . . . . 147

7.1.3 Simple functors in Coh(K)/ann(X1) . . . . . . . . . . . . . . . 150

7.2 The Ziegler spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.2.1 Cantor-Bendixson Rank 0 . . . . . . . . . . . . . . . . . . . . . 151



7.2.4 Classification of indecomposable pure-injective complexes . . . . 152

8 Indecomposable objects in the homotopy category of a derived-discrete

algebra 153

8.1 Indecomposable objects when gldim Λ =∞ . . . . . . . . . . . . . . . . 156

8.2 Indecomposable objects when gldim Λ <∞ . . . . . . . . . . . . . . . . 158

Bibliography 166

Word count 31,015

5

The University of Manchester

Rosanna LakingDoctor of PhilosophyString Algebras in Representation TheoryAugust 4, 2016

The work in this thesis is concerned with three subclasses of the string algebras:domestic string algebras, gentle algebras and derived-discrete algebras (of non-Dynkintype). The various questions we answer are linked by the theme of the Krull-Gabrieldimension of categories of functors.

We calculate the Cantor-Bendixson rank of the Ziegler spectrum of the category ofmodules over a domestic string algebra. Since there is no superdecomposable moduleover a domestic string algebra, this is also the value of the Krull-Gabriel dimensionof the category of finitely presented functors from the category of finitely presentedmodules to the category of abelian groups.

We also give a description of a basis for the spaces of homomorphisms between pairsof indecomposable complexes in the bounded derived category of a gentle algebra. Wethen use this basis to describe the Hom-hammocks involving (possibly infinite) stringobjects in the homotopy category of complexes of projective modules over a derived-discrete algebra.

Using this description, we prove that the Krull-Gabriel dimension of the categoryof coherent functors from a derived-discrete algebra (of non-Dynkin type) is equal to2. Since the Krull-Gabriel dimension is finite, it is equal to the Cantor-Bendixsonrank of the Ziegler spectrum of the homotopy category and we use this to identifythe points of the Ziegler spectrum. In particular, we prove that the indecomposablepure-injective complexes in the homotopy category are exactly the string complexes.

Finally, we prove that every indecomposable complex in the homotopy category ispure-injective, and hence is a string complex.

6

Declaration

The published paper constituting Chapter 5 was previ-

ously submitted by Kristin Krogh Arnesen in support of

an application for a doctoral degree at the Norwegian

University of Science and Technology in 2015. No other

portion of the work referred to in this thesis has been

submitted in support of an application for another de-

gree or qualification of this or any other university or

other institute of learning.

7

Copyright Statement

i. The author of this thesis (including any appendices and/or schedules to this thesis)

owns certain copyright or related rights in it (the “Copyright”) and s/he has given

The University of Manchester certain rights to use such Copyright, including for

administrative purposes.

ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic

copy, may be made only in accordance with the Copyright, Designs and Patents

Act 1988 (as amended) and regulations issued under it or, where appropriate, in

accordance with licensing agreements which the University has from time to time.

This page must form part of any such copies made.

iii. The ownership of certain Copyright, patents, designs, trade marks and other intel-

lectual property (the “Intellectual Property”) and any reproductions of copyright

works in the thesis, for example graphs and tables (“Reproductions”), which may

be described in this thesis, may not be owned by the author and may be owned by

third parties. Such Intellectual Property and Reproductions cannot and must not

be made available for use without the prior written permission of the owner(s) of

the relevant Intellectual Property and/or Reproductions.

iv. Further information on the conditions under which disclosure, publication and com-

mercialisation of this thesis, the Copyright and any Intellectual Property and/or

Reproductions described in it may take place is available in the University IP Policy

(see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any rele-

vant Thesis restriction declarations deposited in the University Library, The Univer-

sity Library’s regulations (see http://www.manchester.ac.uk/library/aboutus/regul-

ations) and in The University’s Policy on Presentation of Theses.

8

Acknowledgements

First and foremost, I would like to thank my supervisor Mike Prest for all of the

interesting and helpful mathematical discussions, as well as for his constant advice

and support. Without the latter I would almost certainly not have completed even

the first year of my PhD.

Understanding mathematics can be challenging, but stimulating conversations with

like-minded mathematicians makes this challenge an extremely enjoyable and satisfy-

ing one. I have been lucky enough to have had many such conversations during my

time in Manchester and I would like to take this opportunity to thank those people

who have made that possible. In particular, I would like to thank all of Mike’s other

PhD students and postdocs (past and present): Amit, Sam, Mike, Isaac, Harry, Lorna

and David. I would also like to thank Inga, Alex, Tahel, Malte and Liz for all the

chats and fun office lunches.

By some happy coincidence, many of the projects I have worked on during my PhD

have been collaborative. Sharing the highs and lows of research with my collaborators

has been a fantastic experience and I would like to thank Gena, Mike, David and

Kristin for working with me.

My final word of thanks belongs to all the friends, family and loved-ones who have

helped me to balance my work with the rest of life!

9

Introduction

A unifying theme in this thesis is the interplay between a topological space known as

the Ziegler spectrum, whose points are particular indecomposable objects in a fixed

category C (e.g. a module category), and a corresponding category of functors. In

particular, we consider the category of finitely presented functors from a ‘generating’

subcategory G of C (e.g. the category of finitely presented modules) to the category of

abelian groups; we will refer to this as the functor category. On the face of it, there is

no reason to suppose that these two mathematical structures are related but, in fact,

it is very often the case that some aspect of the structure of one can be seen in the

structure of the other. The points of the Ziegler spectrum are isomorphism classes

of indecomposable pure-injective objects of C. Originally the topology was given in

terms of model theory, but it was later shown that the each closed set is given by the

intersection of the kernels of a (possibly infinite) set of objects in the functor category.

Moreover, the relative topology on a given closed set corresponds to some localisation

of the functor category and vice versa. Here we study this interplay for categories C

arising in the representation theory of the class of string algebras, as well as that of

two interesting subclasses: the gentle algebras and derived-discrete algebras. In these

cases, the similarities between the structure of the Ziegler spectrum and the structure

of the functor category are even stronger: the isolated points in any closed set are in

one-to-one correspondence with the simple objects in the related localisation of the

functor category. By exploiting this interaction, we are able to calculate the Cantor-

Bendixson rank of the Ziegler spectrum (which is given by iterated removal of isolated

points) both for the module category of a domestic string algebra and for the homotopy

category of complexes of projective modules over a derived-discrete algebra. In these

categories, the Cantor-Bendixson rank coincides with the Krull-Gabriel dimension of

the functor category (which is given by iterated localisation).

10

11

Despite the fact that we do not explicitly refer to any model theoretic notions in

this thesis, it is worth noting that many of the structures arising from the model theory

of modules are implicitly present in the work. We therefore take this opportunity to

give a sketch of how some of these model theoretic structures fit into the picture.

Since Ziegler’s landmark paper [60], the main object of study in the model theory of

modules has been the Ziegler spectrum and it is sometimes said to contain most of

the model theoretic information in the relevant category. For every object M , there

exists a direct sum N of indecomposable pure-injective objects that is elementarily

equivalent to M (i.e. M and N satisfy the same sentences in the language) so, in this

sense, the points of the spectrum relate to the elementary equivalence classes of the

category. Moreover, the closed sets in the topology are given by (possibly infinite)

sets of pairs of positive primitive (pp) formulas. When we consider modules over an

algebra (over an infinite field), it is a consequence of the Baur-Monk theorem [9, 37]

that these correspond exactly to theories in the language. In practice, one may study

the Ziegler spectrum via lattices of pp formulas (any interval in the lattice is defined

by one such pair of pp formulas). Indeed, in [36] we present the work contained in

Chapter 3 in terms of the lattice of pp formulas in one free variable. Since every pair of

pp formulas defines an object in the functor category (and vice versa), we have chosen

to present these arguments in the setting of the functor category; in fact, this was the

original approach of the author. The functor category approach to the model theory

is well-developed and this perspective has been extremely fruitful. We are thus able

to present the work in this thesis using only algebraic notions and choose to do so in

order to avoid unnecessary background requirements.

The functor category, and hence the Ziegler spectrum of a category C, are closely

related to the morphisms in the subcategory G of C. In the first part of the thesis we

take C to be R-Mod for a ring R (or, more generally, a small preadditive category)

and the subcategory G is the category R-mod of finitely presented modules. In the

second part, we take C to be a compactly generated category T and the subcategory

G is the triangulated subcategory Tc of compact objects. In both cases, the objects F

in the functor category are part of an exact sequence of the form

Hom(N,−)→ Hom(M,−)→ F → 0

where M and N are objects in G. By Yoneda’s lemma, the functor F is the cokernel

12

of a natural transformation Hom(f,−) : Hom(N,−)→ Hom(M,−) where f : M → N

is a morphism in G. Thus, studying the functor category and studying the morphisms

in G can be seen as equivalent tasks. Indeed, this is the perspective we take in the

forthcoming chapters. For example, in order to calculate the Krull-Gabriel dimension

in Chapter 3, we make heavy use of the well-understood basis for the vector space

HomA(M,N) where A is a string algebra and M and N are indecomposable finite-

dimensional A-modules. In the second half of the thesis we work in a triangulated

category T where Tc is equivalent to the bounded derived category Db(A-mod) of

a gentle algebra A. We wish to apply similar techniques to this category and so in

Chapter 5 we describe an explicit basis for HomDb(A-mod)(C,D) where C and D are

indecomposable complexes. Again, we then use the close relationship between the

functor category, the Ziegler spectrum and morphisms in Tc to describe the corre-

sponding Ziegler spectrum, Cantor-Bendixson rank and Krull-Gabriel dimension for a

special family of gentle algebras known as the derived-discrete algebras.

In general, the functor category is difficult to describe explicitly and so the idea

of the Krull-Gabriel dimension is that it enables us to compare functor categories; in

some sense the Krull-Gabriel dimension measures the complexity of the structure of

the functor category. The idea of the Cantor-Bendixson rank of a topological space is

similar; a rank is assigned to each point and this is a measurement of ‘how isolated’

this point is. In the examples we consider, the Krull-Gabriel dimension of the functor

category coincides with the Cantor-Bendixson rank of the Ziegler spectrum and so if we

can calculate one then we may calculate the other. We see examples of this idea going

in both directions. In Chapter 5, we combine the recent classification of the points of

the Ziegler spectrum of the module category of a domestic string algebra [48] with the

description of the morphisms in the category to calculate the Cantor-Bendixson rank.

This then allows us to deduce the value of the Krull-Gabriel dimension. Conversely,

the structure of the morphisms in the bounded derived category of a derived-discrete

algebra are well-understood (see, for example, [15]) and so we use this in Chapter 7 to

calculate the Krull-Gabriel dimension. We can then use this to identify the points of

the Ziegler spectrum and to calculate the Cantor-Bendixson rank.

The string algebras are an important class of tame algebras. Their representation

theory is highly combinatorial in nature, making it possible to answer many questions

13

for string algebras that are much more difficult to answer for general tame algebras. As

such, the class plays the role of a ‘test class’ for conjectures and, moreover, they enable

one to develop ones intuition for tame algebras in general via direct computation of

examples. The indecomposable finite-dimensional modules have been classified (see

[49, 23, 20, 16, 58]) and can be described by ‘strings’ and ‘bands’. Furthermore, Ringel

extends the definitions of string and band modules to define certain indecomposable

infinite-dimensional pure-injective modules [50] and, in the same paper, he conjectures

that, if the string algebra has domestic representation type, then the modules he lists

form a complete list of the points of the Ziegler spectrum. In a recent paper [48],

Prest and Puninski confirm this conjecture. This is the starting point for the work

contained in the first part of this thesis.

By definition, a string algebra is the path algebra of a quiver with relations that

satisfies certain conditions designed to restrict the structure of the indecomposable

projective modules. This definition is a slight specialisation of the definition of a special

biserial algebra, for which the restrictions on the quiver mean that all indecomposable

projective modules are biserial. In general, the combinatorics for string algebras is less

complicated than for special biserial algebras, and so we consider this smaller class for

the sake of the clarity of our arguments.

By adding further restrictions on the quiver one obtains the definition of a gentle

algebra. Gentle algebras were introduced in the context of determining the derived

equivalence classes of path algebras of quivers of type A and A [3, 4]. The derived

category of a gentle algebra is similar in nature to the module category of a string

algebra; the indecomposable complexes in the bounded derived category are described

by ‘homotopy strings’ and ‘homotopy bands’ [10]. As such, the gentle algebras form

an effective test class for the derived category in the same way that string algebras do

for the module category. For this reason, as well as to provide a stepping stone to the

results contained in Chapters 7 and 8, we develop the general theory of the derived

category of a gentle algebra by giving a combinatorial description of the morphisms in

the category; this can be found in Chapter 5.

Gentle algebras also arise in the classification of the derived equivalence classes

of the so-called derived-discrete algebras [13]. The definition of a discrete derived

category was given by Vossieck in [57]; the idea is that such a derived category is

14

somewhere between the derived category of a hereditary algebra of finite representation

type and the derived category of a hereditary algebra of infinite representation type.

It turns out that the representatives of the derived equivalence classes are of a specific

form (see Section 5.3) and, moreover, they are gentle. In Chapter 7 we describe the

Ziegler spectrum of these algebras. The case where the algebra is derived-discrete

and self-injective with infinite global dimension has been investigated by Zhe Han

in his PhD thesis [59]. He is able to show that every indecomposable object in the

homotopy category of such an algebra is pure-injective and can be described by an

infinite homotopy string. In Chapter 8 we are able to extend this result to arbitrary

derived-discrete algebras.

Summary of content

The content of this thesis is divided into two parts; the work in the first part takes place

in an abelian setting (specifically module categories, and more generally, categories

of functors) and the second part includes triangulated categories (derived categories

and homotopy categories), though the abelian setting continues to play a role. Both

sections include a significant amount of background material. We have chosen to

include this background explicitly in an attempt to produce a self-contained document,

as well as to fix the notation used in the subsequent proofs. Still, despite the extent

of the background given, we have assumed that the reader has some knowledge of the

representation theory of finite-dimensional algebras and basic category theory. In this

section we will outline the contents of the thesis and we will also give some indication

of the background we have assumed for each section.

Chapter 1 is an overview of the techniques we will use in later chapters; that is, the

functor category and the Ziegler spectrum. In this chapter we have assumed that the

reader has knowledge of basic category theory and, in particular, of abelian categories.

We begin by introducing the functor category and we give a detailed account of how

the structure of the lattice of subfunctors of a finitely presented functor relates to a

lattice of (pointed) morphisms. The calculation of the Krull-Gabriel dimension and

Cantor-Bendixson rank given in Chapter 3 will rely heavily on this description of

the finitely presented functors. We then go on to define the Ziegler spectrum and

15

describe an important example: the Ziegler spectrum of k[T, T−1]-Mod. Section 1.3

is a summary of the strong connections between the Ziegler spectrum and the functor

category. The end of Section 1.3 and Section 1.4 contains some open questions and

conjectures that we will later confirm for string algebras. Finally, in Section 1.5 we

define elementary duality; we will make use of this definition in Section 3.1.4 in which

we describe the elementary duals of all points of the Ziegler spectrum of a domestic

string algebra.

In Chapter 2 we give a summary of the relevant background on string algebras. In

this chapter we assume that the reader is familiar with the basics of representation

theory of finite-dimensional algebras, in particular path algebras of quivers with rela-

tions, as well as some Auslander-Reiten theory. We define all the points of the Ziegler

spectrum of a domestic string algebra (the finite-dimensional points are defined in

Sections 2.3.1 and 2.3.2 and the infinite-dimensional points are defined in Sections

2.3.3 and 2.3.4). In Section 2.4 we describe the Auslander-Reiten structure and the

morphisms between indecomposable modules. Finally, in Section 2.5, we summarise

what is known about the Ziegler spectrum of a domestic string algebra.

In Chapter 3 we calculate the Cantor-Bendixson rank of the Ziegler spectrum of a

domestic string algebra as well as the Krull-Gabriel dimension of the functor category.

The idea of the proof is that we combine the description of the morphisms in the

module category (see Section 2.4.5) with the description of subfunctors of finitely

presented functors (see Section 1.1.1) in order to calculate the Cantor-Bendixson rank

of each point of the Ziegler spectrum. Since the points are completely classified (see

Section 2.5) we may deduce the Cantor-Bendixson rank of the entire space, and we do

so in Section 3.2. Moreover, in light of the connections between the Ziegler spectrum

and the functor category (see Section 1.3.4), this is also the value of the Krull-Gabriel

dimension of the functor category.

In Chapter 4 we give the relevant background on the structure of the bounded

derived category of a gentle algebra. We assume the reader is familiar with the def-

initions of derived categories, homotopy categories and also with the basic theory of

triangulated categories. We also introduce some new notation for homotopy bands

and strings (we refer to this as the ‘unfolded diagram’ of a homotopy string or band);

this notation is extremely important in understanding our approach to the proofs in

16

Chapter 5.

In Chapter 5 we describe a basis for the morphisms between indecomposable ob-

jects in the bounded derived category of a gentle algebra. Section 5.1 deals with the

morphisms between string complexes and the band complexes that sit at the base of

homogeneous tubes in the Auslander-Reiten quiver. Section 5.2 then deals with the

remaining band modules. The approach in these two sections is the same: we first

establish a basis in the category of complexes of finite-dimensional projective modules

and then consider homotopy equivalences between basis elements. The final section

in Chapter 5 then applies our results to the homotopy category of a derived-discrete

algebra. This section also contains various results that are already known on structure

of the category; this structure is central to the arguments in Chapters 7 and 8.

The definitions and results analogous to those in Chapter 1 can be given for com-

pactly generated triangulated categories and in Chapter 6 we describe this setting.

In fact, in Section 6.3 we show that the definitions of purity in such a triangulated

category correspond to purity in a localisation of a functor category.

In Chapter 7 we consider the Ziegler spectrum and functor category associated

to the homotopy category of projective modules over a derived-discrete algebra. The

morphisms in the homotopy category are very well understood and the dimension of

the space of homomorphisms between two indecomposable compact objects is at most

two (see Section 5.3). Given the description of lattices of subfunctors (see Section

1.1.1) we use this to explicitly calculate the Krull-Gabriel dimension of the functor

category. Moreover, since there is a bijection between simple objects and points of the

Ziegler spectrum (see Section 1.3.4), we use this to describe the Ziegler spectrum of the

homotopy category and also to calculate the Cantor-Bendixson rank. This approach

is very similar to that of Chapter 3.

In the final chapter we extend a result from [59]; that is, we prove that every inde-

composable complex in the homotopy category of projective modules over a derived-

discrete algebra is pure-injective. We deal with the infinite and finite global dimension

cases separately. In Section 8.1 we consider the case where the algebra has infinite

global dimension and we prove that every indecomposable object is Σ-pure-injective;

we do this by considering the definable subcategory generated by an arbitrary in-

decomposable object and showing that it is contained in the definable subcategory

17

generated by a Σ-pure-injective object. In Section 8.2 we consider the case where the

algebra has finite global dimension. In this case we explicitly use the structure of cer-

tain representable functors to show that an arbitrary indecomposable complex in the

homotopy category must in fact be isomorphic to an indecomposable pure-injective

object contained in the definable subcategory it generates.

Summary of my contribution to joint work

The work contained in this thesis falls into three distinct collaborative projects, each

of which has been written up into one of the following papers: [36], [1], [2]. This work

was carried out via various styles of collaboration and this has influenced how the work

from each paper is presented. In this section I will outline my contribution to each

project, as well as how far its presentation here deviates from that of the corresponding

paper.

The paper [36] is joint with Mike Prest and Gena Puninski and the work contained

in this paper can be found in Chapter 3. With the exception of the final details, I

produced these results independently of my coauthors. On the one hand, Prest and

Puninski described neighbourhood bases for points of the Ziegler spectrum in terms

of pp formulas and used them to calculate the Cantor-Bendixson rank. On the other

hand, I used arguments based on the structure of the morphisms in the category to

identify simple functors and isolated points and thus calculate the Cantor-Bendixson

rank. During Puninski’s visit to Manchester in 2014, it came to light that we had

the same results (and almost the same arguments, but different lexicons). Following

several discussions, during which we fine-tuned our arguments, we decided to publish

the results together. For the paper we decided to present the results in terms of pp

formulas; I have rewritten them for this thesis in the way I originally proved them,

that is, in terms of morphism between finite-dimensional modules.

The work in [1] is joint with Kristin Arnesen and David Pauksztello and can be

found in Chapter 5. This work was carried out in close collaboration during Arnesen’s

visit to Manchester in April 2014, as well as, a reciprocal visit to Trondheim in June

2014. All results contained in this chapter were worked out with equal contributions

from each of us. I have reworded many of the arguments for this thesis but the content

18

is the same; I hope to have made some of the exposition clearer in this reworking. In

particular, the presentation of the work in Section 5.2 is substantially different from the

paper and I have used the term ‘maximal graph homotopy’ rather than ‘quasi-graph

map’.

Finally, the paper [2] is joint with Kristin Arnesen, David Pauksztello and Mike

Prest. This work is divided between Chapters 7 and 8. The project began with detailed

discussions between all four authors during Arnesen’s second visit to Manchester in

October 2014 where the main ideas were established. We then continued to work on

the details of the proofs both independently and in various subsets of the authors.

The work contained in Chapter 7 was discussed during Arnesen’s visit and, following

this, I worked out the details and wrote them up for the paper. For this reason, the

exposition of this section is almost identical to what is contained in the paper. The

main ideas for the proofs in Chapter 8 were due to Prest but the details required a lot

of work and this was done, in close collaboration, by myself, Pauksztello and Prest.

Moreover, the write up was carried out jointly by myself and Pauksztello and so the

content of Chapter 8 does not deviate far from the final section of [2].

Part I

The module category

19

Chapter 1

The functor category and Ziegler

spectrum

In this first chapter, we will introduce a setting that has its origins in the model

theoretic approach to representation theory. As the theory has developed, it has

became apparent that almost all of the definitions and results can be stated without

reference to the usual model theoretic notions, such as formulas and theories, since

these structures can be understood algebraically by viewing them as structures within

certain categories of functors. The results in this thesis are presented entirely in terms

of functor categories and localisation but could have just as easily been written entirely

in terms of lattices of pp formulas. This choice was made arbitrarily and is not intended

to reflect any preference for one approach or the other; greater insight can be achieved

by understanding both viewpoints, however sticking to one setting rather than two

should make the exposition clearer.

We will require a more general notion of module than is usually given; that is, we

will consider “modules over rings with many objects”. By this we mean a covariant

functor M : R → Ab where R is a skeletally small preadditive category and Ab is

the category of abelian groups. Such a functor is a right R-module. Similarly, a

contravariant functorN : R → Ab (or, equivalently, a covariant functorN : Rop → Ab)

is a right R-module.

Throughout this chapter the notation R will refer to an arbitrary skeletally small

additive category. Let R-Mod denote the category of left R-modules and let R-mod

denote the full subcategory of finitely presented modules. Let Mod-R and mod-R

20

21

denote the analogous categories of right R-modules.

Remark 1.0.1. We will make statements for the category of left R-modules. Since

Mod-R ' Rop-Mod, all statements about categories of left R-modules will apply

equally well to categories of right R-modules and vice versa.

Example 1.0.2. When R has a single object ∗ the endomorphisms R = End(∗) form

a ring with addition coming from the preadditive structure and multiplication coming

from composition of morphisms. In this case, right and left R-modules are R-modules

in the usual sense.

We begin by noting some properties of the category R-Mod that will be important

in later proofs. For each module X in R-Mod, consider the associated (covariant)

representable functor

Hom(X,−) : R-mod→ Ab.

We will denote this functor by (X,−) and, for each f : X → Y in R-Mod, we denote

the natural transformation Hom(f,−) : (Y,−)→ (X,−) by (f,−).

Similarly, the (contravariant) representable functor Hom(−, X) will be de-

noted (−, X) and the natural transformation Hom(−, f) : (−, X) → (−, Y ) will be

denoted (−, f).

We do not give the definitions for finitely presented and finitely generated for a

general category here (see, for example, [44, App. E]). The definition for a functor

category is given below. If R = R is a ring (see Example 1.0.2 above), then a module

M in R-Mod is finitely generated if there is an exact sequence Rn → M → 0 for

some n ∈ N and finitely presented if there is an exact sequence Rm → Rn →M → 0

for some n,m ∈ N.

A category C is locally finitely presented if the full subcategory Cfp of finitely

presented objects is skeletally small, C is complete (i.e. has products and kernels) and

every object in C is a direct limit of a directed system of objects from Cfp.

Theorem 1.0.3 (see, for example, [40, Thm. 3.4.2] or [44, Ex. E.1.20]). For any small

preadditive category R, the category R-Mod is abelian and locally finitely presented.

It is therefore also Grothendieck.

22 CHAPTER 1. THE FUNCTOR CATEGORY AND ZIEGLER SPECTRUM

An object M in a category C is coherent if every finitely generated subobject is

finitely presented. Then C is locally coherent if the full subcategory Ccoh of coherent

objects is skeletally small and every object in C is a direct limit of a directed system

of objects from Ccoh. The local coherence of R-Mod depends on R.

Proposition 1.0.4 ([51, Prop. 2.2]). Let C be a locally finitely presented abelian cat-

egory. Then C is locally coherent if and only if Cfp is an abelian subcategory of C. In

particular, if R = R is a ring, then R-Mod is locally coherent if and only if R is a

coherent ring.

1.1 The functor category

In this section we will consider both the category (R-mod,Ab) of functors fromR-mod

to Ab and its full subcategory (R-mod,Ab)fp

of finitely presented functors. We will

focus of the structure of (R-mod,Ab)fp

and, in particular, the relationship between the

subfunctors of finitely presented functors and the morphisms inR-mod. Finally we will

define extensions of the objects in (R-mod,Ab)fp

to functors on R-Mod; these functors

are central in the definition of the topology on the Ziegler spectrum, a topological space

that will be introduced in the next section.

Since R-mod is a small preadditive category, we may treat (R-mod,Ab) as a mod-

ule category. For example, we may apply Theorem 1.0.3 to obtain that (R-mod,Ab)

is locally finitely presented and abelian. Moreover, every finitely presented object is

coherent (see [44, Cor. 10.2.3]) and so (R-mod,Ab) is also locally coherent. It follows

from Proposition 1.0.4 that (R-mod,Ab)fp

is abelian.

1.1.1 The category of finitely presented functors

It is well-known that the representable functors are exactly the finitely generated pro-

jective objects in (R-mod,Ab). So every finitely presented functor F has a presentation

by representable functors, and this will provide us with an explicit understanding of

the structure of the lattice of finitely presented subfunctors of F and also a direct link

with the morphisms in R-mod.

A functor F in (R-mod,Ab) is finitely presented if there exists an exact sequence

1.1. THE FUNCTOR CATEGORY 23

of functors

(Y,−)τ→ (X,−)→ F → 0

where X, Y are objects in R-mod. By Yoneda’s lemma τ = (f,−) for some f : X → Y

in R-mod. Let im(f,−) denote the image of (f,−) in (X,−). That is, for all modules

M in R-mod, we have im(f,M) = {hf | h : Y →M}. Thus F ∼= (X,−)/ im(f,−).

Let X be a module in R-mod. If f : X → M and g : X → N are morphisms in

R-mod, then let f ≥ g if there is some h : M → N such that g = hf .

Xf //

g��

M

h~~N

Then f ∼ g if and only if f ≥ g and g ≥ f .

Remark 1.1.1. The set of equivalence classes LX of morphisms starting at X is a

modular lattice with f ∨ g =[fg

]: X → Y ⊕ Z and f ∧ g is the pushout of f and g.

The proof of this can be found in [44, Lem. 3.1.2].

If F is an object (R-mod,Ab)fp

, then let L(F ) denote its (modular) lattice of

subobjects in (R-mod,Ab)fp

.

Lemma 1.1.2 ([44, Cor. 3.1.5]). Let X be a module in R-mod. Then

LX ∼= L((X,−))

where the morphism of lattices is given by f 7→ im(f,−).

For each f : X → Y , let Lf denote the interval in LX between im(f,−) and

im(1X ,−) = (X,−)

Corollary 1.1.3. Let F = (X,−)/ im(f,−) be a functor in (R-mod,Ab)fp

. Then

Lf ∼= L(F )

and the morphism is given by g 7→ im(g,−)/ im(f,−).

We will use the following easy lemmas in several places in later chapters; similar

results can be found in [47].


Lemma 1.1.4. Let F = (X,−)/ im(f,−) be a functor in (R-mod,Ab)fp

and suppose

G is a non-zero finitely presented subfunctor of F , then G = im([ gf

],−)/ im(f,−) for

some g ≥ f in LX .

Proof. By Corollary 1.1.3, G = im(g,−)/ im(f,−) for some g ≥ f . It suffices to

show that g ∼[ gf

]. This is clear since f = hg for some h so g = [ 1 0 ]

[ gf

]and[ g

f

]= [ 1

h ] g.

Lemma 1.1.5. If g =n∑i=1

gi with f ≥ g1 in LX , then

im([ gf

],−) = im

([n∑i=2

gi

f

],−).

Proof. By Lemma 1.1.2, it suffices to show that[ gf

]∼[

n∑i=2

gi

f

]. Since g1 = hf for

some h, we have [ 1 h0 1 ]

[n∑i=2

gi

f

]=[ gf

]and [ 1 −h

0 1 ][ gf

]=

[n∑i=2

gi

f

].

Lemma 1.1.6. Suppose g =

[ g1...gn

]: M →

n⊕i=1

Ni and g1 ≥ gi for all 1 ≤ i ≤ n. Then

im(g,−) = im(g1,−).

Proof. By Lemma 1.1.2, it suffices to show that g ∼ g1. This is clear since we have

[ 1 0 ··· 0 ] g = g1 so g ≥ g1. For the other inequality, note that, since g1 ≥ gi, there

exists some fi such that gi = fig1 for each 1 ≤ i ≤ n. Then

[f1...fn

]g1 = g and so g1 ≥ g

as required.

1.1.2 Extensions of functors along direct limits

For every functor F in (R-mod,Ab) there is a unique extension of F to a functor from

R-Mod to Ab which commutes with direct limits. If M is a module in R-Mod, then,

as R-Mod is locally finitely presented, M = lim−→Mi for some directed system (Mi)i∈I

of finitely presented modules. We then define F (M) := lim−→F (Mi). If F is finitely

presented, then the extension of F to R-Mod will be referred to as a pp functor.

In order to avoid cumbersome notation, we will often write F both for a functor in

(R-mod,Ab) and its extension to R-Mod.

Such functors can be characterised by the properties in the following proposition.

Proposition 1.1.7. Let R be a small preadditive category and let F : R-Mod→ Ab.

Then the following conditions are equivalent:

1.2. THE ZIEGLER SPECTRUM 25

1. F is a pp functor.

2. F commutes with direct limits and products.

3. There exist M and N in R-mod such that

(M,−)→ (N,−)→ F → 0

is exact when evaluated at any module in R-Mod.

Proof. The fact that 1 and 3 are equivalent follows directly from the definitions, since

taking limits is right exact and (M,−) commutes with direct limits. The equivalence

between 1 and 2 is [19, Sec. 2.1].

1.2 The Ziegler spectrum

Next we introduce a topological space known as the Ziegler spectrum of R-Mod.

The space was originally introduced by M. Ziegler in his landmark paper [60] and the

definitions of purity as well as the topology were given in terms of pairs of pp formulas.

As we have already mentioned, subsequent work relating these definitions to objects

in the functor category allow us to define the Ziegler spectrum as it is presented here.

A monomorphism f : M → N in R-Mod is pure if the natural transformation

−⊗R f : −⊗RM → −⊗R N

is a monomorphism in (mod-R,Ab). A module L is pure-injective if it is injective

over all pure monomorphisms. That is, for every pure monomorphism f : M → N

and every morphism g : M → L, there exists a morphism h : N → L such that the

following diagram commutes.

Mf //

g��

N

h~~L

Equivalently, L is pure-injective if and only if − ⊗R L is an injective object in

(mod-R,Ab).

The Ziegler spectrum Zg(R-Mod) (which we will write as RZg when R = R is a

ring as in Example 1.0.2) is a topological space with points given by the isomorphism


classes of indecomposable pure-injective modules in R-Mod. Sets of the form

(F ) := {M ∈ Zg(R-Mod) | F (M) 6= 0}

where F is a pp functor form a basis of (compact) open sets.

A definable subcategory of R-Mod is a full subcategory with objects given by

a set of the form

{M | Fi(M) = 0 for all i ∈ Φ}

where {Fi | i ∈ Φ} is a family of pp functors. For a set of objects X, consider the

functors FX = {F | F (M) = 0 for all M ∈ X}. Then the definable subcategory

generated by X is the full subcategory with objects

{N | F (N) = 0 for all F ∈ FX}

and is denoted 〈X〉. This is the smallest definable subcategory (with respect to in-

clusion) containing X. We use the notation 〈M〉 when X = {M} for some module

M .

The closed subsets of Zg(R-Mod) are exactly those of the form D ∩ Zg(R-Mod)

for a definable subcategory D. For each module M , we will refer to the closed subset

Supp(M) := 〈M〉 ∩ Zg(R-Mod) as the support of M .

Example 1.2.1 ([22, Sec. 5]). Let k be an algebraically closed field. In this example

we will describe the points of the Ziegler spectrum of k[T, T−1]-Mod. The modules

are representations of the quiver • Tyy

where the k-linear map is an automorphism.

Since k is algebraically closed, the finite-dimensional indecomposable representations

are of the form Mn,k = (kn, Jn,k) where

Jn,k =

k 1 0 . . . 0 0

0 k 1 . . . 0 0...

...... · · · ...

...

0 0 0 . . . k 1

0 0 0 . . . 0 k

is the n× n Jordan block associated to k ∈ k∗ and n ∈ N.

For each k ∈ k∗ there is a homogeneous tube in the Auslander-Reiten quiver of

k[T, T−1]-mod of the following form.

1.3. DIMENSIONS 27

M1,k

ι1 ,,M2,k

−ι2 ,,

π1ll M3,k

ι3 **

−π2ll · · ·

π3ll

where ιn : Mk,n →Mk,n+1 and πn : Mk,n+1 →Mk,n are given by the following matrices.

ιn =

0 0 0 . . . 0 0

1 0 0 . . . 0 0

0 1 0 . . . 0 0...

...... · · · ...

...

0 0 0 . . . 0 1

πn =

1 0 0 . . . 0 0 0

0 1 0 . . . 0 0 0...

...... · · · ...

......

0 0 0 . . . 0 1 0

Then the direct limit Mk,∞ = lim−→Mk,n of the {ιn}n∈N is known as the Prufer mod-

ule associated to k and is indecomposable and pure-injective. Similarly the inverse

limit Mk,−∞ = lim←−Mk,n of the {πn}n∈N is known as the adic module associated to

k and it is also indecomposable and pure-injective.

The modules

{Mk,i | i ∈ N ∪ {∞,−∞}, k ∈ k∗}

together with the generic module G = k(T ) comprise a complete list of the points of

Zg(k[T, T−1]-Mod).

1.3 Dimensions

In this section we introduce three notions of dimension: the Cantor-Bendixson rank

of a topological space, the Krull-Gabriel dimension of an abelian category and the

m-dimension of a lattice. The definition of each dimension takes a particular form:

a “collapsing” process is carried out iteratively on a mathematical structure and the

dimension measures when (or if) this process produces something trivial.

If we take the topological space to be Zg(R-Mod), the abelian category to be

(R-mod,Ab)fp

, and the lattice to be a lattice of finitely-presented subfunctors, then

the dimensions are very closely related (and in certain cases will coincide). We discuss

this relationship in Section 1.3.4 and in Section 1.3.5 we discuss a further connection

with localisations of (R-mod,Ab).


Finally, in Section 1.3.6, we look at how the above relates to powers of certain

ideals in R-mod when R = R is an Artin algebra. In particular, we will consider

the Jacobson radical radR of R-mod. There is another notion of dimension here: the

iterative process is taking transfinite powers of the ideal radR and the process stops

when radαR = radα+1R for some α. The connection with the other dimensions is not so

clear in this case.

