The Ziegler and Zariski Spectra of Some Domestic String Algebras

Algebras and Representation Theory 5: 211–234, 2002.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

211

The Ziegler and Zariski Spectra of SomeDomestic String Algebras

KEVIN BURKE and MIKE PRESTDepartment of Mathematics, University of Manchester, Manchester M13 9PL, U.K.e-mail: [email protected]

(Received: March 2000)

Presented by A. Verschoren

Abstract. It was a conjecture of the second author that the Cantor–Bendixson rank of the Zieglerspectrum of a finite-dimensional algebra is either less than or equal to 2 or is undefined. Here werefute this conjecture by describing the Ziegler spectra of some domestic string algebras wherearbitrary finite values greater than 2 are obtained. We give a complete description of the Zieglerand Gabriel–Zariski spectra of the simplest of these algebras. The conjecture has been independentlyrefuted by Schröer who, extending his work (1997) on these algebras, computed their Krull–Gabrieldimension.

Mathematics Subject Classifications (2000): Primary: 16G20; secondary: 03C60, 16D70, 16G70,18E15.

Key words: string algebra, domestic representation type, Ziegler spectrum, pure-injective module,Cantor–Bandixon rank, Krull–Gabriel dimension, functor categories.

1. Indecomposable Pure-injectives over Domestic String Algebras

Let R be a domestic string algebra over an arbitrary field k. Modules will generallybe left R-modules: the category of these we denote by R-Mod.

Let F be a finitely presented functor in (R-mod,Ab), the category of additivefunctors from R-mod, the category of finitely presented R-modules, to Ab, thecategory of Abelian groups, having the property that for every finite-dimensionalband module N one has F(N) = 0 and for every finite-dimensional indecom-posable string module N , dimk F (N) � 1 (see [3] for string and band modules).Such a functor is locally simple in the terminology of [7]. We do not assert thatsuch a functor F necessarily exists (though in all examples that we have lookedat it does). We make the further assumption that F is actually a subfunctor of theforgetful functor: thus, if M � N and if a ∈ FN then the assertion ‘a ∈ M’ ismeaningful. For the applications later in this paper, the functor F will have the formF(M) = γM for some γ ∈ R. In this section we will describe the indecomposablepure-injective modules over those domestic string algebras R for which there issuch a functor.

212 KEVIN BURKE AND MIKE PREST

We will make use of the following notion from the model theory of modules:if M is any module and a ∈ M then the pp-type of a in M is ppM(a) = {F �(R,−): F is finitely presented and a ∈ FM}, is the set (in fact, filter in the latticeof finitely presented subfunctors) of all finitely presented subfunctors F of theforgetful functor (R,−) such that a ∈ FM. (This is equivalent to the originaldefinition, which is in terms of pp formulas.)

We let RZg, respectively RZar, denote the left Ziegler, respectively Gabriel–Zariski, spectra of the ring R. The underlying set of both these spaces is the set ofisomorphism classes of indecomposable pure-injective (= algebraically compact)left R-modules. The sets of the kind (F ) = {N ∈ RZg : (F,− ⊗ N) �= 0} forF a finitely presented functor in (mod-R,Ab) form a basis of open subsets for theZiegler topology and their complements (we denote the complement of (F ) by [F ])form a basis of open subsets for the Zariski topology on this set. In practice (seelater), we specify such sets by a variety of means. For more on these topologicalspaces see, for instance, [16, 11, 5]. We also use D to denote the duality betweenthe functor categories (mod-R,Ab)fp and (R-mod,Ab)fp.

At this point we should also say something about the proofs in this section. Ouroriginal proofs made free use of ideas, terminology and results from the modeltheory of modules. The proofs given here are essentially rephrasings, in terms ofthe functors defined by pp formulas, of the original proofs.

Throughout this section we assume that R is a domestic string algebra overan arbitrary field k and that F is a finitely generated subfunctor of the forgetfulfunctor (R,−) such that FN = 0 for all finitely generated band modules N anddimkFN � 1 for all indecomposable finitely generated string modules N .

PROPOSITION 1.1. Let N ∈ RZg with F(N) �= 0. Then N is a direct summandof a direct product

∏λ Nλ of indecomposable finite-dimensional (string) modules

Nλ each of which is such that F(Nλ) �= 0.Proof. We know that N is a direct summand of a direct product

∏ Nλ of in-

decomposable finite-dimensional modules (e.g., [9]). Denote the inclusion of N in∏λ Nλ by j and the projections of

∏λ Nλ to its factors Nλ by πλ. Choose a ∈ FN ,

a �= 0. Set aλ = F(πλj)(a) ∈ FNλ: thus, since F commutes with products,a = (aλ)λ. Let ′ = {λ ∈ : aλ �= 0} and let π denote the canonical projectionfrom

∏ Nλ to

∏ ′ Nλ.

If G ∈ ppN(a) then, for all λ, aλ = πλj (a) ∈ GNλ and so πja = (aλ)λ ∈ ′ ∈G(

∏ ′ Nλ). Conversely, if πja ∈ G(

∏ Nλ) then each aλ, λ ∈ ′ is in G(Nλ)

and hence, for all λ ∈ , aλ ∈ G(Nλ). Therefore ja ∈ G(∏

Nλ). But j is apure embedding and so a ∈ GN . Therefore ppN(a) = pp

∏ ′ Nλ(πja). Recall, e.g.,

[11] that there is a natural identification of FN with (DF,− ⊗N) and, under thisidentification, elements such as a in FN are regarded as morphisms from DF tothe indecomposable injective functor − ⊗ N . Specifically, the embedding F →(RR,−) dualises to a surjection (RR,−) − ⊗ R → DF . Then, given MR ∈mod-R, the component of the natural transformation a at M is the map DF(M) →

THE ZIEGLER AND ZARISKI SPECTRA OF SOME DOMESTIC STRING ALGEBRAS 213

M ⊗R N given by m ∈ DF(M) �→ m⊗ a ∈ M ⊗N where m is any inverse imageof m in − ⊗ R (this is well-defined since a ∈ FN). See [11] (or [2]) for moredetail. The equality above then implies that the composition − ⊗ πj restricted toim(a) is an embedding. Then, since im(a) is essential in − ⊗ N , the morphism− ⊗ πj : (− ⊗ N) → (− ⊗ (

∏ ′ Nλ)) is an embedding and so N → ∏

′ Nλ ispure, hence split, as required. ✷PROPOSITION 1.2. Let N be as in Proposition 1.1 and let a, b be nonzero ele-ments of F(N). Then ppN(a) = ppN(b).

Proof. By 1.1 we have that N is a direct summand (denote the inclusion by j ) ofa product

∏ Nλ where each Nλ is an indecomposable string module and, retaining

the notation from the proof above, aλ �= 0 for each λ. Let bλ = F(πλj)(b) (sothat b = (bλ)λ∈ ), let ′ = {λ ∈ : bλ �= 0} and set π to be the canonicalprojection from

∏ Nλ to

∏ ′ Nλ. As in the proof of 1.1, πj preserves the pp-

type of b and hence is a pure embedding. In particular ppπjN(πja) = ppN(a) andppπjN(πjb) = ppN(b) and we also now have F(π ′

λπj)(a) �= 0 and F(π ′λπj)(b) �=

0 for all λ ∈ ′ (where π ′λ denote the coordinate projections from

∏ ′ Nλ). So,

without loss of generality, we may assume that aλ �= 0 and bλ �= 0 for all λ ∈ .Since dimk(FNλ) = 1 for all λ we have, for each λ ∈ , some nonzero µλ ∈

k such that bλ = µλaλ. Note that µλ ∈ EndNλ. Since µλ �= 0 for all λ, wehave

∏λ µλ ∈ Aut(

∏λ Nλ) and this map takes a to b and hence, since ppN(a) =

pp∏

Nλ(a) and ppN(b) = pp∏

Nλ(b), we have ppN(a) = ppN(b), as required. ✷LEMMA 1.3. Let N be as in Proposition 1.1 and let a, b be nonzero elements ofFN . Then there is a finitely generated subfunctor H of F 2 such that (a, b) ∈ HN

and HN ∩ (N ⊕ 0) = 0 = HN ∩ (0 ⊕ N) - such a functor is called a linkingfunctor between a and b.

Proof. Existence of a subfunctor H of (R2,−) such that (a, b) ∈ HN , HN �FN ⊕ FN and (a, 0) /∈ HN (equivalently (0, b) /∈ HN) is immediate fromindecomposability of N (and, hence, uniformity of N⊗−) and the fact that (a, b) ∈FN ⊕ FN (see [10, 4.11]). For the other condition, suppose that (c, 0) ∈ HN .Since, therefore, c ∈ FN it follows, by 1.2, that if c �= 0 then ppN(a) = ppN(c)and hence (a, 0) ∈ HN – contradiction. ✷PROPOSITION 1.4. If the field k is finite and N is as in 1.1 then dimFN = 1.

Proof. By 1.1, 1.2 it follows that N is in the Ziegler closure of {Nλ}λ for someindecomposable string modules Nλ. Let a, b ∈ FN with a �= 0. Then, with nota-tion as in the proof of 1.1, for each λ, bλ is a scalar multiple of aλ. Therefore eachNλ satisfies

F 2(Nλ) ⊆⋃

µ∈k{(x, y) ∈ F(Nλ) ⊕ F(Nλ) : y = µx}

and, hence, from the definition of Ziegler-closure, the same is true of N . In partic-ular, b = µa for some µ ∈ k, as required. ✷


If k is infinite we have to work rather harder.

THEOREM 1.5. Let N ∈ RZg. Then dimFN � 1.Proof. We have supposed that this is true for finite-dimensional pure-injectives,

so suppose that N is an (infinite-dimensional) indecomposable pure-injective withFN �= 0. Let a, b ∈ FN be nonzero and let the functor H be as in 1.3.

By the usual argument (as in [10, proof of 13.6]) there is an integer d = dHsuch that for any module M and any a′, b′ ∈ M such that (a′, b′) ∈ HM, there isM ′ � M containing a′, b′, with dimM ′ � d and with (a′, b′) ∈ HM ′.

Note (e.g., [17]) that there are just finitely many indecomposable string modulesof dimension � d.

Let N1, . . . , Nr be the indecomposable string modules Ni such that there isN ′′ � N with dimN ′′ � d, with (a, b) ∈ HN ′′ and such that there is a splitepiπ : N ′′ → Ni with πa �= 0.

First note that, retaining this notation, we have πb �= 0: for (πa, πb) ∈ H(Ni)

and so if πb = 0 then (πa, 0) ∈ H(Ni). But there is an inclusion Ni � N ′′ �N and so we would have (πa, 0) ∈ HN and hence (by choice of H ), πa = 0– contradiction. This argument shows that if N ′′ � N with a, b ∈ N ′′ and ifπ : N ′′ → N ′ is a split epi then πa = 0 iff πb = 0.

Still retaining the notation, for each i = 1, . . . , r, let µi ∈ k \ {0} be suchthat πb = µiπa. Suppose we have some N ′′

1 � N with a, b ∈ N ′′1 and a split

epiπ1: N ′′1 → Ni to the same Ni with, say, π1b = µ′π1a �= 0. Let λ ∈ k be

such that πa = λπ1a (using again that F(Ni) is one-dimensional). Then we have(πa, πb), (π1a, π1b) ∈ H(Ni), hence (πa, λπ1b) ∈ H(Ni), and hence (0, πb −λπ1b) ∈ H(Ni). Since Ni is embeddable in N we have H(Ni)∩ (0 ⊕Ni) = 0 andso πb = λπ1b, that is, µiπa = λµ′π1a = µ′πa and hence µi = µ′.

Now suppose that N1 � N is finite-dimensional and (a, b) ∈ H(N1). WriteN1 = N ′ ⊕ N0 where N ′ is a direct sum of string modules and N0 a direct sum ofband modules – so F(N0) = 0 and hence a, b ∈ N ′.

Decompose N ′ as a direct sum of indecomposables: N ′ = N ′1 ⊕ · · · ⊕ N ′

s

and write (a, b) = (a′1, b

′1) + · · · + (a′

s , b′s) accordingly. Let N ′′ � N ′ be of

dimension � d with (a, b) ∈ H(N ′′) and, again without loss of generality, sup-pose that N ′′ is a direct sum of string modules: N ′′ = Ni1 ⊕ · · · ⊕ Nit withNil ∈ {N1, . . . , Nr}, l = 1, . . . , t . Write (a, b) = (a1, b1) + · · · + (at , bt ) ac-cordingly. We have (al, bl) ∈ H(Nil ) and so, as shown above, bl = µil al for someµil ∈ {µ1, . . . , µr } (by arguments already used we can suppose that al, bl �= 0 forl = 1, . . . , t).

Choose j ∈ {1, . . . , s} and let πj : N ′ → N ′j be the projection. We have a′

j =πja = ∑{πjal : l = 1, . . . , t} so choose some l ∈ {1, . . . , t} with πjal �=0. Then we have the two embeddings, followed by the canonical projection πj

Nil → N ′′ → N ′ → N ′j composing to a map between indecomposable string

modules which takes al to a nonzero element πjal = λa′j , say, of F(N ′

j ). We have(a′

j , b′j ) ∈ H(N ′

j ) (since N ′j is a direct summand of N ′ and (a, b) ∈ H(N ′)), so


(πjal, λb′j ) ∈ H(N ′

j ). Also (al, bl) ∈ H(Nil ) and so (πjal, πjbl) ∈ H(N ′j ). We

conclude, as above, that λb′j = πjbl = πjµil al = µil λa

′j and so b′

j = µil a′j .

Thus we have shown that for every finite-dimensional N1 � N containing a, b

and with (a, b) ∈ H(N1) there are g1, . . . , gr ∈ EndN1 (projections in fact) suchthat (a, b) = ∑

i gi(a, b) and for each i = 1, . . . , r, gib = µigia.Next let p = ⋂{F : F ∈ ppN(a, b)} – a not necessarily finitely generated

subfunctor of (R,−)2. We use Ziegler’s criterion (in effect, the fact that EndN islocal) to show that b = µia for some i ∈ {1, . . . , r}. In fact, the main work ofthe proof is done and what follows is a standard type of argument from the modeltheory of modules (e.g., see [10, 4.30]) which we have expressed, perhaps in asomewhat stilted fashion, in functorial terms.

Let " = "(x0, y0, x1, y1, . . . , xr , yr) � (R,−)2+2r be the functor which is theintersection of p1+r (regarded in the obvious way as a subfunctor of (R,−)2+2r)with the subfunctor # of (R,−)2+2r defined by the conditions (x0, y0) =∑r

i=1(xi, yi) and yi = µixi, i = 1, . . . , r. We claim that any finitely generatedsubfunctor $ of (R,−)2+2r which contains " satisfies $N �= 0.

Any such $ is easily seen to contain a functor of the form (G1+r )∩# where Gis a finitely generated subfunctor of (R,−)2 in p. We have (a, b) ∈ GN ∩HN andhence (e.g., [10, proof of 13.6]) there is a finitely generated submodule N1 of Nwith (a, b) ∈ (G∩H)(N1). Then, by the above, we have g1, . . . , gr ∈ EndN1 suchthat (a, b) = ∑

i(gia, gib) and for each i = 1, . . . , r, gib = µigia. We also have(gia, gib) ∈ G(N1) and hence (gia, gib) ∈ GN for i = 1, . . . , r. Thus we haveproved the claim: namely (a, b, a1, b1, . . . , ar , br ) ∈ ((G1+r ) ∩ #)(N) � $N .