1.3.1 Cantor-Bendixson rank

Let T be a compact topological space. Define the first Cantor-Bendixson derivative

T ′ to be the closed subset of T consisting of the non-isolated points. The set T ′ is

a topological space with the subspace topology and may have new isolated points.

Define T (0) := T , T (α+1) := (T (α))′ for any ordinal α and T (λ) :=⋂β<λ T

(β) when λ is

a limit ordinal.

If a point p is isolated in T (α) for some α, then we say p has Cantor-Bendixson

rank α. In this case we write CB(p) = α. If p is not isolated in T (α) for any ordinal

α, then we say that the Cantor-Bendixson rank of p is undefined and this is denoted

CB(p) =∞.

Let T (∞) =⋂α T

(α). If T (∞) = ∅, then the Cantor-Bendixson rank of T is the

least ordinal β such that T (β+1) = ∅. This is denoted CB(T ) = β. If T (∞) 6= ∅, then we

say that the Cantor-Bendixson rank of T is undefined. This is denoted CB(T ) =∞.

Remark 1.3.1. 1. Since T is a set, we must have T (∞) = T (α) for some ordinal α.

2. If T (λ) = ∅ for some limit ordinal, then T (β) = ∅ for some β < λ. This follows

because T = (T (λ))c =⋃β<λ(T

(β))c and so, since T is compact, there is some

finite set β1, · · · , βn < λ such that T =n⋃i=1

(T (βi))c. So the intersection⋂ni=1 T

(βi)

is empty, that is T (βj) = ∅ where βi ≤ βj for all 1 ≤ i ≤ n.

3. If CB(T ) = β for some ordinal β, then the isolated points of T are dense in T .

Otherwise there is some open set ∅ 6= U ⊆ T containing no isolated points and

U ⊆ T (∞), a contradiction.

1.3. DIMENSIONS 29

1.3.2 Krull-Gabriel dimension

Let A be an abelian category. A Serre subcategory S of A is a full subcategory

such that for all exact sequences

0→ F → G→ H → 0

in A, we have that F,H ∈ S if and only if G ∈ S. We can define the (Serre)

localisation A/S of A by S to be the category with the same objects as A and for

F,G we have

HomA/S(F,G) := lim−→ HomA(F ′, G/G′)

where the direct limit is over all subobjects F ′ of F such that F/F ′ ∈ S and G′ of G

such that G′ ∈ S.

Given an abelian category A, let A0 denote the Serre subcategory generated by the

simple objects inA (that is, consisting of all finite length objects inA). If F ∈ A0, then

we say that F has Krull-Gabriel dimension 0. Then let Aα+1 be the Serre subcategory

generated by objects of A that become finite length or zero in A/Aα for any ordinal

α and Aλ :=⋃β<λAβ when λ is a limit ordinal. For an ordinal α, let qα : A → A/Aα

be the canonical localisation functor.

We say that F ∈ A has Krull-Gabriel dimension α + 1 if F ∈ Aα+1\Aα. This

is denoted KG(F ) = α if such an ordinal exists, otherwise we say the Krull-Gabriel

dimension of F is undefined and this is denoted KG(F ) =∞.

If Aα = A for some ordinal α, then we say that the Krull-Gabriel dimension

of A is the least such α. This is denoted KG(A) = α. If no such α exists, then we say

that the Krull-Gabriel dimension of A is undefined and we denote this KG(A) =∞.

1.3.3 m-dimension

Suppose L is a modular lattice with bottom and top elements. Then, for x, y ∈ L,

let x ∼ y if and only if there are only finite length chains between x ∧ y and x ∨ y.

This defines a congruence relation on L and so the equivalence classes L/∼ have a

modular lattice structure. Let L0 := L, for any ordinal α let Lα+1 = Lα/∼ and let

Lλ = lim−→ α<λLα for any limit ordinal λ. We define the m-dimension of L to be

the least ordinal α such that Lα+1 = 0 if such an ordinal exists. This is denoted


mdim(L) = α. If no such α exists then we say that the m-dimension of L is undefined

and we write mdim(L) =∞.

Remark 1.3.2. If λ is a limit ordinal, then Lλ = 0 if and only if Lα = 0 for some

successor ordinal α < λ. To see this let πα : L → Lα be the canonical surjection of

lattices. Then if Iα = {x ∈ L | πα(x) = 0} for each ordinal α, then Iλ =⋃α<λ Iα.

Since Lα = 0 if and only if 1 ∈ Iα, the assertion follows.

The following proposition linking m-dimension and Krull-Gabriel dimension follows

directly from the definitions (or see, for example, [32, Lem. 1.1]). If F is an object in

an abelian category A, then let LA(F ) denote its (modular) lattice of subobjects in

A.

Proposition 1.3.3. Let A be an abelian catgory.

1. For all F ∈ A and every ordinal α, we have LA/Aα(qα(F )) ∼= LA(F )α.

2. For all non-zero F ∈ A, the m-dimension of LA(F ) is α+ 1 if and only if qα(F )

is non-zero and has finite length in A/Aα (the m-dimension will not be a limit

ordinal, see Remark 1.3.2). In particular, we have KG(F ) = mdim(LA(F )).

We will also need the following characterisation of when Krull-Gabriel dimension

is defined.

Proposition 1.3.4. The Krull-Gabriel dimension of A is defined if and only if for

each F ∈ A no subset of LA(F ) forms a densely ordered chain.

1.3.4 Fundamental bijections

Next we outline some correspondences that exist between subcategories of R-Mod

and subcategories of (R-mod,Ab)fp

. The close relationship between particular sub-

categories in these structures have led to some interesting questions, many of which

remain open and have given rise to some conjectures which we will state at the end of

the section and later in Section 1.4.

Let X be a set of objects or a full subcategory in R-Mod. Then we define the

annihilator of X to be the set

ann(X) := {F ∈ (R-mod,Ab)fp | F (M) = 0 for all M ∈ X}.

1.3. DIMENSIONS 31

Let S be a set of objects or a full subcategory in (R-mod,Ab)fp

. Then we define the

annihilator of S to be the set

ann(S) := {M ∈ R-Mod | F (M) = 0 for all F ∈ S}.

In both definitions F is identified with its unique extension to a pp functor.

The following bijections represent the deep connections between the functor cate-

gories, the module category and the Ziegler spectrum. The bijection between the sets

given in 1 and 2 is by definition; between the sets given in 1 and 3 is essentially [31,

Prop. 4.5].

Proposition 1.3.5. For a small preadditive category R there are bijections between

the following.

1. The definable subcategories of R-Mod.

2. The closed subsets of Zg(R-Mod).

3. The Serre subcategories of (R-mod,Ab)fp

.

The mutually inverse bijections are given as follows.

{The definable subcategories of R-Mod} ↔ {The closed subsets of Zg(R-Mod)}

X 7→ X ∩ Zg(R-Mod) 〈X〉 ←[ X.

{The definable subcategories of R-Mod} ↔ {The Serre subcategories of (R-mod,Ab)fp}

X 7→ ann(X ) ann(S)←[ S.

{The closed subsets of Zg(R-Mod} ↔ {The Serre subcategories of (R-mod,Ab)fp}

X 7→ ann(X) ann(S) ∩ Zg(R-Mod)←[ S.

Consider the Krull-Gabriel analysis described in Section 1.3.2 when A is taken to

be the functor category (R-mod,Ab)fp

. By Proposition 1.3.5, the Serre subcategory

Aα, for each ordinal α, corresponds to a definable subcategory Xα+1 := ann(Aα) and

a closed subset Xα+1 := Xα+1 ∩ Zg(R-Mod).


Remark 1.3.6. In this remark we will give a full explanation of the fact that

Zg(R-Mod)(α+1) ⊆ Xα+1 for all successor ordinals α in the hope that it will eluci-

date the connections between the Ziegler spectrum and the functor category. We

will shorten the notation Zg(R-Mod) to Zg for the duration of the remark and let

A = (R-mod,Ab)fp

. We refer the reader to Sections 1.3.1 and 1.3.2 for the remaining

notation.

We argue inductively. For the base case, let α = 0. The Serre subcategory A0

of A consists of the finite length functors. Let M ∈ (Xα+1)c (where c refers to the

set theoretic complement of Xα+1). By definition, there must be some F ∈ A0 such

that F (M) 6= 0. Since F is finite length it has a finite length composition series and

at least one of the simple composition factors must be non-zero on M . Thus we may

assume that F is simple. But then M is isolated by F in Zg (see, for example, [44,

Cor. 5.3.3]), so M ∈ (Zg(1))c as claimed.

Next fix a successor ordinal α and suppose that for all successor ordinals β < α

we have that (Xβ+1)c ⊆ (Zg(β+1))c. Let M ∈ (Xα+1)c. Then there is a functor F ∈ A

such that F is finite length or zero in A/Aα−1. If F is zero in A/Aα−1, then there is

some successor β < α such that F ∈ Aβ. By our induction hypothesis, we have that

M ∈ (Xβ+1)c ⊆ (Zg(β+1))c ⊆ (Zg(α+1))c

as required. If F is finite length in A/Aα−1, then, as before, we may assume that F

is simple in A/Aα−1 and so M is isolated in Xα (again, by [44, Cor. 5.3.3]). By the

induction hypothesis, (Xα)c ⊆ (Zg(α))c and so either M is not in Zg(α) or M is isolated

in Zg(α). In either case, we may conclude that M ∈ (Zg(α+1))c as required.

It follows easily from this that there is a connection between the Krull-Gabriel

dimension of (R-mod,Ab)fp

and the Cantor-Bendixson rank of Zg(R-Mod). This

connection is first established in [60].

Proposition 1.3.7. Let R be a skeletally small preadditive category. If the Krull-

Gabriel dimension of A := (R-mod,Ab)fp

is defined, then the Cantor-Bendixson rank

of Zg(R-Mod) is defined.

Proof. For all successor ordinals α, we have Zg(R-Mod)(α+1) ⊆ ann(Aα)∩Zg(R-Mod).

If the Krull-Gabriel dimension of (R-mod,Ab)fp

is some successor ordinal α then

1.3. DIMENSIONS 33

Aα = (R-mod,Ab)fp

and ann(Aα) contains only the zero module. Thus we have

ann(Aα) ∩ Zg(R-Mod) = ∅ and so Zg(R-Mod)(α+1) = ∅. The Cantor-Bendixson rank

of Zg(R-Mod) is therefore less than or equal to α.

Open Question. If the Cantor-Bendixson rank of Zg(R-Mod) is defined, does it

follow that the Krull-Gabriel dimension of (R-mod,Ab)fp

is defined?

Clearly, if Zg(R-Mod)(α+1) = Xα+1 for all α, then the answer to the above question

is positive. An equivalent way of stating this is that for every isolated point M in

Zg(R-Mod)(α+1), there is a functor F whose image is simple in (R-mod,Ab)fp/Aα

such that (F ) ∩ Zg(R-Mod)(α+1) = {M}. So the simple functors at each stage of the

Krull-Gabriel analysis are in a one-to-one correspondence with the isolated points at

each stage of the Cantor-Bendixson analysis.

We say that the isolation condition holds for a closed subset Z of Zg(R-Mod)

(considered as a topological space with the relative topology) if the following is satis-

fied.

(IC) For every closed subset X of Z and every isolated point M in X there is a functor

F whose image is simple in (R-mod,Ab)fp/ann(X) such that (F ) ∩X = {M}.

Proposition 1.3.8 ([41, Thm. 10.19] or [60, Thm. 8.6]). Let R be a small preadditive

category and suppose the (IC) holds for a closed subset Z of Zg(R-Mod). Then the

Krull-Gabriel dimension of F = (R-mod,Ab)fp/ann(Z) is equal to α if and only if the

Cantor-Bendixson rank of Z is equal to α.

Moreover, there is a bijective correspondence

{M ∈ Z | CB(M) = α} ←→ {F ∈ F/Fα | F is simple}

where Fα is the Serre subcategory defined at the αth stage of the Krull-Gabriel analysis.

There are certain situations where (IC) is known to hold and we will use the following

in later sections.

Proposition 1.3.9 ([41, Sec. 10.16]). Let Z be a closed subset of Zg(R-Mod). If

the Krull-Gabriel dimension of (R-mod,Ab)fp/ann(Z) is defined, then the isolation

condition holds for Z.


A pure-injective module is said to be superdecomposable if it has no (non-zero)

indecomposable summands.

Proposition 1.3.10 ([60]). If there is no superdecomposable pure-injective module

over a ring R, then the isolation condition holds for RZg.

Open Question. For which rings R does the Isolation Condition hold for RZg? Does

the isolation condition hold for the Ziegler spectrum of every ring R?

1.3.5 Localisation with respect to a hereditary torsion pair

Here we summarise some of the relevant definitions and results on torsion-theoretic

localisation that will be used in later sections; we refer the reader to [44, Ch. 11] for

a more detailed account. Throughout this section we take C to be a Grothendieck

abelian category.

Localisation takes place with respect to a hereditary torsion pair (T ,F) in C. The

idea is that the objects in T are “torsion” and will become zero in the localisation,

while the objects in F are “torsion-free” and the injective cogenerators of this class

will be unchanged in the localisation.

A subclass T of the objects in C is a torsion class if it is closed under epimorphic

images, extensions and arbitrary direct sums. It is hereditary if it is also closed

under subobjects in C. Similarly, we say that a subclass F of the objects in C is a

tosion-free class if it is closed under subobjects, extensions and arbitrary products.

Such a class is hereditary if it is also closed under taking injective hulls.

For any torsion class T , the class

T ⊥ := {B ∈ C | (A,B) = 0 for all A ∈ T }

is a torsion-free class. Similarly, for any torsion-free class F , the class

⊥F := {A ∈ C | (A,B) = 0 for all B ∈ F}

is a torsion class. Moreover T = ⊥(T ⊥) and F = (⊥F)⊥. Thus, any pair (T ,F) of

subclasses with T = ⊥F a torsion class (or equivalently with F = T ⊥ a torsion-free

class) is called a torsion pair. For any (T ,F), the torsion class T is hereditary if and

only if the torsion-free class F is hereditary; in this case we call (T ,F) a hereditary

torsion pair.

1.3. DIMENSIONS 35

Associated to any hereditary torsion pair (T ,F) there is a left exact subfunctor t

of the identity functor on C. We define this functor on an object C to be

tC :=∑{D | D is a subobject of C and D ∈ T }.

That is, tC is the largest torsion subobject of C. The action of t on morphisms is

restriction/corestriction.

Remark 1.3.11. Each of T , F and t completely determine the others (see [44, 11.1.1]).

In particular, F = {C | tC = 0} and T = {C | tC = C}.

From the above data we define the localisation C/T of the category C at the

torsion pair (T ,F): for an object C in C define

qt(C) := π−1(t(E(C ′)/C ′))

where C ′ := C/tC, E(C ′) denotes the injective hull of C ′ and π is the cokernel of the

canonical embedding of C ′ into E(C ′). For the objects in C/T , we take the collection

of objects qt(C) (as C runs over objects of C) and we take the obvious morphisms

induced by the definition of the objects.

Remark 1.3.12. Clearly qt : C → C/T defines a functor. We refer to qt as the local-

isation functor and we note the following properties:

1. The canonical embedding it : C/T → C is a right adjoint to qt.

2. If E is a torsion-free injective object then E ∼= itqt(E).

3. If D is a subobject of C, then C/D ∈ T if and only if qt(D) = qt(C). In this

case we say that D is T -dense in C.

Given a class of injective objects E in C we say that a torsion pair (T ,F) is

cogenerated by E if T = {C | (C,E) = 0 for all E ∈ E}; in this case we will denote

the torsion pair by (TE ,FE). Since E is a class of injective objects, T is a hereditary

torsion class. The torsion-free class F consists of objects cogenerated by E , that is, F

contains those objects of C which embed in an arbitrary product of copies of objects

in E .

We will be interested in hereditary torsion theories corresponding to definable

subcategories and these will be those of finite type, meaning t commutes with direct

limits. We note the following property of such torsion theories:


Proposition 1.3.13. [44, 11.1.29] Let C be a locally finitely presented abelian category

and suppose that (T ,F) is a hereditary torsion pair of finite type on C. Then (T ,F)

is cogenerated by the set of indecomposable injective objects in F .

We will be concerned with localisation in functor categories and we note that

R-Mod is locally coherent. The following result links the notions of localisation at a

Serre subcategory and at a torsion pair in this setting. If C is a locally coherent abelian

category, let C fpdenote the full subcategory of finitely presented objects. Moreover,

given S ⊆ C then let−→S denote the full subcategory of C containing all objects lim−→i

Ai

with Ai ∈ S.

Theorem 1.3.14 ([26],[30]). Let C be a locally coherent abelian category.

1. There is a bijective correspondence between Serre subcategories of C fpand hered-

itary torsion theories of finite type in C. The correspondence is given by

S 7→(−→S ,−→S ⊥)

and (T ,F) 7→ T ∩ C fp

.

2. The image of the restriction of qt : C → C/T to C fp(where t is the torsion

functor associated to (T , T ⊥)) is the same as the localisation of C fpat the Serre

subcategory T ∩ C fp.

Remark 1.3.15. By Theorem 1.3.14, the hereditary torsion pairs of finite type in

(R-mod,Ab) are also in bijection with the sets in Proposition 1.3.4.

1.3.6 Artin algebras and fp idempotent ideals

When R is an Artin algebra, there are also correspondences between the sets listed

in Proposition 1.3.4 and certain ideals of morphisms in R-mod. Let I be an ideal in

R-mod and define the annihilator of I to be

ann(I) = {F ∈ (R-mod,Ab)fp | F (f) = 0 for all f ∈ I}.

Then we say that I is fp-idempotent if ann(I) is closed under extensions. Note that

an ideal J in R-mod is idempotent if and only if the class of functors in (R-mod,Ab)

vanishing on J is closed under extensions; an idempotent ideal is fp-idempotent but

1.3. DIMENSIONS 37

the converse is not true in general. For a set X or a full subcategory of R-Mod we

define the ideal [X] of morphisms in R-mod that factor through a module in add(X).

The bijections below are due to Krause; Prest refined the result for closed subsets

by showing that [X ] = [X ∩ Zg(R-Mod)]. Indeed, for any subset Z that is dense in

X ∩ Zg(R-Mod), we have [X ] = [Z].

Proposition 1.3.16 ([34, 43]). Let R be an Artin algebra. Then the fp-idempotent

ideals in R-mod are also in bijection with the sets listed in Proposition 1.3.5. The

relevant mutually inverse bijections are as follows.

{Definable subcategories of R-Mod} ↔ {Fp-idempotent ideals of R-mod}

X 7→ [X ] {M ∈ R-Mod | F (M) = 0 for all F ∈ ann(I)} ←[ I.

{Closed subsets of Zg(R-Mod)} ↔ {Fp-idempotent ideals of R-mod}

X 7→ [X] {M ∈ Zg(R-Mod) | F (M) = 0 for all F ∈ ann(I)} ←[ I.

{Serre subcategories of (R-mod,Ab)fp} ↔ {Fp-idempotent ideals of R-mod}

S 7→ {f ∈ R-mod | F (f) = 0 for all F ∈ S} ann(I)←[ I.

In [34], Krause uses the above correspondences to connect the transfinite powers of

the Jacobson radical of R-mod with the Serre subcategories arising in the Krull-Gabriel

analysis of (R-mod,Ab)fp

.

First we define what we mean by the transfinite powers of an ideal. Let I be an

ideal in R-mod. Then the finite powers of I are defined in the usual way:

In := {m∑i=1

fin . . . fi1 | fij ∈ I and m < ω}.

Then for any limit ordinal λ let Iλ =⋂β<λ Iβ and for any ordinal α := λ + n

where λ is a limit ordinal and n < ω let Iα := (Iλ)n+1. We refer to the ideals Iα for

ordinals α as the transfinite powers of I. Also let I∞ :=⋂α Iα.

The notation in the following comes from Sections 1.3.1, 1.3.2 and 1.3.4. If R is

an Artin algebra and M,N are indecomposable finite-dimensional R-modules, then


define radR(M,N) to be the set of non-isomorphisms from M to N . Then radR is the

ideal in mod-R generated by the union⋃M,N

radR(M,N) where M and N range over

all indecomposable finite-dimensional R-modules.

Proposition 1.3.17 ([34, Thm. 8.12] ). Let R be an Artin algebra and α a successor

ordinal. Then

radωαR ⊆ {f ∈ R-mod | F (f) = 0 for all F ∈ Sα−1} = [Xα].

Corollary 1.3.18. Suppose the Cantor-Bendixson rank of Zg(R-Mod) is defined for

an Artin algebra R. For each successor ordinal α, every element of radωαR factors

through a finite direct sum of indecomposable pure injective modules with Cantor-

Bendixson rank equal to α.

Proof. Since the Cantor-Bendixson rank is defined, the isolation condition holds. This

means that Xα = Zg(α) where this denotes the closed subset of Zg(R-Mod) containing

points with Cantor-Bendixson rank greater than or equal to α. The result follows

immediately from this since, in any topological space where the Cantor-Bendixson

rank is defined, the isolated points in the space are dense (see, for example, [44,

Lem. 5.3.57]).

Corollary 1.3.19 ([34, Cor. 8.4]). If R is an Artin algebra with Krull-Gabriel dimen-

sion equal to α, then radωα+nR = 0 for some n ∈ N.

Open Question. If radωα+nR = 0 for some ordinal α, does it follow that the Krull-

Gabriel dimension of (R-mod,Ab)fp

is less than or equal to α?

1.4 Representation type

We saw in the previous section that there are many notions of dimension that measure

the complexity of structures relating to the category R-mod. When R = R is a finite-

dimensional k-algebra, the representation type of R is considered to be an indicator of

the complexity of R-mod. Accordingly, there appears to be some connection between

representation type and the notions of dimension discussed in the previous section.

In this section we outline what is known, as well as some conjectures that make this

1.4. REPRESENTATION TYPE 39

connection more precise. Later in the thesis we will prove that these conjectures hold

for string algebras.

If R and S are finite-dimensional k-algebras, then a functor F : R-Mod→ S-Mod

is a representation embedding if it is a k-linear, exact functor that reflects isomor-

phism and preserves direct sums, direct products and indecomposability.

Let A be a finite-dimensional algebra over a field k. Then A has wild represen-

tation type if there is a representation embedding from k〈X, Y 〉-Mod to A-Mod.

We define a one-parameter family of dimension n to be a family of finite-

dimensional R-modules

FB = {B ⊗k[T ] S | S is a simple left k[T ]-module }

where B is an (A,k[T ])-bimodule which is free of rank n over k[T ]. Then A has

tame representation type if, for every n ≥ 1, there exist one-parameter families

FB1 , . . . ,FBm of dimension n such that all but finitely many of the n-dimensional

A-modules are contained in⋃mi=1FBi .

The following is the famous Tame and Wild Theorem, first proved by Drozd in [21]

but strengthened using different methods in [17].

Theorem 1.4.1. [17, 21] Let A be a finite-dimensional algebra. Then A has tame or

wild representation type but not both.

The class of tame algebras can be further sub-divided. Firstly, we say that an alge-

bra A has finite representation type if there are only finitely many indecomposable

finite-dimensional A-modules up to isomorphism.

Proposition 1.4.2. [7, 6] Let A be an finite-dimensional algebra over a field k. Then

the following are equivalent.

1. The algebra A has finite representation type.

2. The Krull-Gabriel dimension of (A-mod,Ab)fp

is zero.

3. The ideal radωA = 0.

4. The Cantor-Bendixson rank of ZgA is zero.


Proposition 1.4.3. [34] Let A be a finite-dimensional algebra. If A has wild repre-

sentation type, then rad∞A 6= 0 and the Krull-Gabriel dimension of (A-mod,Ab)fp

is

undefined.

Following these results, it is natural to ask whether the various dimensions we

have defined detect the dividing line between tame and wild. This is in fact not the

case, indeed, non-domestic string algebras are examples of tame algebras where the

Krull-Gabriel dimension of (A-mod,Ab)fp

and the Cantor-Bendixson rank of AZg are

undefined and also rad∞A 6= 0.

It is conjectured that the dimensions may determine whether a tame algebra is

domestic or non-domestic. Let µ(n) the number of one parameter families required to

cover almost all of the modules of dimension n. Then if A is a tame algebra such that

there exists some µ such that µ(n) < µ for all n ≥ 1, then we say that A has (tame)

domestic representation type.

Conjecture 1.4.4 ([44, Conj. 9.1.15]). Let A be a finite-dimensional algebra. Then


1. The algebra A has domestic representation type.


is defined.


is finite.

Conjecture 1.4.5 ([44, Conj. 9.1.16]). Let A be a finite-dimensional algebra. Then


1. The algebra A has domestic representation type.

2. The ideal rad∞A = 0.

3. The ideal radω2

A = 0.

In [53] Schroer makes a more precise conjecture of what the connection between

Krull-Gabriel dimension and the transfinite powers of the radical might be.

Conjecture 1.4.6. Let n ≥ 2 and let A be a finite-dimensional algebra. Then the


is equal to n if and only if radω(n−1)A 6= 0

and radωnA = 0.

1.5. ELEMENTARY DUALITY 41

These conjectures have been confirmed for cycle-finite algebras [56] and in Chapter

3 we will show that this also holds for string algebras.

1.5 Elementary duality

Let R be a ring. We begin by introducing a functor that will turn out to be a duality

between (R-mod,Ab)fp

and (mod-R,Ab)fp

. It was first introduced in [24] and also

in [8]. Let F be in (R-mod,Ab)fp

, then dF is the functor in (mod-R,Ab) defined as

follows.

• For every module A in mod-R, let dF (A) := (F,A⊗−).

• For every morphisms g : A→ B in mod-R, let dF (g) : (F,A⊗−)→ (F,B ⊗−)

be given by τ 7→ (g ⊗−)τ .

Let σ : F → G be a morphism in (R-mod,Ab)fp

. Then dσ : dG → dF maps τ in

(G,M ⊗−) to τσ in (F,M ⊗−) for each M in mod-R.

Theorem 1.5.1 (See [44, Thm. 10.3.4]). Let F be a functor in (R-mod,Ab)fp

, then dF

is finitely presented in (mod-R,Ab). Moreover, d : ((R-mod,Ab)fp

)op → (mod-R,Ab)

fp

is an equivalence of categories.

Let N ∈ ZgR. We say that N is reflexive if there is a unique point D(N) ∈ RZg

such that, for every F in (R-mod,Ab)fp

, we have N ∈ (F ) if and only in D(N) ∈ (dF ).

Refer to D(N) as the elementary dual of N .

Theorem 1.5.2 ([25, Thm. 4.10]). Let X be a closed subset of Zg(R-Mod) and N ∈

Zg(R-Mod) such that N is isolated in X by a functor F such that the image of F in

(R-mod,Ab)fp/ann(X) is simple. Then N is reflexive.

Corollary 1.5.3. If every point of X is reflexive, then X and

D(X) := {D(M) |M ∈ X}

are homeomorphic via elementary duality.

Remark 1.5.4. If R is a finite-dimensional algebra over a field k, then every finite-

dimensional module M is isolated in Zg(R-Mod) by a simple functor and D(M) is

equal to the k-dual given by Homk(M,k). Denote Homk(M,k) by M∗.


The following is true for any ring with a suitable dual replacing (−)∗, see [44,

Sec. 1.3.3].

Theorem 1.5.5 ([61, Prop. 3]). Let R be a finite-dimensional algebra over a field k

and let M be a left R-module. Then for every F ∈ (R-mod,Ab)fp

F (M) = 0 if and only if dF (M∗) = 0.

Corollary 1.5.6. Let R be a finite-dimensional algebra. If M ∈ RZg and M∗ ∈ ZgR,

then D(M) = M∗ and D(M∗) = M .

Chapter 2

String algebras

In this chapter we will outline some relevant background material about

finite-dimensional string algebras. We begin by defining combinatorial objects known

as strings and bands consisting of words made up of letters arising from arrows in

the quiver. These combinatorial objects allow us to construct some indecomposable

modules whose structure reflects that of the corresponding string or band. In Section

2.3 we will describe the construction of these modules.

The finite-dimensional modules over a string algebra are known to be exactly the

finite-dimensional string and band modules (the proof of this can be found in [20]

or [16]; Butler and Ringel credit their method to [23]). The classification of the

indecomposable representations has led to many other results to do with the category

of finitely presented modules. In Section 2.4 we give a survey of some of these results.

In a recent paper [48], Prest and Puninski proved that if we include infinite strings,

then the strings and bands also parametrise the points of the Ziegler spectrum of a

domestic string algebra. We describe their result in Section 2.5, as well as some related

results regarding the Cantor-Bendixson rank and the Krull-Gabriel dimension of the

functor category.

2.1 Definition of a string algebra

Let Q = (Q0, Q1) be a finite quiver and I an admissible ideal of the path algebra

kQ over a field k. The algebra A = kQ/I is a monomial algebra if the ideal I is

generated by paths of length at least two. For each arrow a ∈ Q1, let s(a) ∈ Q0 denote

43

44 CHAPTER 2. STRING ALGEBRAS

the starting point of a and let e(a) ∈ Q0 denote the ending point of a.

Definition 2.1.1. A monomial algebra A = kQ/I is a string algebra if the following

conditions hold:

1. For all x ∈ Q0, there are at most two arrows a, b ∈ Q1 such that s(a) = s(b) = x.

2. For all x ∈ Q0, there are at most two arrows c, d ∈ Q1 such that e(c) = e(d) = x.

3. For all b ∈ Q1, there is at most one arrow a ∈ Q1 such that ab /∈ I.

4. For all b ∈ Q1, there is at most one arrow c ∈ Q1 such that bc /∈ I.

If the ideal I is generated by paths of length 2 and we replace the last two conditions

with the following conditions.

3’. For all b ∈ Q1, there is at most one arrow a ∈ Q1 such that ab /∈ I and at most

one a′ ∈ Q1 such that s(a′) = e(b) and a′b ∈ I.

4’. For all b ∈ Q1, there is at most one arrow c ∈ Q1 such that bc /∈ I and at most

one c′ ∈ Q1 such that s(b) = e(c′) and bc′ ∈ I.

then A is said to be a gentle algebra.

For each string algebra, it will be useful to fix functions σ, ε : Q1 → {1,−1} with

the following properties:

1. If a 6= b are arrows with s(a) = s(b) then σ(a) = −σ(b).

2. If c 6= d are arrows with e(c) = e(d) then ε(c) = −ε(d).

3. If a and b are arrows with s(a) = e(b) and ab /∈ I then σ(a) = −ε(b).

2.2 Strings and bands

Throughout this section A = kQ/I will be a monomial algebra. For each a ∈ Q1 we

introduce a formal inverse a− and extend the functions s and t so that s(a−) = e(a)

and e(a−) = s(a). Let Q−1 := {a− | a ∈ Q1}. We will refer to the elements of the set

Q1 tQ−1 as letters. A letter u contained in Q1 will be referred to as a direct letter

and a letter u contained in Q−1 will be referred to as an inverse letter. Also if u = a−

where a is a direct letter then u− = a.

2.2. STRINGS AND BANDS 45

2.2.1 Finite strings

Definition 2.2.1. A sequence of letters u = un . . . u1 is a string (of length n) if the

following conditions hold:

1. For all 1 ≤ i < n, we have that e(ui) = s(ui+1).

2. For all 1 ≤ i < n, we have that ui 6= u−i+1.

3. For all 1 ≤ i < i+ k ≤ n, neither the sequence ui+k . . . ui nor the sequence

u−i . . . u−i+k is contained in I.

For such a string u = un . . . u1, we extend the definitions of inverse, starting and

ending in the obvious ways: u− = u−1 . . . u−n , s(u) = s(u1) and e(u) = e(un). We will

refer to u1 as the first letter of u and to un as the last letter of u.

For each vertex x ∈ Q0 we introduce two strings 1(x,1), 1(x,−1) of length zero. For

i ∈ {1,−1} we have that 1−(x,i) = 1(x,−i) and s(1(x,i)) = e(1(x,i)) = x. (In the second half

of the thesis we will only consider one string of length zero for every vertex x ∈ Q0; it

is purely for technical reasons that we require two such strings here). If u = un . . . u1

and v = vm . . . v1 are strings then the concatenation of u and v is defined to be

uv = un . . . u1vm . . . v1 provided the sequence un . . . u1vm . . . v1 satisfies the conditions

in Definition 2.2.1. Otherwise the concatenation of u and v is not defined.

If every letter of a string u is direct then we say that u is a direct string. If every

letter of u is inverse then we say that u is an inverse string. The strings of length

zero are considered to be both direct and inverse.

If A is a string algebra, we extend the functions σ, ε to the set of strings in the

obvious way: For an inverse letter l ∈ Q−1 , let σ(l) = ε(l−) and ε(l) = σ(l−); For a

string u = ln . . . l1 of length n > 0, let σ(u) = σ(l1) and ε(u) = ε(ln); For x ∈ Q0, let

σ(1(x,i)) = −i and ε(1(x,i)) = i.

If uv is defined then σ(u) = −ε(v) but, unless A is a gentle algebra, the converse is

not true since uv must also satisfy the third condition of Definition 2.2.1. We say that

the concatenation 1(x,i)u is defined and equal to u if e(u) = x and ε(u) = i. Similarly,

the concatenation u1(x,i) is defined and equal to u if s(u) = x and σ(u) = −i.


2.2.2 Bands

A string b of length n ≥ 1 is cyclic if s(b) = e(b). If, in addition, there is no string u

such that the m-fold concatenation u . . . u is equal to b then b is a primitive cyclic

string. If a primitive cyclic string b = bn . . . b1 is such that bm 6= 0 for all m ≥ 1 and

also b1 is an inverse letter and bn is a direct letter, then b is a band.

2.2.3 Infinite strings

Consider a sequence of letters u = . . . u2u1 indexed by N and unbounded to the left.

Then u is a (left) N-string if un . . . u1 is a string for every n ∈ N. If u is a left N-string,

then s(u) = s(u1) and σ(u) = σ(u1). A sequence v = v−1v−2 . . . is a (right) N-string

if v−1 . . . v−n is a string for every n ∈ N. Also e(v) = e(v−1) and ε(v) = ε(v−1). A

Z-indexed sequence of letters v = . . . v2v1v0v−1 . . . is a Z-string if vi . . . vj is a string

for every pair of integers i ≥ j. We will refer to N- and Z-strings as infinite strings.

We extend the definition of v− to infinite strings in the obvious way.

Let u be a left N-string. The string u is periodic if there exists some primitive

cyclic string c such that u = . . . cc. We will use the notation u = ∞c. A left N-string u

is almost periodic if it is periodic or if there is some primitive cyclic string c and some

finite string v = v0 · · · v−m such that u = ∞cv and ∞cv0 is not periodic. We say that

∞c is the maximal periodic substring of u. We say that an almost periodic string

u = ∞cv is expanding if the last letter of c is direct. We say that u is contracting

if the last letter of c is inverse. The definitions periodic, almost periodic, expanding

and contracting apply to a right N-string u if they apply to the left N-string u−.

A Z-string v is periodic if there is some primitive cyclic string c with v = ∞c∞.

A Z-string v is almost periodic if it is not periodic but there exist almost periodic

N-strings u and w such that v = wu. Suppose w = . . . v2v1 and u = v0v−1v−2 . . . , then

w is a left maximal periodic substring if it is periodic and wv0 is not periodic. If

v = wu has such a left maximal periodic substring w = ∞c, then v is left expanding

if the last letter of c is direct and v is left contracting if the last letter of c is inverse.

An almost periodic Z-string v is

• expanding if both v and v− are left expanding;

• contracting if both v and v− are left contracting; and

2.2. STRINGS AND BANDS 47

• mixed if one of v and v− is left expanding and the other is left contracting.

Example 2.2.2. Consider the algebra given by the quiver

• beea 99

with relations ab = b2 = a2 = 0. Then

• the string ∞(ba−)b− is an expanding left N-string with maximal cyclic substring

∞(ba−);

• the string ba−(ab−)∞ is a contracting right N-string; and

• the string v = ∞(ba−)(ab−)∞ is the unique almost periodic Z-string. The string

∞(ba−) is its left maximal periodic substring and so v is left expanding. The

string v− is left contracting since ∞(a−b) is its maximal periodic substring, there-

fore v is a mixed Z-string.

2.2.4 Equivalence relations

Finite strings We say that strings u∼−1v for strings u, v if and only if u = v− or

u = v . Let St denote a fixed set of representatives of ∼−1-equivalence classes of

finite strings.

Bands We say that b∼rc for bands b, c if and only if we have b = bn . . . b1 and we

have c = bi . . . b1bn . . . bi+1 for some 1 ≤ i < n. Let Ba denote a fixed set of

representatives of ∼r-equivalence classes of bands.

We say that b∼bc if and only if b∼−1c or b∼rc. Then let Ban denote the set of

∼b-equivalence classes.

Infinite strings Let Con be a fixed set of representatives of ∼−1-equivalence classes

of contracting N- and Z-strings.

Let Exp be a fixed set of representatives of ∼−1-equivalence classes of expanding

N- and Z-strings.

Let Mix be a fixed set of representatives of ∼−1-equivalence classes of mixed

Z-strings. We take the representative of each equivalence classes to be the left

expanding one.


2.3 Some indecomposable modules

Let A = kQ/I be a monomial algebra. In this section we describe the construction of

string and band modules over A. The infinite-dimensional string modules were first

described in [50].

2.3.1 Finite-dimensional string modules

Let u = un . . . u1 be a string of length n ≥ 1. Define a function

t : {0, 1, . . . , n} → Q0 where t(i) :=

s(u1) if i = 0,

e(ui) if i 6= 0.

Then for each x ∈ Q0, let Ix := {i | t(i) = x}. From this data we define the string

module M(u) over A as follows:

• For each x ∈ Q0 take the vector space Mx =⊕

i∈Ix Ki where Ki∼= k for each

i ∈ Ix.

• For each a ∈ Q1 with s(a) = x and e(a) = y take the matrix

Ma :⊕i∈Ix

Ki →⊕j∈Iy

Kj

with entries

(Ma)i,j :=

idk if j = i+ 1 and uj = a,

idk if i = j + 1 and ui = a−,

0 otherwise.

It will be useful to fix a basis for the vector spaces Mx so let zi denote the basis

element spanning Ki for 0 ≤ i ≤ n. We refer to this as a canonical basis element

of M(u). We may also refer to zi as the basis element between ui and ui+1 when

0 < i < n.

Example 2.3.1. Consider the algebra

• beea 99

2.3. SOME INDECOMPOSABLE MODULES 49

with relations ab = b2 = a2 = 0. Consider the finite string u = ba−b−ab−. We depict

the string module M(u) as follows.

k

1k

b

�� 1k

a

��k k

1k

b

��

k

1k

a

�� 1k

b

��k k

2.3.2 Finite-dimensional band modules

Let b = bn . . . b1 be a band and let M = (U,Φ) be a finite-dimensional k[T, T−1]-

module. That is U is a finite-dimensional vector space and Φ is an invertible linear

endomorphism Φ: U → U corresponding to multiplication by T . For each vertex

x ∈ Q0, let Jx := {j | t(bj) = x}.

We define the band module B(b,M) as follows:

• For each x ∈ Q0 take the vector space Bx :=⊕

j∈Jx Uj where Uj ∼= U for each

j ∈ Jx.

• For each a ∈ Q1 with s(a) = x and e(a) = y take the matrix

Ba :⊕i∈Jx

Ui →⊕j∈Jy

Uj

with entries

(Ba)i,j :=

idU if j = i+ 1 and bj = a,

idU if i = j + 1, i 6= 1 and bi = a−,

Φ if i = 1, j = 2 and b1 = a−,

0 otherwise.

Recall from Example 1.2.1 that when k is algebraically closed the finite-dimensional

k[T, T−1]-modules are of the form Mk,n = (kn, Jk,n) for some k ∈ k∗. We follow Schroer

in [55] and fix a canonical basis {z(j)i | 1 ≤ i ≤ m, 1 ≤ j ≤ n} for the band module

B(b,Mk,n) with band b = bm . . . b1. The action of a ∈ Q1 on the canonical basis is as


follows

a.z(j)1 =

kz(1)m if j = 1, a = bm

kz(j)m + z

(j−1)m if j 6= 1, a = bm

z(j)2 if a = b−1

0 otherwise.

and for i 6= 1

a.z(j)i =

z

(j)i−1 if a = b−i−1

z(j)i+1 if a = bi

0 otherwise.

Example 2.3.2. Consider the Kronecker algebra • •b

hhavv

. There is a unique band

b = ab− (up to equivalence). Let M = (U,Φ) be a finite-dimensional k[T, T−1]-module.

Then we will depict the band module B(b,M) as follows.

U Ub=Φ

ii

a=1Uuu

Now suppose that k is algebraically closed. The above picture can be expanded;

we will depict B(b,M3,k) for some k ∈ k∗.

k k k k1kqq

kmm k1kii

k

hh

1k

vvk

k

ee

1k

yy

1k

gg

2.3.3 Infinite-dimensional string modules

Let u be a left N-string (in which case u = . . . u2u1) or a Z-string (in which case

u = . . . u2u1u0u−1 . . . ). Extend the function given in Section 2.3.1 in the following

way:

If u is a left N-string then let

t : {0, 1, 2 . . . } → Q0 where t(i) :=

s(u1) if i = 0,

e(ui) if i 6= 0.


and if u is a Z-string then let t : Z→ Q0 where t(i) := e(ui).

Then again for each x ∈ Q0, let Ix := {i | t(i) = x}. From this data we define the

direct-sum module M(u) over A as follows:

• For each x ∈ Q0 take the vector space Mx =⊕


i ∈ Ix.

• For each a ∈ Q1 with s(a) = x and e(a) = y take the (infinite) matrix

Ma :⊕i∈Ix

Ki →⊕j∈Iy

Kj

with entries

(Ma)i,j :=



0 otherwise.

Similarly, we define the direct-product module N(u) over A as follows:

• For each x ∈ Q0 take the vector space Nx =∏


i ∈ Ix.


Na :∏i∈Ix

Ki →∏j∈Iy

Kj

with entries

(Na)i,j :=



0 otherwise.

If u is a right N-string, then define M(u) := M(u−) and N(u) := N(u−). We can,

of course, adjust the above constructions to directly construct a direct sum module

and a direct product module associated to u; the modules M(u) and N(u), as we

have defined it, is isomorphic to the direct sum module and the direct product module

constructed in this way.

Whenever u is a Z-string we define the (left) mixed module L(u) over A to be

the submodule of N(u) given as follows:


• For each x ∈ Q0 take the subspace Lx of Nx containing elements (ki)i∈Ix such

that ki = 0 for i� 0 .


La : Lx → Ly

with entries

(La)i,j :=



0 otherwise.

We define the (right) mixed module R(u) to be L(u−).

As in the finite-dimensional case, we will let zi denote a fixed basis element spanning

each vector space Ki.

Theorem 2.3.3 ([50]). Let A be a monomial algebra and let u be an almost periodic

infinite string. Then the following modules are pure injective:

1. M(u) when u is a contracting N- or Z-string.

2. N(u) when u is an expanding N- or Z-string.

3. L(u) when u is a mixed Z-string such that u left expanding.

4. R(u) when u is a mixed Z-string such that u is left contracting.

For each almost periodic infinite string u over a monomial algebra, we will denote

the corresponding indecomposable pure-injective by C(u).

2.3.4 Infinite-dimensional band modules

Let b = bn . . . b1 be a band and let M = (U,Φ) be a infinite-dimensional indecompos-

able pure-injective k[T, T−1]-module (see Example 1.2.1). That is, U is the underlying

vector space of a Prufer, adic or generic module and Φ is the automorphism of U given

by u→ T.u. We define the band module B(b,M) as in Section 2.3.2.

The construction described in Section 2.3.2 can be used to define a functor

Fb : k[T, T−1]-Mod → A-Mod and in this section we will give a short argument es-

tablishing that this functor is a representation embedding.


Recall, from Section 1.4, that if R and S are finite-dimensional k-algebras, then a

functor F : R-Mod → S-Mod is a representation embedding if it is a k-linear, exact

functor that reflects isomorphism and preserves direct sums, direct products and inde-

composability. An equivalent characterisation is that F preserves indecomposability,

reflects isomorphisms and F ∼= M ⊗R − for M an (S,R)-bimodule that is finitely

generated as a right R-module and such that k acts centrally on M .

Proposition 2.3.4. Let b be a band over a monomial algebra A. Then the functor

Fb : k[T, T−1]-Mod→ A-Mod given by M 7→ B(b,M) is a representation embedding.

Proof. Suppose b = bn . . . b1. It is clear from the construction that Fb both preserves

indecomposability and reflects isomorphism. It remains to show that Fb is isomorphic

to an appropriate tensor product functor.

Let v = ∞b∞ and label the letters of v = . . . v2v1v0v−1 . . . so that vi+jn = bi for all

1 ≤ i ≤ n and j ∈ Z. Consider the direct sum module M(v). We follow the notation

from Sections 2.3.2 and 2.3.3. M(v) is an (A,k[T, T−1])-bimodule with T acting on

basis elements of M(v) as T.zt = zt+n; this is the shift endomorphism described in

[50]. It is clear from this that M(v) is finitely-generated as a k[T, T−1]-module and k

acts centrally on M(v).

Next we show that Fb is isomorphic to the functor M(v)⊗k[T,T−1] −. Given that

M(v) ∼= M(v)⊗k[T,T−1] k[T, T−1] we can explicitly describe an isomorphism between

Fb(k[T, T−1]) and M(v)⊗k[T,T−1] k[T, T−1] as follows:

Ψ: Fb(k[T, T−1])→M(v)

where for each u = k−tT−t + · · · + k0 + k1T + · · · + kmT

m ∈ Ui ∼= k[T, T−1] we have

that Ψ(u) =∑m

j=−t kjzi+jn. Since Fb and M(v)⊗k[T,T−1] − are both right exact and

commute with finite direct sums, it follows that Fb(M) ∼= M(v)⊗k[T,T−1] M for any

finitely presented module M . Moreover, both functors commute with direct limits so

the isomorphism holds for arbitrary M .

Theorem 2.3.5 ([42, Thm. 7]). Suppose that R and S are rings and

Φ: S-Mod→ R-Mod

is a representation embedding. Then Φ induces a homeomorphism between SZg and a

closed subset of RZg.


It follows that the set of isomorphism classes of (finite- and infinite-dimensional)

band modules is a closed subset of AZg.

2.4 The module category of a string algebra

The fact that all the indecomposable modules in A-mod can be described so explicitly

means that many questions that are hard to answer in general become tractable when

restricted to the class of string algebras. In particular, we have a description of the

Auslander-Reiten structure of A-mod [16] and, more surprisingly, we also have an ex-

plicit description of a nice basis for HomA(M,N) when M and N are any indecompos-

able modules. This description can be found in a series of papers by Crawley-Boevey

and Krause [18, 19, 29].

In this section we will first give an account of the construction of the Auslander-

Reiten sequences in A-mod. Next we will define the basic morphisms between finite-

dimensional string and band modules and use this to describe the hammock posets

introduced by Schroer [52]. We will prove that the morphisms between indecomposable

finite-dimensional modules and the indecomposable infinite-dimensional pure-injective

modules are determined by where the associated infinite string or band fits into the

hammock poset. This observation is a central component of the argument given in the

next section to determine the Cantor-Bendixson rank of Ziegler spectrum.

2.4.1 Auslander-Reiten sequences for string algebras

Let A = kQ/I be a string algebra. In this section we present results from [16] in which

Butler and Ringel describe the Auslander-Reiten sequences for a string algebra. An

important property of string algebras is that the possible number of middle terms in

an Auslander-Reiten sequence is bounded above by two.

We begin by describing the Auslander-Reiten sequences with one middle term. For

each arrow a ∈ Q1, let la = b−l . . . b−1 be the inverse string of greatest length such that

laa is a string. Similarly, let ra = c−r . . . c−1 be the inverse string of greatest length such

that ara is a string. Finally, let ua = laara.

Since la is an image substring of ua and ra is a factor substring of ua, there are

induced graph maps f : M(la)→M(ua) and g : M(ua)→M(ra).

2.4. THE MODULE CATEGORY OF A STRING ALGEBRA 55

Proposition 2.4.1. For each a ∈ Q1, the sequence

0→M(la)f→M(ua)

g→M(ra)→ 0

is an Auslander-Reiten sequence.

If u is of the form la for some a ∈ Q1, we will also denote ua by u+ and f will be

denoted f+ : M(u)→M(u+).

Example 2.4.2. Consider the algebra Λ2 given by the quiver

1b

66a((2 c // 3

e66

d((4

with relations cb = ec = 0. Then for the arrow e, we have the Auslander-Reiten

sequence 0 → M(le)f→ M(ue)

g→ M(re) → 0 where M(le) is given by the following

diagram.

k1k

a��k

1kc��k

1k

d ��k

which embeds into M(ue) which is given by the following.

k1k

a��k

1kc��k

1k

d ��

k1k

e��

1k

d ��k k

which projects onto M(re) as in the following diagram.

k1k

d ��k


Next we describe the Auslander-Reiten sequences with two middle terms. Let u be

a string. Assume that M(u) is not injective and u is not of the form la for any a ∈ Q1.

The following algorithm produces an Auslander-Reiten sequence starting at M(u).

Consider the start of u: 1. Suppose there exists an arrow b ∈ Q1 such that ub is

a string. Then let u+ := ubv where v is the inverse string of greatest length

such that ubv is a string.

2. Suppose there is no arrow b ∈ Q1 such that ub is a string. Then let u+ := w

where u = wd−v where v is a direct string.

In the first case u is a substring of u+ and in the second case u+ is a factor substring

of u. Thus, denote the induced graph map by f+ : M(u)→M(u+).

Consider the end of u: 1. Suppose there exists an arrow b ∈ Q1 such that b−u is

a string. Then let +u := vb−u where v is the direct string of greatest length

such that vb−u is a string.

2. Suppose there is no arrow b ∈ Q1 such that b−u is a string. Then let

+u := w where u = vdw where v is an inverse string.

In the first case u is a substring of +u and in the second case +u is a factor substring

of u. Thus, denote the induced graph map by f+ : M(u)→M(+u).

Example 2.4.3. Consider the algebra Λ2 as in Example 2.4.2 and consider the string

u = c−d−e. Then M(u) is given by the following diagram.

k1k

c��k

1k

d ��

k1k

e��

k

If we apply the first algorithm to M(u) then we produce the following module M(u+).

k1k

c��k


When we apply the second algorithm we obtain the module M(+u) given by the

following diagram.

k1k

b��

1ka��

k k1k

c��k

1k

d ��

k1k

e��

k

The third term of the Auslander-Reiten sequence is therefore the module M(+u+)

given by the following diagram.

k1k

b��

1ka��

k k1k

c��k

Proposition 2.4.4. Let u be a string such that M(u) is not injective and u is not of

the form la for any a ∈ Q1. Then

0→M(u)

(f++f

)−→ M(u+)⊕M(+u)

( +g −g+ )−→ τ−1(M(u))→ 0

is an Auslander-Reiten sequence. Moreover, we have that τ−1(M(u)) = M(+u+) and

g+ : M(u+)→M(+u+) and +g : M(+u)→M(+u+) are the obvious graph maps.

Theorem 2.4.5 ([16]). Let A be a string algebra. The Auslander-Reiten sequences in

A-mod are those described in Propositions 2.4.1 and 2.4.4, together with those of the

form

0→ Fb(L)→ Fb(M)→ Fb(N)→ 0

where b is a band and 0 → L → M → N → 0 is an Auslander-Reiten sequence in

k[T, T−1]-mod.


2.4.2 Morphisms between string modules

In this and the next few sections we describe a basis for the morphisms between finite-

dimensional string and band modules over a monomial algebra A. The description

was originally given by Crawley-Boevey and Krause in a series of papers: [18], [19]

and [29].

Let u = un . . . u1 be a string and v = ui . . . uj a substring of u (so 1 ≤ j ≤ i ≤ n.

Then we say that v is an image substring (of u) if ui+1 is inverse or i = n and also

uj−1 is direct or j = 1. Similarly, if ui+1 is inverse and ui is direct for some 1 ≤ i ≤ n

then the string 1(e(ui),ε(ui)) is also an image substring of u.

Dually we say that v is a factor substring (of u) if ui+1 is direct or i = n and

also uj−1 is inverse or j = 1. If ui+1 is direct and ui is inverse for some 1 ≤ i ≤ n then

the string 1(e(ui),ε(ui)) is also a factor substring of u. If u is of length zero then u itself

is its unique factor substring and its unique image substring.

If v is an factor substring of u, then there is an epimorphism M(u)→M(v). Also,

if v is an image substring of w , then there is a monomorphism M(v) → M(w). A

composition f : M(u) → M(w) of two such morphisms is known as a graph map

(between string modules) and v will be referred to as the string associated to f .

Example 2.4.6. Consider the algebra given by the quiver • beea 99 with relations

ab = b2 = a2 = 0. Let u = ba−ba−b−a and let w = a−ba−b. Then the string

v = a−ba− is a factor substring of u and an image substring of w . Thus we have a

morphism M(u) → M(w) that factors as an epimorphism onto M(v) followed by a

monomorphism. These morphisms can be visualised as follows.

k1k

b��1k

a ��

1k

��

k

1k

��

1k

b��1k

a ��k k

1k

��

k

1k

��

1k

b ��

k1k

a��k

k

1k

��

1k

a ��

k

1k

��

1k

a ��

1k

b��k

1k

��

k

1k

��k 1k

a ��

k 1k

a ��

1k

b��

k1k

b��k k


Remark 2.4.7. The definition of a graph map f as it stands fixes the orientation of

the string v associated to f as a substring of u and w . Since M(v) ∼= M(v−) we will

include in the collection of graph maps M(u)→M(w) compositions of the form

M(u)→M(v)∼=−→M(v−)→M(w)

Proposition 2.4.8 ([18]). Let A be a monomial algebra and let u and v be finite

strings. If G is the collection of graph maps M(u)→M(w), then G is a k-linear basis

for HomA(M(u),M(w)).

2.4.3 Morphisms between string and band modules

Let b be a band and M = (U,Φ) a finite-dimensional k[T, T−1]-module. If v is a

finite factor substring of ∞b∞ and g ∈ Homk(U,k) then we can define a morphism

B(b,M)→ M(v). The composition f : B(b,M)→ M(w) of such a morphism with a

graph map M(v) → M(w) between string modules will be known as a graph map

(from a band module to a string module) and v will be known as the string

associated to f .

Example 2.4.9. Consider the Kronecker algebra given by the quiver •a ))

b

55 • . If

M = (U,Φ) is an indecomposable finite-dimensional k[T, T−1]-module and b = ba−,

then the corresponding band module B(b,M) is given by the following

UΦ = a))

1U = b55 U .

Fix g ∈ Homk(U,k) and consider the factor substring u = a−b of ∞b∞. The following

depicts a graph map f : B(b,M)→M(u).

UΦ=a ))

1U=b

55

g

##

gΦ−1 ��

U

gΦ−1

{{k b

1k &&k

k

a

1k

88

Similarly, if v is a finite image substring of ∞b∞ and g ∈ Homk(k, U) then we can

define a morphism M(v) → B(b,M). The composition f : M(w) → B(b,M) of such

a morphism with a graph map M(w)→M(v) will be known as a graph map (from

a string module to a band module) and v the associated string.


Example 2.4.10. Consider the Kronecker algebra (as in Example 2.4.9) and fix g ∈

Homk(k, U) and consider the image substring v = ba−ba− of ∞b∞. A graph map

f : M(v)→ B(b,Φ) is shown below.

k

Φ2g

zz

k

1k=a 88

1k=b

&&

Φg

��

k

Φg

��

k

1k=a 88

1k=b

&&

g

��

k

g

��U

Φ=a ))

1U=b

55 U

Remark 2.4.11. Let f : M(w) → B(b,M) be a graph map from a string module to

a band module with v the associated string. There are, in general, several possible

morphisms k → U that could be used to define f . For instance, in Example 2.4.10

we could use g, Φg or Φ2g. Thus, for each graph map f , we fix the representative of

the equivalence class of v and say that f is determined by the element of Homk(k, U)

corresponding to the (left) end of v . So in Example 2.4.10, we choose v = ba−ba− (as

opposed to v− = ab−ab−) and f is determined by g.

Proposition 2.4.12 ([29]). Let A be a monomial algebra. Suppose b is a band and

M = (U,Φ) a finite-dimensional k[T, T−1]-module.

1. Fix a basis BL for the finite-dimensional k-vector space Homk(U,k) and let GLbe the collection of graph maps B(b,M)→M(w) determined by the elements of

BL. Then GL is a k-linear basis for the vector space HomA(B(b,M),M(w)).

2. Fix a basis BR for the finite-dimensional k-vector space Homk(k, U) and let GRbe the collection of graph maps M(w)→ B(b,M) determined by the elements of

BR. Then GR is a k-linear basis for the vector space HomA(M(w), B(b,M)).

2.4.4 Morphisms between band modules

Let b and c be bands, and let M = (U,Φ) and N = (V,Ψ) be finite-dimensional

k[T, T−1]-modules. Consider a finite string u such that u is a factor substring of

∞b∞ and an image substring of ∞c∞. Then, for any g ∈ Homk(U, V ) there exists a


morphism f : B(b,M)→ B(c, N) defined in a similar way to the graph maps between

string and band modules. As before, we say that v is the string associated to f and,

fixing the representative of the equivalence class of v , we say that g ∈ Homk(U, V )

determines f if g is the morphism corresponding to the end of v ; see Remark 2.4.11.


• beea 99

with relations ab = b3 = a3 = 0. Then there is a band b = ab− and a band c = aab−b−.

If we fix a k-linear morphism g ∈ Homk(U, V ), then we have the following graph map

B(b,M)→ B(c, N).

U4

f��

bgga 77

V 2bgga 77

where the actions of a and b on U4 are given by the following matrices.

a =

0 1U 0 0

0 0 0 1U

0 0 0 0

0 0 0 0

b =

0 0 0 Φ

0 0 0 0

0 0 0 0

0 0 1U 0

and the actions of a and b on V 2 are given by the following matrices.

a =

0 1V

0 0

b =

0 Ψ

0 0

and f is given by

f =

0 g 0 Φg

0 0 g 0

.

The structure of this morphism is clearer if we consider the infinite strings ∞b∞

and ∞c∞. In the following diagram, the copies of U and V are denoted Ui and Vj for


1 ≤ i ≤ 4 and 1 ≤ j ≤ 2 and we identify appropriate copies of U and V .

U3

g

��

a

1k}} b

1k!!

· · · U4

b

Φ!!

U2

g

��

a

1k}}

U4

Φg

��

b

Φ!!

U2

a

1k}}

· · ·

U1 U1

· · · V2

a

1k

}} b

Ψ

!!

V2

b

Ψ

!!a

1k

}}

V2

a

1k

}} b

Ψ

!!

· · ·

V1 V1 V1 V1

Such a morphism, together with any morphism of the form Fb(f) for some f in

Homk[T,T−1](M,N), will be referred to as a graph map (between band modules).

Proposition 2.4.14 ([18]). Let A be a monomial algebra, let b and c be bands, and

let M = (U,Φ) and N = (V,Ψ) be finite-dimensional k[T, T−1]-modules. Then

• Fix bases Bk and Bk[T,T−1] for the finite-dimensional k-vector spaces Homk(U, V )

and Homk[T,T−1](M,N) respectively.

• Define Gk to be the set of graph maps B(b,M)→ B(c, N) determined by elements

of Bk.

• Define Gk[T,T−1] to be the set of graph maps of the form Fb(g) for g ∈ Bk[T,T−1] if

b = c and the empty set otherwise.

Then the set G := Gk ∩ Gk[T,T−1] is a k-linear basis for HomA(B(b,M), B(c, N)).

2.4.5 Hammock posets and factorisations of graph maps

Let A = kQ/I be a string algebra over an algebraically closed field k. In this section

we give a brief description of the hammock posets for A introduced by Schroer in

his thesis [52]. The elements of the posets represent (pointed) indecomposable finite-

dimensional modules over A and pairs u < v in the poset represent (pointed) graph

maps between finite-dimensional modules. The elements of intervals [u, v ] in the posets

represent all the possible factorisations f = g2g1 of the graph map f corresponding to

x < y such that g1, g2 are graph maps. These posets are the main tool used by Schroer

to determine for which ordinals α we have radαA = 0.


Let x ∈ Q0. We partition the strings with an endpoint at x into two sets H1(x)

and H−1(x) using the ε and σ functions. We think of the strings in H−1(x) as those

extending to the right of x i.e. ending at x and we think of the strings in H1(x) as

those extending to the left of x i.e. starting at x.

H−1(x) := {v ∈ St | e(v) = x and ε(v) = 1}

We define a linear ordering on H−1(x): For u, v ∈ H−1(x) let u < v if one of the

following holds.

1. v = uax for a ∈ Q1 and x ∈ St.

2. u = vb−1y for b ∈ Q1 and y ∈ St.

3. v = zax and u = zb−1y for a, b ∈ Q1 and x , y , z ∈ St.

Let

H1(x) := {v ∈ St | s(v) = x and σ(v) = −1}

We define a linear ordering on H1(x): For u, v ∈ H1(x) let u < v if v−1 < u−1 in

H−1(x).

Example 2.4.15. Consider the Kronecker algebra given by the quiver 1a ))

b

55 2 and

suppose σ(b−) = −1 and ε(a) = 1 (this determines ε and σ). Then

H−1(2) = {1(2,1) < ab− < ab−ab− < · · · < ab−ab−a < ab−a < a}

and

H1(2) = {1(2,1) < ab− < ab−ab− < · · · < b−ab−ab− < b−ab− < b−}.

Note that, although some strings occur in both posets (with the exception of the trivial

strings) they are pointed at opposite ends.

The definition of ε and σ mean that we can compose strings from H−1(x) with

those from H1(x) unless the relations on A prohibit it. Let

H(x) := {(v ,w) ∈ H1(x)×H−1(x) | u = vw ∈ St}.

We extend the orderings on H−1(x) and H1(x) to an ordering on H(x): For (v ,w),

(y , z) ∈ H(x) let (v ,w) < (y , z) if v ≤ y and w ≤ z (but not both v = y and w = z).


We can think of each element (v ,w) ∈ H(x) as the string u = vw with a dis-

tinguished vertex between w and v . When the location of the distinguished vertex

is obvious or unimportant we will refer to the string u as an element of H(x). The

definition of the ordering means that, if u, v ∈ H(x), then there is a graph map

f : M(u) → M(v) taking the distinguished vertex in u to the distinguished vertex in

v if and only if u ≤ v (or u− ≤ v).

In the arguments given in later sections, we are sometimes given an arbitrary string

module M(u) and we consider the graph maps going to or from M(u) based on the

ordering of the strings. Since M(u) ∼= M(u−), we must consider where both u and

u− occur in H(x) for each x ∈ Q0. Thus, when we consider the elements of H(x) as

strings with distinguished vertices, we will consider a pointed version of u− ∈ H(x) to

be an occurrence of a pointed version of u in H(x).

Let f : M(u)→M(v) be a graph map between string modules such that im(f) has

the simple module S(x) corresponding to the vertex x ∈ Q0 as a composition factor. It

follows that we can find canonical basis elements zu, zv in M(u) and M(v) respectively

such that f(zu) = zv. Thus we may consider u and v as elements of H(x) with u ≤ v

(taking inverse strings if we need to). Moreover, the interval

[u, v ] := {w ∈ H(x) | u ≤ w ≤ v}

corresponds to the proper factorisations f = g2g1 of f such that g1 and g2 are graph

maps between string modules.

Remark 2.4.16. Schroer extends the posets H(x) to include the graph maps to and

from finite-dimensional band modules. We will not use this poset in our arguments

but the idea is to first include the set B(x) of periodic point strings (∞bb2, b1b∞) such

that b2b1 = b. For any finite string u = (v ,w) ∈ H(x) and finite-dimensional band

module B(b,M), there is a graph map M(u)→ B(b,M) if and only if u < (∞bb2, b1b∞)

for some b2b1 = b. Similarly, there is a graph map B(b,M) → M(u) if and only if

(∞bb2, b1b∞) < u.

For each band there is a whole family of tubes containing finite-dimensional mod-

ules and so Schroer extends the poset to reflect this. We do not do this here but refer

the reader to [52, 54].


2.4.6 Hammock posets and morphisms between pure-injective

modules

Let A = kQ/I be a string algebra over an algebraically closed field k. In this section

we will extend the hammock posets for A to include the infinite strings corresponding

to the indecomposable infinite-dimensional pure-injective string modules. We will see

that these extended posets will indicate the existence of non-zero morphisms to and

from these modules. We have also seen that the periodic Z-strings indicate when

there is a non-zero morphism to or from a finite-dimensional band module; we will

see that the position of the periodic Z-strings also indicates the existence of non-zero

morphisms to and from the infinite-dimensional band modules.

Let Hi(x) be the poset obtained by including N-strings in the poset Hi(x) for

i ∈ {−1, 1} with the same ordering as before. Again, we can consider the poset

H(x) := {(v ,w) ∈ H1(x)× H−1(x) | u = vw ∈ St ∪ Exp ∪ Con ∪Mix}.

. For (v ,w), (y , z) ∈ H(x), let (v ,w) < (y , z) if and only if one of the following holds.

• v < x in H1(x) and w < z in H−1(x).

• v is a finite string, v = x and w < y in H−1(x).

• w is a finite string, v < x in H1(x) and w = y .

Remark 2.4.17. To define the ordering on H(x) we have added an extra condition

that rules out graph maps that are supported on infinitely many vertices of the string.

We do this because in Proposition 2.4.18 we will be considering graph maps between an

infinite-dimensional indecomposable module and a finite-dimensional indecomposable

module. The ‘graph maps’ that are supported on infinitely many degrees would give

rise to infinite sums of elements in the codomain and so are not well-defined. For the

duration of this remark we will refer to such a morphism as an infinite graph map.

Note that the only way an infinite graph map could arise is if it were between

a finite-dimensional band module B(b,M) and an infinite-dimensional string module

C(u) where (one of) the maximal periodic substrings of u is equivalent to b∞. Since

we want the ordering to reflect the morphisms between two such modules, we con-

sider the possible combinations and check that we have not excluded any well-defined

morphisms.


First consider infinite graph maps B(b,M) → C(u). Suppose C(u) is a direct

sum module (or a mixed module such that the contracting maximal periodic substring

of u is equivalent to b∞). Then the infinite graph map would not be a well-defined

morphism. Next suppose C(u) is a direct product module (or a mixed module such

that the expanding maximal periodic substring of u is equivalent to b∞). Then, since

the maximal periodic substring is expanding, there can be no infinite graph map

B(b,M)→ C(u).

Similarly, consider such an infinite graph map C(u) → B(b,M). If C(u) is a

direct product module (or a mixed module such that the expanding maximal periodic

substring of u is equivalent to b∞), then this infinite graph map is not well-defined.

But, again, if C(u) is a direct sum module (or a mixed module such that the contracting

maximal periodic substring of u is equivalent to b∞), then there can be no such graph

map since u is contracting string.

Proposition 2.4.18. Let w be an almost periodic string, u a finite string and let C(w)

and M(u) be the associated (indecomposable, pure-injective) string modules.

1. There exists a non-zero morphism M(u)→ C(w) if and only if u < w or u < w−

in H(x) for some x ∈ Q0.

2. There exists a non-zero morphism C(w)→M(u) if and only if w < u or w < u−

in H(x) for some x ∈ Q0.

3. Let c ∈ Ba and let l ∈ k∗, j ∈ N. Then there exists a non-zero morphism

B(c,Ml,j)→ C(w) if and only if (∞cc2, c1c∞) < w or (∞cc2, c1c∞) < w− in H(x)

for some x ∈ Q0 and c = c2c1.


C(w)→ B(c,Ml,j) if and only if w < (∞cc2, c1c∞) or w− < (∞cc2, c1c∞) in H(x)

for some x ∈ Q0 and c = c2c1.

Let M be an indecomposable infinite-dimensional pure-injective k[T, T−1]-module. That

is M = Mi,k for i ∈ {∞,−∞}, k ∈ k∗ or M = G (see Example 1.2.1). Let b ∈ Ba.

5. Let u be a finite string. Then there exists a non-zero morphism M(u)→ B(b,M)

if and only if u < (∞bb2, b1b∞) in H(x) for some x ∈ Q0 and b2b1 = b.

2.5. THE ZIEGLER SPECTRUM OF A DOMESTIC STRING ALGEBRA 67

6. Let u be a finite string. Then there exists a non-zero morphism B(b,M)→M(u)

if and only if (∞bb2, b1b∞) < u in H(x) for some x ∈ Q0 and b2b1 = b.


B(c,Ml,j)→ B(b,M) if and only if one of the following two conditions hold.

(a) We have (∞cc2, c1c∞) < (∞bb2, b1b∞) for some c2c1 = c and b2b1 = b.

(b) We have c = b and M = Ml,∞.


B(b,M)→ B(c,Ml,j) if and only if one of the following two conditions hold.

(a) We have (∞bb2, b1b∞) < (∞cc2, c1c∞) for some c2c1 = c and b2b1 = b.

(b) We have c = b and M = Ml,−∞.

Proof. In each case, it is immediate that, if the strings are ordered as stated, then there

will be a morphism as stated since we may construct a graph map as we described in

Sections 2.4.2, 2.4.3 and 2.4.4.

For the converse, we may use the same arguments as in [29] (or for string modules

[19, Lem. 1.4]) to obtain that if there is a morphism as stated in each case, then there

is an underlying common substring of the associated strings in H(x) for some x ∈ Q0

such that the end-points satisfy the conditions implied by the ordering on strings. In

Remark 2.4.17 we have already seen that this underlying common substring cannot be

infinite.

2.5 The Ziegler spectrum of a domestic string al-

gebra

As before, we will let A = kQ/I be a string algebra over an algebraically closed field

k. We say that A is domestic if there are only finitely many bands over A. Since it

follows from [16] that the one-parameter families of tubes in A-mod are indexed by

the bands, this notion of domestic is the same as the more general definition given in

Section 1.4.


Proposition 2.5.1 ([50, Prop. 2]). Let A be a string algebra. The following conditions

are equivalent:

1. A is domestic.

2. For each arrow a ∈ Q1, there is at most one band b = bn . . . b1 such that bn = a.

3. Any infinite string is almost periodic or periodic.

Following this result, Ringel conjectures that the indecomposable pure-injective

modules described in the preceding sections form a complete list of indecomposable

pure-injective modules over a domestic string algebra. More recently, Prest and Punin-

ski have confirmed this conjecture to obtain the following theorem.

Theorem 2.5.2 ([48, Thm. 5.1]). Let A be a domestic string algebra over an alge-

braically closed field k. Then the following is a complete list of the points of the Ziegler

spectrum AZg of A-Mod.

1. M(u) for all u ∈ St.

2. M(b,Φ) for all b ∈ Ba and each point (U,Φ) of the Ziegler spectrum k[T,T−1]Zg

of k[T, T−1]-Mod.

3. M(v) for all v ∈ Con.

4. N(v) for all v ∈ Exp.

5. L(v) for all v ∈ Mix.

In the same paper Prest and Puninski prove that there is no superdecomposable

pure-injective module over a domestic string algebra.

Theorem 2.5.3 ([48, Thm. 5.6]). Let A be a domestic string algebra over an alge-

braically closed field. Every pure-injective module over A contains an indecomposable

direct summand.

Remark 2.5.4. It follows from Proposition 1.3.10 that the isolation condition holds for

a domestic string algebra A. Thus, by Proposition 1.3.8, the Krull-Gabriel dimension

of (A-mod,Ab)fp

is equal to the Cantor-Bendixson rank of AZg.