Thus if $ is any finitely generated subfunctor of (R,−)2+2r which contains "then (a, b) ∈ π1,2$(N) where π1,2 denotes projection to the first two coordinates.So, since N is pure-injective we have (see [10, 2.8]) (a, b) ∈ π1,2"(N) that is,we have, in N , (a, b) = ∑

i(ci , di) say, with di = µici and ppN(ci, di) containingppN(a, b) for i = 1, . . . , r. From this last and since N is pure-injective, there are([10, 2.8]) f1, . . . , fr ∈ EndN with fi(a, b) = (ci, di). Since EndN is local, oneof the fi is an automorphism and so ppN(a, b) = ppN(ci, di) for some i and, inparticular, since di = µici , we have b = µia, as required. ✷

Now we use this to show that every infinite-dimensional pure-injective moduleN with FN �= 0 is one of the standard modules (see [17]) obtained from a (doublyor singly) infinite string.

So let N be any infinite-dimensional indecomposable pure-injective with FN �=0: choose a ∈ FN, a �= 0.

We have N = lim→ Nλ where each Nλ is a finite-dimensional submodule contain-

ing a. Each inclusion map fλµ: Nλ → Nµ can be written as(gλµ && &

): N ′

λ⊕N ′′λ →

N ′µ ⊕ N ′′

µ where N ′λ and N ′

µ are indecomposables containing a (note that if wewrite Nλ as a direct sum of indecomposables Mi then, by 1.5, exactly one of thesesatisfies F(Mi) �= 0 – take N ′

λ to be this Mi and similarly for Nµ). Since fλµa = a,gλµa = a.


Since for every band module M, FM = 0, both N ′λ and N ′

µ are string modulesand hence, see, e.g., [5], gλµ is a k-linear combination of graph maps. Also sincegλµa = a, we can write gλµ = g∗

λµ+h for some h where g∗λµ is the graph map from

N ′λ to N ′

µ determined by taking a ∈ N ′λ to a ∈ N ′

µ. Note that g∗λµ is uniquely de-

termined by this condition and so g∗λν = g∗

µνg∗λµ when these are defined. Therefore

the system (N ′λ)λ, (g∗

λµ)λ�µ is directed.Let N ′, (g∗

λ: N ′λ → N ′)λ be the direct limit of this system. Set b = g∗

λa. Wehave ppN

′(b) = ⋃

λ ppN′λ(a) = ⋃

λ ppNλ(a) = ppN(a) and hence N , which isthe hull of a (in the sense of [10, p. 74] – that is N ⊗ − is the injective hull of theimage of the morphism a from (R,−) R⊗− to N⊗−) is isomorphic to the hullof b, which is a direct summand of the pure-injective hull of N ′. Therefore everyinfinite-dimensional indecomposable pure-injective with FN �= 0 is obtained as adirect summand of the pure-injective hull of the direct limit of a directed system ofgraph maps between indecomposable string modules. These we now identify. Weretain the notation from above and use notation for strings as, for example, in [17].

Let N ′λ = M(wλ) where the string wλ is lλ−sλ

. . . lλ−1lλ0 . . . l

λtλ

where a is thebasis vector at the 0-position. Set (wλ)i = lλi . Given λ and s′

λ � sλ, t ′λ � tλ, let−s ′

λ(wλ)t ′λ = lλ−s ′

λ. . . lλ−1l

λ0 . . . l

λt ′λ

and, for µ > λ, let nλµ, mλµ be the unique integerssuch that −nλµ(wλ)mλµ

= −nλµ(wµ)mλµwith the former string closed under prede-

cessors in wλ and the latter closed under successors in wµ (that is, corresponding tothe morphism g∗

λµ which factorises as N ′λ → M(−nλµ(wλ)mλµ

) → N ′µ). Therefore

the image of g∗λ: N ′

λ → N ′ is a copy of −nλ(wλ)mλwhere nλ = minµ�λ(nλµ) and

mλ = minµ�λ(mλµ) and hence N ′ = ⋃λ im(g∗

λ) is of the form M(w) where w =lim→ λ

(−nλ(wλ)mλ), that is, w = . . . l−n . . . l0 . . . lm . . . where li = (−nλ(wλ)mλ

)i

if this is defined, for any λ, (and note that it is well defined by definition of thenλ,mλ).

Therefore N ′ is the infinite string module M(w) and N is an indecomposablesummand of the pure-injective hull of M(w). But, since R is domestic and henceevery one-sided infinite word is either expanding or contracting [17], such sum-mands are completely described by [17] and [1] and are (apart from prüfers andadics associated to bands, which we ignore since FN �= 0) just the indecomposablepure-injectives described in [17].

THEOREM 1.6. Let R be a domestic string algebra. Suppose that there is a col-lection of subfunctors Fi of the forgetful functor which are such that for each i,FiN = 0 for every finite-dimensional band module N and dimkFiN � 1 for everyindecomposable finite-dimensional string module N . Suppose also that if N is aninfinite-dimensional point of the Ziegler spectrum of R then either N lies on a band(that is, is in the image of the representation embedding from Mod-k[X] to Mod-Rwhich corresponds to one of the finitely many bands of R) or else FiN �= 0 forsome i. Then the infinite-dimensional points of the Ziegler spectrum of R are thefollowing:


(a) associated to each band, a prüfer and adic module for each quasisimple mod-ule associated to the band, together with a generic module for the band;

(b) to each one-sided or two sided infinite expanding, contracting or mixed word,the corresponding indecomposable pure-injective in the sense of [17].

Proof. Any representation embedding from Mod-k[X] to Mod-R induces ([12])a homeomorphic embedding of Zgk[X] into ZgR so, applying the description ofZgk[X] from [22] (or see [13]), we obtain the first type of module listed above (formore detail, see [13, 18]). The remaining modules are then, by assumption and theresults above, infinite string modules as described. ✷2. The Ziegler Spectrum of �2

Let R be the path algebra over the field k, where, for simplicity, we assume k to bealgebraically closed, of the quiver 2.

In the previous section we showed (take the functor F to be that given by M �→γM) that every infinite-dimensional point of the Ziegler spectrum of R is eitherassociated to one of the two bands or is a string module in the sense of [17]. In thissection we describe the Ziegler topology by giving a neighbourhood basis at everypoint. In the subsequent section we will describe the Zariski topology.

First we recall (see [3, 20]) something of the description of the Auslander–Reiten quiver (and hence of the isolated points of RZg). Since R is a string algebra,every indecomposable finite-dimensional module is either a string or band mod-ule. All but one of the Auslander–Reiten components are obtained as images ofAuslander–Reiten components of A1, the quiver of which is shown, under thefollowing two functors.


Fix F12: A1-Mod → R-Mod to be the ‘obvious’ functor taking an A1-moduleto the corresponding R-module which has support on e1, e2 and with a, b beingreplaced by α, β respectively. Similarly define F34: A1-Mod → R-Mod replacinga, b by ε, δ respectively. Note that these are representation embeddings. In fact F12

is an equivalence of A1-Mod with the full subcategory whose objects are thosemodules M with e3M = 0 = e4M and similarly for F34.

Under these functors, each component of the Auslander–Reiten quiver of A1

maps to a complete component, with the exception of the preinjective componentand one regular component under F12 and the preprojective component and oneregular component under F34. These exceptional components are sent to two raysand two co-rays of the remaining exceptional component E of the Auslander–Reiten quiver of R, which is an A∞∞ sheet with a ‘hole in the middle’.

By [12] it follows that there are induced maps between Ziegler spectra.

LEMMA 2.1. The functors F12,F34 induce homeomorphic embeddings of A1Zg

into RZg with respective images C12, C34, say. These sets are Ziegler-closed andZariski-open.

Proof. The only point not covered by [12] is that the images are Zariski-open.But C12 is defined by the condition Hom(S3,−) = 0 = Hom(S4,−), where Si isthe simple module at vertex i and similarly for C34. ✷

Denote by V the (Ziegler-open, Zariski-closed) set ZgR \ C where C = C12 ∪C34.

The infinite-dimensional points in C we will denote as follows where λ ∈P

1(k) = k ∪ {∞} and where the λ-indexing of regular A1-modules correspondsto the action of a − λb (a − ∞b being interpreted as (−)b):

712,λ – the image under F12 of the λ-adic module over A1;812,λ – the image under F12 of the λ-prüfer module over A1;G12 – the image under F12 of the generic module over A1;734,λ – the image under F34 of the λ-adic module over A1;834,λ – the image under F34 of the λ-prüfer module over A1;G34 – the image under F34 of the generic module over A1.

We will use the following notations for the infinite string modules (we definethem making use of the notations M(w), M(w) and M+(w) from [17]):

One-sided infinite: for each n � 0:∞7γ8n – the module M(∞(β−1α)γ ε(δ−1ε)n);∞7γ7n – the module M(∞(β−1α)γ (εδ−1)n);n8γ8∞ – the module M(n(β−1α)γ ε(δ−1ε)∞);n7γ8∞ – the module M(n(αβ−1)αγ ε(δ−1ε)∞).


Two-sided infinite:∞7γ8∞ – the module M+(∞(β−1α)γ ε(δ−1ε)∞).

These modules all are indecomposable pure-injectives (by 1.6 and [17]).We use CB to denote the Cantor–Bendixson rank of a topological space (see,

e.g., [16]) and of the points in it.

THEOREM 2.2. The above is a complete list of the nonisolated points of 2Zg(1).(1) The points of Cantor–Bendixson rank 1 are:

each one-sided infinite word;

712,∞;812,λ λ �= ∞;734,λ λ �= ∞;834,∞.

(2) The points of Cantor–Bendixson rank 2 are:

the two-sided infinite word ∞7γ8∞;

712,λ λ �= ∞;812,∞;734,∞;834,λ λ �= ∞.

(3) The points of Cantor–Bendixson rank 3 are:

G12, G34.

Hence CB(RZg) = 3.Proof. Completeness of the list is by the last section and [17] – since all infinite

words have been included. The remainder of this section is devoted to the proof ofthe remainder of the theorem. The proof will result in an explicit description of aneighbourhood basis for each nonisolated point. (Recall [10, p. 268] that isolatingneighbourhoods for the finite-dimensional points are given directly by Auslander–Reiten sequences and these may be obtained from [3].) ✷COROLLARY 2.3 (also [21]). The Krull–Gabriel dimension of 2 Zg(1) is equalto 3.

Proof. In fact, this is an immediate corollary, by [22], only in the case that Ris countable (see [16] for a discussion of the relation between Cantor–Bendixsonrank and Krull–Gabriel dimension). In the general case, it is a consequence ofour descriptions of neighbourhood bases that the isolation condition, see [16],(‘condition (∧)’ of [10]), holds and so we get that the Krull–Gabriel dimensionand Cantor–Bendixson ranks are equal. ✷


We need some results from [1] which give the decomposition of certain pure-injective modules associated with infinite words. Recall that Ringel [17] associatedto each infinite word over a domestic string algebra a module which is indecom-posable pure-injective. In the case of a contracting word w the associated moduleM(w) is the ‘direct sum module’; in the case of an expanding word w, the associ-ated module N(w) (Ringel writes this M(w) but, below, we avoid this because ofthe notational clash with the use of M to denote pure-injective hull) is the ‘directproduct module’ and, in the case of a doubly infinite word which is expanding inone direction and contracting in the other, the associated module M+(w) is a ‘directproduct module on the expanding side and direct sum module on the contractingside’. The results from [1] describe the product module N(w) for a contractingword and the pure-injective hull of the direct sum module M(w) for an expandingword.

THEOREM 2.4 ([1, 3.4]). Let R be a monomial algebra and let w be an ex-panding one-sided infinite word of the form uy∞ where y is a primitive cyclicword. Let Fy denote the functor from Mod-k[X,X−1] to Mod-R correspondingto the band y. Then the pure-injective hull of the direct sum module M(w) isN(w) ⊕ ∏

P Fy(7P ) = N(w) ⊕ p.i.(⊕

P Fy(7P)) where P ranges over theprimes of k[X,X−1], ‘p.i.’ denotes pure-injective hull and 7P denotes the P -adick[X]-module (the P -adic completion of k[X]).

THEOREM 2.5 ([1, 4.4]). Let R be a monomial algebra and let w be a contractingone-sided infinite word of the form uy∞ where y is a primitive cyclic word. Let Fy

denote the functor from Mod-k[X,X−1] to Mod-R corresponding to the band y.Then the direct product module N(w) is M(w) ⊕ ⊕

P Fy(8P ) ⊕ G(κ)y where P

ranges over the primes of k[X,X−1], 8P denotes the P -prüfer k[X]-module, Gy

is the generic module associated with the band y and κ is some cardinal number.

The results in [1] also cover doubly infinite words (mixed and unmixed) but thisis all we need here.

Denote by RZg0 the set of isolated (= finite-dimensional) points of RZg and letRZg(1) denote the set of nonisolated points.

First we consider the string modules. These all belong to the Ziegler-open set Vand so neighbourhoods and Cantor–Bendixson ranks of these points may be com-puted working entirely within V . From the description of the Auslander–Reitenquiver of R we see that V ∩ RZg0 is a subset of the exceptional component E. Alsonote that V is the Ziegler-open set defined by the condition im γ �= 0. We denotethis set (im γ �= 0) and use similar notation throughout.

One-sided infinite points: In each case we show first that the CB-rank is 1by finding an open neighbourhood which (intersected with V ) contains no otherinfinite-dimensional point. Then we look more closely to obtain a neighbourhoodbasis. We will use terminology such as ‘δ is epi’, meaning that, regarded as a map


from e3M to e4M, δ is epi. Let us emphasise that, all the time, we are workingwithin V , even where this is not said explicitly. Also, we use & to denote an ar-bitrary, finite or infinite, possibly empty, string or replacement symbol(s) (such asn7) for a string.

∞7γ8n: V ∩ (δ not epi) = {&γ8m : m � 0} by inspection. On each suchmodule we have the definable map given by ρ(x, y) which is: x = e4x ∧ ∃z(εz =x∧ δz = y) – that is, the map from e4M to e4M which is δε−1 (this is well-definedsince ε is monic and epi in these modules).

Now we use the fact that &γ8n satisfies, on the one hand, (δε−1)n+1 = 0 and,on the other hand, im(δε−1)m � ker(δε−1) for each m � n. We also observe that

V ∩ (δ not epi) ∩ (ker(δε−1)

> ker(δε−1) ∩ im(δε−1)n+1) = {&γ8k : k � n}and

V ∩ (δ not epi) ∩ (e4M > ker(δε−1)n) = {&γ8k : n � k}.Therefore, intersecting these, we obtain a neighbourhood of ∞7γ8n which

contains no other infinite-dimensional point and hence which isolates ∞7γ8n inRZg(1) (so CB(∞7γ8n) = 1).

The finite-dimensional modules in this neighbourhood are exactly those of theform m7γ8n and m8γ8n where m � 0. We consider the modules of the latterform first.

Recall the following results on closure under direct and inverse limits.

THEOREM 2.3. (a) Let C be a closed subset of the Ziegler spectrum of any ringR. Then the class of modules with support (see, e.g., [16]) contained in C is closedunder direct limits. In particular if a point N of ZgR is a direct limit of a set X ofpoints of ZgR then it lies in the Ziegler-closure of X.

(b) Let R be an Artin algebra and let N ∈ ZgR be an indecomposable sum-mand of the inverse limit of a set X of points of ZgR which are Hom-duals of leftR-modules. Then N lies in the Ziegler-closure of X.