2.5. THE ZIEGLER SPECTRUM OF A DOMESTIC STRING ALGEBRA 69

2.5.1 The transfinite radical and a lower bound for the Krull-

Gabriel dimension of a string algebra

We saw in Section 1.3.6 that there is some connection between the Krull-Gabriel

dimension of (R-mod,Ab)fp

where R is an Artin algebra and the transfinite powers

of the Jacobson radical radR of R-mod. In [54, 55] Schroer establishes the powers of

radA that are zero for a string algebra A and thus provides a lower bound for the


. The first result distinguishes between the

powers of the radical of domestic and non-domestic string algebras.

Theorem 2.5.5 ([54, Thm. 2]). Let A be a string algebra. The following are equivalent.

1. A is non-domestic.

2. The radical of A-mod is such that rad∞A 6= 0.

Let A be a domestic string algebra. In [54] (see also [52]) Schroer proves that the

ordinals α for which radαA = 0 are determined by the bridge quiver QBa defined below.

The idea of the bridge quiver of a domestic string algebra is that it describes the form

of the possible strings in terms of where copies of the bands may occur in a string.

Example 2.5.6. This example illustrates the idea of the bridge quiver. Consider the

quiver

•

w

��•uoo

v

��

with relations w2 = v2 = wuv = 0. Let the set Ba (defined in Section 2.2.4) consist of

the following bands:

b = uvu−1w−1

b−1 = wuv−1u−1.

All of the strings over the corresponding string algebra are substrings of a string of

the form b∞uvu−1b−∞ or b∞uv−1u−1b−∞. We then define the bridge quiver to have

vertices given by the bands and arrows given by the possible strings connecting them.

So in this example, the bridge quiver is the following.

buvu−1

++

uv−1u−1

33 b−1


Let A be a domestic string algebra and let b, c ∈ Ba. We say that a finite string u

is a bridge from b to c if the following conditions are satisfied.

1. The concatenation buc is a string.

2. There is no primitive cyclic word d such that u = v2v1 (where v1 and v2 are

strings of positive length) and v2dv1 is a well-defined string.

We consider the quiver QBa = (QBa0 , QBa

1 ) of A where the vertex set QBa0 is given by

a fixed set of representatives of Ba. For b, c ∈ QBa0 , there is an arrow b → c in QBa

1

whenever there is a bridge u from b to c. This quiver is known as the bridge quiver

of A.

Theorem 2.5.7 ([54, Thms 3.10, 4.3]). Let A be a domestic string algebra. Let n be

the length of a maximal path in QBa, then

radω(n+1)A 6= 0 and rad

ω(n+2)A = 0.

Combined with Corollary 1.3.19, this gives us a lower bound for the Krull-Gabriel

dimension and hence the Cantor-Bendixson ranks of A.

Corollary 2.5.8. Let A be a domestic string algebra and let n be the length of a

maximal path in QBa, then the Krull-Gabriel dimension of A is equal to the Cantor-

Bendixson rank of A and is at least n+ 1.

Chapter 3

The Cantor-Bendixson rank and

Krull-Gabriel dimension of a

domestic string algebra

Throughout this chapter A = kQ/I will denote a domestic string algebra over an

algebraically closed field k and Zg will denote the Ziegler spectrum ZgA of A-mod. We

follow Prest and Puninski [48] in assuming that field is algebraically closed because

our results depend on their classification of the points of Zg, however it is not clear

that this is assumption is required for their classsifcation. the We have already seen

(Remark 2.5.4) that the Cantor-Bendixson rank of Zg is equal to the Krull-Gabriel

dimension of (A-mod,Ab)fp

and also that the simple functors at each stage of the Krull-

Gabriel analysis are in a one-to-one correspondence with the isolated modules at each

stage of the Cantor-Bendixson analysis. Moreover, we have a complete list of the

indecomposable pure-injective modules in A-Mod (see Theorem 2.5.2). We therefore

take the following strategy: we consider each module M ∈ Zg in turn and identify a

family of morphisms such that each F in the corresponding family of finitely presented

functors isolates that module at some stage in the Cantor-Bendixson analysis. We may

then conclude that the functor F is simple at the corresponding stage in the Krull-

Gabriel analysis and we may also deduce the Cantor-Bendixson rank M .

The results in this chapter are contained in paper in [36] which is joint work with

G. Puninski and M. Prest. The methods used here are exactly the same in the paper

but the language in which they are expressed is different. To translate between the

71

72 CHAPTER 3. CB-RANK AND KG-DIMENSION

two the reader should refer to [44, Sec. 3.1].

3.1 The Cantor-Bendixson rank of points

It is well-known that, over a finite-dimensional algebra, the simple functors in

(A-mod,Ab)fp

correspond to the functors F = (M,−)/ im(−, f) where M is an in-

decomposable finite-dimensional module and f : M → N is the left-hand side of an

almost split sequence (see [5, Sec. 2]). Thus the points in Zg with Cantor-Bendixson

rank 0 are exactly the indecomposable finite-dimensional modules.

Proposition 3.1.1. The points of the Ziegler spectrum Zg with Cantor-Bendixson

rank 0 are exactly the finite-dimensional string and band modules.

3.1.1 The Cantor-Bendixson rank of N-string modules

Let y be an N-string. We will identify certain serial functors F such that

(F ) ∩ Zg(α) = {C(y)}

for some ordinal α. A functor F in (A-mod,Ab)fp

is serial if its lattice L(F ) of finitely

presented subfunctors is a linearly ordered. The functors we describe here correspond

to the intervals of the pp lattice described in [46, Lem. 5.3]

Let u be a finite string such that M(u) is not injective and such that u 6= la for any

a ∈ Q1 (see Section 2.4.1 for the definition of la). Let x = s(u).

1. If σ(u) = −1 then suppose u < w in H1(x) and define

fu,w =[f1f2

]: M(u)→M(u+)⊕M(w)

where

• f1 : M(u) → M(u+) is the irreducible morphism described in Section 2.4.1

and

• f2 : M(u) → M(w) is the graph map induced by the ordering on H1(x)

described in Section 2.4.5.

3.1. THE CANTOR-BENDIXSON RANK OF POINTS 73

2. If σ(u) = 1 then ε(u−) = 1 so suppose u− < w in H−1(x) and let

fu,w =[f1f2

]: M(u)→M(+u)⊕M(w)

where

• f1 : M(u) → M(+u) is the irreducible morphism described in Section 2.4.1

and

• f2 : M(u) → M(w) is the graph map induced by the ordering on H−1(x)

described in Section 2.4.5.


x beea 99

with relations ab = b2 = a2 = 0. The string u = ba−b−ab−a is not injective or of the

form ld for any d ∈ Q1. We choose w = a−ba−b−ab−a since u < w in H1(x). The

morphism fu,w is then given by the morphism f1 that may be depicted as follows.

k

1k

��

1k

b��1k

a ��k

1k

��

k

1k

��

1k

b ��

k

1k

��

1k

a��

1k

b ��

k1k

a��k

1k

��

k

k 1k

a ��

1k

b��k k 1k

b ��

k1k

a��k

The component f2 corresponds to the following diagram.

k

1k

��

1k

b��1k

a ��k

1k

��

k

1k

��

1k

b ��

k

1k

��

1k

a��

1k

b ��

k1k

a��

1k

��

k

1k

��

k

1k

��

k 1k

a ��

k1k

b��

1k

a ��k k 1k

b ��

k1k

a��

1k

b ��

k1k

a��k k

Proposition 3.1.3. Let u,w and f = fu,w be as above. Then the functor

F = (M(u),−)/ im(f,−)


is a serial functor and the finitely presented subfunctors of F correspond to

• the interval [u,w) = {y ∈ H1(x) | u ≤ y < w} in H1(x) whenever σ(u) = −1

and

• the interval [u−,w) = {y ∈ H−1(x) | u− ≤ y < w} in H−1(x) whenever σ(u) = 1.

Proof. We give a proof for the case where σ(u) = −1, the case where σ(u) = 1 is

completely analogous.

Let G ⊆ F be a finitely presented subfunctor of F . Then, by Corollary 1.1.3,

G = im([ g

f

],−)/ im(f,−) for some g ≥ f in LM(u). That is, there is some morphism

t such that f = tg. We will show that[ gf

]∼[hf

]in LM(u) where h : M(u)→M(y) is

the graph map induced by u ≤ y < w in H1(x). Clearly any two distinct morphisms

of this from will yield distinct functors and so it suffices to show that all subfunctors

G are of this form.

Let g =n∑i=1

kigi where 0 6= ki ∈ k and gi is a graph map (such a decomposition

must exist by the results in Section 2.4.3) for each 1 ≤ i ≤ n.

First we show that we may assume that gi is not a morphism from a string module

to a band module for any 1 ≤ i ≤ n. By Lemma 1.1.5, it suffices to show that f ≥ gi

for any such gi : M(u) → B(b,M). This is immediate from the description of the

graph maps from string modules to band modules. Suppose v is the string associated

to gi. Then v is a factor substring of u and an image substring of ∞b∞. If u is an

image substring of u+ (i.e. we are in the first case considered in the construction of

Auslander-Reiten sequences with two middle terms in Section 2.4.1), then v is a factor

substring of u+ and so gi ≤ f1 ≤ f in LM(u). If u+ is a factor substring of u, then by

the definition of u+, v must still be a factor substring of u+ and gi ≤ f1 ≤ f . We may

then apply Lemma 1.1.5.

So, without loss of generality, we may assume that M =m⊕j=1

M(uj ) where uj is a

finite string for each 1 ≤ j ≤ m. Next we show that for any gi : M(u) → M(uj) with

1 ≤ i ≤ n and 1 ≤ j ≤ m, either gi is induced by uj ≤ u in H1(x) or gi ≤ f . So

suppose gi is not induced by the ordering in H1(x) and let v be the string associated

to gi. If u = v2vv1 with the length of v1 non-zero, then an argument similar to the

argument given for band modules yields that gi ≤ f1 ≤ f . If u = v2v and uj = y2vy1,

then the length of y1 must be non-zero since we have assumed that gi is not induced by


the ordering on H1(x). As v is an image substring of uj, the last letter of y1 is direct.

Therefore u+ must be obtained from u by adding a hook, but then there must be a

factor substring x of u+ containing v such that x is an image substring of uj. That is

gi ≤ f1 ≤ f .

It follows that for each gi : M(u) → M(uj) we have that uj ∈ H1(u) and gi is the

graph map induced by the ordering u ≤ uj. Clearly, if w ≤ uj, then gi ≤ f2 ≤ f so, by

Lemma 1.1.5, we may assume uj ∈ [u,w). Moreover, the graph map M(u) → M(uj)

induced by u ≤ uj is unique and so we may assume that m = n and g =

[k1g1

...kngn

].

Finally, sinceH1(x) is a linear ordering, there is a minimum element ui of {u1, . . . , un}

with respect to this ordering and so gi ≥ gj for 1 ≤ j ≤ n. Without loss of gener-

ality, suppose u1 is the minimum element. Then it follows from Lemma 1.1.6 that[ gf

]∼[ g1f

]in LM(u) as required.

Corollary 3.1.4. Let F be as in Proposition 3.1.3, then

• (F ) = {C(y) | y ∈ H1(x) such that u ≤ y < w} whenever σ(u) = −1 and

• (F ) = {C(y) | y ∈ H−1(x) such that u− ≤ y < w} whenever σ(u) = 1

where C(y) is the indecomposable pure-injective module associated with the string y.

Proof. Combine the argument in the proof of Propositon 3.1.3 with Lemma 2.4.18.

Let u be a left N-string or a Z-string, then u = u2u1 with u2 = ∞b for some b ∈ Ba.

This band b is uniquely determined by u. As b is a vertex in QBa, we define the left

indent of u to be the length of the maximal length path in QBa ending at b. If u is

a right N-string or Z-string, then define the right indent of u to be the left indent

of u−. If u is a left (respectively right) N-string then we will use the term indent to

refer to the left (respectively right) indent of u.

Theorem 3.1.5. Let y be an N-string with indent n, then the Cantor-Bendixson rank

of C(y) is n + 1 where C(y) is the indecomposable infinite-dimensional pure-injective

A-module associated to y.

Proof. Without loss of generality we assume that y is a left N-string with σ(y) = −1

and let x = s(y). There is some b ∈ Ba such that y = ∞bx for some finite string x .

Let u = bx and w = ◦bx where ◦b is the string obtained from b by removing the last


letter. Since b is a band, this last letter is direct and u < w in H1(x). The interval

contains all strings of the form vu where v is a (possibly infinite) string.

We proceed by induction on the indent n of y . Suppose n = 0. Then for every

vu ∈ [u,w ], the string y is finite unless vu = y . Otherwise there is a path in QBa starting

at b contradicting our assumption that n = 0. Thus, if F = (M(u),−)/ im(fu,w ,−),

then by Corollary 3.1.4 and Proposition 3.1.1 we have

(F ) ∩ Zg(1) = {C(y)}.

Now suppose the statement is true for all string modules C(x ) where x has indent

less than n and suppose y has indent n. Then the infinite strings in [u,w ] are of the

form ∞cvu for some band c ∈ Ba and finite string v . Then, unless ∞cvu = y , the

indent of ∞cvu must be strictly less than n and so, by our induction hypothesis, the

Cantor-Bandixson rank of C(∞cvu) is less than or equal to n. Moreover, all strings

of this form are in [u,w ] so there are infinitely many of Cantor-Bendixson rank n. It

follows that we have

(F ) ∩ Zg(n+1) = {C(y)}

since otherwise we would contradict the compactness of Zg(n) and so the Cantor-

Bendixson rank of C(y) is n+ 1.

Corollary 3.1.6. 1. Let F = (M(u),−)/ im(fu,w ,−) where u < w in H1(x) for

some x ∈ Q0. Then the m-dimension of the interval [u,w ] in H1(x) is equal to

the Krull-Gabriel dimension of F and is m+ 1 where m is the maximum indent

of a string in [u,w ].

2. Let F = (M(u),−)/ im(fu,w ,−) where u− < w in H−1(x) for some x ∈ Q0. Then

the m-dimension of the interval [u−,w ] in H−1(x) is equal to the Krull-Gabriel

dimension of F and is m + 1 where m is the maximum indent of a string in

[u−,w ].

Example 3.1.7. The morphism fu,w defined in Example 3.1.2 gives rise to a functor

F as in the proof of Proposition 3.1.3. Then the interval [u,w ] in H1(x) is the linear

poset

ba−b−ab−a < (ba−)2b−ab−a < (ba−)3b−ab−a < · · · < a−(ba−)2b−ab−a < a−ba−b−ab−a.


Then it is clear that y = ∞(ba−)b−ab−a is the only infinite string in the interval [u,w ]

in H1(x). Thus (F ) ∩ Zg(1) = {C(y)} and so CB(C(y)) = 1.

3.1.2 The Cantor-Bendixson rank of Z-string modules

Consider an almost periodic Z-string y = ∞bv2v1c∞ where ∞b is the left maximal

periodic substring of y and c∞ is the right maximal periodic substring of y . Let

u = bv2v1c and w = ◦bv2v1c◦ where ◦b is the string obtained from b by removing the

last letter and c◦ is the string obtained from c by removing the first letter. Since c

and b are bands, the last letter of b is direct and the first letter of c is inverse so, if

x = e(v1) = s(v2), then bv2 <◦bv2 in H1(x) and v1c < v1c◦ in H−1(x). That is, u < w

in H(x).

Let

gy = [ g1g2 ] : M(u)→M(◦bv2v1c)⊕M(bv2v1c◦)

where g1 and g2 are induced by the posets H1(x) and H−1(x) respectively.

Proposition 3.1.8. Let y , u,w and g = gy be as above. Then the finitely presented

subfunctors of G = (M(u),−)/ im(g,−) are of the form

im(h,−)/ im(g,−)

for h =

[ h1...hm

]: M(u)→

k⊕j=1

M(xj) where xj = x2jx1j for x2j ∈ [bv2,◦bv2) in H1(x) and

x1j ∈ [v1c, v1c◦) in H−1(x) for each 1 ≤ j ≤ m.

Proof. Let H ⊆ G be a finitely presented subfunctor of G. Then by Lemma 1.1.4

H = im([

hg

],−)/ im(g,−) for some h ≥ g in LM(u). So let h =

n∑i=1

kihi : M(u)→ M

be the unique decomposition of h into a linear combination of pairwise different graph

maps hi with 0 6= ki ∈ k for each 1 ≤ i ≤ n.

First note that, by Lemma 1.1.5, we can assume that the finite string associated

to hi is u since otherwise f ≥ hi. If hi is a graph map from a string module to a band

module B(d , N), then ∞d∞ contains the string b1br where b = br . . . b1 and c1cs where

c = cs · · · c1 because u must be closed under successors in ∞d∞. But this contradicts

Proposition 2 in [50, Sec. 11].

So, without loss of generality, assume that M =m⊕j=1

M(xj) for finite strings xj with

1 ≤ j ≤ m and each hi : M(u) → M(xj) has u as its associated string. Suppose for


some xj there is some 1 ≤ i1 < i2 ≤ n such that we have hi1 : M(u) → M(xj) and

hi2 : M(u) → M(xj). Since hi1 and hi2 are distinct graph maps, xj must contain two

distinct copies of u. But then there exists a string x 6= bi for any i > 0 with bxb a

well-defined string. By [54, Lem. 4.2], this is impossible. Thus h =

[ k1h1...

kmhm

]and by

[47, Lem. 2.5], h ∼[ h1

...hm

]as required.

Corollary 3.1.9. Let y , u, w and G be as in Proposition 3.1.8. Then

(G) = {C(x ) | u is an image substring of x }

where C(x ) is the indecomposable pure-injective module associated with the (possibly

infinite) string x .

Proof. Combine the argument in the proof of Proposition 3.1.8 with Lemma 2.4.18.

Theorem 3.1.10. Let y be a Z-string over A with left indent m and right indent

n. The Cantor-Bendixson rank of the module C(y) is m + n + 2 where C(y) is the

indecomposable infinite-dimensional pure-injective A-module associated to y.

Proof. Let y , u,w and G be as in Proposition 3.1.8. We proceed by induction on m+n.

Suppose m = n = 0. Then the strings corresponding to modules in (G) are either

equal to y , finite or N-strings with indent 0. By Theorem 3.1.5

(G) ∩ Zg(2) = {C(y)}

since there are infinitely many N-string modules C in (G) with CB(C) = 1 so if

CB(C(y)) = 1, then this would contradict the compactness of Zg(1). And so we have

CB(C(y)) = 2.

Suppose the statement is true for all Z-strings with left indent p and right indent

q such that p + q < m + n. Then by Theorem 3.1.5 all N-string modules in (G) have

Cantor-Bendixson rank less than or equal to max{m,n}+ 1 < m+ n+ 2. Moreover,

since every string associated to an module in (G) contains bv2v1c, we have that any

Z-string module in (G) is either equal to C(y) or has Cantor-Bendixson rank equal to

p + q + 2 < m + n + 2 for some left indent p ≤ m and right indent q ≤ n such that

not both p = m and q = n hold. Since Zg(m+n+1) is compact and there are infinitely

many Z-string modules with Cantor-Bendixson rank m+ n+ 1, we have

(G) ∩ Zg(m+n+2) = {C(y)}


and so CB(C(y)) = m+ n+ 2.

Corollary 3.1.11. Let y , u, w and G be as in Proposition 3.1.8 and suppose that y has

left indent m and right indent n. Then the Krull-Gabriel dimension of G is m+n+2.

Example 3.1.12. Consider the algebra Λ3 given by the quiver

1b1

66a1((2

c1 // 3b2

66a2((4

c2 // 5b3

66a3((6

with relations c1b1 = b2c1 = c2b2 = b3c2 = 0. Let y = ∞(b2a−2 )c−2 a

−3 (b−3 a

−3 )∞. The

bridge quiver looks like

b1a−1

c−1 a−2 // b2a

−2

c−2 a−3 // b3a

−3

a3b−3

a3c2 // a2b−2

a2c1 // a1b−1

so the left indent of y is 1 and the right indent of y is 0.

By constructing the functor G as in the proof of Theorem 3.1.10 we obtain an

open set of Zg consisting of the string modules in which the finite string module

M(b2a−2 c−2 a−3 b3a

−3 ) embeds. We will only consider the Z-string modules in (G) since

they will attain the greatest rank. There is an infinite family

{xn = ∞(b1a−1 )c−1 a

−2 (b2a

−2 )nc−2 a

−3 (b3a

−3 )∞ | n ∈ N}

such that C(xn) ∈ (G) for each n ∈ N. These correspond to maximal paths in QBa

and so CB(C(xn)) = 2 for each n ∈ N. The only other Z-string module in (G) is C(y)

and so CB(C(y)) = 3.

3.1.3 The Cantor-Bendixson rank of Prufer modules

Next we consider the Prufer modules in Zg. Let k ∈ k∗. Consider the morphism

fn : B(b,Mk,1) → B(b,Mk,n) taking the canonical basis element z(1)i in B(b,Mk,1) to

the canonical basis element z(1)i in B(b,Mk,n) for 1 ≤ i ≤ m. For n > 1, this is the

compositions fn = Fb(ιn−1) · · ·Fb(ι1) where ιi : Mk,i → Mk,i+1 is defined in Example

1.2.1 for i ∈ N.

Let v = v2bv1 be the substring of ∞b∞ of maximal length such that v1 is direct and

v2 is inverse. Let h : B(b,Mk,1) → M(v) be the (unique) graph map with z(1)m 7→ zv

where zv is the canonical basis element of M(v) between b and v1.


Proposition 3.1.13. Let h : B(b,Mk,1)→M(v) be as above and consider the functor

H = (B(b,Mk,1),−)/ im(h,−). The finitely presented subfunctors of H are of the form

G = im ([ gh ] ,−) / im(h,−)

where g = [ g1g2 ] : B(b,Mk,1)→ N1 ⊕N2 with

1. the module N1 =m⊕i=1

M(ui) where ui contains v as a factor substring for 1 ≤ i ≤

m and g1 =n∑j=1

kjg1j with g1j : B(b,Mk,1) → M(ui) is induced by ∞b∞ < ui < v

in H(s(b)); and

2. the module N2 = B(b,Mk,n) for some n ≥ 1 and g2 = fn.

Proof. By Lemma 1.1.4, every finitely presented subfunctor G ⊆ H is of the form

G = im ([ gh ] ,−) / im(h,−)

for some g : B(b,Mk,1) → N with g ≤ h. Let g = [ g1g2 ] : B(b,Mk,1) → N1 ⊕N2 where

N ∼= N1⊕N2 and N1 is a direct sum of string modules and N2 is a direct sum of band

modules.

We begin by considering g2. Let g2 =t2∑i=1

kig2i be its decomposition into a linear

combination of graph maps. If there exists some g2i : B(b,Mk,1)→ B(c, N) that is not

in the image of Fb , then there is some finite factor substring of u of ∞b∞ associated

to g2i. Moreover, it must contain v as a factor substring since ∞b∞ < u < v but this

means that c = b or c = b− because there is at most one band starting with b1 (see

[50]). But then u is a factor substring and an image substring of ∞b∞ containing b

which is impossible. So every g2i is in the image of Fb and, moreover, we must have

g2 =

[k1g21

...ktg2t2

]since g2i = fni for some ni ≥ 1 for each 1 ≤ i ≤ t2 is the unique

morphism B(b,Mk,1) → B(b,Mk,n) (up to scalar multiplication) in the image of Fb .

Let ni be the least integer such that g2i = fni . Then g2i ≤ kjg2j for all 1 ≤ j ≤ t2 and

so by Lemma 1.1.6, we may assume that g2 = g2j as required.

Now consider g1 : B(b,Mk,1) → N1 and let g1 =t1∑i=1

kig1i be the decomposition

into a linear combination of graph maps. Suppose N1 =m1⊕j=1

M(uj) and consider

g1i : B(b,Mk,1) → M(uj). Then by Lemma 1.1.5, we may assume g1i < h and so

∞b∞ ≤ uj ≤ v and uj must therefore contain v as a factor substring. Thus g1 is of the

form described in (1).


Corollary 3.1.14. Let H be as in Proposition 3.1.13. Then

(H) = {B(b,Mk,i) | i ≥ 1} ∪ {B(b,Mk,∞)} ∪ {C(u) | ∞b∞ < u < v in H(s(b))}.

where C(u) is the indecomposable pure-injective module associated to the (possibly

infinite) string u.

Proof. Combine the proof of Proposition 3.1.13 with Proposition 2.4.18.

Let u = ∞bv be a left N-string or a Z-string. Then the left upper-indent of u is

the maximal length of a path un . . . u1 in QBa ending at b such that b < u1 in H1(x)

and x = s(b) = s(u1). The left lower-indent of u is the maximal length of a path

un . . . u1 in QBa ending at b such that u1 < b in H1(x).

Let u be a right N-string or a Z-string. The right upper-indent and right

lower-indent of u is defined to be the left upper-indent and left lower indent of u−

respectively.

Theorem 3.1.15. The Cantor-Bendixson rank of B(b,Mk,∞) for a band b is n+m+1

where n is the left upper-indent of ∞b∞ and m is the right upper-indent of ∞b∞.

Proof. Consider (H) and note that in the preceding sections we have calculated the

Cantor-Bendixson rank of all the modules in this open set apart from the Prufer

module. The modules of highest Cantor-Bendixson rank (apart from B(b,Mk,∞)) are

Z-string modules with left indent n − 1 and right indent m − 1. By Theorem 3.1.10,

these Z-string modules have Cantor-Bendixson rank m+ n. Since there are infinitely

many of them and Zg(m+n) is compact, it follows that (H)∩Zg(m+n+1) = {B(b,Mk,∞)}

and so the CB(B(b,Mk,∞)) = m+ n+ 1.

3.1.4 The Cantor-Bendixson rank of adic modules

We will use elementary duality to apply the results about Prufer modules directly to

the adic modules. We have been considering left modules over the path algebra A

of the bound quiver (Q, I). To consider right A-modules we can consider left Aop

-

modules. The algebra Aop

is obtained by taking the path algebra over the opposite

quiver Qop

with Qop

0 = Q0 and Qop

1 = {aop: j → i | a : i → j ∈ Q1}; the relations are

reversed in the obvious way. Then Aop

is a domestic string algebra so all of the results

we have proved so far can be applied to right A-modules.


Let u = um . . . u1 be a string and define the opposite string uop= u

op

1 . . . uop

m by

extending the (−)op

operation to formal inverses of arrows. Note that if b is a band,

then bopis a cyclic rotation of a band over A

op.

The definitions and results referred to in the following proposition can be found in

Section 1.5.

Proposition 3.1.16. Elementary duality gives a homeomorphism D : AZg → ZgA.

The following list shows how D acts on infinite-dimensional modules in AZg.

1. Let b be a band. The Prufer modules over b are dual to the adic modules over

a band cyclically equivalent to bopand the adic modules over b are dual to the

Prufer modules over a band cyclically equivalent to bop. The generic module over

b is dual to the generic module over a band cyclically equivalent to bop.

2. Let u ∈ Con. Then M(u) is dual to N(uop) and uop

is an expanding string.

3. Let u ∈ Exp. Then N(u) is dual to M(uop) and uop

is a contracting string.

4. Let u ∈ Mix. Then L(u) is dual to the mixed module R(uop).

Proof. 1. Let b be a band and let 0 6= k ∈ k. An easy calculation yields that

B(b,Mi,k)∗ ∼= B(c,Mi,k−1) for each i ≥ 1 where c is a band over A

opthat is some

cyclic rotation of bop. Consider the left A-module B(b,M∞,k) i.e. the Prufer

module associated to b. ThenB(b,M∞,k) = lim−→ iB(b,Mi,k) and since Homk(−,k)

sends colimits in A-Mod to limits in Mod-A, it follows that B(b,M∞,k)∗ is equal

to B(c,M−∞,k−1). So, by Corollary 1.5.6, D(B(b,M∞,k)) = B(c,M−∞,k−1) and

D(B(c,M−∞,k−1)) = B(b,M∞,k).

Finally, modules M with finite endolength are always reflexive and D(M) also

has finite endolength (see, for example, [44, Cor. 5.4.17]) and so, if we consider

the closed subset given by the image of Fb , we can apply Corollary 1.5.3 and

conclude that D(B(b, G)) = B(c, G) and D(B(c, G)) = B(b, G) where G is the

generic module over k[T, T−1].

2. It is immediate that if u = ∞bv is a contracting N-string, then uop is an expanding

N-string. Also, it follows directly from the definitions that M(biv)∗ ∼= M((biv)op

)


for each i ≥ 1. Since Homk(−,k) sends colimits to limits, we have

M(u)∗ ∼= lim←− iM((biv)op

) ∼= N(uop

).

Therefore, by Corollary 1.5.3, D(M(u)) = N(uop).

Similarly u = ∞bvc∞ then M(u) = lim−→ iM(ibvci) and so by the same argument

D(M(u)) = N(uop) and D(N(uop

)) = M(u).

3. This follows from the arguments in the proof of 2..

4. Let u = ∞bvc∞ ∈ Mix. Then L(u) = lim−→ iN(∞bvci) so,

L(u)∗ = lim←− iN(∞bvci)∗ = lim←− iM((∞bvci)op

) = R(uop

).

Again, it follows that D(L(u)) = R(uop) and D(R(uop

)) = L(u).

Theorem 3.1.17. The Cantor-Bendixson rank of B(b,M−∞) for a band b is m+n+1

where n is the left lower-indent of ∞b∞ and m is the right lower-indent of ∞b∞.

Proof. We consider the Prufer module B(b ′, N∞) over Aop

such that B(b ′, N∞)∗ is

equal to B(b,M−∞). Then

(H) = {B(c, Ni) | i ≥ 1} ∪ {B(c, N∞)} ∪ {C(u) | ∞(c)∞ < u < v in H+(s(c))}

where C(u) is the indecomposable pure-injective module associated to the (possibly

infinite) string u and v = v2cv1 as defined at the beginning of the previous section. It

is immediate from Proposition 3.1.16 that

(dH) = {B(b,Mi) | i ≥ 1} ∪ {B(b,M−∞)} ∪ {C(uop

) | vop

< uop

< ∞b∞ in H+(s(b))}.

Thus the modules in (dH) with greatest Cantor-Bendixson rank are Z-string modules

C(w) with left lower-indent m−1 and right lower-indent n−1, so by Theorem 3.1.10,

CB(C(w)) = m+ n. Since there are infinitely many such modules, it follows that the

Cantor-Bendixson rank of B(b,M−∞) is m+ n+ 1.


3.1.5 The Cantor-Bendixson rank of generic modules

Let b ∈ Ba, u = b and v = v2bv1 as described in Section 3.1.3. Let g : M(u) → M(v)

be the graph map induced by u < v in H(x) for some x ∈ Q0.

Proposition 3.1.18. Let k be as above and consider K = (M(u),−)/ im(g,−). Then

the finitely presented subfunctors F of K are of the form F = im([

fg

],−)/ im(g,−)

where f =n∑i=1

kifi with 0 6= ki ∈ k and fi a graph map with associated string u. Also

(K) = {B(b,Ml,i) | i ∈ N∪{∞,−∞}}∪{M(w) | u ≤ w < v in H(s(b))}∪{B(b, G)}.

Proof. By Lemma 1.1.4 and Lemma 1.1.5, the finitely presented subfunctors F of K

are of the form F = im([

fk

],−)/ im(k,−) where f ≥ k and if f =

n∑i=1

kifi is the

unique decomposition of f into a linear combination of graph maps with ki 6= 0 for

all 1 ≤ i ≤ n, then fi ≥ k for each 1 ≤ i ≤ n. The description of graph maps implies

that the graph maps with fi ≥ k must have u as the associated string.

Similarly, the description open set (K) is immediate from the description of graph

maps. In particular, there can be no band modules B(c,M) with c 6= b and c 6= b−

since b cannot be an image substring of ∞(c)∞.

Theorem 3.1.19. Let b ∈ Ba. The Cantor-Bendixson rank of the generic module

B(b, G) is m+ 2 where m is the maximal length of a path in QBa passing through b.

Proof. If l is the left indent of ∞b∞ and r is the right indent of ∞b∞, then m = r + l.

The modules of highest Cantor-Bendixson rank in (K) (apart from B(b, G)) are Z-

string modules with left indent l and right indent r − 1 (or with left indent l − 1

and right indent r). By Theorem 3.1.10, these modules have Cantor-Bendixson rank

r + l + 1 = m+ 1. Since there are infinitely many of them and Zg(m+1) is compact, it

follows that CB(B(b, G)) = m+ 2.

3.2 The Cantor-Bendixson rank of the Ziegler spec-

trum and Krull-Gabriel dimension of the func-

tor category

It follows immediately from the previous sections that we have the following theorem.

3.2. THE CB RANK AND KG DIMENSION OF A STRING ALGEBRA 85

Theorem 3.2.1. Let A be a domestic string algebra and let m be the maximum length

of a path in the bridge quiver QBa. Then the Krull-Gabriel dimension of (A-mod,Ab)fp

is equal to the Cantor-Bendixson rank of AZg and is equal to m+ 2.

Proof. By Theorem 2.5.2 and Remark 2.5.4 the Cantor-Bendixson rank and the Krull-

Gabriel dimension are equal. Moreover, this value is given by the maximum rank of

the points in AZg. Thus it follows from Theorems 3.1.5, 3.1.10, 3.1.15, 3.1.17 and

3.1.19 that the Krull-Gabriel dimension and Cantor-Bendixson rank are both equal to

m+ 2.

Corollary 3.2.2. Conjecture 1.4.6 holds for domestic string algebras.

Proof. Compare the above theorem with Theorem 2.5.7.

Part II

The derived category

86

Chapter 4

The bounded derived category of a

gentle algebra

In this chapter we will describe the indecomposable objects in the bounded derived

category of a gentle algebra A. For the definition of a gentle algebra see Section 2.1.

The classification was first given in [10]; we follow Bekket and Merklen in describing

the indecomposable objects in Kb,−(A-proj) and then referring to the following well-

known equivalence of categories (see, for example, [28, Prop. 6.3.1]).

Kb,−(A-proj) ' Db(A-mod).

As with the module category of a string algebra, the indecomposable objects in

Kb,−(A-proj) correspond to combinatorial objects that resemble strings and bands

and the structure of the complex reflects the structure of the ‘string’ or ‘band’. We

begin this chapter by defining homotopy strings and homotopy bands; as before, these

combinatorial objects have letters that arise from arrows in the quiver but they also

carry data that indicates which degree of the corresponding complex the letter should

sit in. The remainder of the chapter will be dedicated to describing the construction

of the indecomposable objects in Kb,−(A-proj), that is string and band complexes.

Remark 4.0.3. In this second part of the thesis we look at certain triangulated cate-

gories associated to the categories of modules over a ring R; that is, derived categories

and homotopy categories. In practice we will only ever be working directly in homotopy

categories of projective modules. These categories have, as their objects, complexes

of projective modules and, as their morphisms, morphisms of chain complexes up to

87

88 CHAPTER 4. THE DERIVED CATEGORY OF A GENTLE ALGEBRA

homotopy equivalence. To reduce the amount of cumbersome notation, we will make

use of the following abbreviations:

• D := D(R-Mod) denotes the derived category of R-Mod.

• Db := Db(R-mod) denotes the bounded derived category of R-mod.

• K := K(R-Proj) denotes the (unbounded) homotopy category of R-Proj.

• Kb := Kb(R-proj) denotes the homotopy category of bounded complexes of

finitely generated projective R-modules.

• Kb,− := Kb,−(R-proj) denotes the homotopy category of right bounded com-

plexes of finitely generated projective R-modules whose cohomology is bounded.

• Cb,− := Cb,−(R-proj) denotes the category of right bounded complexes of finitely

generated projective R-modules whose cohomology is bounded.

Clearly, the drawback of such abbreviations is that the ring R underlying each

category is no longer indicated; we will fix the ring within each chapter. For the

duration of Chapter 4 and the first part of Chapter 5 (up to and including Section

5.2) we will be considering an arbitrary gentle algebra A. From Section 5.3 until the

end of the thesis we will be considering a fixed derived-discrete algebra Λ of infinite

representation type.

4.1 Homotopy strings and homotopy bands

Throughout this section A = kQ/I will be a gentle algebra over an algebraically closed

field. The letters of the homotopy strings and homotopy bands will be given by direct

and inverse strings over A. For the relevant definitions, see Section 2.2.

4.1.1 Homotopy strings

The definition of a homotopy string can be easier to process when the reader un-

derstands the construction of the string complex to which it corresponds. Therefore

we will include a running example that gives an indication of the construction of the

corresponding string complex; the formal definition is given in Section 4.2.

4.1. HOMOTOPY STRINGS AND HOMOTOPY BANDS 89

Example 4.1.1. Consider the gentle algebra given by the quiver

1

b

��

3f

��0

a

^^

d ��2

c

@@

4

e

OO

with relations ba = cb = ac = df = fe = ed = 0.

We say that a triple (w ,m, n) is a homotopy letter (of positive length) if w

is a direct or inverse string and m,n ∈ Z such that n =

m+ 1 if w is direct

m− 1 if w is inverse.

The key observation for understanding these definitions is the following.

Observation 4.1.2. Let A be a gentle algebra. Then there is a natural bijection

{Paths p : x y in Q} ←→ {Basis elements of HomA-mod(P (y), P (x))}

where P (x), P (y) denote the projective covers of the simple at x and y in Q0 respec-

tively.

Recall that there are bases of P (x) and P (y) corresponding to paths starting at

x and y respectively. Let u ∈ P (y) be such a basis element, then up ∈ P (x). The

bijection is given by taking the path p to the morphism p 7→ up. For the remainder of

this document we will consider these paths to be direct strings and we will identify the

morphism p 7→ up with the string p. Moreover, paths of length zero will be represented

by 1 (this should be thought of as multiplication by the identity in k).

Remark 4.1.3. Since we read paths from right to left, we will compose morphisms

between projective modules from left to right. That is, we will use the following

convention for the presentation of matrices and matrix multiplication: all matrices

will be transposed and if A and B are such matrices, then AB will mean “do A

then do B”. For example if we consider the algebra given in Example 4.1.1. Then

the morphisms between projective modules in the following are written as transposed

matrices.

P (1)⊕ P (4)( ad )−→ P (0)

( f c )−→ P (3)⊕ P (2)

To see why this notation is preferable consider that the multiplication is then written

as ( ad ) ( f c ) =(af acdf dc

)=(af 00 dc

).


Example 4.1.4. Consider the algebra given in Example 4.1.1. Then (af, 0, 1) is a

homotopy letter. The integers m and n refer to degrees of the corresponding string

complex. For example, the homotopy letter (af, 0, 1) gives rise to a complex which is

only non-zero in degrees 0 and 1 and there is one non-zero differential map given by

the path af .

A homotopy string (of length b > 0) is a triple W = (wb · wb−1 · . . . · w1,m, n)

where w1, . . . ,wb are direct or inverse strings and m,n ∈ Z such that W is produced

as a result of the following induction: First we take homotopy letters to be homotopy

strings and, if W = (wb · wb−1 · . . . · w1,m, n) and U = (ua · ua−1 · . . . · u1, i, j) are

homotopy strings, then their concatenation W ◦U = (wb · . . . · w1 · ua · . . . · u1,m, j) is

a homotopy string whenever

1. i = n;

2. s(w1) = e(ua);

3. if w1 and ua are both direct (respectively both inverse) then w1ua ∈ I (respec-

tively (w1ua)− ∈ I; and

4. if one of w1 and ua is direct and the other is inverse, then u 6= w− where u is the

last letter of ua and w is the first letter of w1.

Example 4.1.5. Consider the algebra given in Example 4.1.1. Then (b · af, 0, 3) and

(e− · f− · c, 2, 1) are both homotopy strings. Again, the integers indicate degrees of the

complex. For example, the complex arising from (b · af, 1, 3) is the following.

P (2) b // P (1)af // P (3)

where P (2) is in degree 1, P (1) is in degree 2 and P (3) is in degree 3. The non-zero

differentials are given by ‘composition with the paths b and af ’; thus the relation

between b and af means that the differential squares to zero.

Similarly, the complex arising from (e− · f− · c, 2, 1) has the following non-zero

degrees 0, 1 and 2.

P (0)( c f ) // P (2)⊕ P (3)

( 0e )// P (4)


We also introduce homotopy strings (of length zero). These are triples (1t,m,m)

where m ∈ Z and 1t is the string of length zero associated to the vertex t ∈ Q0

(from now on we will only consider one string of length zero for each vertex in

the quiver). We extend the functions s and e to homotopy strings in the obvious

way. That is s(wb · · · · · w1,m, n) = s(w1), e(wb · . . . · w1,m, n) = e(wb) and finally

s(1t,m,m) = e(1t,m,m) = t.

Next we define composition of homotopy strings. Let

W = (wb · wb−1 · . . . · w1,m, n) and U = (ua · ua−1 · . . . · u1, i, j)

be homotopy strings such that s(W ) = e(U) and n = i. Then the composition WU

is a defined to be W ◦ U as above, or WU = (wb · . . . · w1ua · . . . · u1,m, j) where w1ua

is either a direct or an inverse string. If U = (ua · ua−1 · . . . · u1, i, j) is a homotopy

string, then we can compose (1t,m,m) to the left of U if i = m and e(U) = t. We

can compose (1t,m,m) to the right of U if j = m and s(U) = t. In both cases, the

resulting homotopy string is U.

For a homotopy string W = (wb · . . . · w1,m, n) of length b > 0, we define the

inverse of W to be the homotopy string W − := (w−1 · . . . · w−b , n,m). If (1t,m,m)

is a homotopy string of length zero, then the inverse homotopy string is defined to be

(1t,m,m)− := (1t,m,m).

To each homotopy string W = (wb · . . . · w1,m, n) of length b > 0, we attach the

diagram below; we will refer to it as the unfolded diagram of W .

mb+1 mb m2 m1

•w∗b • · · · •

w∗1 •

Where w∗i is the direct string out of {wi,w−i }, mb+1 = m, m1 = n and

mi+1 =

mi − 1 if wi is direct

mi + 1 if wi is inverse.

The line •w∗i • represents an arrow •

w∗i // • when wi is direct and an arrow

• •w∗ioo when wi is inverse.

Example 4.1.6. Consider the algebra given in Example 4.1.1. The homotopy string

(e− · f− · c · b · af, 2, 3) is the concatenation of the strings given in Example 4.1.5 and


has the following unfolded diagram.

2 1 0 1 2 3

• •eoo •foo c // • b // • af // •

The string complex given by (e− · f− · c · b · af, 2, 3) can be found in Example 4.2.2.

To each homotopy string (1t,m,m) of length zero, we associate the unfolded dia-

gram below.

m•

4.1.2 Homotopy bands

A homotopy string W = (wb · . . . · w1,m, n) of length b > 0 is called a homotopy

band if

1. s(W ) = e(W );

2. m = n;

3. one of w1 and wb is direct and the other is inverse; and

4. W is not a proper power of another homotopy string satisfying (1) and (2).

The extra restrictions on the band mean that the first letter and the last letter end

in the same vertex Q, but also that the homotopy string starts and ends in the same

degree of the complex. This means that we can construct a complex in which the ends

of the string are identified This can be seen in the following example.

Example 4.1.7. Consider the algebra given in Example 4.1.1. Then the homotopy

string (d− · e− ·d− · c · b ·a, 0, 0) is a homotopy band. The corresponding string complex

is the following (the matrices are transposed, see Remark 4.1.3).

P (0)( c f ) // P (2)⊕ P (3)

( b 00 e )// P (1)⊕ P (4)

( a 00 d )// P (0)⊕ P (0)

The corresponding band complex is given in Example 4.2.6; the definition of a ho-

motopy band allows us to identify the copies of P (0) and to adjust the differential

accordingly.


To each homotopy band W = (wb · . . . ·w1,m, n) we associate the following infinite

repeating unfolded diagram.

mb+1 mb m2 m1

. . .w∗1 •

w∗b • · · · •w∗1 •

w∗b. . .

For each 1 ≤ i ≤ b, we take mi, w∗i and •w∗i • to be as in the definition of an

unfolded homotopy string above.

Example 4.1.8. Consider the algebra given in Example 4.1.1 and the homotopy band

(d− · e− · d− · c · b · a, 0, 0). The unfolded diagram is as follows.

0 −1 −2 −3 −2 −1

. . . a // • •doo •eoo •doo c // • b // • a // . . .

4.1.3 Resolvable homotopy strings

The homotopy strings and homotopy bands we have seen so far give rise to bounded

complexes. The final class of homotopy strings give rise to complexes that are bounded

in cohomology but are unbounded to the left.

Let C(A) be the set of arrows a ∈ Q1 such that there is a path aa1 . . . at in Q with

s(at) = e(a), aa1 ∈ I and aiai+1 ∈ I for all 1 ≤ i < t. Note that C(A) is non-empty if

and only if A has infinite global dimension.

Let W = (wb · . . . · w1,m, n) be a homotopy string of length b > 0 such that W is

the concatenation of the following homotopy letters:

(wb,m = mb, nb), (wb−1,mb−1, nb−1), . . . , (w1,m1, n = n1).

Then W is left resolvable if wb is direct, m ≤ mi, ni for all 1 ≤ i ≤ b and there

exists a ∈ C(A) such that (a ·wb · . . . ·w1,m− 1, n) is a homotopy string. We say that

a resolves W . If, in addition, wb is a string of length > 1, then we say that W is

primitive left resolvable. We say that W is (primitive) right resolvable if W −

is (primitive) left resolvable. If W is both (primitive) left resolvable and (primitive)

right resolvable then we say that W is (primitive) two-sided resolvable.

For each primitive left resolvable string W = (wb · . . . · w1,m, n), there is a unique

cyclic path al...a1 with ai+1ai ∈ I for each 1 ≤ i < l such that a1 resolves W . Consider

the sequence of homotopy strings

A1 = (a1,m− 1,m)


A2 = (a2 · a1,m− 2,m)

...

Al = (al · . . . · a1,m− l,m)

Al+1 = (a1 · al · . . . · a1,m− l − 1,m)

...

Then AiW is defined for each i ≥ 1 and we let ∞W denote the combinatorial limit of

this sequence. The corresponding unfolded diagram is below.

mb+3 mb+2 mb+1 mb m2 m1

. . . •a3 a2 • a1 •w∗b • · · · •

w∗1 •

Where mb+2 = mb+1− 1 and mb+3 = mb+1− 2 etc. We can define the strings W∞ and

∞W∞ for primitive right and two-sided resolvable strings in an analogous way.

Example 4.1.9. Consider the algebra given in Example 4.1.1. The homotopy string

W = (dc · b,−2, 0) is primitive left resolvable by e. The unfolded diagram of ∞W is

the following.

−5 −4 −3 −2 −1 0

. . . e // • d // • f // • e // • dc // • b // •

An explanation of the terminology ‘resolvable’ will be given in Section 4.2.1.

4.1.4 Equivalence relations

The homotopy strings and homotopy bands give rise to indecomposable complexes in

Kb,−. To obtain a parametrisation of the indecomposable objects in Kb,−, we must

consider equivalence classes of homotopy strings and homotopy bands.

Homotopy strings We say W∼−1U for finite homotopy strings U,W if and only

if W = U− or W = U. Let HSt be a fixed set of ∼−1-equivalence classes of

homotopy strings.

Homotopy bands We say that W∼rU for homotopy bands U,W if and only if W

is the concatenation of the following homotopy letters:

(wb,m = mb, nb), (wb−1,mb−1, nb−1), . . . , (w1,m1, n = n1)

4.2. INDECOMPOSABLE COMPLEXES 95

and U is the concatenation of the following homotopy letters:

(wi,mi, ni), . . . , (w1,m1, n1), (wb,mb, nb), . . . , (wi+1,mi+1, ni+1)

for some 1 ≤ i < b.

We say that W∼bU for homotopy bands W , U if and only if W∼rU or W∼−1U.

Let HBa be a fixed set of representatives of ∼b-equivalence classes of homotopy

bands.

Resolvable homotopy strings Let L tRt T be the disjoint union of fixed sets of

representatives of ∼−1-equivalence classes of primitive left, right and two-sided

resolvable strings respectively. Then let

HRes := {W∞ | W ∈ R} ∪ {∞W | W ∈ L} ∪ {∞W∞ | W ∈ T }.

4.2 Indecomposable complexes

In this section we will describe the indecomposable objects in the bounded derived cat-

egory Kb,−(A-proj) where A is a gentle algebra. In analogy with the module category

of a string algebra, the objects will correspond to homotopy strings and homotopy

bands. We will reach the following theorem due to Bekkert and Merklen [10].

Theorem 4.2.1. There is a bijection

{Indecomposable objects in Db(A-mod)} ←→ HSt t (HBa× k∗ × N) t HRes.

Moreover, the perfect complexes (i.e. those contained in the canonical copy of Kb(A-proj))

are exactly those corresponding to the elements of HSt t (HBa× k∗ × N).

4.2.1 String complexes

Let W = (wb · . . . · w1,m, n) be a homotopy string of length b > 0 such that W is the

concatenation of the following homotopy letters:

(wb,m = mb, nb), (wb−1,mb−1, nb−1), . . . , (w1,m1, n = n1).


Then, for each i ∈ Z, let

Ii =

{r | mr = i} t {0} if i = n

{r | mr = i} otherwise.

The string complex PW associated to W is the complex with P iW :=

⊕r∈Ii

P (θW (r))

in degree i where θW (r) = t(wr) for r > 0 and θW (0) = s(w1) and zero in all other de-

grees. The differentials are matrices where the entries are determined by the homotopy

letters of W : for each direct wr there is a non-zero entry P (θW (r))w∗r−→ P (θW (r − 1))

and for every inverse wr there is a non-zero entry P (θW (r − 1))w∗r−→ P (θW (r)). The

remaining entries in the matrices are zero.

Example 4.2.2. Consider the algebra given in Example 4.1.1 and consider the ho-

motopy string given in Example 4.1.6 with the following unfolded diagram.

2 1 0 1 2 3

• •eoo •foo c // • b // • af // •A conceptually useful intermediate diagram is the following.

0 1 2 3

• b // • af // ••

c 44

f** • e // •

Then the string complex arising from this homotopy string is the following.

P (0)( c f ) // P (2)⊕ P (3)

( b 00 e )// P (1)⊕ P (4)

(af0

)// P (3)

The non-zero degrees are 0, 1, 2 and 3.

If W is primitive left resolvable then the set Ii is finite for each i ∈ Z and so the

string complex P∞W is defined in exactly the same way. Similarly, we can define PW∞

and P∞W∞ for primitive right and two-sided resolvable strings W respectively.

Example 4.2.3. Consider the algebra given in Example 4.1.1 and consider the left

resolvable string with the following unfolded diagram.

−5 −4 −3 −2 −1 0

. . .e // • d // • f // • e // • dc // • b // •

Then the string complex associated to this infinite string is the following.

. . . e // P (4) d // P (0)f // P (3) e // P (4) dc // P (2) b // P (1)


Remark 4.2.4. The terminology ‘resolvable’ refers to the fact that such strings are

projective resolutions of certain complexes. For example, the string complex in Ex-

ample 4.2.3 is the projective resolution of the complex

ker(dc) // P (4) dc // P (2) b // P (1)

Let U = (1t,m,m) be a homotopy string of length 0. Then the string complex

PU associated to U is the complex with P (t) in degree m and zeroes elsewhere.

Remark 4.2.5. Let W be an infinite length homotopy string (that does not necessarily

arise from a resolvable string). As long as the set Ii is finite for every i ∈ Z, the above

construction makes sense. We can therefore extend the above definition of a string

complex to incule PW for such homotopy strings W .

4.2.2 Band complexes

Next we describe the construction of band complexes. The construction is similar to

that of a string complex, but as with the analogous modules, we introduce a parameter

so there is a k∗-indexed family of band complexes for every band.

Let W = (wb · . . . ·w1,m,m) be a homotopy band such that W is the concatenation

of the following homotopy letters:

(wb,m = mb, nb), (wb−1,mb−1, nb−1), . . . , (w1,m1,m = n1).

Then, for each j ∈ Z, let Jj = {r | mr = j}. Let k ∈ k∗. The 1-dimensional band

complex BW ,k,1 associated to W and k is the complex with

BjW ,k,1 =

⊕r∈Jj

P (e(wr))

in degree j. If b > 1 then the differential is described in the same way as for a string

complex except that the entry corresponding to w1 is P (e(w1))kw1−→ P (s(w1)) when w1

is direct and P (s(w1))kw−1−→ P (e(w1)) when w1 is inverse. If b = 1 and w1 is direct, then

the only non-zero differential is P (e(w1))kw1+w−2−→ P (s(w1)) in degrees m− 1 and m. If

b = 1 and w1 is inverse, then the only non-zero differential is P (s(w1))kw−1 +w2−→ P (e(w1))

in degrees m and m+ 1.


Example 4.2.6. Consider the algebra given in Example 4.1.1 and consider the band

(d− · e− · f− · c · b · a, 0, 0) as in Example 4.1.7 with the unfolded diagram.

0 −1 −2 −3 −2 −1 0

. . . a // • •doo •eoo •foo c // • b // • a // . . .

As with the string complex, the following diagram is useful in understanding the

construction of the band complex.

−3 −2 −1 0

• b // • a++•

c 33

f ++•

• e// • d

33

Let k ∈ k∗, then the band complex defined by this band is the following.

P (0)( c f ) // P (2)⊕ P (3)

( b 00 e )// P (1)⊕ P (4)

( kad )// P (0)

Let Dj : BjW ,k,1 → Bj+1

W ,k,1 be the jth differential map in BW ,k,1 for each j ∈ Z and

let r > 1. Then the r-dimensional band complex associated to W and k ∈ k∗ is

the complex with the direct sum (BjW ,k,1)(r) of r copies of Bj

W ,k,1 in degree j. The jth

differential map is given by the matrix

DW ,k,r =

Dj Aj 0 . . . 0

0 Dj Aj . . . 0...

.... . . . . .

...

0 . . . Dj Aj

0 0 . . . 0 Dj

where, if we write djm,n for the (m,n)th entry of Dj, then we define the (m,n)th entry

of Aj to be

ajm,n =

w1 if djm,n = kw1

w−1 if djm,n = kw−1

0 otherwise.

for b > 1. If b = 1, then

ajm,n =

w1 if djm,n = kw1 + w−2

w−1 if djm,n = kw−1 + w2

0 otherwise.

Note that all matrices here are transposed (see Remark 4.1.3).


Example 4.2.7. Let everything be as in Example 4.2.6. Then the 3-dimensional band

complex is the following.

P (0)3 d1 // (P (2)⊕ P (3))3 d2 // (P (1)⊕ P (4))3 d3 // P (0)3

where d1 and d1 are the block matrices with ( c f ) and ( b 00 e ) respectively on the diagonal

and

d3 =

ka a 0

d 0 0

0 ka a

0 d 0

0 0 ka

0 0 d

To each r-dimensional band complex we associate the following unfolded diagram

to BW ,k,r:

Layer r: · · · kw1 //w1

$$

• wb •wb−1 · · · w2 • kw1 //

w1

##

• wb · · ·

Layer r − 1: · · · kw1 // • wb •wb−1 · · · w2 • kw1 // • wb · · ·

......

Layer 2: · · · kw1 //w1

$$

• wb •wb−1 · · · w2 • kw1 //

w1

##

• wb · · ·

Layer 1: · · · kw1 // • wb •wb−1 · · · w2 • kw1 // • wa · · ·

We refer to the arrows labelled v1 that go between layers as link arrows.

Example 4.2.8. The unfolded diagram for the complex given in Example 4.2.7 is the

following.

Layer 3: · · · ka //a

$$

• •doo •e

oo •foo c // • b // • ka //

a

$$

· · ·

Layer 2: · · · ka //a

$$

• •doo •e

oo •foo c // • b // • ka //

a

$$

· · ·

Layer 1: · · · ka // • •doo •e

oo •foo c // • b // • ka // · · ·

Chapter 5

Morphisms between

indecomposable objects in the

bounded derived category of a

gentle algebra

In this chapter we describe a basis for the k-vectorspace HomKb,−(M,N) where M and

N are indecomposable objects in Kb,− (note that we are using the same abbreviations

as in Remark 4.0.3). These results are analogous to those described in Section 2.4,

originally found in [18, 29, 19]. The work contained in this chapter is joint work with

D. Pauksztello and K. Arnesen and can be found in [1].

5.1 Morphisms between string complexes and 1-

dimensional band complexes

We begin by considering HomKb,−(M,N) when M and N are string complexes or 1-

dimensional band complexes. The basis we obtain for such complexes contains three

families of morphisms: graph maps, single maps and double maps. We first establish

a basis in Cb,− and then identify homotopy classes of basis morphisms, as a result we

completely describe a basis for HomKb,−(M,N).

100

5.1. MORPHISMS BETWEEN STRING AND BAND COMPLEXES 101

5.1.1 The setup and some preliminary lemmas

Let V ,W ∈ HSt ∪ HBa ∪ HRes. Let

QW =

PW if W ∈ HSt ∪ HRes

BW ,k,1 if W ∈ HBa

where k ∈ k∗ and define QV in the same way (where, if QW and QV are both band

complexes, then let QW = BW ,k,1 and QV = BV ,l,1 for k, l ∈ k∗).

Let f : QV → QW . For each degree i ∈ Z, the modules QiV and Qi

W are finite

direct sums of projective modules and so the morphism f i : QiV → Qi

W is a matrix

with entries given by linear combinations of paths in the quiver Q. We will refer to a

scalar multiple of a path arising in this way as a component of f .

Example 5.1.1. Consider the following quiver with the relation bd = 0.

1c

��−1 a // 0

b

@@

2d

oo

Let V = (dcb · dcba, 0, 2) and W = (a− · dcb · dcb · dcba, 0, 2) and consider the

following morphism f : PV → PW .

P (0)( 1+hdcb 0 )��

dcb // P (0)1��

dcba // P (−1)1��

P (0)( dcb a )

// P (0)⊕ P (−1)( dcb0 )

// P (0)dcba// P (−1)

where h ∈ k∗. Then hdcb : P (0)→ P (0) in degree zero is a component of f .

If we display the complexes as unfolded diagrams and adjust the components of f

accordingly, then the structure of the morphism becomes clearer.

P (0) dcb //

1+hdcb��

P (0)1��

dcba // P (−1)1��

P (−1) P (0)aoodcb// P (0)

dcb// P (0)

dcba// P (−1)

By considering the components separately, we are able to spot linearly independent

“parts” of f . For example, f decomposes as follows.

102 CHAPTER 5. MORPHISMS BETWEEN INDECOMPOSABLE COMPLEXES

P (0) dcb //

1��

P (0)1��

dcba // P (−1)1��

P (−1) P (0)aoodcb// P (0)

dcb// P (0)

dcba// P (−1)

and

P (0) bcd //

hdcb��

P (0) dcba // P (−1)

P (−1) P (0)aoodcb// P (0)

dcb// P (0)

dcba// P (−1)

where the absence of arrows indicates a zero entry in the corresponding matrix.

The following lemma is really about certain possible configurations of morphisms

between indecomposable projective modules in A-mod but the diagram we consider

is deliberately arranged to resemble a component of a morphism between string and

1-dimensional band complexes since this will be where it is applied.

Suppose we have the following set up:

• vL

fL��

(∗)

•(∗∗)

vR

fC��

•fR��

• wL• wR

•

where the • stand for indecomposable projective modules in A-mod and vL, vR, wL,

wR, fL, fR, fC are paths in Q. The lines • • may indicate a morphism from left

to right or from right to left. If this orientation means that any one of vLvR, wLwR,

vRvL or wRwL is a path in Q then we require that it is contained in I.

Lemma 5.1.2. Suppose the above configuration of morphisms between indecomposable

projective modules in A-mod is such that

1. fC is a path of length > 0;

2. vL is direct if and only if wL is direct;

3. vR is direct if and only if wR is direct; and

4. the squares (∗) and (∗∗) commute.

Then at most one of (∗) and (∗∗) has a non-zero commutativity relation.


Proof. Consider the case where all four homotopy letters vL, vR, wL and wR are di-

rect; the arguments for the remaining orientations are completely analogous. Suppose

vLfC = fLwL 6= 0. The paths fC and wL start with the same arrow and since wLwR = 0

it follows that fCwR = 0. Similarly, if fCwR = vRfR 6= 0 then fC and vR start with the

same arrow, but vLvR = 0 and so vLfC = 0.

5.1.2 A basis in Cb,−(A-proj)

The key to understanding the decomposition of morphisms between string and 1-

dimensional band complexes is to “unfold” the morphisms as we did in Example 5.1.1.

We use this method of presentation to define the form of the basic morphisms; we

begin by describing how the unfolded diagrams represent particular morphisms.

Consider the following diagram representing a morphism between the string or 1-

dimensional band complexes QV and QW . The horizontal parts • • represent

sections of the unfolded diagrams of W and V (so, for each 0 ≤ t ≤ p, we have that

vi+t and wj+t are paths in Q and the scalars kt and lt will be equal to 1 unless V and

W are homotopy bands respectively). The arrows f0, . . . , fp+1 represent entries in the

matrices that give the relevant degree of a morphism f : QV → QW :

degrees: tp+1 tp t1 t0

QV : •fp+1 ��

kpvi+p •fp ��

· · · k1vi+1 • k0vi

f1 ��

•f0��

QW : •lpwj+p

• · · ·l1wj+1

•l0wj

•

degrees: tp+1 tp t1 t0

An important point here is that the degrees of the unfolded diagrams must agree

for this to make sense and so the orientation of vi+t and wj+t must agree for each

0 ≤ t ≤ p. Also, the squares in the diagram must commute if it is to correspond to

a well-defined morphism of complexes. See Example 5.1.1 for an illustration of how

such diagrams may be used to construct morphisms QV → QW .

Lemma 5.1.2 allows us to identify linearly independent parts of morphisms from

QV to QW . Next we define three classes of morphism that will turn out to form a

k-linear basis for HomCb,−(QV , QW ).

Definition 5.1.3 (Graph map). Let V and W ∈ HSt ∪ HBa ∪ HRes. A graph map

QV → QW is a morphism whose non-zero components can be displayed in an unfolded


diagram as follows.

degrees: tp tp−1 t1 t0

QV : •fL ��

kLvL •1=mp+1��

kpup •mp��

kp−1up−1· · · k2u2 •m2 ��

k1u1 •m1 ��

kRvR •fR��

QW : •lLwL

•lpup

•lp−1up−1

· · ·l2u2

•l1u1•

lRwR•


where

1. mi ∈ k for each 1 ≤ i ≤ p;

2. wL 6= vL and wR 6= vR;

3. the square

• kiui

mi+1 ��

•mi��

•liui•

commutes for each 1 ≤ i ≤ p;

4. if vL and wL exist and have the same orientation then there exists some scalar

multiple of a path fL such that the square

• kLvL

fL ��

•1��

•lLwL

•

commutes; otherwise we have the following orientation

• •kLvLoo

1��•

lLwL// •

where either of vL and wL may not exist;

5. if vR and wR exist and have same orientation then there exists some scalar

multiple of a path fR such that the square

• kRvR

m1 ��

•fR��

•lRwR

•

commutes; otherwise we have the following orientation

• kRvR //m1 ��

•

• •lRwRoo

where either of vR and wR may not exist.


Remark 5.1.4. The dotted lines in the diagram represent arrows that may not exist;

for example, if V is a homotopy string with the first homotopy letter (u1, t1, t0), then vR

is said not to exist. Moreover, if V ,W ∈ HRes then the sequence of commuting squares

may continue infinitely far to the left or right; if the diagram is left (respectively right)

infinite then vL and wL (respectively vR and wR) are said not to exist.

Example 5.1.5. Again consider the quiver with the relations in Example 5.1.1 and

consider homotopy letters V = (dcb ·dcba, 0, 2) and W = (a− ·dcb ·dcb ·dcba, 0, 2). The

unfolded diagram

P (0) dcb //

1��

P (0)1��

dcba // P (−1)1��

P (−1) P (0)aoodcb// P (0)

dcb// P (0)

dcba// P (−1)

defines the following graph map:

P (0)( 1 0 )��

dcb // P (0)1��

dcba // P (−1)1��

P (0)( dcb a )

// P (0)⊕ P (−1)( dcb0 )

// P (0)dcba// P (−1)

Definition 5.1.6 (Single map). Let V and W ∈ HSt ∪ HBa ∪ HRes. A single map

QV → QW is a morphism whose non-zero component f is a string of length 6= 0 and

can be displayed in an unfolded diagram as follows.

• kLvL • kRvR

f��

•

•lLwL

•lRwR

•

where the following conditions hold for f .

1. If we have • kLvL // • , then vLf = 0 and if we have • •lLwLoo , then fwL = 0.

2. If we have • •kRvRoo , then vRf = 0 and if we have • lRwR // • , then fwR = 0.

Example 5.1.7. Consider algebra given in Example 4.1.1 and consider the homotopy

strings V = (f · e, 0, 2) and W = (c− · f · e, 0, 1). Then there exists the following

unfolded diagram.

• f //

f

��

• e // •

• • e//

coo •

f// •


which gives defines the following single map.

P (0)f //

( 0 f )��

P (3) e // P (4)

P (0)( c f )

// P (3)⊕ P (2) e// P (4)

Definition 5.1.8 (Double map). Let V and W ∈ HSt∪HBa∪HRes. A double map

QV → QW is a morphism whose non-zero components fL and fR are strings of length

6= 0 and can be displayed in an unfolded diagram as follows.

• kLvL • kCvC //fL ��

• kRvR

fR��

•

•lLwL

•lCwC

// •lRwR

•

where

1. the central square has a non-zero commutativity relation i.e. vCfC = fLwC 6= 0;

2. fL satisfies the first condition in the definition of a single map; and

3. fR satisfies the second condition in the definition of a single map.

We will sometimes use the notation (fL, fR) for such a double map.

Example 5.1.9. Consider the algebra given in Example 4.1.1. Then we have the

following unfolded diagram

• dc // • b // • a //

a

��

•f��

• •coo

f// • e // •

which gives rise to the following double map.

P (4) dc // P (2) b // P (1) a //

a

��

P (0)

( 0 f )

��P (0)

( c f )// P (2)⊕ P (3) e // P (1)

Observation 5.1.10. It follows directly from the requirement that the squares in

the unfolded diagrams commute that single, double or graph maps are completely

determined by any non-zero component.


Let V ,W ∈ HSt t HBa t HRes. Let BCV ,W be the set of graph, single and double

maps in HomCb,−(QV , QW ).

Proposition 5.1.11. Let V ,W ∈ HSt t HBa t HRes. Then BCV ,W is a k-linear basis

for HomCb,−(QV , QW ).

Proof. First we show that BCV ,W is a linearly independent set. Let f be a non-zero

element of BCV ,W . Suppose f =

m∑i=1

kifm is some linear combination of elements of BCV ,W

with fi 6= fj for all 1 ≤ i, j ≤ m. Then, by Observation 5.1.10, and the fact that all

relations in a gentle algebra are monomial, it follows that f = fi and ki = 1 for some

1 ≤ i ≤ m and by the same reasoning kj = 0 for all j 6= i.

Next we show that BCV ,W spans HomCb,−(QV , QW ). Let h : QV → QW be a non-zero

morphism and let h′ : P (i)→ P (j) be a non-zero component of h (the term component

is defined at the beginning of Section 5.1.1). Consider the unfolded diagram with this

non-zero component. Since ever projective in QV has at most two non-zero components

of the differential incident with it. It follows that the unfolded diagrams with this non-

zero component determine a scalar multiple of an element of BCV ,W that by Lemma 5.1.2

must be a summand of h and by Observation 5.1.10 is uniquely determined.

5.1.3 A basis in Kb,−(A-proj)

In this section we will consider morphisms QV → QW up to homotopy equivalence.

Let f, g ∈ HomCb,−(QV , QW ), then f and g are homotopy equivalent if there exists

a morphism hi : QiV → Qi−1

W for each i ∈ Z such that f i− gi = diQVhi+1 +hidi−1

QW(where

composition is carried out from left to right as before). Such a family {hi}i∈Z is known

as a homotopy from f to g. In this case we will write f ' g and if g = 0 then we

say that f is null-homotopic.

Let f ∈ BCV ,W and let

H(f) := {g ∈ BCV ,W | f ' kg, k ∈ k}.

This is not the same as the homotopy equivalence class of f but if h : QV → QW

is such that h ' f then h decomposes into a linear combination of elements from

H(f)∪H(0). Thus the set BKV ,W := {H(f) | f ∈ BC

V ,W } is a basis for HomKb,−(QV , QW ).

As we did with morphisms in the previous section, we will consider components of


the homotopy {hi}i∈Z. That is, if f and g are components of homotopy equivalent

morphisms corresponding to the following unfolded diagram:

• kivi •ki+1vi+1

f−g��

•

•ljwj•lj+1wj+1

•

Then there must be components of the homotopy that compose with the paths vi,

vi+1, wj, wj+1 to make up f − g. Of course, this will depend on the orientation of the

diagram. For example, if we have the following orientation:

• kivi // •h1

��

ki+1vi+1//

�� h3��

•h2

��•

ljwj// • •lj+1wj+1

oo

then h1, h2, h3 are components (i.e. scalar multiples of paths) of the homotopy and

f − g = lj(h1 + h2)wj + ki+1vi+1h3.

We begin by showing that graph maps are not null-homotopic and have trivial

homotopy classes.

Proposition 5.1.12. Let f : QV → QW be a graph map. Then H(f) = {f}.

Proof. Let g ∈ H(f) and suppose f is labelled as in Definition 5.1.3. Then consider

the entry gi in the matrix gtp that fits into the unfolded diagram as follows:

tp tp−1

QV : •fL ��

kLvL •gi �� 1��

kpup •mp��

kp−1up−1

QW : •lLwL

•lpup

•lp−1up−1

tp tp−1

We will assume that the diagram is oriented with • kpup // • ; the argument for the

other orientation is similar. Then there are components of the homotopy h1, h2, h3

such that 1 − gi = kpuph1 + kLvLh2 + lLh3wL (where we take h2 = 0 if we have

• kLvL // • and h3 = 0 if we have • •lLwLoo ). The compositions uph1, vLh2 and h3wL

must consist of paths of length > 0 since the differential only contains paths of length

> 0, but no linear combination of such paths can yield a 1. Therefore we must have

that kpuph1 + kLvLh2 + lLh3wL = 0 and gi = 1. By Observation 5.1.10 it follows that

f = g.


Suppose f, g ∈ BCV ,W are not graph maps. We say that there is a graph homotopy

between f and g if the unfolded diagram of QV and Σ−1QW are related as follows.


QV : • kLvL •mp+1��

kpup •mp��

kp−1up−1· · · k2u2 •m2 ��

k1u1 •m1 ��

kRvR •

Σ−1QW : •lLwL

•lpup

•lp−1up−1

· · ·l2u2

•l1u1•

lRwR•


Where mp+1 = 1 and mi ∈ k such that kimi = −mi+1li for 1 ≤ i ≤ p and the following

conditions must be satisfied.

1. If f is a single map then either we have • kLvL // • and f = vL; or we have

• •lLwLoo and f = wL.

2. If f = (fL, fR) is a double map then we have • kLvL // • and fL = vL and we have

• •lLwLoo and fR = wL.

3. If g is a single map then either we have • lRwR // • and g = wR; or we have

• •kRvRoo and g = vR.

4. If f = (fL, fR) is a double map then we have • kLvL // • and fL = vL and we have

• •lLwLoo and fR = wL.

We will sometimes write h : QV Σ−1QW to denote such a graph homotopy.

Example 5.1.13. This example demonstrates how an unfolded diagram such as the

one above describes a homotopy {hi}i∈Z between f and g.

Consider the algebra as in Example 4.1.1 and consider the following single map.