Proof. For (a) see [8] or, in a general setting, [19]. Part (b) seems to be folklore– briefly, for, say, a finite-dimensional k-algebra, if N ′ is the inverse limit of aninverse system of modules Ni each of which has the form Homk(Mi, k) then let Mbe the direct limit of the corresponding direct system of left modules Mi , so thenN ′ = Homk(M, k). By part (a) all indecomposable summands of the pure-injectivehull of M are in the Ziegler-closure of the Mi and so, by elementary duality [6], Nis in the Ziegler-closure of the Ni . ✷

So any infinite set of modules of the form m8γ8n (n fixed) has the infinitestring module ∞8γ8n ‘in its Ziegler-closure’, meaning that any indecomposablesummand of the pure-injective hull of the module ∞8γ8n lies in the Ziegler-closure of these modules. By 2.4 the pure-injective hull of ∞8γ8n is ∞7γ8n ⊕


p.i.(⊕{712,λ : λ �= 0,∞}). Hence every open neighbourhood of ∞7γ8n con-

tains all but finitely many of the modules m8γ8n (m � 0).Next consider the modules m7γ8n: given any infinite set of such modules we

have, by 2.6, that the inverse limit, ∞7γ8n, lies in the Ziegler-closure of thisset since it is an inverse limit of modules which are Hom-duals of right modules.Therefore every open neighbourhood of ∞7γ8n contains all but finitely many ofthe modules m7γ8n.

We conclude that the cofinite subsets of {m7γ8n, m8γ8n : m � 0}∪{∞7γ8n}, those containing ∞7γ8n of course, form a basis of open neighbour-hoods of ∞7γ8n.

∞7γ7n: V ∩ (ε not monic) = {&γ7m : m � 0} by inspection. Within thesemodules, it is easy to compute (δ−1ε)m · ker δ (in particular, this is of dimensionm+1 in &γ7m′

for m′ � m) and we obtain: V ∩ (ε not mono)∩ (((δ−1ε)n ·ker δ)∩ker ε > 0) = {&γ7k : k � n}, for n � 0, and V ∩ (ε not mono) ∩ (e4M >

ε(δ−1ε)n−2 · ker δ) = {&γ7k : k � n} for n � 2 and also V ∩ (ε not mono) ∩(e4M �= 0) = {&γ7k : k � 1}.

Therefore, intersecting these (variously for n = 0, 1, n � 2), we obtain aneighbourhood of ∞7γ7n which contains no other infinite-dimensional point andhence which isolates ∞7γ7n in RZg(1).

The finite-dimensional modules in this neighbourhood are exactly those of theform m7γ7n and m8γ7n where m � 0 and we continue just as in the previ-ous case, concluding that the cofinite subsets of {m7γ7n,m 8γ7n : m � 0} ∪{∞7γ7n} which do not omit ∞7γ7n form a basis of open neighbourhoods of∞7γ7n.

n7γ8∞: V ∩ (β not monic) = {m7γ & : m � 0} by inspection. Within thesemodules we have the definable map given by ρ(x, y) which is: x = e2x ∧∃z(αz =x∧βz = y) – that is, the map from e2M to e2M which is βα−1 (this is well-definedsince α is monic and epi in these modules).

Now we use that n7γ8∞ satisfies, on the one hand, n+1(βα−1) = 0 and, on theother hand, im(m(βα−1)) � ker(βα−1) for each m � n. We have

V ∩ (β not monic) ∩ (ker(βα−1)

> ker(βα−1) ∩ im(n+1(βα−1)) = {k7γ & : k � n}and

V ∩ (β not monic) ∩ (e2M > kern(βα−1)) = {k7γ & : n � k}.Therefore, intersecting these, we obtain a neighbourhood of n7γ8∞ which

contains no other infinite-dimensional point and hence which isolates n7γ8∞ inRZg(1).

The finite-dimensional modules in this neighbourhood are exactly those of theform n7γ8m and n7γ7m where m � 0.

Any infinitely many of the modules n7γ8m have, as a direct limit, n7γ8∞which is, therefore, in their Ziegler-closure.


Any infinite set of modules n7γ7m has, as an inverse limit, the product modulen7γ7∞. By 2.5 this pure-injective module decomposes as n7γ8∞ ⊕ ⊕{834,λ :λ �= 0,∞} ⊕ G

(κ)34 (for a suitable cardinal κ) and, as before (by 2.6), the indecom-

posable summands are in the Ziegler-closure of the given set.We conclude that those cofinite subsets of {n7γ7m,n 7γ8m : m � 0} ∪

{n7γ8∞} which do not omit n7γ8∞ form a basis of open neighbourhoods ofn7γ8∞.

n8γ8∞: V ∩ (α not epi) = {m8γ & : m � 0} by inspection. We consider the(only partially defined) action of αβ−1 and βα−1 and observe:

V ∩ (α not epi) ∩ (im γ > γ · n(αβ−1) · imα)

= {k8γ & : k � n} for n � 0

and

V ∩ (α not epi) ∩ ((n−2(βα−1) · imβ) ∩ imα �= 0)

= {k8γ & : k � n} for n � 2

and

V ∩ (α not epi) ∩ (e1M �= 0) = {l8γ & : l � 1}.Therefore, intersecting these (variously for n = 0, 1, n � 2), we obtain a

neighbourhood of n8γ8∞ which contains no other infinite-dimensional point andhence which isolates n8γ8∞ in RZg(1).

The finite-dimensional modules in this neighbourhood are exactly those of theform n8γ8m and n8γ7m where m � 0 and we continue just as before, conclud-ing that those cofinite subsets of {n8γ8m, n8γ7m : m � 0} ∪ {n8γ8∞} whichdo not omit n8γ8∞ form a basis of open neighbourhoods of n8γ8∞.

That concludes the discussion of the one-sided infinite modules.

Two-sided infinite point: After removing the one-sided infinite points from theopen set V we are left just with the point ∞7γ8∞ which must, therefore, haveCB-rank � 2. On the other hand this module is the direct limit of the modules∞7γ8m (m � 0), these have CB-rank 1 and so CB(∞7γ8∞) � 2, givingCB(∞7γ8∞) = 2, as claimed.

As for open neighbourhoods of ∞7γ8∞, we cannot omit from any of its neigh-bourhoods infinitely many of the one-sided infinite points – otherwise we wouldcontradict compactness of the set V ∩ RZg(1). On the other hand, we can excludeany finitely many. For instance, V ∩ (e4M > ker(δε−1)n) is a neighbourhood of∞7γ8∞ which omits all those modules of the form &γ8k with k < n. Similarlyfor the other three series of one-sided infinite points, using various of the open setsintroduced above.

Therefore a neighbourhood basis in the relative topology on RZg(1) consists ofthose cofinite subsets of V which contain ∞7γ8∞. As for a neighbourhood basis


in RZg, recall first that V ∩ RZg0 is contained in the exceptional component E. Theopen sets of the above paragraph may be used to omit any finitely many rays andcorays of E.

We must show that if X ⊆ E is not contained in any finite union of rays andcorays: then ∞7γ8∞ is in the Ziegler-closure of X.

Let U ⊆ V be an open neighbourhood of ∞7γ8∞. Let N1, . . . , Nt be thefinitely many one-sided-infinite points in V (1) \ U , where V (1) = V ∩ RZg(1),and for each i = 1, . . . , t let Ui be the open set introduced earlier which consistsof Ni and its finite-length ‘approximants’ – for example if Ni = ∞7γ8n thenUi = {Ni} ∪ {m7γ8n, m8γ8n : m � 0}. Let U ′ = U ∪ (

⋃t1 Ui). If Y = V \ U ′

(a subset of RZg0, note) were infinite then we would contradict compactness of thebasic Ziegler-open set V . Hence V (1) \ U is contained in finitely many rays andcorays of E.

Therefore a basis of open neighbourhoods of ∞7γ8∞ is given by V \(finitelymany rays and corays of the exceptional component E, each together with itsassociated one-sided infinite (limit or colimit) point).