P (0)f //

( 0 c )

��

P (3) e // P (4) dc // P (2)

P (0)( f c ) // P (3)⊕ P (2)

( e0 )// P (4) dc // P (2) b // P (1)

We will refer to this single map as c. Consider the following graph homotopy from

c.

0 1 2 3

• f //

1��

•−1��

e // •1��

dc // •−1��

• • f //coo • e // • dc // • b // •1 0 1 2 3 4


We will show how this can be used to construct a family of basis morphisms that

are homotopy equivalent to c. First note that we have the following list of graph

homotopies starting at c that are contained in the above maximal graph homotopy.

• f //

1��

• e // • dc // •

• • f //coo • e // • dc // • b // •

• f //

1��

•−1��

e // • dc // •

• • f //coo • e // • dc // • b // •

• f //

1��

•−1��

e // •1��

dc // •

• • f //coo • e // • dc // • b // •These graph homotopies give rise to the following homotopies.

P (0)f //

( f c )

��

1

xx

P (3) e // P (4) dc // P (2)

P (0)( f c )

// P (3)⊕ P (2)( e0 )

// P (4)dc// P (2)

b// P (1)

P (0)f //

( 0 c )

��1xx

P (3)

(−1 0 )xx−e��

e // P (4) dc // P (2)

P (0)( f c )

// P (3)⊕ P (2)( e0 )

// P (4)dc// P (2)

b// P (1)

P (0)f //

( 0 c )

��1xx

P (3)

(−1 0 )

xx

e // P (4)

1{{dc��

dc // P (2)

P (0)( f c )

// P (3)⊕ P (2)( e0 )

// P (4)dc// P (2)

b// P (1)

And finally the maximal graph homotopy gives rise to the following homotopy.

P (0)f //

( 0 c )��1xx

P (3)

(−1 0 )

xx

e // P (4)

1{{

dc // P (2)

−1{{−b��

P (0)( f c )

// P (3)⊕ P (2)( e0 )

// P (4)dc// P (2)

b// P (1)

So c ' −f ' e ' −dc ' b.

Lemma 5.1.14. If there is a homotopy starting at f ∈ BCV ,W containing a component

given by a path of length > 0, then f is null-homotopic.


Proof. By Propositon 5.1.12, we may assume that f is a single or a double map; we

will only give the argument for a single map since the argument for a double map is

similar.

Without loss of generality, suppose the component given by a path of length > 0

is adjacent to a non-zero component f of f since otherwise there will be a homotopy

consisting of identities between f and some other single or double map with this

property. Suppose we have the following orientation of wj and there is component h

of a homotopy such that h is a path of length > 0.

• kivi •h

��

ki+1vi+1

f��

•

lj−1wj−1•

ljwj// •lj+1wj+1

•

This defines a null-homotopy because all other entries can be taken to be zero and, if

we have, • kivi // • (respectively • •ki+1vi+1oo ), then vih = 0 (respectively vi+1h) because

vif = 0 and vi+1f = 0 by definition of a single map. Moreover, if we have • •lj−1wj−1oo ,

then hwj−1 = 0 since hwj 6= 0 and the algebra is gentle. All other orientations will

yield a similar result.

Proposition 5.1.15. Let f, g ∈ BCV ,W be distinct morphisms and suppose neither f ' 0

nor g ' 0. Then f ' g if and only if there is a graph homotopy between g and f .

Proof. Clearly if there is a graph homotopy between f and g then it can be used to

construct a homotopy {hi}i∈Z giving us that f ' g.

So suppose that f ' g. Since g and f are distinct, it follows from Proposition

5.1.12 that they are not graph maps. We will consider the case where f and g are

single maps; the case where either (or both) of them is a double map is similar.

First we suppose that the non-zero component of f and g (which we will denote by

f and g respectively from now on) are in the same position on the unfolded diagram.

By Observation 5.1.10, we have f 6= g . We are in the following situation.

• kLvL • kRvR

f ��g��

•

•lLwL

•lRwR

•

As we have noted before Proposition 5.1.12, there must be some non-zero components

of the homotopy that compose with vL, vR,wL or wR to give f − g . By Lemma 5.1.14,


if any of these component of the homotopy is a path of length > 0, then it implies

that f ' g ' 0. So all components must be given by scalar multiplication. Without

loss of generality, suppose g is equal to either vR or wR and f is equal to either vL or

wL (otherwise invert one of the homotopy strings and relabel).

We consider the case where we have • kRvR // • , g = vR and there is a component

of the homotopy given by a scalar multiple m ∈ k.

• kLvL • kRvR

f ��g��

•myy•

lLwL•lRwR

•

If we consider some non-zero components the homotopy is forced to have under this

assumption, it follows that we have the following unfolded diagram (where u and w

have the indicated orientation if they exist) with m1, . . . ,mp ∈ k.

QV : • kRvR •m��

kpup •mp��

kp−1up−1· · · k2u2 •m2 ��

k1u1 •m1 ��

kv // •

Σ−1QW : • •lpup•lp−1up−1

· · ·l2u2

•l1u1• •

lwoo

If we construct a homotopy from this diagram (as we did in Example 5.1.13) then

we produce a null-homotopy from g and so f ' g ' 0, contradicting our initial

assumption. In the case where • •lRwRoo , g = wR a similar null-homotopy is produced.

We may therefore assume that f and g are in different positions in the unfolded

diagrams of QV and QW . Without loss of generality we again assume that the non-zero

component of the homotopy incident with g is m as above. If we again consider the

non-zero components the homotopy is forced to have to the right of g , then it is clear

that we must produce a graph homotopy from g to f . In particular, the components

of the homotopy induced by m must either satisfy the condition (5) in the definition

of a graph map QV → Σ−1QW , in which case they will define a null-homotopy, a

contradiction, or f will satisfy condition (3) in the definition of a graph homotopy

from g to f .

Definition 5.1.16 (Maximal graph homotopy). Consider a graph homotopy {hi}i∈Zfrom f to g such that vL 6= wL and vR 6= wR. The associated unfolded diagram

shall be referred to as a maximal graph homotopy. The graph homotopy that

corresponds to the identity when QV∼= ΣQW is a band complex is also a maximal

graph homotopy.


Remark 5.1.17. In [1], we use the term quasi-graph map rather than maximal graph

homotopy. This terminology is explained by the fact that a maximal graph homotopy

can be thought of as analogous to a graph map QV → Σ−1QW except conditions (4)

and (5) of the definition must not be satisfied.

Remark 5.1.18. Consider a maximal graph homotopy and suppose it is labelled as

in the definition of a graph homotopy. As we saw in Example 5.1.13, this defines

a homotopy from a morphism f that is either a single map with component vL or

wL or a double map (vL,wL). The homotopy is from f to a single or double map g

with component(s) corresponding to vR and/or wR. There is also a family of graph

homotopies from single maps with component ui to single maps with component uj

(where 1 ≤ i < j ≤ p) given by “sub-diagrams” of the maximal graph homotopy.

Thus H(f) = {f, g, u1, . . . , up} (where we identify the components ui with the single

maps themselves).

It follows that the classes H(f) 6= H(0) with more than one element correspond to

maximal graph homotopies between single and double maps.

Corollary 5.1.19. The classes H(f) where f ∈ BCV ,W is not null-homotopic and

H(f) 6= {f} are in one-to-one correspondence with maximal graph homotopies.

Finally we must consider single and double maps that are not null-homotopic and

are not homotopy equivalent to any other elements of BCV ,W . That is, the single and

double maps that arise in such a way that they cannot be part of a graph homotopy.

A simple application of the definitions gives rise to the following list.

Corollary 5.1.20. 1. A non-null-homotopic single map f ∈ BCV ,W is such that

H(f) = {f} if and only if the corresponding unfolded diagram is one of the

following (up to inverting one of the homotopy strings).

(a)

• kLvL •f��•lRwR

•

(b)

• kLvL •kRffR //

f��

•

•lRwR

•


(c)

• kLvL •f��

•lLfLf

// •lRwR

•

(d)

• kLvL •kRffR //

f��

•

•lLfLf

// •lRwR

•

where fR and fL are paths of length > 0 and none of vL, vR,wL or wR are equal

to f .

2. A double map f = (fL, fR) ∈ BCV ,W with unfolded diagram

• kLvL • kCvC //fL ��

• kRvR

fR��

•

•lLwL

•lCwC

// •lRwR

•

is not null-homotopic and H(f) = {f} if and only if there exists a path f of

length > 0 such that vC = fLf and wC = ffR.

We have now established a basis for HomKb,−(QV , QW ).

Theorem 5.1.21. Let V ,W ∈ HSt t HBa t HRes. Then there is a k-linear basis for

HomKb,−(QV , QW ) consisting of

1. all graph maps QV → QW ;

2. a representative f for each H(f) arising from a maximal graph homotopy; and

3. all single and double maps of the form described in Corollary 5.1.20.

Example 5.1.22. This example demonstrates the Theorem allows us to easily identify

a basis for HomKb,−(QV , QW ) given any two strings or bands V and W .

Consider the quiver with relations given in Example 5.1.1 and consider the two

homotopy strings given in the same example. Let A be the path algebra associated to

the bound quiver, then we can identify the following basis in Cb,−(A-proj).

P (0) dcb //

1��

P (0)1��

dcba // P (−1)1��

P (−1) P (0)aoodcb// P (0)

dcb// P (0)

dcba// P (−1)

5.2. MORPHISMS BETWEEN R-DIMENSIONAL BAND COMPLEXES 115

P (0) bcd //

dcb��

P (0) dcba // P (−1)

P (−1) P (0)aoodcb// P (0)

dcb// P (0)

dcba// P (−1)

P (0) bcd // P (0)dcb��

dcba // P (−1)

P (−1) P (0)aoodcb// P (0)

dcb// P (0)

dcba// P (−1)

P (0)a��

bcd // P (0) dcba // P (−1)

P (−1) P (0)aoodcb

// P (0)dcb

// P (0)dcba// P (−1)

P (0)dcba��

bcd // P (0) dcba // P (−1)


// P (0)dcb

// P (0)dcba// P (−1)

We also have the following maximal graph homotopy and null-homotopy.

P (0) dcb //

1xx dcb��a

ss

P (0)

1xx dcb��

dcba // P (−1)

P (−1) P (0)aoodcb// P (0)

dcb// P (0)

dcba// P (−1)

P (0)dcba ��

bcd // P (0)aww

dcba // P (−1)


// P (0)dcb

// P (0)dcba// P (−1)

So dim HomKb,−(PV , PW ) = 2.

5.2 Morphisms between r-dimensional band com-

plexes

Let V ,W ∈ HBa, k, l ∈ k and r, s ∈ N. In this section we consider morphisms

BV ,k,r → BW ,l,s. We will employ the same strategy as in the previous section; that

is we will establish a basis in Cb,− and then consider homotopy equivalence classes of

basis elements.


5.2.1 A basis in Cb,−(A-proj)

As with morphisms between strings and 1-dimensional band complexes we will consider

the structure of morphisms in terms of unfolded diagrams. Let f ∈ BCV ,W and suppose

f is not the identity. Then for 1 ≤ m ≤ r and 1 ≤ n ≤ s, we have that f uniquely

determines a morphism f (m,n) : BV ,k,r → BW ,l,s in Cb,− obtained by placing a copy of f

from the indecomposable projective modules in layer m of BV ,k,r to the indecomposable

projective modules in layer n of BW ,l,s. As in Section 5.1.2, the squares in the unfolded

diagrams must commute and so, if some component of the copy of f has a non-zero

composition with a link arrow, then f (m,n) will acquire additional components in order

to satisfy the necessary commutativity relations. As we have assumed that f is not

the identity, the additional components of f (m,n) will at most occur between layers

m+ 1 and n and between layers m and n− 1. We call f (m,n) the lift of f to layers

(m,n).

Example 5.2.1. This example demonstrates what happens when you lift a graph

map that has a component that composes non-trivially with a link arrow. Consider

the gentle algebra given by the quiver

1 a // 2

0 d //

b

OO

3

c

OO

e // 4

6

h

OO

f// 5

g

OO

with relations ab = cd = gf = 0. Consider the bands V = (e · c− · a · bh · f− · g−, 0, 0)

and W = (d− · c− · a · b, 1, 1). Then there is the following graph map BW ,l,1 → BV ,k,1.

. . . •doo

1��

•coo

1��

a // • lb //

1��

•1lh

��

. . .doo

. . . e// • •coo

a// •

bh// • . . .

foo

When we lift this morphism to layers (2, 2) of BW ,l,3 and BV ,k,3, we are forced to add


components to ensure that the correct commutativity relations hold in the complex.

. . . •

1l

��

doo •

1l

��

coo a // •

1l

��

lb //b

%%

• . . .doo

. . . •

1

��

doo •

1

��

coo a // •

1

��

lb //b

%%

•

1lh

��

. . .doo

. . . •doo •coo a // • lb // • . . .doo

. . . e// • •coo

a// •

bh// • . . .

foo

. . . e// • •coo

a// •

bh// • . . .

foo

. . . e// • •coo

a// •

bh// • . . .

foo

The lift id(m,n)BV ,k,1

: BV ,k,r → BV ,k,s of the identity idBV ,k,r ∈ BCV ,V to layers (m,n)

behaves slightly differently. The additional components required will occur from layer

m − i to layer n − i for 0 ≤ i ≤ n − 1 and also from layer m + j to layer n + j for

0 ≤ j ≤ r−m; that is, the top-most layers of BV ,k,r will be embedded in the bottom-

most layers of BV ,k,s. It follows that there are exactly min{r, s} lifts of idBV ,k,r ∈ BCV ,V

to morphisms BV ,k,r → BV ,k,s.

Example 5.2.2. This example demonstrates what happens to a lift of the identity

morphism. Consider the algebra given in Example 5.2.1 and the band W = (d− · c− ·

a · b, 0, 0). The following diagram depicts the lift of the identity morphism idBW ,l,1 to

the layers (2, 2) of BW ,l,3 and BW ,l,4.

. . . •

��

doo •

��

coo a // •

��

lb //b

((

•

��

. . .doo

. . . •

��

doo •

��

coo a // •

��

lb //b

((

•

��

. . .doo

. . . •

��

doo •

��

coo a // •

��

lb // •

��

. . .doo

. . . •doo •coo a // • lb //b

((

• . . .doo

. . . •doo •coo a // • lb //b

((

• . . .doo

. . . •doo •coo a // • lb //b

((

• . . .doo

. . . •doo •coo a // • lb // • . . .doo

where all of the components of the lifted morphism are identities.

Lemma 5.2.3. Let BCV ,W (r, s) be the set of possible lifts of elements of BC

V ,W to mor-

phisms BV ,k,r → BW ,l,s. Then BCV ,W (r, s) is a basis for HomCb,−(BV ,k,r, BW ,l,s). In

particular, if d = dim HomCb,−(BV ,k,1, BW ,l,1), then dim HomCb,−(BV ,k,r, BW ,l,s) is as

follows.


1. If BV ,k,1 is isomorphic to BW ,l,1, then

dim HomCb,−(BV ,k,r, BW ,l,s) = rs(d− 1) + min{r, s}.

2. If BV ,k,1 and BW ,l,1 are not isomorphic, then we have

dim HomCb,−(BV ,k,r, BW ,l,s) = rs(d).

Proof. Consider some component of a morphism f : BV ,k,r → BW ,l,s. The requirement

that the squares commute in the unfolded diagrams and Lemma 5.1.2 will ensure

that this component is a scalar multiple of an element of BCV ,W (r, s) and that this

morphism is a summand of f . Repeating this reasoning will yield that BCV ,W (r, s)

spans HomCb,−(BV ,k,r, BW ,l,s).

By construction, an element of BCV ,W (r, s) that has been lifted to layers (m,n) is

completely determined by any non-zero component between layers m and n. Then, by

the same argument as in the first half of the proof of Proposition 5.1.11, the elements

of BCV ,W (r, s) are linearly independent.

5.2.2 A basis in Kb,−(A-proj)

As in the previous section, we now consider the homotopy equivalence classes of the

elements f of the basis BCV ,W (r, s). We can define the lift of a graph homotopy

to layers (m,n) in the same way as the lift of a morphism. The same reasoning

as the previous section mean that any homotopy from f to some other morphism

g must decompose into a linear combination of lifts of graph homotopies (and null-

homotopies). Therefore, we can completely determine the homotopy equivalence class

of f by describing the morphisms that are homotopy equivalent to f via a lift of a

graph homotopy. The following lemma is immediate from these observations.

Lemma 5.2.4. Let BV ,k,r and BW ,l,s be band complexes and let f ∈ BCV ,W . If we have

H(f) = {f}, then for every pair (m,n) with 1 ≤ m ≤ r and 1 ≤ n ≤ s the lifted

morphism f (m,n) is not homotopy equivalent to any other lifted basis morphism.

Again, the subtlety arises when the components of the graph homotopies compose

with link arrows. It is clear that if a lifted graph homotopy does not compose with a

link map then we can construct a homotopy between lifted single or double maps as

we did in the previous section.


Lemma 5.2.5. Let BV ,k,r and BW ,l,s be band complexes. If a lift h(m,n) of a maximal

graph homotopy h : BV ,k,1 Σ−1BW ,l,1 does not have any component that composes

with a link arrow and if H(f) is the set of homotopy equivalent basis morphisms cor-

responding to h (see Section 5.1.3), then the lift of h to layers (m,n) defines a family

of homotopies between lifts g(m,n)1 and g

(m,n)2 where g1, g2 ∈ H(f).

Proof. This is clear by the same reasoning as in the previous section. See Example

5.1.13.

It remains to consider graph homotopies where there is some non-zero component

that composes with a link arrow. We will begin by considering the case where BV ,k,1

is not isomorphic to Σ−1BW ,l,1 (see Example 5.2.11 for a demonstration of why this is

a special case).

Example 5.2.6. This example demonstrates what happens when a component of a

graph homotopy composes with a link arrow. Consider the algebra given in Example

5.2.1 and consider the bands V = (a ·bh ·f− ·g− ·e ·c−, 0, 0) and W = (d− ·c− ·a ·b, 1, 1).

The following diagram shows a double map and a single map BV ,k,1 → BW ,l,1.

1 0 1 0 1 2 1

. . . •goo e //e��

•d

rr

•kcoo a // • hb //

b��

• . . .foo

. . . •d

oo •coo

a// •

lb// • •

doo . . .c

oo

1 0 −1 0 1 0 −1

The following diagram shows a maximal graph homotopy that exists between the

double map (e, d) and the single map b.

1 0 1 0 1 2 1

. . . •goo e // •1��

•−k��

kcoo a // •k��

hb // • . . .foo

. . .lb

// • •d

oo •coo

a// •

lb// • . . .

doo

1 2 1 0 1 2 1

If we lift these morphisms and the graph homotopy to the layers (2, 2) in the complexes

BV ,k,3 and BW ,l,3, the we see that we obtain a homotopy from the double map (e, d)

to a linear sum of a pair of lifts of the single map b as well as a lift of the single map c.

This is displayed on the diagram below; the graph homotopy is shown as solid arrows

and the morphisms are shown as dashed arrows.


. . . •goo e // • •

c

{{

kcooc

vv

a // • hb // • . . .foo

. . . •

e

��

goo e // •

d

��

1

��

•

−k

��

kcooc

vv

a // •

klb

��

kb

��

k

��

hb // • . . .foo

. . . •gooe

// • •kcoo a // • hb // • . . .foo

. . . •d

oo •coo

a// •

lb//

b ((

• •d

oo . . .coo

. . . •d

oo •coo

a// •

lb//

b ((

• •d

oo . . .coo

. . . •d

oo •coo

a// •

lb// • •

doo . . .c

oo

So (e, d)(2,2) ' −(klb(2,2) + c(3,2) + kb(2,1)).

We will choose a homotopy equivalence class representative of the additional sum-

mands we acquire. The following example shows how we can always write a lifted

single map as a linear combination of single maps corresponding to the right-most link

arrow.

Example 5.2.7. Consider the homotopy given in Example 5.2.6. There exists the

following lifted graph homotopy between c(3,2) and lb(3,2) + b(3,1).

. . . •goo e // • •

c

{{

1

��

kcooc

vv

a // •

−1

��

−lb

��

−b

��

hb // • . . .foo

. . . •goo e // • •kcooc

vv

a // • hb // • . . .foo

. . . •gooe

// • •kcoo a // • hb // • . . .foo

. . . •d

oo •coo

a// •

lb//

b ((

• •d

oo . . .coo

. . . •d

oo •coo

a// •

lb//

b ((

• •d

oo . . .coo

. . . •d

oo •coo

a// •

lb// • •

doo . . .c

oo

If we consider the homotopy given by the sum of this lifted graph homotopy and the one

given in Example 5.2.6, then we have that (e, d)(2,2) ' −(klb(2,2)+lb(3,2)+b(3,1)+kb(2,1)).

Lemma 5.2.8. Let BV ,k,r and BW ,l,s be band complexes such that BV ,k,1 is not isomor-

phic to Σ−1BW ,l,1. Suppose we have a maximal graph homotopy h : BV ,k,1 ΣBW ,l,1

from f to g for f, g ∈ BCV ,W such that some lift of h has a component that composes

with a link morphism. Then any lift g(m,n) : BV ,k,r → BW ,l,s of g ∈ H(f) is homotopy

equivalent to a linear combination of lifts of the right-most link arrow that composes

with a component of h.


Remark 5.2.9. To clarify what we mean in the statement of the lemma, consider the

case where h is as follows.

• vL •1��

ut · · · •±1 ��

kup// •±k ��

up−1· · ·uq+1 • uq //

±k��

•±kl��

· · · •±kl ��

vR•

• vL• ut · · · • up

// • up−1· · ·uq+1

•luq// • · · · • vR

•

Then, for any g ∈ H(f) and 1 ≤ m ≤ r, 1 ≤ n ≤ s, we have g(m,n) is homotopy

equivalent to a linear combination of lifts of uq.

Proof. Without loss of generality we will assume that the maximal graph homotopy

we are considering is given by the diagram above. Note that this fixes the orientation

of uq and up, as well as the fact that the link arrow in BV ,k,r is to the left of the link

arrow in BW ,l,s. The proof requires only minor modifications for the other cases.

Then H(f) = {f, g, u1, · · · , up, · · · , uq, · · · , ut} where we have identified the single

maps arising from h with their non-zero component (see Remark 5.1.18). We will

consider lifts of elements of H(f) to (m,n).

First consider the ui such that 1 ≤ i ≤ q. Then there is a graph homotopy that

can be lifted to layers (m,n) such that u(m,n)q ' ±u(m,n)

i (just take a scalar multiple of

the section of the above homotopy between ui and uq). Similarly, u(m,n) ' g(m,n).

Next suppose q < i < p. Then there is a lift of a graph homotopy giving us that

u(m,n)i ' ±(lu(m,n)

q +u(m,n−1)q ) (again, take the lift of the section of the above homotopy

between ui and uq).

Finally consider g or ui such that p ≤ i ≤ t. We argue for ui, the argument for g

is the same. Then there is a lifted graph homotopy giving ui ' ±(lu(m,n)q + u(m,n−1)

q +

u(m+1,n)p ) (this is given by lifting the section of the above graph homotopy between

ui and uq). But then if we lift the same section of the above homotopy to the layers

(m+ 1, n) we have that ui ' ±(lu(m,n)q + u(m,n−1)

q ± (lu(m+1,n)q + u(m+1,n−1)

q + u(m+2,n)p )).

We can then lift the same section of the homotopy to layers (m + 2, n) and so on.

Finally note that the section of the homotopy lifted to layers (r, n) does not acquire

any additional up term and so we are done.

Lemma 5.2.10. Let h, BV ,k,r and BW ,l,s be as in Lemma 5.2.8. Then the lifts u(m,n)q

for 1 ≤ m ≤ r, 1 ≤ n ≤ s are linearly independent in HomKb,−(BV ,k,r, BW ,l,s).

Proof. Suppose∑i,j

ki,ju(i,j)q ' 0 where i and j range over 1 ≤ i ≤ r, 1 ≤ j ≤ s. We have

already observed that this null-homotopy must decompose into a linear combination of


lifts of graph homotopies. The only possible such null-homotopy is a linear combination

of lifts of a graph homotopy without end points (i.e. one lifted from the identity

BV ,k,1 Σ−1BW ,l,1). But we have assumed that BV ,k,1 is not isomorphic to Σ−1BW ,l,1

and so we must have∑i,j

ki,ju(i,j)q = 0. By Lemma 5.2.3, it follows that ki,j = 0 for all

1 ≤ i ≤ r, 1 ≤ j ≤ s.

Next we consider the special homotopy arising when BV ,k,1∼= Σ−1BW ,l,1.

Example 5.2.11. Consider the algebra given in Example 5.2.1 and consider the band

V = (a · bh · f− · g− · e · c−, 0, 0). This example demonstrates what happens when

we lift the graph homotopy that is supported on the whole band. Below we show the

lifted maps kc(3,1) : BV ,k,3 → ΣBV ,k,3 and kc(2,1) and the homotopy between them. This

homotopy is obtained by lifting the ‘identity’ graph homotopy to the layers (2, 2).

. . . •goo e // • •

−c

{{

kcooc

vv

a // • hb // • . . .foo

. . . •

1

��

goo e // •

−1

��

•

1

��

c

{{

kcooc

vv

a // •

−1

��

hb // •

1

��

. . .

−1

��

foo

. . . •gooe

// • •kcoo a // • hb // • . . .foo

. . .e // • •kcoo

c

vv

a // • hb // • •foo . . .goo

. . . e // • •kcooc

vv

a // • hb // • •foo . . .goo

. . . e// • •kcoo a // • hb // • •foo . . .

goo

Consider the following graph homotopy h : BV ,k,1 BV ,k,1.

· · ·kv1 •±1 ��

vb •∓1 ��

vb−1· · · • vq

∓1��

•±1��

· · · v2 • kv1

∓1��

•±1��

vb · · ·

· · ·kv1• vb• vb−1· · · • vq

• · · · v2•kv1• vb· · ·

For each 1 ≤ t ≤ b, let vt : BV ,k,1 → ΣBV ,k,1 denote associated single maps in H(v1).

It follows from Lemma 5.2.8 that all lifts v (m,n)t for 1 ≤ t ≤ b can be written as

a linear combination of lifts v (i,j)q of vq. It remains to determine how many linearly

independent non-null homotopy classes are lifted from H(vq). For this we will require

the following lemma.

Lemma 5.2.12. Consider the homotopy h given above. For 1 ≤ i < r and 1 < j ≤ s,

we have v (i,j−1)1 ' −v (i+1,j)

1 . Moreover, when i > 1, we have v (i,1)1 ' 0 and, when j < s,

v (r,j)1 ' 0.


Proof. The homotopy equivalences are obtained by lifting h to each pair of layers

(i, j).

Repeated application of Lemma 5.2.12 yields the following homotopy equivalences:

v (1,1)1 ' −v (2,2)

1 ' v (3,3)1 ' · · · ' ±v (s,s)

1 ;

v (1,2)1 ' −v (2,3)

1 ' v (3,4)1 ' · · · ' ±v (s−1,s)

1 ;

...

v (1,s)1 .

where we interpret v (i,j) as 0 whenever i > r or j > s. So if r ≥ s then above

homotopy equivalences do not produce any null-homotopy classes. In this case there

are s non-null homotopy classes.

If r < s, then the first s − r homotopy classes are null-homotopic. In this case

there are s− (s− r) = r non-null homotopy classes.

All lifts of v1 that are not listed above can be seen to be null-homotopic with

another application of Lemma 5.2.12. If r ≥ s then we have 0 ' v (i,1)1 ' −v (i+1,2)

1 '

v (i+2,3)1 ' · · · ' ±v (r,s+2−i)

1 for 2 ≤ i ≤ r. If r < s then we have 0 ' v (i,1)1 ' −v (i+1,2)

1 '

v (i+2,3)1 ' · · · ' ±v (r,r+1−i)

1 for 2 ≤ i ≤ r.

We have shown that the lifts of this exceptional graph homotopy yield min{r, s}

linearly independent homotopy classes in HomKb,−(BV ,k,r,ΣBV ,k,s). We are now ready

to prove the main theorem of this section.

Theorem 5.2.13. Let r, s ∈ N, V ,W ∈ HBa, and k, l ∈ k∗. Also let

d = dim HomKb,−(BV ,k,1, BW ,l,1).

1. If BV ,k,1 is not isomorphic to BW ,l,1 nor Σ−1BW ,l,1, then

dim HomKb,−(BV ,k,r, BW ,l,s) = rs(d).

2. If BV ,k,1∼= BW ,l,1 or BV ,k,1

∼= Σ−1BW ,l,1, then

dim HomKb,−(BV ,k,r, BW ,l,s) = rs(d− 1) + min{r, s}.

Proof. By Lemma 5.2.3, the set BCV ,W spans HomKb,−(BV ,k,r, BW ,l,s). Moreover we

have observed that all homotopies between morphisms in HomKb,−(BV ,k,r, BW ,l,s) will


decompose into a linear combination of lifts of graph homotopies. Thus it remains

to determine how many linearly independent homotopy classes are produced by lifts

of maximal graph homotopies. We consider the cases set out in the statement of the

lemma.

1. The fact that every class H(f) in HomKb,−(BV ,k,1, BW ,l,1) gives rise to rs linearly

independent homotopy classes in HomKb,−(BV ,k,r, BW ,l,s) is immediate from Lem-

mas 5.2.4, 5.2.5, 5.2.8 and 5.2.10.

2. If BV ,k,1∼= BW ,l,1 then, again, the result is immediate from Lemma 5.2.3 (2) and

the combination of Lemmas 5.2.4, 5.2.5, 5.2.8 and 5.2.10.

If BV ,k,1∼= Σ−1BW ,l,1 then the result follows from Lemmas 5.2.4 and 5.2.5 as well

as Lemma 5.2.12 and the discussion that follows it.

Finally, we note that slight adaptations of the above arguments where r = 1 or s =

1 will yield analogous results for morphisms between r-dimensional band complexes

and string complexes. We may therefore extend the above theorem as follows.

Theorem 5.2.14. Let r ∈ N, V ∈ HBa, W ∈ HSt ∪ HRes and let k ∈ k∗. Then

dim HomKb,−(BV ,k,r, PW ) = rs dim HomKb,−(BV ,k,1, PW )

and

dim HomKb,−(PW , BV ,k,r) = rs dim HomKb,−(PW , BV ,k,1).

5.3 Application: The homotopy category of a derived-

discrete algebra

In this final section of Chapter 5, we will apply our results to string complexes in the

homotopy category of gentle algebras whose derived category is discrete.

If A is a finite-dimensional algebra and X is an object in Db(A-mod) then let

DimX := (dimH i(X))i∈Z where dimH i(X) is the dimension vector of H i(X) for each

i ∈ Z. That is, DimX ∈ K0(A)(Z) where K0(A) is the Grothendieck group of A-mod.

5.3. HOMOTOPY CATEGORY OF A DERIVED-DISCRETE ALGEBRA 125

In [57] Vossieck defines Db(A-mod) to be a discrete derived category if every

positive x ∈ K0(A)(Z) there are only finitely many isomorphism classes of indecom-

posable objects X in Db(A-mod) with DimX = x. In this case we say that A is a

derived-discrete algebra.

In [13] the derived-discrete algebras are classified up to derived equivalence. That

is, a finite-dimensional algebra A over an algebraically closed field k has a discrete

derived category if and only if it is derived equivalent to either

1. an algebra of finite representation type i.e. A is piece-wise hereditary of Dynkin

type; or

2. an algebra Λ(r, n,m), for 1 ≤ r ≤ n and m ≥ 0, given by the following quiver

with relations indicated by dotted lines.

−m −m+ 1 · · · −1 0a−m a−1

1

n− 1

2

n− 2

n− r

n− r + 2

n− r + 1

b0

b1

bn−r

cn−r+1cn−1

cn−2

In the remainder of this thesis we will use the term derived-discrete algebra to

refer to algebras with quivers Λ(r, n,m) since they will be our primary object of study.

We do not consider the piece-wise hereditary algebras of Dynkin type since their

representation theory is well-understood.

In Chapter 7 and Chapter 8 we will prove a series of results about the homotopy

category K(Λ-Proj) for Λ = Λ(r, n,m), 1 ≤ r ≤ n and m ≥ 0. The full subcategory

K(Λ-Proj)c of compact objects is equivalent to Db(Λ-mod) and so it follows that our

results will apply to K(A-Proj) for any derived-discrete algebra A.

Let 1 ≤ r ≤ n and m ≥ 0. We will eventually show that all of the indecomposable

objects in K(Λ-Proj) are string complexes of the kind described in Section 4.2.1. The

form of the quiver means that there are no homotopy bands over Λ and the form of

the homotopy strings is quite limited.

Lemma 5.3.1. Let 1 ≤ r ≤ n and m ≥ 0. (Up to equivalence) the homotopy strings

over the derived-discrete algebra Λ(r, n,m) are shifted copies of subwords of the fol-

lowing strings.


1. • ◦a−1...a−moo Vk // ◦ cn−1 // . . .cn−1+1 // •

bn−r...b0a−1...a−m// •

2. • ◦a−1...a−moo V // ◦ V // ◦ V // ◦ //

3. // ◦ V // ◦ V // ◦ cn−1 // . . .cn−1+1 // •

bn−r...b0a−1...a−m// •

4. // ◦ V // ◦ V // ◦ V // ◦ //

where V is the homotopy string with unfolded diagram • cn−1 // . . .cn−1+1 // •

bn−r...b0// • and

Vk is the concatenation of k copies of V . When r = n we set b0 = c0.

Remark 5.3.2. Recall Remark 4.2.5 in which we consider infinite string complexes.

For Λ(r, n,m) the condition that there are only finitely many indecomposable sum-

mands in each degree is satisfied by all the possible infinite homotopy strings W and

so we can consider the associated string compexes PW . Moreover, the results in the

previous chapter hold for morphisms from and to such infinite string complexes.

Proposition 5.3.3. Let Λ = Λ(r, n,m) for some 1 ≤ r ≤ n and m ≥ 0.

1. If Λ has finite global dimension, then the indecomposable compact objects in

K(Λ-Proj) are the string complexes PW where W is a finite homotopy string.

2. If Λ has infinite global dimension, then the indecomposable compact objects in

K(Λ-Proj) are the string complexes PW such that W is either a right infinite

substring of (2) in Lemma 5.3.1 or a finite homotopy string.

Proof. In [27] Jørgensen describes the compact objects for the category K(R-Proj)

where R is a coherent ring where all flat modules have finite projective dimension.

Since Λ is a finite-dimensional algebra it satisfies these conditions: it is coherent and

every flat module is projective. We consider Jørgensen’s construction for Λ.

Let (−)∗ = HomΛ(−,Λ) be the well-known duality between left and right projective

Λ-modules. Let M be a finite-dimensional left Λ-module and consider the projective

resolution PM of M∗. This can be considered as complex in Kb,−(Λop

-proj). We can

also consider the complex P ∗M .

Let G := {∑i P ∗M | M ∈ Λ

op-mod, i ∈ Z} and let C be the thick subcategory of

K(Λ-Proj) generated by G. If D is the thick subcategory of K(Λop

-Proj) generated by


{G∗ | G ∈ G}, then by [27, Thm. 3.2]

C(−)∗ // D(−)∗oo

are quasi-inverse equivalences of triangulated categories. Moreover, it is shown that D

is the thick subcategory generated by the set of projective resolutions PN of modules

N in Λop

-mod. That is D = Kb,−(Λop

-proj) and the indecomposable complexes are

exactly string complexes PU where U is either a finite string or U = ∞W where W is a

primitive left resolvable string over Λop

. The duality (−)∗ sends the string complexes

PU to the string complexes described in the statement of the proposition.

5.3.1 A bound on the dimension of Hom(PV , PW )

As an application of the results in this section, we are able to use homotopy string

combinatorics to extend a result found in [15], that is, the bound on the dimension

of the spaces of morphisms between indecomposable objects in Kb,−(Λ-proj) when

Λ = Λ(r, n,m). We extend this result to the infinite global dimension case and also

to include morphisms in K(Λ-Proj) to and from string complexes PV when V may be

an infinite string.