(12)-band points: Given λ ∈ P1(k) let Tλ denote the tube in A1-mod consisting

of modules in which a − λb is not an isomorphism and let T12,λ, T34,λ denote theimages of this tube under F12 respectively F34. With the exceptions of T12,∞ andT34,∞ these are tubes in the Auslander–Reiten quiver of R. The exceptional tubesare found in E, together with F12P and F34I where P, I denote the preprojective,respectively preinjective, components of A1-mod. We use the description of thepoints and topology of A1

Zg ([13, 18]) without special comment.812,λ: (α − λβ not monic) ∩ C = {812,λ} ∪ T12,λ ∪ F12I. Consider (α −

λβ not monic) ∩ V : this does not contain ∞7γ8∞, as is easily checked (seebelow) by considering the effect of α and β on, say, the right-most coordinateof any element of ker(α − λβ). It follows that CB(812,λ) � 2. In the same wayone checks easily that no module lies in (α− λβ not monic)∩V provided λ �= ∞.Hence CB(812,λ) = 1 provided λ �= ∞. On the other hand, V ∩ (β not monic) ={n7γ & : n � 0, & arbitrary}. Consider the direct system of maps between then7γ8∞ defined by mapping n7γ8∞ onto M(αn(β−1α)) and then embeddingthis into (the ‘left-most segment’ of) n+17γ8∞. This direct system has limit 812,∞which, therefore, has CB-rank 2.

In RZg(1) each point of the form 812,λ, apart from 812,∞, is isolated. The latterpoint has, for a neighbourhood, (β not monic) which contains only the n7γ8∞and so a basis of neighbourhoods of 812,∞ in RZg(1) is given by those cofinitesubsets of {n7γ8∞ : n � 0} ∪ {812,∞} which contain 812,∞.

As for neighbourhoods in RZg, we now give the proof that, provided β �= ∞,we have (α−λβ not monic)∩V = ∅. For otherwise, let M be a point in this set andlet m ∈ kerM(α − λβ). Suppose for simplicity that M is finite-dimensional. NowM is a string module so we may express m as µ1z1 + · · · + µnzn (µi ∈ k) wherez1, . . . , zn is a standard basis of e1M and (w0, )w1, . . . , wn is the correspondingbasis of e2M. Let i be maximal with µi �= 0. Then αm = µ1w1 + · · · + µiwi and


λβm = λµ1w0 + · · · + λµiwi−1 and so, unless β = ∞, these cannot be equal. Inthe case λ = ∞ we have (α−λβ)m = βm = λµ1w0 +· · ·+λµiwi−1. This can bezero only if M has the form n7γ & – so w0 is not there – and m = µ1z1. Any set ofmodules of the form n7γ & where infinitely many values of n occur has 812,∞ as adirect limit (argue as with the n7γ8∞ above). On the other hand, any set of suchmodules where the indices n which occur are bounded is disjoint from an open setcontaining 812,∞ (use the neighbourhood (e2M > ker(n(βα−1))) which appearedwhen we considered the one-sided infinite string n7γ8∞). Therefore we obtainthe following.

A basis of open neighbourhoods of 812,λ for λ �= ∞ consists of those cofinitesubsets of T12,λ ∪ F12I ∪ {812,λ} which contain 812,λ.

A basis of open neighbourhoods of 812,∞ consists of those sets which can beexpressed as {812,∞} ∪ {m7, m7γ & : & arbitrary,m � n} ∪ X for some n and forsome cofinite subset X of F12I.

712,λ: (α−λβ not epi)∩C = {712,λ}∪T12,λ∪F12P. Consider (α−λβ not epi)∩V : provided β �= ∞, this does not contain ∞7γ8∞. For, given any elementof e∞

2 7γ8∞, it is easy to construct, inductively, the coordinates of an inverseimage of this element under α − λβ. Also, 712,∞ ∈ (ker γ > ker γ ∩ imβ)

whereas ∞7γ8∞ does not lie in this open set. Therefore, for every λ, CB(712,λ) �2. Now, (ker γ > ker γ ∩ im β) ∩ V has no infinite-dimensional points and soCB(712,∞) = 1. On the other hand, we have seen already that, by 2.4 and 2.6,each 712,λ with λ �= 0,∞ lies in the Ziegler-closure of the set {n8γ8∞}n andhence CB(712,λ) = 2 for λ �= 0,∞. As for 712,0, consider the directed systemof maps between the n8γ8∞ defined by mapping n8γ8∞ to M(n(β−1α)) andthen embedding this into (the left-most segment of) n+18γ8∞. This direct systemhas limit M(∞(αβ−1)). Note that ∞(αβ−1) is an expanding word and so, again by2.4, we have p.i.(M(∞(αβ−1))) = 712,0 ⊕ p.i.(

⊕{712,λ : λ ∈ k, λ �= 0}) and soCB(712,0) = 2.

In RZg(1) the point 712,∞ is isolated. Next, in any module of the form &7γ &,the map α−λβ (λ �= ∞) is easily seen to be epi. This leaves the modules n8γ8∞and, as in the paragraph above, 712,λ (λ �= ∞) is in the closure of any infinite set ofthese. Therefore a basis of neighbourhoods of each point 712,λ (λ �= ∞) in RZg(1)

is given by the cofinite subsets of {n8γ8∞ : n � 0}∪{712,λ} which contain 712,λ.Turning to neighbourhoods in RZg, first consider λ = ∞. We have (ker γ >

ker γ ∩ imβ)∩V = ∅ and hence a basis of open neighbourhoods of 712,∞ consistsof those cofinite subsets of T12,∞ ∪ F12P ∪ {712,∞} which contain 712,∞.

Now suppose λ �= ∞. We have seen that (α−λβ not epi)∩V contains no pointsof the form &7γ &, leaving the points n8γ & in E to be considered. Any set of suchpoints where the values of n occurring is not bounded has, as argued above, themodule M(∞(αβ−1)) as a direct limit and hence has each module 712,λ λ �= ∞in its Ziegler-closure. Conversely, any set of points n8γ & where the values of noccurring are bounded is disjoint from an open set containing each module 712,λ

(argue as for the case 812,λ).


So a basis of open neighbourhoods of 712,λ (λ �= ∞) consists of those setswhich can be expressed as {712,λ} ∪ {m8, m8γ & : m � n, & arbitrary} ∪ X forsome n and for some cofinite subset X of T12,λ.

The Cantor–Bendixson ranks and neighbourhood bases for the remaining points834,λ, 734,λ may be computed as above but are more easily found using dualitytogether with what we have already.

Recall [6] that for a ring R with KG(R) < ∞ elementary duality gives a homeo-morphism N �→ DN between the right and left Ziegler spectra RZg and ZgR. For afinite-dimensional algebra, on finite-dimensional points this duality coincides withthe usual duality Homk(−, k) and this is also true for some infinite-dimensionalpoints. One checks (see [1] or the remark of Krause on p. 430 of [17]) that if w isany word (finite or infinite) then Homk(M(w), k) = N(w) where the latter is thecorresponding product right module. Thus, for example, D(γ8∞) = γ7∞. Wehave that Mod-R is equivalent to Rop-Mod and that Rop is obtained by reversingthe arrows of R – so that, for right R-modules, the (12)-position corresponds to acontracting word and the (34)-position to an expanding word (so, for example, thetwo-sided infinite point in ZgR is ∞8γ7∞).

It is clear that the functors F12,F34 ‘commute’ with duality and so, from theknown effect of duality on the right and left Ziegler spectra of A1 (see [13, 18])one obtains the following list of duals:

D(712,λ) = 812,λ (the module on the right being a right module),D(812,λ) = 712,λ,D(G12) = G12,and similarly for (34).

The right module 812,λ is analogous to the left module 834,λ since the rolesof expanding and contracting words have been interchanged in moving from leftto right. Similarly the left module 834,λ is analogous to the right module 812,λ

which is the dual of the left module 712,λ and so we deduce that CB(834,λ) = 2for λ �= ∞ and CB(834,∞) = 1. Furthermore, a basis of open neighbourhoods of834,∞ consists of those cofinite subsets of T34,∞ ∪ F34I ∪ {834,∞} which contain834,∞ and a basis of open neighbourhoods of 834,λ (λ �= ∞) consists of those setswhich can be expressed as {834,λ} ∪ {&γ7m,7m : m � n, & arbitrary} ∪ X forsome n and some cofinite subset X of T34,λ.