Lemma 5.3.4. If r > 1, we have dim HomΛ(P (i), P (j)) ≤ 1 for all −m ≤ i, j ≤ n−1.

If r = 1 then the bound is as above except we have dim HomΛ(P (0), P (j)) = 2 for all

−m ≤ j ≤ 0.

Proof. This follows directly from the form of the quiver and Observation 4.1.2.

As in previous sections, we use the terminology “v does not exist” to mean that

the string ends before this homotopy letter; in the diagram the dotted lines indicate

that the homotopy letter may not exist.

Lemma 5.3.5. Let V and W be (possibly infinite) homotopy strings and consider the

following unfolded diagram.

v ′L • vL

fL��

•1��

up •1��

up−1 · · · u2 •1 ��

u1 •1 ��

vR •v ′R

fR��

w ′L• wL

• up • up−1· · · u2 • u1 • •wR w ′R


1. Suppose the diagram represents a maximal graph homotopy.

(a) If vL exists, then at least one of v ′L or w ′L does not exist.

(b) If vR exists, then at least one of v ′R or w ′R does not exist.

2. Suppose the diagram represents a graph map. Then at least one of v ′L and w ′L

does not exist and at least one of v ′R and w ′R does not exist.

Proof. 1. We give the argument for statement (a); the argument for (b) is com-

pletely dual. By definition of a maximal graph homotopy, if vL exists then

vL 6= wL. If wL does not exist then clearly w ′L does not exist. If wL does exist,

then the form of the strings mean that vL or wL (or both) must be a substring

of bn−r · · · b0a−1 . . . a−m and in this case v ′L or w ′L respectively cannot exist.

2. We only consider the left end of the string; the argument for the right end is

similar. If fL exists then vL and wL are oriented in the same direction but are

not equal. Again vL or wL (or both) must be a substring of bn−r · · · b0a−1 . . . a−m

and so at least one of w ′L and v ′L does not exist. If fL is zero, then we must have

that vL anf wL are oriented in different directions or one or both of them does

not exist; in the latter case clearly at least one of v ′L or w ′L does not exist. If

both are non-zero then one of them is oriented in the opposite direction to up.

By Lemma 5.3.1, this only occurs at the end of a string, so at least one of v ′L or

w ′L does not exist.

Remark 5.3.6. The idea of this lemma is that, for any graph map PV → PW or

maximal graph homotopy between single and double maps PV → PW , the common

part of the relevant unfolded diagrams spans every degree where PV and PW are non-

zero. The same will be true for single and double maps and so Lemma 5.3.4 combined

with Observation 5.1.10 will imply the bound on the dimension of the hom-spaces.

The diagram in the lemma shows only the case where the common part of the

unfolded diagrams is finite. The proof of the lemma deals with endpoints only so the

result in fact holds for graph maps that are supported in infinitely many degrees. Note

that infinite graph homotopies are always null-homotopies.


Theorem 5.3.7. Let Λ = Λ(r, n,m) be a derived-discrete algebra and let V and W be

(possibly infinite) homotopy strings. Then the following hold.

1. If r > 1, then dim HomK(PV , PW ) ≤ 1.

2. If r = 1, then dim HomK(PV , PW ) ≤ 2.

Proof. If r > 1 then, by Lemma 5.3.4 and Lemma 5.3.5 combined with Observation

5.1.10, if there exists a graph map or a maximal graph homotopy, then this is the

unique basis morphism PV → PW .

When r = 1, the above considerations apply but, by Lemma 5.3.4, we may also

have that there is a graph map and a homotopy class of single maps given by a maximal

graph homotopy. By the same reasoning as above there can be no other basis maps.

Next we consider the case where there is a single map f : PV → PW such that

H(f) = {f}. Then the form of the stings mean that the only possibility is that f falls

into case (a) of Corollary 5.1.20. Clearly this can be the only such basis morphism.

Finally, the only possible double maps g with H(g) = {g} arise when either V or W

consist of a single homotopy letter i.e. PV or PW is a two-term complex. Again, it is

clear that g can be the only basis morphism from PV to PW in this case.

Example 5.3.8. This example shows the r = 1 case where the upper bound is at-

tained. Consider the derived-discrete algebra Λ = Λ(1, 1, 3). That is, the algebra given

by the quiver

1

b1��

−1a−1 // 0

b0

@@

2b2oo

with the relation b2b0 = 0. If we take the homotopy strings V = (b2b1b0 ·b2b1b0a−1, 2, 4)

and W = (a−1 · b2b1b0 · b2b1b0 · b2b1b0a−1, 2, 4). Then we have the following graph map.

• b2b1b0 //1��

•b2b1b0a−1//

1��

•1��

• •a−1

oob2b1b0

// •b2b1b0

// •b2b1b0

// •

As well as the following maximal graph homotopy.

• b2b1b0 //1��

•b2b1b0a−1//

1��

•

• •a−1

oob2b1b0

// •b2b1b0

// •b2b1b0

// •


Corollary 5.3.9. Let Λ be a derived-discrete algebra and let V be a (possibly infinite)

homotopy string. The complex PV indecomposable.

Proof. If V is a finite homotopy string, then PV is known to be indecomposable by

[10]. So assume that V is infinite. From the proof of Theorem 5.3.7, we have that

dim HomK(Λ-Proj)(PV , PV ) = 1 unless there is both a maximal graph homotopy and a

graph map PV → PV . But as the string is infinite the graph homotopy arising in this

way will be supported in infinitely many degrees and so will be a null-homotopy. Thus

HomK(Λ-Proj)(PV , PV ) = k and since the category K(Λ-Proj) is idempotent complete,

it follows that PV is indecomposable.

5.3.2 The category when gldim Λ <∞

The algebra Λ = Λ(r, n,m) has finite global dimension if and only if r < n [13].

The bounded derived category in the finite and infinite global dimension cases have

slightly different structure; in this section we describe the structure of Db(Λ-mod) for

gldim Λ <∞ and in the next section we describe Db(Λ-mod) for gldim Λ =∞.

In [13], the authors describe the form of the components of the Auslander-Reiten

quiver of Db(Λ-mod). Here we extend this description to the category add(StrΛ)

where StrΛ is the collection of all indecomposable string objects in K(Λ-Proj) and

add(StrΛ) is the closure of StrΛ under finite direct sums and summands. In particular

we describe a quiver where the vertices are indecomposable string complexes and the

arrows are the irreducible morphisms between them. This description is obtained via

an application of [1, Sec. 6] where the irreducible morphisms between string complexes

are described; this argument is due to K. Arnesen.

Remark 5.3.10. Let A be a gentle algebra. A description of the AR triangles in

Kb(A-proj) is first given in [12]. In [1] our description of morphisms between string

complexes in Kb,−(A-proj) is used to extend this description to include unbounded

complexes in Kb,−(A-proj). The argument actually applies to all indecomposable

string complexes in K(Λ-Proj) (since they are finite-dimensional in each degree, see

Remark 5.3.2).

The Auslander-Reiten quiver of add(StrΛ) consists of 8r components. For each


k ∈ {0, 1, · · · , r − 1} we have components

X k−∞,X k

∞,Yk−∞,Yk∞

of type A∞∞ and Zk∞ of type A1, all containing infinite string complexes. Moreover the

finite string complexes are contained in components

X k,Yk

of type ZA∞ and Zk of type ZA∞. The components can be seen to fit together as

follows.

Y0 X 1 Y2

X 0 Y1 X 2

Z0 Z1 Z2Zr−1

The A∞∞ component to the left of X 0 is X 0−∞ and to the right is X 0

∞. Similarly, the

A∞∞ component to the left of Y0 is Y0−∞ and to the right is Y0

∞. The component of

type A1 (that is, the single vertex) between X 0 and Y0 is Z0∞. The configuration for

k > 0 is similar (though ‘upside-down’ for odd k).

It is also necessary to fix an indexing of the indecomposable objects of add(StrΛ).

The indexing we choose is consistent with [13] and also [15]. First we fix how the

objects in specific components are labelled:

Xi,j ∈ X k with i, j ∈ Z and j ≥ i;

Y ki,j ∈ Yk with i.j ∈ Z and j ≤ i;

Zi,j ∈ Zk with i, j ∈ Z;

Xi,∞ ∈ X k∞ with i ∈ Z;

X−∞,i ∈ X k−∞ with i ∈ Z;

Yi,∞ ∈ Yk∞ with i ∈ Z;


Y−∞,i ∈ Yk−∞ with i ∈ Z;

Zk∞ ∈ Zk∞.

The following diagram indicates how the indexing looks in the Auslander-Reiten

quiver.

Y k(−1,−1)

��

Y k(0,0)

��

Y k(1,1)

��. . .

��

...

??

Y k(0,−1)

��

??

Y k(1,0)

��

??. . . ...

Y k(−1,−∞)

��

...

??

Y k(1,−1)

��

??. . . Y k

(∞,1)

??

Y k(0,−∞)

��

...

??. . . Y k

(∞,0)

??. . .

��

...

Y k(1,−∞)

��

Y k(∞,−1)

??. . .

��

Zk(−1,1)

��

??

...

. . . ...

??. . .

��

Zk(−1,0)

��

??

Zk(0,1)

��

??

...

Zk∞ Zk

(−1,−1)

��

??

Zk(0,0)

��

??

Zk(1,1)

��

??

... . . .

��

... Zk(0,−1)

��

??

Zk(1,0)

��

??. . .

Xk(−∞,1)

??

Xk(−1,∞)

��

... Zk(1,−1)

��

??. . .

Xk(−∞,0)

??. . .

��

... Xk(0,∞)

��

...

??. . .

Xk(−∞,−1)

??. . .

��

Xk(−1,1)

��

??

... Xk(1,∞)

��...

??. . .

��

Xk(−1,0)

��

??

Xk(0,1)

��

??

... . . .

Xk(−1,−1)

??

Xk(0,0)

??

Xk(1,1)

??

The morphisms between these objects all travel from left to right and ‘wrap around’

so that the components form a cylinder or a mobius strip (depending on the parity

of r). In [15] the so-called ‘Hom-hammocks’ in the category Db(Λ-mod) (for Λ with

finite global dimension) are described.

Remark 5.3.11. The hammock poset H(x) (for x ∈ Q0) defined in Section 2.4.5 is

the forward Hom-hammocks of the projective cover P (x) of the simple module S(x).

In the next result we describe these hom-hammocks and also extend them to include

the infinite string complexes; the proof that these are indeed the hom-hammocks (for


both the infinite and finite global dimension case) can be found in [2, Sec. 3] and the

argument is due to D. Pauksztello.

For M ∈ StrΛ, we define the Hom-hammocks

H+∞(M) := {X ∈ StrΛ | HomK(M,X) 6= 0} the forward Hom-hammock of A;

H−∞(M) := {X ∈ StrΛ | HomK(X,M) 6= 0} the backward Hom-hammock of A.

Proposition 5.3.12 ([2, Prop. 3.12]). Suppose Λ is derived-discrete with finite global

dimension. Let a, b ∈ Z and 0 ≤ k < r. The forward Hom-hammocks of objects of

StrΛ are given by:

H+∞(Xk

a,b) = H+(Xka,b) ∪ {Xk

i,∞ | a ≤ i ≤ b} ∪ {Xk+1−∞,j | a′ − 1 ≤ j ≤ b′ − 1};

H+∞(Y k

a,b) = H+(Y ka,b) ∪ {Y k

∞,j | b ≤ j ≤ a} ∪ {Y k+1i,−∞ | b′′ − 1 ≤ i ≤ a′′ − 1};

H+∞(Zk

a,b) = H+(Zka,b) ∪ {Xk+1

−∞,j | j ≥ a′ − 1} ∪ {Xk+1i,∞ | i ≤ a′ − 1}

∪ {Zk+1∞ } ∪ {Y k+1

i,−∞ | i ≥ b′′ − 1} ∪ {Y k+1∞,j | j ≤ b′′ − 1};

H+∞(Xk

a,∞) = {Xki,∞ | i ≥ a} ∪ {Zk

i,j | i ≥ a and j ∈ Z} ∪ {Xk+1−∞,j | j ≥ a′ − 1}

∪ {Xk+1i,j | i ≤ a′ − 1 and j ≥ a′ − 1};

H+∞(Xk

−∞,b) = {Xk−∞,j | j ≥ b} ∪ {Xk

i,j | i ≤ b and j ≥ b} ∪ {Xki,∞ | i ≤ b}

∪ {Y k∞,j | j ∈ Z} ∪ {Zk

i,j | i ≤ b and j ∈ Z} ∪ {Zk∞};

H+∞(Y k

∞,b) = {Y k∞,j | j ≥ b} ∪ {Zk

i,j | i ∈ Z and j ≥ b} ∪ {Y k+1i,−∞ | i ≥ b′′ − 1}

∪ {Y k+1i,j | i ≥ b′′ − 1 and j ≤ b′′ − 1};

H+∞(Y k

a,−∞) = {Y ki,−∞ | i ≥ a} ∪ {Y k

i,j | i ≥ a and j ≤ a} ∪ {Y k∞,j | j ≤ a}

∪ {Xki,∞ | i ∈ Z} ∪ {Zk

i,j | i ∈ Z and j ≤ a} ∪ {Zk∞};

H+∞(Zk

∞) = {Xki,∞ | i ∈ Z} ∪ {Y k

∞,j | j ∈ Z} ∪ {Zki,j | i, j ∈ Z}.

where for a ∈ Z we set

a′ =

a+ r +m if k = r − 1;

a otherwise,a′′ =

a+ r − n if k = r − 1;

a otherwise.


These sets can be best understood via the following diagrams.

X

Z

Y0 X 1 Y2

X 0 Y1 X 2

Z0 Z1 Z2

X0a,∞ X2

−∞,b

Y0 X 1 Y2

X 0 Y1 X 2

Z0 Z1 Z2

We can also easily identify which strings arise in the various components of the

quiver. Let X denote the subcategoryr−1⋃k=0

X k etc. Then the finite strings are contained

in X ∪Y ∪Z; the (right) infinite substrings of string (2) in Lemma 5.3.1 are contained

in X∞ ∪ Y−∞; the (left) infinite substrings of string (3) in Lemma 5.3.1 are contained

in X−∞ ∪Y∞; and the shifts of the (two-sided) infinite string (4) are contained in Z∞.

5.3.3 The category when gldim Λ =∞

Next we give a similar description of the Auslander-Reiten quiver for Λ = Λ(r, n,m)

with gldim Λ = ∞ i.e. when r = n. The description of the form of the Auslander-

Reiten components of Db(Λ-mod) ' Kb,−(Λ-proj) can be found in [13]. Again we

extend this description to add(StrΛ) via [1, Sec. 6].

The Auslander-Reiten quiver of StrΛ consists of 3r components. For each k ∈

{0, · · · , r − 1} there is a component X k of type ZA∞. There is also a ‘ladder-type’

component Zk:


��

��

Zki−1

��Xki−1,∞

??

��

Zki

��Xki,∞

��

??

Zki+1

��Xki+1,∞

��

??

We will label the right-hand ‘beam’ of the ladder by Zk and the left-hand ‘beam’

by X k∞. Finally there is a component Zk∞ of type A1. The components X k and Zk

make up the Auslander-Reiten quiver of Db(Λ-mod).

The morphisms in the category mean that components can be seen to fit together

as follows

X 0

X 1

X 2

X 3

X 4

X r−1

The ladder Z0 is above and to the right of X 0; the remaining ladder-type compo-

nents for k > 0 are labelled similarly but are ‘upside-down’ for odd k. Again, there

are morphisms from the components labelled by r−1 into the components labelled by

0 and so the global structure of the category resembles a cylinder or a mobius strip,

depending on the parity of r.

As before, we fix an indexing on the objects of StrΛ and it is consistent with [13].

For each 0 ≤ k ≤ r − 1, we have:

Xki,j ∈ X k with i, j ∈ Z and j ≥ i;

Xki,∞ ∈ X k

∞ with i ∈ Z;

Zki ∈ Zk with i ∈ Z;

Zk∞ ∈ Zk∞ with i ∈ Z.


The indexing fits into the Auslander-Reiten quiver as follows.

Zk∞

. . .

��. . .

��

Zki−1

��Xki−1,∞

??

��

Zki

��. . .

��

... Xki,∞

��

??

Zki+1

��. . .

��

Xki−1,i+1

??

��

... Xki+1,∞

��

??

. . .

. . .

��

Xki−1,i

??

��

Xki,i+1

??

��

... . . .

Xki−1,i−1

??

Xki,i

??

Xki+1,i+1

??

Proposition 5.3.13 ([2, Prop. 3.16]). Suppose Λ is derived-discrete with infinite global

dimension. Let a ≤ b ∈ Z and 0 ≤ k < r. The forward Hom-hammocks of objects of

StrΛ are given by:

H+∞(Xk

a,b) = H+(Xka,b) ∪ {Xk

i,∞ | a ≤ i ≤ b};

H+∞(Zk

a ) = H+(Zka ) ∪ {Zk+1

∞ } ∪ {Xk+1i,∞ | i ≤ a′ − 1};

H+∞(Xk

a,∞) = H+(Zka ) ∪ {Zk+1

∞ } ∪ {Xki,∞ | i ≥ a}.

Moreover, for 0 ≤ k < r:

H−∞(Xka,∞) = H+

∞(Zk−1a ) and H−∞(Zk

a ) = H−(Zka ) ∪ {Zk

∞} ∪ {Xki,∞ | i ≤ a},

where a =

a− r −m if k = 0;

a otherwise.

As before, these regions can be understood more easily via the following diagrams.

X 0

X 1

X 2

X 3

X 4

X r−1

X 0

X 1

X 2

X 3

X 4

X r−1


Again we can identify which strings occur in which component of the Auslander-

Reiten quiver. Let X denoter−1⋃k=0

X k etc. Complexes given by finite strings are con-

tained in X ; the (right) infinite substrings of string (3) in Lemma 5.3.1 are contained

in Z; the (left) infinite substrings of string (2) are contained in X∞; and the shifts of

the two-sided infinite string (4) are the indecomposable complexes in Z∞.

The bound on the dimension of the Hom-spaces as well as the structure of the

Hom-hammocks mean that the mesh relations within the components extend across

components in the obvious way. This will be an incredibly useful result. Again, the

argument for this property is due to D. Pauksztello.

Proposition 5.3.14 ([2, Prop. 31.6]). Let A,B,C ∈ StrΛ. If B is in H+∞(A) and C

is in H+∞(A) ∩ H+

∞(B), then any map f : A→ C factors as A→ B → C.

Chapter 6

The Ziegler spectrum of a

compactly generated triangulated

category

In this chapter we outline how the techniques described in Chapter 1 may be applied

to the setting of a compactly generated triangulated category. The definitions for

purity and the Ziegler spectrum in a compactly generated triangulated category were

first given by Krause in [33] and also Beligiannis in [11]. We will show here that the

functor category used in these definitions is equivalent to a localisation of the usual

functor category (that is, as we defined it in Chapter 1). Moreover, it follows there is a

homeomorphism between the Ziegler spectrum of a compactly generated triangulated

category and a closed subset of the Ziegler spectrum of a module category. We may

then apply all the results stated in Chapter 1 to the triangulated setting.

6.1 Compactly generated triangulated categories

Let T be a triangulated category with suspension functor Σ: T→ T. Then we say an

object C in T is compact if the canonical morphism

HomT(C,⊕i∈I

Di)→⊕i∈I

HomT(C,Di)

is an isomorphism for any set {Di | i ∈ I} of objects in T. We denote the full

(triangulated) subcategory of compact objects in T by Tc. Suppose Tc is skeletally

138

6.2. THE ZIEGLER SPECTRUM OF T 139

small and for every non-zero object D in T, there is a non-zero morphism C → D for

some compact object C, then we say that T is compactly generated.

Example 6.1.1. Let R be a ring. The category K(R-Proj) is compactly generated

and the subcategory of compact objects K(R-Proj)c is equivalent to Db(mod-R)op

(see

[39, Thm. 1.1] which extends the results in [27]).

Example 6.1.2. Let R be a ring and let D(R-Mod) be the derived category of R.

Then D(R-Mod) is a compactly generated triangulated category and the compact

objects are exactly those contained in the canonical copy of Kb(R-proj).

6.2 The Ziegler spectrum of a compactly generated

triangulated category

Let T be a compactly generated triangulated category with full subcategory Tc of

compact objects. Consider the category

Mod-Tc = ((Tc)op

,Ab)

of contravariant functors from Tc to the category Ab of abelian groups. Then let

Y : T→ Mod-Tc be the functor that takes objects N ∈ T to (−, N)∣∣Tc and morphisms

f : M → N to (−, f)∣∣Tc . We refer to the functor Y as the restricted Yoneda

functor. If the context is clear, we will simply write (−, N) for (−, N)∣∣Tc and (−f)

for (−, f)∣∣Tc .

We say that a monomorphism in T is pure if Y (f) = (−, f) is a monomorphism

in Mod-Tc. If an object M ∈ T is injective over all pure monomorphisms, then we

say that N is pure-injective. That is, for every pure monomorphism f : M → L

and every morphism g : M → N , there exists a morphism h : L → N such that the

following diagram commutes.

Mf //

g��

L

h~~N

In fact, there are several equivalent ways of characterising the pure-injective objects

in T, we give a summary in the following proposition.

140 CHAPTER 6. MODEL THEORY IN A TRIANGULATED CATEGORY

Proposition 6.2.1 ([33, Sec. 1.4]). Let M be an object in a compactly generated

triangulated category T. Then the following conditions are equivalent:

1. The object M is pure-injective in T.

2. The functor (−,M) is injective in Mod-Tc.

3. For every object N in T, the induced morphism (N,M) → ((−, N), (−,M)) is

an isomorphism.

Moreover, the second condition leads to a slightly stronger statement.

Proposition 6.2.2 ([33, Cor. 1.9]). Let T be a compactly generated triangulated cat-

egory. Then the restricted Yoneda functor induces an equivalence of categories

Pinj(T) ' Inj(Mod-Tc)

where Pinj(T) is the full subcategory of T consisting of pure-injective objects and

Inj(Mod-Tc) is the full subcategory of Mod-Tc consisting of injective objects.

As we did in Section 1.2 of Part I in the module category, we will define a topology

on the isomorphism classes of indecomposable pure-injective objects in T. Let F be

a covariant functor from T to Ab. We say that F is coherent if there exists a exact

sequence of functors

(B,−)→ (A,−)→ F → 0

where A and B are compact objects (and (B,−) and (A,−) are the unrestricted

covariant hom-functors).

We then define the Ziegler spectrum Zg(T) of T to be the topological space

with points given by isomorphism classes of pure-injective objects in T and a basis of

(compact) open sets

(F ) := {M ∈ Zg(T) | F (M) 6= 0}

for each coherent functor F .

As before, the closed subsets of Zg(T) correspond to the definable subcategories of

T. A full subcategory D of T is said to be definable if there is a family {Fi | i ∈ Φ}

of coherent functors such that

D = {M ∈ T | Fi(M) = 0 for all i ∈ Φ}.

The closed sets of Zg(T) are exactly those of the form D∩Zg(T) (see [35, Prop. 7.3]).

6.3. HOMEOMORPHISM BETWEEN ZG(T) AND ZG(ABS-TC) 141

6.3 Homeomorphism between Zg(T) and Zg(Abs-Tc)

In this section we will show that there is a homeomorphism between Zg(T) and the

closed subset of Zg(Mod-Tc) containing only absolutely pure objects (in fact, this is

exactly the collection of indecomposable injective functors). This means that we can

apply the results stated in Part I Section 1.2 to Zg(T).

For any skeletally small additive category R, an object F ∈ Mod-R is called

absolutely pure if every monomorphism F → G in R-Mod is a pure monomorphism.

Let Abs-Tc denote the full subcategory of Mod-Tc consisting of the absolutely pure

objects. The following equivalent properties are well-known, see for example [44,

Prop. 2.3.1].

Proposition 6.3.1. Let R be a skeletally small preadditive category. Then, for each

F in Mod-R, the following are equivalent.

1. F is absolutely pure.

2. F is injective over all monomorphisms τ : G → H such that coker(τ) is finitely

presented. An object with this property is known as fp injective.

3. Ext1(G,F ) = 0 for all G in R-mod.

Since Tc is triangulated, we have the following additional characterisation of absolutely

pure objects in Tc-Mod.

Proposition 6.3.2 ([33, Lem.2.7]). A functor F in Mod-Tc is absolutely pure if and

only if it is flat. In particular, it is the direct limit of representable functors.

Lemma 6.3.3 ([35, Lem. 7.2]). Let T be a compactly generated triangulated category.

Then there is an equivalence of categories

(mod-Tc)op ∼−→ Coh(T)

taking a functor F to the functor F∨ where F∨(M) := (F, (−,M)) for each object M

in T.

Since mod-Tc is abelian, we have that Mod-Tc is locally coherent (by Proposition

1.0.4). It therefore follows that Abs-Tc is definable (see [44, Thm. 3.4.24]). Thus let

Zg(Abs-Tc) denote the closed subset Abs-Tc ∩ Zg(Mod-Tc) with the subset topology.


Theorem 6.3.4. There is a homeomorphism Zg(T)∼→ Zg(Abs-Tc) given by the fol-

lowing assignment N 7→ (−, N).

Proof. By Proposition 6.2.2, the assignment N 7→ (−, N) is a bijection between the

points of Zg(T) and Zg(Abs-Tc). We will show that this bijection is a homeomorphism.

The basic open sets of Zg(Abs-Tc) are (G) where G is an object of the localisation

(mod-Tc,Ab)fp/ann(Z) and Z = Zg(Abs-Tc) (see Section 1.3.4 for notation). By [44,

Thm. 18.1.4], the definable subcategory Abs-Tc is equivalent to the category of exact

functors from (mod-Tc,Ab)fp/ann(Z) to Ab and the latter is determined by Abs-Tc

up to natural equivalence. But in [45, Thm. 7.2] it is shown that the category of

exact functors from (mod-Tc)op

to Ab is equivalent to Abs-Tc. It therefore follows

that (mod-Tc)op

is equivalent to (mod-Tc,Ab)fp/ann(Z). We can use this equiva-

lence to characterise the open sets (G) in terms of isomorphism classes of objects in

mod-Tc. It is implicit in [45] that A ∈ (mod-Tc)op

is taken to the image of (A,−) in

(mod-Tc,Ab)fp/ann(Z) (since (A,−) is dual to A⊗Tc −). It follows that we can take

the basic open sets of Zg(Abs-Tc) to be

{(−, N) ∈ Zg(Abs-Tc) | (A, (−, N)) 6= 0}

for A in mod-Tc.

The equivalence in Lemma 6.3.3 gives us that for all F in Coh(T), there is a A in

mod-Tc such that F = A∨. The corresponding open set (F ) is

{N ∈ Zg(T) | F (N) 6= 0} = {N ∈ Zg(T) | (A, (−, N)) 6= 0}.

It is immediate from this that N 7→ (−, N) is a homeomorphism.

Recall that a typical object G in Coh(T) has a presentation

(D,−)→ (C,−)→ G→ 0

where C,D ∈ Tc. Since the Yoneda functor is full, there must be some f : C → D such

that G = coker(f,−); then we will denote G by Ff . Using the above, we ca explicitly

describe the form of F ∈ Tc-mod when F∨ = Ff .

Corollary 6.3.5. Consider the equivalence of Lemma 6.3.3 above. Suppose F∨ = Ff

for some f : C → D in Tc. Then there is an exact sequence

0 −→ F −→ (−, C)(−,f)−→ (−, D).

6.4. Σ-PURE-INJECTIVE OBJECTS 143

Proof. Suppose, under the equivalence of Lemma 6.3.3, we have F∨ = Ff for some

f : C → D in Tc. We have that (−, D)∨ = (D,−) whenever D is a compact object

since ((−, D), (−,M)) ∼= (D,M) for any M in T. We can rewrite the exact sequence

(D,−)(f,−)−→ (C,−) −→ Ff −→ 0 as (−, D)∨

(−,f)∨−→ (−, C)∨ −→ F∨ −→ 0

and so 0 −→ F −→ (−, C)(−,f)−→ (−, D) is also exact.

6.4 Σ-pure-injective objects

In this section we summarise some background on Σ-pure-injective objects in trian-

gulated categories. Given the above homeomorphism we may apply some standard

results for Σ-pure-injective modules to this setting. These results will be central to

the arguments in Section 8.1.

An object of T is Σ-pure-injective if the coproduct N (I) is pure-injective for any

(possibly infinite) set I. Equivalently, N is Σ-pure-injective if and only if for each

C ∈ Tc, the EndT(N)-module HomT(C,N) satisfies the descending chain condition on

EndT(N)-submodules.

In practice it is often easier to prove the stronger property of having finite en-

dolength (of course, only when such a property applies). An object M in a compactly

generated triangulated category T has finite endolength if the left End(M)-module

HomT(C,M) is finite length for all compact objects C in T (see [35]).

Remark 6.4.1. The equivalences described in Section 6.3 allow us to make use of the

following standard results about Σ-pure-injective modules.

• Any object of finite endolength is Σ-pure-injective [44, Cor. 4.4.24].

• A direct sum of finitely many Σ-pure-injective objects is Σ-pure-injective [44,

Lem. 4.4.26].

• If M is a Σ-pure-injective object, then every object in 〈M〉 is Σ-pure-injective

[44, Prop. 4.4.27].

The bound on the dimension of the Hom-spaces between indecomposable string

complexes (Theorem 5.3.7) imply that the string complexes PV are pure-injective.


Corollary 6.4.2. Let Λ be a derived-discrete algebra and let V be a (possibly infinite)

homotopy string. The complex PV has finite endolength and is therefore pure-injective

and indecomposable.

Proof. This is immediate from Corollary 5.3.9, the bounds given in Theorem 5.3.7 as

well as the description of the compact objects given in Proposition 5.3.3.

6.5 Localisation with respect to closed subsets

The connections we have made between the various categories involved in this set up

all interact well with the localisations described in Section 1.3.5. In this section we

will summarise the bijections implicit in the previous section; starting with a closed

subset X of Zg(T) we will list the subcategories determined by X.

• A closed subset X of Zg(T).

• The closed subset {(−, X) | X ∈ X} of Zg(Abs-Tc).

• The Serre subcategory {F | F ((−, X)) = 0 for all X ∈ X} of the category

(mod-Tc,Ab)fp/ann(Z) where Z = Zg(Abs-Tc).

• The Serre subcategory S = {A ∈ mod-Tc | (A, (−, X)) = 0 for all X ∈ X} of

mod-Tc.

• The Serre subcategory

{F ∈ Coh(T) | F = A∨ where A ∈ S} = {F ∈ Coh(T) | F (X) = 0 for all X ∈ X}

in Coh(T).

• The Hereditary torsion pair (−→S , (−→S )⊥) of finite type in Mod-Tc. This is the

torsion pair cogenerated by the set {(−, X) | X ∈ X} of indecomposable injective

objects in Mod-Tc.

Chapter 7

The Ziegler spectrum of the

homotopy category of a

derived-discrete algebra

In this chapter we will be looking at the compactly generated category K(Λ-Proj)

where Λ is a derived discrete algebra. As before we will denote K(Λ-Proj) by K. The

full subcategory of compact objects will be denoted Kc (see Proposition 5.3.3).

In Section 1.3.4, we saw that the if the isolation condition holds for a closed subset

Z of Zg(Mod-Kc) then the Krull-Gabriel analysis of (mod-Kc,Ab)fp/ann(Z) and the

Cantor-Bendixson analysis of Z work in parallel. We begin by showing that the isola-

tion condition holds for Zg(Abs-Kc), the closed subset given by the absolutely pure ob-

jects of Mod-Kc. By Proposition 1.3.9, it is enough to show that the Krull-Gabriel di-

mension of (mod-Kc,Ab)fp/ann(Z) is defined. Given the equivalences of categories de-

scribed in Section 6.3, this is equivalent to the analogous result for Zg(K) and Coh(K);

the definitions made in Chapter 1 can be given in this context, for example, if X is

a closed subset of Zg(K), then ann(X) = {F ∈ Coh(K) | F (M) = 0 for all M ∈ X}.

We will use the notation set up in Section 1.3.

Remark 7.0.1. The proof of [44, Lem. 10.2.2] gives us that we have a lattice iso-

morphism analogous to the one described in Corollary 1.1.3. That is, all coherent

subfunctors of Ff have the form im(h,−)/ im(f,−) for some factorisation f = gh of

f in Kc.

145

146 CHAPTER 7. THE ZIEGLER SPECTRUM

Lemma 7.0.2. Let Λ be a derived-discrete algebra. Then the Krull-Gabriel dimension

of Coh(K) is defined.

Proof. Let F ∈ Coh(K) and consider the lattice L(F ) of coherent subfunctors of F .

By Remark 7.0.1 and the descriptions of the Hom-hammocks in Propositions 5.3.12

and 5.3.13, it is clear that L(F ) has no densely ordered subset. By Proposition 1.3.4,

it follows that the Krull-Gabriel dimension of Coh(K) is defined.

Corollary 7.0.3. The isolation condition holds for Zg(K) and so the Cantor-Bendixson

rank of Zg(K) is defined and there is a natural bijection

{M ∈ Zg(K) | CB(M) = α} 1−1←→ {F ∈ Coh(K)/Coh(K)α | F is simple}.

The correspondence takes a complex M to the image of a functor that isolates it in

Xα.

We have that Coh(K)α = ann(Xα) where Xα is the closed subset of Zg(K) con-

taining points with Cantor-Bendixson rank greater than or equal to α. For each α, let

qα : Coh(K)→ Coh(K)/ann(Xα) be the corresponding localisation functor.

7.1 Krull-Gabriel analysis of Coh(K)

In this section we will describe the simple functors at each stage of the Krull-Gabriel

analysis of Coh(K). As a result we will prove the following theorem.

Theorem 7.1.1. Let Λ be a derived-discrete algebra. The Krull-Gabriel dimension of

Coh(K(Λ-Proj)) is 2.

7.1.1 Simple functors in Coh(K)

It is well-known that the simple functors in Coh(K) correspond to the Auslander-

Reiten triangles comprised of compact objects in K; see [5, §2].

Proposition 7.1.2. The simple objects in Coh(K) are exactly those of the form Ff

where Xf→ Y

g→ Z → ΣX is an Auslander-Reiten triangle.

7.1. KRULL-GABRIEL ANALYSIS OF COH(K) 147

7.1.2 Simple functors in Coh(K)/ann(X0)

We begin by identifying a family of morphisms that give rise to simple functors in

Coh(K)/ann(X0) and then we will go on to show that, up to an equivalence relation,

these are in fact all of the simple functors in Coh(K)/ann(X0).

Definition 7.1.3. Let Λ be a derived-discrete algebra.

1. Suppose gldim Λ < ∞. For 0 ≤ k < r the following morphisms will be referred

to as 1-simple morphisms:

h : Xki,j → Xk

i+1,j ⊕ Zki,t h : Y k

i,j → Y ki,j+1 ⊕ Zk

t,j

h : Zki,j → Zk

i+1,j ⊕Xk+1t,i′−1 h : Zk

i,j → Zki,j+1 ⊕ Y k+1

j′′−1,t

where h =(h1h2

)with h1 6= 0 and h2 6= 0.

2. Suppose gldim Λ = ∞. For 0 ≤ k < r the following morphisms will be referred

to as 1-simple morphisms.

h : Xki,j → Xk

i+1,j ⊕ Zki h : Zk

j → Zkj+1 ⊕Xk+1

i,j′−1

where h =(h1h2

)with h1 6= 0 and h2 6= 0.

From Remark 7.0.1 and our description of morphisms, it is clear that q0(Fh) is a

simple functor if h is a 1-simple morphism. Since these functors are simple, it fol-

lows from Corollary 7.0.3 that there are corresponding indecomposable pure-injective

objects with Cantor-Bendixson rank equal to 1.

Proposition 7.1.4. Let Λ be a derived-discrete algebra.

1. If gldim Λ = ∞ then the elements of ind(Z) are indecomposable, pure-injective

complexes of Cantor-Bendixson rank 1.