The same reasoning gives us CB(734,λ) = 1 for λ �= ∞ and CB(734,∞) = 2,with the cofinite subsets of T34,λ ∪ F34P ∪ {734,λ} which contain 734,λ giving aneighbourhood basis of 734,λ and with a basis of neighbourhoods of 734,∞ beinggiven by those sets which can be expressed as {734,∞} ∪ {&γ8m,8m : m �n, & arbitrary} ∪ X for some n and some cofinite subset X of F34P.

One further checks the following:

D(∞7γ8n) = ∞8γ7n;D(∞7γ7n) = ∞8γ8n;D(n8γ8∞) = n7γ7∞;


D(n7γ8∞) = n8γ7∞;D(∞7γ8∞) = ∞8γ7∞.

For instance, the dual of ∞7γ8n may be computed via Homk(−, k) or asfollows. The open set (δ not epi), that is (x = e4x/δ | x) (meaning the set ofpoints where the functor (x = e4x )/( δ | x) is nonzero), contains, of the infinite-dimensional points, just the ∞7γ8m. The dual of this open set is (xδ = 0/xe4 =0), that is, (δ not monic) which, of the infinite-dimensional points, contains just the∞8γ7m. Then the particular value of n is separated out by making use of the dualof the relation ρ(x, y) used for ∞7γ8n – which is the relation ‘xε = yδ’ on Me4.

Generic points: Clearly each of G12, G34 is in a Ziegler-open set which does notcontain the other, each is in the Ziegler-closure of points of CB-rank 2 and thereare no other points of rank � 2. Hence CB(G12) = 3 = CB(G34).

To compute neighbourhoods of these points in RZg1 we note the following.Gij is in the Ziegler-closure of every infinite-dimensional (ij)-band point (since

we have this already over A1).Both G12 and G34 are in the Ziegler-closure of ∞7γ8∞: the functors

Hom(S1,−) and Hom(S4,−) define Ziegler-open sets (Hom(S1,−) �= 0) and(Hom(S4,−) �= 0) with G12 in the first and not in the second and vice versa forG24. Furthermore, each of these functors is of infinite length when evaluated on∞7γ8∞. Hence ([10, Section 10.3]) each functor is nonzero on some other pointin the Ziegler-closure of the point ∞7γ8∞.

G12 is in the Ziegler-closure of each point ∞7γ&; G34 is in the Ziegler-closureof each point &γ8∞ (by the argument just used for ∞7γ8∞).

G12 is in the Ziegler-closure of any set of points n7γ & where the values of n areunbounded – for we have already seen that 812,∞ and hence G12 lies in the closureof such a set – but is not in the Ziegler-closure of any set where these values arebounded (by arguments already seen). Similarly for G34.

Hence in RZg(1) a basis of open neighbourhoods for G12 is given by the sets ofthe form {G12} ∪ {∞7γ&} ∪ {712,λ}λ ∪ {812,λ}λ ∪ {m7γ & : m � n} for some n.A basis of open neighbourhoods for G34 is given by the sets of the form {G34} ∪{&γ8∞} ∪ {734,λ}λ ∪ {834,λ}λ ∪ {&γ8m : m � n} for some n. (In each case λ runsover P

1(k).)Then, using arguments as before, we obtain a basis of open neighbourhoods

for G12 and G34 in RZg. We may describe, for example, a neighbourhood basisfor G12 as follows: choose a set M of finitely many finite-dimensional modulessatisfying M = e1M ⊕ e2M and define UM to be the set of all R-modules M suchthat e1M⊕e2M (i.e. M/(e3M⊕e4M)) is not in M. Then the sets UM, as M varies,form a neighbourhood basis for G12.


3. The Zariski Topology on �2

Now we will describe the Zariski topology by giving a basis of open neighbour-hoods at each point. Recall [8, 14] that the finitely generated points are all Zariski-isolated and that any Zariski-isolated point must be of finite endolength.

Each representation embedding F12, F34 induces [12] a homeomorphism of

A1Zg with its image and hence of A1

Zar with its image (recall that the Zariski topol-ogy is defined in terms of the Ziegler topology since it has, for a basis of open sets,the complements of the compact Ziegler-open sets). The image of each of thesefunctors, restricted to a map of Zariski spectra, is a Zariski-open subset of RZar:F12(A1

Zar) = [e3M = 0] ∩ [e4M = 0] for instance. Therefore a neighbourhoodbasis for each element of C = C12 ∪C34 is just the image of a neighbourhood basisfor its preimage in A1

Zar (for a description of which see [14] or [15]).This gives the following. Each regular, adic or prüfer point N of A1

Zar is as-sociated to a scalar (possibly ∞) λ = λ(N). For L any finite subset of P

1(k), setU(L) = {N ∈ A1

Zar : N /∈ P ∪ I, N �= G, λ(N) /∈ L} – a Zariski-open set.Then neighbourhood bases are the following (L will always denote a finite subsetof P

1(k)):

for 712,λ – {G12} ∪ {712,λ} ∪U12(L) where L is any finite subset of P1(k) and

U12(L) = F12(U(L));for 812,λ – {G12} ∪ {812,λ} ∪ U12(L) where L is any finite subset of P

1(k);for G12 – {G12} ∪ U(L) where L is any finite subset of P

1(k).Similarly for the (34)-modules.

The following observations will be useful in computing neighbourhoods of theinfinite string modules.

Let E12: R-Mod → A1-Mod be defined by taking M to M/(e3M ⊕ e4M) withthe obvious A1-action and let E34: R-Mod → A1-Mod be defined by taking M toe3M ⊕ e4M with the obvious A1-action. Note that these induce maps from RZar to

A1Zar. This is a slight lie in that for some N we have E12N = 0: so let us formally

regard 0 as lying in every closed set.In fact, these induced maps (we use the same notation for them) are continu-

ous. For let F ∈ (A1-mod,Ab)fp. Define E&12F ∈ (R-mod,Ab) by E&

12F(M) =F(E12M). It is easily checked that E&

12F ∈ (R-mod,Ab)fp. Just from the definitionwe have E−1

12 .(F ) = (E&12F) and E−1

12 [F ] = [E&12F ] (a point N with E12N = 0 we

regard as having image in [F ]). Thus E12 and, similarly, E34 are continuous for theZiegler and Zariski topologies.

Consider ∞7γ8n. We have E12(∞7γ8n) = 7∞ (the adic A1-point with

parameter ∞) and E34(∞7γ8n) = 8n where 8n the last denotes a certain finite-

dimensional A1-module. Pulling back one of the above basic Zariski-open neigh-bourhoods of 7∞ (that is, one of the form {G} ∪ {7∞} ∪ U(L)) we obtain, by theabove paragraph, a neighbourhood C34 ∪{∞7γ8n}∪ {G12}∪ {712,∞}∪U12(L) of∞7γ8n. Intersecting with [ker γ = imβ] removes 712,∞. Then, pulling back the


open neighbourhood {8n}, we obtain the open neighbourhood C12∪{8n}∪{&γ8n :& arbitrary} of ∞7γ8n: since 8n is clopen we remove it. Now intersect theseneighbourhoods to obtain the open neighbourhood {G12} ∪ {∞7γ8n} ∪ U12(L) –the net effect having been to replace 712,∞ by ∞7γ8n in a typical neighbourhoodof 712,∞. We have to check that these do form a basis around ∞7γ8n.

So let U be any open subset of one of these already determined open neighbour-hoods of ∞7γ8n. Then U ∩C12 is a Zariski-open subset of C12. If it is nonemptythen it contains one of our given open neighbourhoods. If this intersection wereempty then ∞7γ8n would be Zariski-isolated – but it is not of finite endolength,contradiction, as required.