2. If gldim Λ <∞ then the elements of ind(X∞ ∪ X−∞ ∪ Y∞ ∪ Y−∞) are indecom-

posable, pure-injective complexes of Cantor-Bendixson rank 1.

Proof. For each of these complexes there is a 1-simple morphism f such that the

open set corresponding to the functor (X,−)/ im (f,−), where X is the domain of f ,

contains that complex and complexes of Cantor-Bendixson rank 0. So this is imme-

diate from Proposition 7.0.3 together with the description of the Hom-hammocks in

Propositions 5.3.12 and 5.3.13.


We now wish to show that we have identified the full list of simple functors in

Coh(K)/ann(X0); in order to do this we will first require some preliminary lemmas.

Definition 7.1.5. If h : A→ B1⊕B2 and h′ : C → D1⊕D2 are 1-simple morphisms,

then we say that h ∼ h′ if and only if A and C are contained the same ray and B2

and D2 are contained in the same coray of the AR quiver. This defines an equivalence

relation on the set of 1-simple morphisms.

From the description of the Hom-hammocks in Propositions 5.3.12 and 5.3.13, we

see that (Fh) and (Fg) contain the same pure-injective with Cantor-Bendixson rank 1

exactly when h ∼ g. So the following corollary is immediate from Corollary 7.0.3 and

inspection of the Hom-hammocks.

Corollary 7.1.6. Let h and g be 1-simple morphisms. Then, q0(Fh) = q0(Fg) if and

only if h ∼ g.

Lemma 7.1.7. Let T be a compactly generated triangulated category and let f : A→ B

be a morphism in Tc. If q : Coh(T) → Coh(T)/ann(X) is a localisation functor and

q(Ff ) is simple, then there exists some g : C → D in Tc such that q(Ff ) = q(Fg) and

C is indecomposable.

Proof. Suppose A = A′ ⊕ A′′ where A′, A′′ ∈ Tc are nonzero. Since

(A,−) ∼= (A′,−)⊕ (A′′,−),

we have Ff ∼=((A′,−)⊕ (A′′,−)

)/ im(f,−) where we identify im(f,−) with its image

in (A′,−)⊕ (A′′,−). Consider the subfunctors of Ff

(im(f,−)+A)/im(f,−)∼= (A′,−)/im(f,−)∩(A′,−) = H ′

and

(im(f,−)+A)/im(f,−)∼= (A′′,−)/im(f,−)∩(A′′,−) = H ′′.

The sum of these subfunctors is Ff so, since q(Ff ) is simple, the image of at least one

of them under q equals q(Ff ). Hence either q(Ff ) ∼= q(H) or q(Ff ) ∼= q(H ′).

Now, im(f,−) ∩ (A′,−) ⊆ (A′,−) is a finitely generated subfunctor so, by Re-

mark 7.0.1, there exists some g : A′ → B′ in Tc such that im(g,−) ∼= im(f,−)∩(A′,−)

and similarly with A′′ in place of A. As Tc is Krull-Schmidt, the result follows.

7.1. KRULL-GABRIEL ANALYSIS OF COH(K) 149

Proposition 7.1.8. Suppose q0(Ff ) is simple in Coh(K)/ann(X0), then there exists

a 1-simple morphism h such that q0(Ff ) = q0(Fh).

The proof for the finite global dimension case can be found in [14]; our proof applies

to both infinite and finite global dimension.

Proof. If f = gh, then Fh is a factor of Ff . Thus, for any such h, we have that

q0(Fh) ∼= q0(Ff ) if and only if q0(Fh) 6= 0. We have already observed that if h is

a 1-simple morphism, then q0(Fh) 6= 0 so it remains to show that we always have

a factorisation f = gh where h is 1-simple. By Lemma 7.1.7 we may assume that

f : A→⊕n

i=1Bi where A,B1, . . . , Bn are indecomposable objects in Kc.

We observe that the Hom-hammock structure, combined with Proposition 5.3.14,

implies that there is a 1-simple morphism through which f factors, except in the

following cases:

1. A = Xki,j for some 0 ≤ k < r and i ≤ j and Bl = Xk

i,j+t for some t ≥ 0 and

1 ≤ l ≤ n.

2. A = Y ki,j for some 0 ≤ k < r and j ≤ i and Bl = Y k

i+t,j for some t ≥ 0 and

1 ≤ l ≤ n.

3. A = Zki,j for some 0 ≤ k < r and i, j ∈ Z and Bp = Zk

i+t,j, Bq = Zki,j+s for some

t, s ≥ 0 and 1 ≤ p, q ≤ n.

We argue, by contradiction, that none of these cases arise. In each of these cases,

by inspection of the Hom-hammocks, the set of indecomposable compact objects C

for which Ff (C) 6= 0 is finite. By Corollary 7.2.1, the open set (Ff ) contains only

finitely many isolated points of Zg(K). It follows from Lemma 7.0.2, that the Cantor-

Bendixson rank of Zg(K) is defined and so the isolated points are dense (see, for

example, [44, Lem. 5.3.36]). But each isolated point, being of finite endolength, is

closed (see [44, Thm. 5.1.12]) so there are no other points in (Ff ). Since these are the

only points on which Ff is nonzero and since their direct sum is of finite endolength,

it follows that Ff is finite length and q0(Ff ) = 0 which is a contradiction.

Corollary 7.1.9. The simple objects in Coh(K)/ann(X0) are in one-to-one corre-

spondence with the ∼-equivalence classes of 1-simple morphisms.


Corollary 7.1.10. Let Λ be a derived-discrete algebra.

1. If gldim Λ = ∞ then the indecomposable, pure-injective complexes of Cantor-

Bendixson rank 1 are exactly the elements of ind(Z).

2. If gldim Λ < ∞ then the indecomposable, pure-injective complexes of Cantor-

Bendixson rank 1 are exactly the elements of ind(X∞ ∪ X−∞ ∪ Y∞ ∪ Y−∞).

Proof. Combine Proposition 7.1.4 with Corollary 7.0.3.

7.1.3 Simple functors in Coh(K)/ann(X1)

We obtain Coh(K)/ann(X1) by localising Coh(K) at the Serre subcategory consisting

of the functors F such that q0(F ) has finite length.

Remark 7.1.11. Since the isolation condition holds and using Proposition 7.1.10, we

may apply a similar argument to the one contained in the proof of Proposition 7.1.8

to obtain that an object q0(F ) in Coh(K)/ann(X0) is finite length if and only if (F )

contains finitely many string complexes of Cantor-Bendixson rank 1.

Proposition 7.1.12. Let Λ be a derived-discrete algebra of either finite or infinite

global dimension. An object q1(F ) in Coh(K)/ann(X1) is simple if and only if we

have that q1(F ) = q1((Zkj ,−)) for some j, k.

Proof. We give an argument for the case where Λ has infinite global dimension; the

case where Λ has finite global dimension can be found in [14].

Note that, by Remark 7.1.11 and the description of Hom-hammocks in Propo-

sition 5.3.13, the objects q1((Xki,j,−)) = 0 and q1((Zk

j ,−)) 6= 0 for all i, j ∈ Z and

0 ≤ k < r. It follows from this and Lemma 7.1.7 that any simple object will be of

the form q1(Ff ) where f : Zkj → B for some compact object B, j ∈ Z and 0 ≤ k < r.

It remains to show that q1((Zkj ,−)) is simple. Any coherent subobject of q1((Zk

j ,−))

will (by Remark 7.0.1) be the image under q1 of some im(f,−) ⊆ (Zkj ,−) where

f : Zkj →

⊕ni=1Bi with B1, . . . , Bn indecomposable. Note that im(f,−) is the sum of

the im(πif,−) where πi is the projection to Bi. If Bi = Zkt for any 1 ≤ i ≤ n and

t ≥ j, then Remark 7.1.11 and Proposition 5.3.13 gives us that q1(Ff ) = 0 and so

q1((Zkj ,−)) = q1(im(f,−)). So consider the case where Bi ∈ X k+1 for each 1 ≤ i ≤ n.

Then there is an epimorphism (⊕n

i=1Bi,−)→ im(f,−) and so q1(im(f,−)) = 0.

7.2. THE ZIEGLER SPECTRUM 151

This completes the proof of Theorem 7.1.1 since it is clear that everything is finite

length in Coh(K)/ann(X1); that is ann(X2) = Coh(K)2 = Coh(K).

7.2 The Ziegler spectrum

By Corollary 7.0.3, the list of simple functors we have produced at each stage of the

Krull-Gabriel analysis are in bijection with the isolated points at the corresponding

stage of the Cantor-Bendixson analysis. In this section we list the pure-injective com-

plexes of each Cantor-Bendixson rank. We repeat results from the previous sections

for the sake of providing a full list of the points of the Ziegler spectrum.

By Proposition 6.4.2, the infinite string complexes are indecomposable pure-injective

objects in K, that is, they are points of Zg(K). Thus we can determine their Cantor-

Bendixson by matching them to a functor that isolates them. This is not hard to do

given the descriptions of the Hom-hammocks given in Section 5.3 and the resulting

ranks are listed in Sections 7.2.1 to 7.2.3.

The question remains as to whether there are any more indecomposable pure injec-

tive objects in K. The answer is in fact that there are not; this follows from the fact

that every simple object at each stage in the Krull-Gabriel analysis isolates a string

complex and so, again by Corollary 7.0.3, the set of string complexes in K are exactly

the set of indecomposable pure-injective objects in K.

7.2.1 Cantor-Bendixson Rank 0

Proposition 7.2.1. Let Λ be a derived-discrete algebra

1. If gldim Λ =∞ then ind(X ) is the set of isolated points in Zg(K).

2. If gldim Λ <∞ then ind(X ∪ Y ∪ Z) is the set of isolated points in Zg(K).

Proof. This now follows from Corollary 7.0.3 and the description of Auslander-Reiten

triangles in [13] from which it follows that it is the indecomposable bounded complexes

which begin Auslander-Reiten sequences.


Proposition 7.2.2. Let Λ be a derived-discrete algebra.


1. If gldim Λ = ∞ then the indecomposable, pure-injective complexes of Cantor-

Bendixson rank 1 are exactly the elements of ind(Z).

2. If gldim Λ < ∞ then the indecomposable, pure-injective complexes of Cantor-

Bendixson rank 1 are exactly the elements of ind(X∞ ∪ X−∞ ∪ Y∞ ∪ Y−∞).

Proof. Combine Proposition 7.1.4 with Corollary 7.0.3.


Proposition 7.2.3. Let Λ be a derived-discrete algebra of either finite or infinite

global dimension. The set Z∞ is a complete list of all indecomposable, pure-injective

complexes of Cantor-Bendixson rank 2.

Proof. We have already seen that the objects Zk∞ are indecomposable and pure-

injective and it is clear from the Hom-hammocks that the Hom-functors (Zkj ,−) isolate

these points in X2.

7.2.4 Classification of indecomposable pure-injective complexes

Theorem 7.2.4. Let Λ be a derived-discrete algebra. Then the set of indecompos-

able pure-injective complexes in K(Λ-Proj) is exactly the set of string complexes in

K(Λ-Proj).

Proof. By Proposition 6.4.2, every string complex is indecomposable and pure-injective.

In Sections 7.2.1 to 7.2.3 we have matched every simple object at each stage of the

Krull-Gabriel analysis with a string complex. It follows from Corollary 7.0.3 that we

have a full list of the indecomposable pure-injective complexes.

Chapter 8

Indecomposable objects in the

homotopy category of a

derived-discrete algebra

In this chapter we will use the results of the previous chapter to prove our main result:

that is, every indecomposable complex over a derived-discrete algebra is pure-injective

and therefore is a string complex.

Remark 8.0.5. Although every indecomposable complex over a derived-discrete alge-

bra is pure-injective, there are complexes which are neither pure-injective nor a direct

sum of indecomposable objects, see [11, Thm. 9.3] or [33, Thm. 2.10].

As in the previous section we will let Λ = Λ(r, n,m) be a derived-discrete algebra

and the homotopy category K(Λ-Proj) will be denoted by K.

By the results in Section 6.3, the Ziegler spectrum of K can be seen in two ways:

• The points are indecomposable pure-injective objects of K and the topology is

given by coherent functors F in Coh(K).

• The points are indecomposable injective objects of Abs-Kc (i.e. the functors

(−, N) where N is an indecomposable pure-injective object of K) and the topol-

ogy is given by A in mod-Kc.

The Krull-Gabriel analysis in the last chapter was carried out via iterated local-

isation of Coh(K) ' (mod-Kc)op

with respect to Serre subcategories. As we saw in

153

154 CHAPTER 8. INDECOMPOSABLE OBJECTS IN K(Λ-PROJ)

Section 1.3, there are parallel and compatible localisations of Mod-Kc with respect

to hereditary torsion theories. In order to make use of the torsion theoretic setting,

we will often work in Mod-Kc and then restrict back to Coh(K) ' (mod-Kc)op via

Theorem 1.3.14 and Corollary 6.3.5.

In order to prove that all indecomposable objects in K are pure-injective we will

consider the case where Λ has finite global dimension and the case where Λ has infinite

global dimension separately. Before we consider the separate cases we will prove some

preliminary lemmas. The main idea is that we consider the definable subcategory

generated by an arbitrary indecomposable complex M (see Section 1.2) and show that

if it contains any indecomposable compact object C, then M = C. This reduces the

problem to considering arbitrary indecomposable complex whose support in the Ziegler

spectrum do not contain any compact objects. The terminology used in the following

lemma is that of Section 1.3.5.

Lemma 8.0.6. Let (T ,F) be a finite-type hereditary torsion pair in Mod-Tc and let

q : Mod-Tc → Mod-Tc/T

be the localisation functor with right adjoint given by the canonical embedding functor

i : Mod-Tc/T → Mod-Tc. Suppose G ∈ F and that G is fp-injective. Then G ∼= iq(G).

Proof. Proving the statement amounts to proving that G is injective over all T -dense

embeddings i.e. over embeddings F ′ → F such that F/F ′ ∈ T . Since Mod-Tc is locally

finitely presented, it is enough to check all T -dense embeddings F ′ → F where F is

finitely presented.

So let f : F ′ → F be such an embedding and let g : F ′ → G be an arbitrary

morphism. We must show that there exists some h : F → G such that hf = g. Since

T is a finite-type torsion class, there is a finitely generated T -dense subfunctor F ′′ of

F contained in F ′ (see [44, Prop. 11.1.14]), which must therefore be T -dense in F ′. Let

g′ : F ′′ → G and f ′ : F ′′ → F be the restrictions of g and f to F ′′ respectively. Then

f ′ has a finitely presented cokernel so, since G is fp-injective, there is some h : F → G

such that g′ = hf ′. In other words, g′ − hf ′ = (g − hf)|F ′′ = 0 and so there is an

induced morphism g − hf : F ′/F ′′ → G. Since F ′/F ′′ is torsion and G is torsion-free

we must have g − hf = 0. It therefore follows that g = hf .

155

Lemma 8.0.7. Let M be an object in a compactly generated triangulated category T

and denote the support supp(M) := 〈M〉 ∩ Zg(T) of M by X. Suppose the image of a

functor F in Coh(T)/ann(X) is simple with (F ) ∩X = {C} for some compact object

C. Then C is a direct summand of M .

Proof. Let q : Coh(T)→ Coh(T)/ann(X) and let i : Coh(T)/ann(X)→ Coh(T) be the

canonical embedding (that is, i is right adjoint to q). We will also denote by q and i

the localisation and embedding functors associated to the localisation of Mod-Tc with

respect to the corresponding torsion class T .

Since q(F ) is non-zero in Coh(T)/ann(X), we must have F (M) 6= 0 and also, since

C ∈ (F ) we must have that F (C) 6= 0. Let G be the functor in mod-Tc such that

F = G∨. Since (−)∨ is an equivalence, q(G) must be simple in Mod-Tc/T and, by

definition, 0 6= F (M) = (G, (−,M)) and 0 6= F (C) = (G, (−, C)).

Since the objects (−,M) and (−, C) are torsion-free, we must have that

(q(G), q(−,M)) 6= 0 and (q(G), q(−, C)) 6= 0. Since q(G) is simple, there are em-

beddings

k : q(G)→ q(−,M) and j : q(G)→ q(−, C).

Since C is indecomposable and compact, it is an indecomposable pure-injective object

of T and so q(−, C) is an indecomposable injective object of Mod-Tc/M (see, for

example, [44, Prop. 11.1.31]). It follows that q(−, C) is the injective hull of q(G) and

so j(q(G)) is the simple socle of q(−, C). The morphism j : q(G)→ q(−, C) is between

finitely presented objects so the cokernel is also finitely presented. Since q(−,M) is fp-

injective there is some h : q(−, C)→ q(−,M) such that k = hj. It follows that h must

be a monomorphism because q(G) is the simple essential socle of q(−, C). Moreover,

by Lemma 8.0.6, we have (−, C) ∼= iq(−, C) and (−,M) ∼= iq(−,M). Therefore if we

consider the image of h under i (we shall still denote this by h), we can regard it as a

monomorphism from (−, C) to (−,M). By Yoneda’s lemma (which we may apply as C

is compact) there must be some h′ : C →M and this must be a pure monomorphism.

But C is pure-injective so C must be a direct summand of M .

Corollary 8.0.8. Let M be an indecomposable object in K and suppose there is a

compact object C ∈ X = supp(M). Then C = M .

Proof. First we argue that if C is isolated in X, then C = M . By Corollary 7.0.3,


the isolation condition holds for Zg(K). So there is some F ∈ Coh(K) such that that

{C} = (F )∩X and the image of F in Coh(K)/ann(X) is simple. So by Lemma 8.0.7,

C is a direct summand of M , that is, C = M .

Next we argue that if C ∈ X then C is isolated in X. If C is isolated in Zg(K)

then it is also isolated in X. Otherwise, without loss of generality, supposed X is

contained in the set Zg(K)′ of non-isolated points in Zg(K). In particular, we must

be in the case where Λ has infinite global dimension. By Proposition 7.2.2, the point

C has Cantor-Bendixson rank 1 and so is isolated in Zg(K)′. But then C must also

be isolated in X.

8.1 Indecomposable objects when gldim Λ =∞

We will make use of the notation introduced in Section 5.3.3 for string objects in the

homotopy category of a derived-discrete algebra Λ when gldim Λ =∞.

Let Λ be a derived-discrete algebra with gldim Λ =∞. Recall that, for 0 ≤ k < r,

there is a sequence of irreducible morphisms between noncompact, right-infinite string

complexes Xki,∞ (forming the lefthand beams of the ladder type AR components Zk).

· · · // Xki−1,∞

tki−1 // Xki,∞

tki // Xki+1,∞

// · · ·

We define Xk∞ :=

⊕i∈ZX

ki,∞ and write X :=

⊕r−1k=0X

k∞.

The main result of this section will be obtained as a corollary of the fact that

X is Σ-pure-injective. We will consider an arbitrary indecomposable object M in K

and show that the definable subcategory it generates is contained in the definable

subcategory generated by a Σ-pure-injective complex. By Remark 6.4.1, it follows

that M must be Σ-pure-injective. See Section 6.4 for the relevant definitions and

terminology.

Lemma 8.1.1. Let Λ be a derived-discrete algebra with infinite global dimension. If

C is a string complex determined by a finite homotopy string, then HomK(C,Xk∞)

satisfies the descending chain condition on EndK(Xk∞)-submodules.

Proof. Since C is compact, HomK(C,Xk∞) ∼=

⊕i∈Z HomK(C,Xk

i,∞). The descrip-

tion of the Hom-hammocks given in Proposition 5.3.13 implies that there are only

8.1. INDECOMPOSABLE OBJECTS WHEN gldim Λ =∞ 157

finitely many non-zero direct summands in⊕

i∈Z HomK(C,Xki,∞). Moreover, each

HomK(C,Xki,∞) is a finite-dimensional k-vector space by Proposition 5.3.7. It follows

that HomK(C,Xk∞) satisfies the descending chain condition on EndK(Xk

∞)-submodules.

We now consider the case where C is a right-infinite string complex. That is, C is

compact but is not contained in Kb(Λ-proj) (see Section 5.3.3).

Lemma 8.1.2. Let Λ be a derived-discrete algebra with infinite global dimension and

suppose C = Z`j for some 0 ≤ ` < r and j ∈ Z. Then HomK(C,Xk

∞) satisfies the

descending chain condition on EndK(Xk∞)-submodules.

Proof. By Proposition 5.3.7, we have dim HomK(C,Xki,∞) ≤ 1. The description of the

Hom-hammocks given in Proposition 5.3.13 gives that there exists an N ∈ Z such

that HomK(C,Xkl,∞) = 0 for all l > N . Also, by Proposition 5.3.7, we have that

dim HomK(C,XkN,∞) = 1 so fix a basis element bN ∈ HomK(C,Xk

N,∞). By Propo-

sition 5.3.14, we can choose a basis element bi ∈ HomK(C,XkN,∞) for each i ≤ N

such that bj = tkj−1 · · · tki bi for all i < j ≤ N . Moreover, since C is compact,

we have HomK(C,Xk∞) ∼=

⊕i∈Z HomK(C,Xk

i,∞) and so {bi | i ≤ N} is a basis for

HomK(C,Xk∞).

Let M be a proper EndK(Xk∞)-submodule of HomK(C,Xk

∞). If bi ∈M for some i,

we must have bi+1 = tki bi ∈M . Since we have assumed that M is a proper submodule,

it follows that {i | bi ∈ M} has a minimal element and therefore M must be finite-

dimensional. It follows that HomK(C,Xk∞) has the descending chain condition on

EndK(Xk∞)-submodules as required.

Proposition 8.1.3. Let Λ be a derived-discrete algebra with infinite global dimension.

Then the module X is Σ-pure-injective.

Proof. By Lemmas 8.1.1 and 8.1.2 and the fact that HomK(−, Xk∞) commutes with

finite direct sums, we have that HomK(C,Xk∞) satisfies the descending chain condition

for each 0 ≤ k < r and each compact object C. That is, each module Xk∞ for 0 ≤ k < r

is Σ-pure-injective. Then X is Σ-pure-injective by Remark 6.4.1.

Corollary 8.1.4. Let Λ be a derived-discrete algebra with infinite global dimension.

If an indecomposable object M in K is such that supp(M) contains no compact object,

then M is Σ-pure-injective.


Proof. Recall from the proof of Corollary 6.4.2 that every string complex has finite

endolength and is therefore Σ-pure-injective. Consider Z := X ⊕ (r−1⊕k=0

Zk∞) and note

this is the direct sum of all noncompact indecomposable pure-injective objects in K.

Since it is a finite direct sum of Σ-pure-injective modules it is Σ-pure-injective. Then

〈M〉 ⊆ 〈Z〉 and by Remark 6.4.1, we must have that M is Σ-pure-injective.

Theorem 8.1.5. Let Λ be a derived-discrete algebra with infinite global dimension.

Then the indecomposable objects of K = K(Λ-Proj) are exactly the (possibly infi-

nite) string complexes StrΛ. In particular, every indecomposable object in K is pure-

injective.

Proof. Let M be an arbitrary indecomposable object in K. Then if supp(M) contains

a compact object, it follows that M is compact. By Proposition 5.3.3, in this case

M is a string complex. If supp(M) does not contain any compact objects then by

Corollary 8.1.4 we have that M is Σ-pure-injective. The classification of the pure-

injective objects in K given in Theorem 7.2.4 gives us that M is an indecomposable

string complex.

8.2 Indecomposable objects when gldim Λ <∞

At the beginning of this chapter we reduced the problem to considering an arbitrary

indecomposable object M in K such that supp(M) does not contain a compact object.

Next we will reduce the problem further so that we need only consider M where

supp(M) contains at least one module of Cantor-Bendixson rank 1 but no compact

objects.

Lemma 8.2.1. Let Λ be a derived-discrete algebra with finite global dimension and let

M be an indecomposable object in K. If supp(M) contains only points with Cantor-

Bendixson rank 2, then M is Σ-pure-injective.

Proof. Let N be the direct sum of one copy of each object in supp(M). By Corollary

7.2.3, there are only finitely many points with Cantor-Bendixson rank 2. It follows

that N is the direct sum of finitely many Σ-pure-injective complexes and hence is Σ-

pure-injective (by Remark 6.4.1). The definable category 〈N〉 therefore contains only

Σ-pure-injective objects and, in particular, we conclude that M is Σ-pure-injective.

8.2. INDECOMPOSABLE OBJECTS WHEN gldim Λ <∞ 159

Consider the closed set X0 of Zg(K) consisting of all points of Cantor-Bendixson

rank greater than or equal to 1. Then let T0 denote the hereditary torsion class of

finite type in Mod-Kc corresponding to X0. Then let q : Mod-Kc → Mod-Kc/T0 be

the localisation functor with right adjoint i : Mod-Kc/T0 → Mod-Kc given by the

canonical inclusion.

Throughout this section we will consider an arbitrary indecomposable object of K,

which we will denote by M . By Corollary 8.0.8 and Lemma 8.2.1, we may assume that

supp(M) does not contain any compact objects and that there is at least one object

with Cantor-Bendixson rank 1.

Let N be an object of Cantor-Bendixsion rank 1 contained in supp(M). Then,

according to Proposition 7.2.2, we have N is contained in one of the components

X k∞,X k

−∞,Yk∞,Yk−∞ for some 0 ≤ k < r. So, if N lies in X k∞ or Yk∞ for some k, there

is a sequence of indecomposable complexes

N = N1α1−→ N2

α2−→ N3α3−→ N4

α4−→ · · · . (8.1)

and we shall write βn = αn · · ·α2α1 for each n ≥ 1.

If N lies in X k−∞ or Yk−∞ for some k, then there is a sequence of indecomposable

complexes and irreducible morphisms

L1α1−→ L2

α2−→ L3α3−→ L4

α4−→ · · · . (8.2)

in X k∞ or Yk∞ such that there is a non-zero morphism N

γ0−→ L1. We shall write

γn = αn · · ·α1γ0 for each n ≥ 1.

We can embed these sequences in Mod-Kc via the restricted Yoneda embedding to

obtain:

(−, N) = (−, N1)(−,α1)−→ (−, N2)

(−,α2)−→ (−, N3)(−,α3)−→ (−, N4)

(−,α4)−→ · · · .

and

(−, N)(−,γ0)−→ (−, L1)

(−,α1)−→ (−, L2)(−,α2)−→ (−, L3)

(−,α3)−→ · · · .

Lemma 8.2.2. Let Λ be a derived-discrete algebra with finite global dimension and let

M,N be as above. Then the subfunctors

ker(−, β1)→ ker(−, β2)→ · · · or ker(−, γ0)→ ker(−, γ1)→ · · ·

of (−, N) are such that⋃n≥1 ker(−, βn) = (−, N) or

⋃n≥0 ker(−, γn) = (−, N).


Proof. The functors will be zero on any indecomposable compact object outside of

H−∞(N) and so we only consider these objects (for the notation, see Section 5.3.2,

noting that the backward hom-hammocks are easily deduced from Proposition 5.3.12).

The case where N is contained in X k∞ or Yk∞ for some k and the case where N lies in

X k−∞ or Yk−∞ for some k behave slightly differently. In both cases, the lemma follows

easily from an examination of the Hom-hammocks.

The following diagrams illustrate the calculation required in the case where N is

contained in X k∞ or Yk∞.

N

Nt

Ns

(8.3)

The darkest shaded region shows the objects C such that (C,N) 6= 0 but where

there are non-zero morphisms f : C → N such that f ∈ ker(C, βt+1). The intermediate

shaded region together with the darkest shaded region shows those objects such that

(C,N) 6= 0, and where there are non-zero morphisms f : C → N such that f is in

ker(C, βs+1). The three shaded regions together show objects C such that (C,N) 6= 0.

As we continue to consider the C such that ker(C, βj) 6= 0 for successively larger j,

it is clear that the region covered approaches the region containing all C such that

(C,N) 6= 0.

In exactly the same way, we can consider the case where N is contained in X k−∞ or

Yk−∞. The picture to consider is below.

N

L1

Lt

(8.4)


As each (−, Ni) and each (−, Li) is isolated in to closed subset of Zg(Abs-Kc)

corresponding to X0, it follows that each q(−, Ni) and q(−, Li) is the injective hull of

a simple object in Mod-Kc/T0 and, in particular, has a simple socle. We will identify

the functors corresponding to these simple socles and use them to prove the main

result.

Lemma 8.2.3. Let M , N be as above. Then for n ≥ 1 we have

q(ker(−, βn))

q(ker(−, βn−1))∼= q(ker(−, αn)) or

q(ker(−, γn))

q(ker(−, γn−1))∼= q(ker(−, αn))

and q(ker(−, αn)) is a simple functor in Mod-Kc/T0.

Proof. Note that in both cases, there is a morphism

ker(−, βn)

ker(−, βn−1)

ϕ−→ ker(−, αn) orker(−, γn)

ker(−, γn−1)

ϕ−→ ker(−, αn)

and so there are embeddings imϕ→ ker(−, αn).

Comparing the Hom-hammocks, we see that ker(−, αn)/ imϕ is only non-zero on

finitely many indecomposable compact objects (in the first case, they points lying on

the blue dashed coray between the shaded block and the mouth of the component;

and in the second case on the red dotted coray from the red dashed ray to the mouth

of the component). But this implies that ker(−, αn)/ imϕ is a finite length functor

(see Remark 7.1.11) and hence q(ker(−, αn)/ imϕ) = 0 and the first claim follows.

We will now prove the second claim. By Proposition 7.1.9 and Corollary 6.3.5,

we are interested in finding an equivalence class H of 1-simple morphisms such that

for each compact C there exists some h ∈ H such that there exists an embedding

ker(C, αn) → ker(C, h). As the localised functor q(ker(−, αn)) is non-zero, it follows

that q(ker(−, αn)) = q(ker(−, h)) and so q(ker(−, αn)) is simple as claimed. An inspec-

tion of the Hom-hammocks confirms that the following classes fulfil these requirements

(the definitions depend on the notation used in Proposition 5.3.12).

• if N = Xka,∞ take H = {Zk+1

a,j → Zk+1a+1,j ⊕Xk+1

t,a′−1 | j ∈ Z and t ≤ a′ − 1};

• if N = Y k∞,b take H = {Zk+1

i,b → Zk+1i,b+1 ⊕ Y

k+1b′′−1,t | i ∈ Z and t ≥ b′′ − 1};

• if N = Xk−∞,b and L1 = Y k

∞,d take

H = {Zki,d → Zk

i,d+1 ⊕ Y k+1d′′−1,t | i ∈ Z and t ≥ d′′ − 1};


• if N = Y ka,−∞ and L1 = Xk

c,∞ take

H = {Zkc,j → Zk

c+1,j ⊕Xk+1t,c′−1 | i ∈ Z and t ≤ c′ − 1}.

Remark 8.2.4. The proof of the main theorem in this section will require us to

construct a morphism from a infinite string complex to an arbitrary indecomposable

complex. Before we begin this proof we will outline how this construction will work.

Let N be a one-sided infinite string complex. Without loss of generality, suppose that

N comes from a right infinite string (that is, the string (2) in Lemma 5.3.1). Then

there is a ray or coray in the Auslander-Reiten quiver such that, for any subsequence,

C0α0 // C1

α1 // C2α2 // · · ·

of the indecomposable finite string complexes on this ray or coray (where αi is a

composition of irreducible morphisms) we have N = holim−−−→Ci (see [38] for the definition

of homotopy colimit). Moreover, this sequence can be chosen so that

• for each i ≥ 0, the morphism αi is a graph map;

• the left end of each string is fixed and agrees with the left end of the string giving

N ; and

• the number of degrees n in which αni is non-zero strictly increases with i.

Example 8.2.5. This example illustrates what we mean in the above explanation.

Consider the algebra given by the quiver

1c

��−1 a

// 0

b

@@

2d

oo


with relations bd = dc = 0. Then we have the following sequence of irreducible maps.

0 d //

1��

2 cba //

1��

−1

1��

2 cb //

1��

0 d //

1��

2

1��

cba // −1

1��

−1 0aoo d //

1��

2 cb //

1��

0 d //

1��

2 cba //

1��

−1

1��

0 d //

1��

2 cb //

1��

0 d //

1��

2 cba //

1��

−1

1��...

......

......

Let M be an arbitrary object in K. If we have a family {hi | hi : Ci →M}i∈N such

that hi+1αi = hi for each i ≥ 0, then clearly we can construct a morphism h : N →M .

That is, for each degree n ∈ Z, let hn = hni for some i ∈ N where αni 6= 0.

Theorem 8.2.6. Let Λ be a derived-discrete algebra with finite global dimension. Then

the indecomposable objects in K = K(Λ-Proj) are exactly the (possibly inifinte) string

complexes StrΛ. In particular, every indecomposable object is pure-injective.

Proof. Suppose for a contradiction, that M is an indecomposable object in K but that

M is not a string complex. Then, by Corollary 8.0.8 and Lemma 8.2.1, we have that

supp(M) does not contain any compact objects and there is at least one object with

Cantor-Bendixson rank 1. As before, let N ∈ supp(M) denote such an object with

Cantor-Bendixson rank 1.

By Lemmas 8.2.2 and 8.2.3, we have q(ker(−, α1)) = q(ker(−, β1)) is the simple

socle of (−, N). Let S denote ker(−, β1). Then S∨ is a coherent functor and S∨(N) 6= 0

and so S∨(M) = (S, (−,M)) 6= 0 since N ∈ 〈M〉. As q(S) is simple, there is an

embedding q(S)→ q(−,M). Moreover, q(−,M) is fp-injective so by Lemma 8.2.3 we

have

q(S) = q(ker(−, β1)) //

��

q(ker(−, β2)) //

∃

uu

q(ker(−, β3)) //

∃

rr

· · · ,

q(−,M).

We can take the direct limit of these embeddings and so there exists an embedding

q(−, N) → q(−,M). Since supp(M) does not contain any isolated points and N has


Cantor-Bendixson rank 1, the functors (−,M) and (−, N) are torsion-free and fp-

injective, so by Lemma 8.0.6, we have iq(−,M) ∼= (−,M) and iq(−, N) ∼= (−, N).

Thus there is an embedding τ : (−, N)→ (−,M).

As N is pure-injective, the functor (−, N) is injective and there must exist some

σ : (−,M) → (−, N) such that στ = 1(−,N). But then, since N is pure-injective,

we have (M,N) ∼= ((−,M), (−, N)), that is, there exists some pure epimorphism

f : M → N such that σ = (−, f). The final step of the proof is to construct some

h : N →M such that fh = 1N since then we may conclude that N ∼= M .

Next we will use the construction in Remark 8.2.4 to produce such an h. Let

C0α0 // C1

α1 // C2α2 // · · ·

be a sequence of compact objects as in Remark 8.2.4. Since the Ci are strictly increas-

ing in length, we must have a morphism ιi : Ci → N for each i ≥ 0 such that each

ιi : Ci → N factors as

Ciαi //

ιi ""

Ci+1.

ιi+1

��N.

. We may then apply the functors τ and σ to this sequence to obtain the following

commutative diagram where τi := τ(Ci) for each i ≥ 0.

(C0,M)(C0,f)

11 (C0, N)τ0qq

(C1,M)(C1,f)

11

(α0,M)

OO

(C1, N)τ1qq

(α0,N)

OO

(C2,M)(C2,f)

11

(α1,M)

OO

(C2, N)τ2qq

(α1,N)

OO

From the commutativity of the squares in the tower above, we get

Ciαi //

τi(ιi) ""

Ci+1

τi(ιi+1)��M

and thus we have a family of morphisms {τi(ιi) | τi(ιi) : Ci →M}i∈N with the proper-

ties described in Remark 8.2.4. Let h : N →M be the morphism constructed from this


family. Since the ιi are graph maps and στ = 1(−,N), it follows that for each degree

n ∈ Z such that Nn 6= 0, we have (fh)n = 1Nn and hence fh = 1N as desired.

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STRING ALGEBRAS IN REPRESENTATION THEORY · over a domestic string algebra, this is also the value of the Krull-Gabriel dimension of the category of nitely presented functors from