Therefore, a basis of open neighbourhoods of ∞7γ8n consists of the sets ofthe form {G12}∪{∞7γ8n}∪U12(L) for L a finite subset of P

1(k). We also deducethat a basis of open neighbourhoods of ∞7γ8n in Zar(1) is the collection of setsof the form {G12} ∪ {∞7γ8n} ∪ {712,λ, 812,λ : λ /∈ L} as L ranges over finitesubsets of P

1(k).An almost identical analysis replaces ∞7γ8n by ∞7γ7n throughout and a

similar analysis applies to give that a basis of open neighbourhoods of n8γ7∞(respectively, n7γ7∞) consists of the sets of the form {G12}∪{n8γ7∞}∪U34(L)

(resp. {G12} ∪ {n7γ7∞} ∪ U34(L)) for L a finite subset of P1(k).

Notice that this kind of argument could also be used to replace some of the moreexplicit descriptions of Ziegler-open sets in the previous section.

Finally, if any Zariski-open set containing ∞7γ8∞ had empty intersectionwith, say, C12 then, by duality, the corresponding right module ∞8γ7∞ wouldhave a neighbourhood having empty intersection with the right (12)-points andhence, using an earlier argument, ∞7γ8∞ would also have a neighbourhoodhaving empty intersection with C34. But then ∞7γ8∞ would be isolated – con-tradiction since it does not have finite endolength. Therefore we obtain, arguing asabove, that a basis of open neighbourhoods of ∞7γ8∞ consists of the sets of theform {G12} ∪ U12(L) ∪ {∞7γ8∞} ∪ U34(L) ∪ {G34} where L is any finite subsetof P

1(k).It follows that the dimension of RZar(1) is 1, the points of rank 1 being G12 and

G34. Here we mean ‘dimension’ in, at least, an informal algebraic-geometric sense.

COROLLARY 3.1. The Ziegler topology cannot be recovered purely in terms ofthe Zariski topology.

Proof. Let θ be the bijection on RZar which interchanges the points in C12

associated to 0 and ∞ (that is, interchange points of T12,∞ with the correspondingpoints of T12,0 and similarly with the four infinite-dimensional points). We claimthat this is a homeomorphism of the Zariski spectrum of R. One checks this bynoting that if U is any of the basic open neighbourhoods that we listed above thenso is θU . But it is not a homeomorphism of RZg since it does not preserve CB-rankin RZg. ✷


Thus the Zariski topology, unlike the Ziegler topology, does not ‘see’ the dif-ference between the ∞-labelled points and the others.

The algebra 2 destroyed the conjecture on Krull–Gabriel dimension/Cantor–Bendixson rank but we see, arising from the ashes, a transformed conjecture.

CONJECTURE. If R is a finite-dimensional algebra of infinite representation typethen the (algebraic-geometric) dimension of Zar(1)R is 1 or ∞ (according to whetherR is of domestic type or not).

COROLLARY 3.2. Every point apart from G12 and G34 is a closed point in theZariski topology.

Proof. A point N is not closed iff there is a point N ′ �= N with N ′ ∈ Zar-cl(N),equivalently, with N ∈ Zg-cl(N ′). The points which are in the Ziegler-closure ofother points are exactly the two generics, as required. ✷

We have:

Zar-cl(G12) = {G12} ∪ {712,λ, 812,λ : λ ∈ P1(k)} ∪ {∞7γ& : & arbitrary} ∪

{∞7γ8∞};Zar-cl(G34) = {G34} ∪ {734,λ, 834,λ : λ ∈ P

1(k)} ∪ {&γ8∞ : & arbitrary} ∪{∞7γ8∞}.

We see, therefore, that RZar(1) is the union of two irreducible closed sets, namelythe closures of the generic points, and that these intersect at the single point∞7γ8∞.

4. Generalisation to �n

What we have done for 2 may be carried through for all the quivers n (n � 2).

We will confine ourselves to computing the Cantor–Bendixson rank of eachpoint in n

Zg, deducing that CB( nZg) = n + 1. Indeed, we will just sketch the

computations but we will be sufficiently precise that it should be clear how to fillin the details. Moreover, using our methods, the generalisation to n of everythingthat we did for 2 should be a routine matter, although quite lengthy and probablyproviding no further illumination.

Set R = n. Let Fi : A1-Mod → R-Mod (i = 1, . . . , n) be the obvious functorwhich takes the arrows a, b of A1 to αi , respectively βi . It is clear that everyindecomposable n-module M either lies in the image of one of these functorsor satisfies γiM �= 0 for some i. Therefore the result of the first section applies to


give us the following complete list of infinite-dimensional points of RZg. We useessentially the same notation as for 2 but use subscripts 1, . . . , n to indicate the‘band’ which supports an infinite (piece of a) module and we use ‘fin’ to denoteany finite word (possibly empty).

One-sided infinite:∞7iγifin (i = 1, . . . , n);fin γj−18

∞j (j = 1, . . . , n).

Two-sided infinite:∞7iγifin8∞

j (i < j).

Band modules:7i,λ,8i,λ (i = 1, . . . , n, λ ∈ P

1(k));Gi (i = 1, . . . , n).

THEOREM 4.1. The Cantor–Bendixson ranks of the infinite-dimensional pointsof n

Zg are as follows.

CB(∞7iγifin) = i,CB(fin γj−18

∞j ) = (n + 1) − j ,

CB(∞7iγifin8∞j ) = i + (n + 1) − j ,

CB(7i,λ) = (n + 1) − i for λ �= ∞,CB(7i,∞) = i,CB(8i,λ) = i for λ �= ∞,CB(8i,∞) = (n + 1) − i,CB(Gi) = n + 1.

In particular CB( nZg) = n + 1.

Proof. The proof is by induction on n. Observe that we have the result for n = 2.We also have to carry, through the induction, descriptions of neighbourhoods ofpoints. These descriptions will be given just at the places where we call on them.

Let E1: R-Mod → A1-Mod and E2: R-Mod → n−1-Mod be given byE1(M) = M/

⊕{eiM : i = 3, . . . , 2n} and E2(M) = ⊕{eiM : i = 3, . . . , 2n}on objects (and the obvious action on morphisms), where E1M, E2M are given thenatural ‘induced’ A1-, respectively, n−1-structure. By the arguments that we usedbefore (when considering the Zariski topology on 2), these functors induce mapsof Ziegler spectra (we use the same notation) which are continuous.

N = ∞71γ1fin: E2N is an isolated point of n−1Zg so pulls back to an openset consisting of just ∞71γ1fin and finite-dimensional points. So we have an openneighbourhood of ∞71γ1fin which contains no other infinite-dimensional point.Hence CB(∞71γ1fin) = 1.

N = ∞7iγifin, i � 2: E2N has, for open neighbourhoods, sets consistingof modules of the form ∞7j & γifin (j < i), fin′γifin (where fin denotes thegiven fixed string and fin′ denotes any finite string) – this is part of our (not ex-plicitly stated) induction-on-n hypothesis. Pulling back such modules to RZg and


discarding finite-dimensional ones we are left with an open neighbourhood of∞7iγifin consisting of modules of the forms ∞7j & γifin (j < i) (those of theform ∞71 & γifin are obtained by pulling back string modules of words of theform m(α2β

−12 )α2 & γifin). By induction on i, these have Cantor–Bendixson ranks

bounded above by i − 1 and, hence, CB(∞7iγifin) � i. On the other hand,by induction on i, CB(∞7i−1γi−1

m(αiβ−1i )αiγifin) = i − 1 for each m � 0.

Now ∞7iγifin is an inverse limit of these modules and hence, by 2.6, is in theirZiegler-closure. We deduce that CB(∞7iγifin) = i.

Now, what we have shown so far applies to Rop and so, by elementary dualitywe have that the Cantor–Bendixson ranks of the modules fin γj−18

∞j also are as

stated.Next consider N = ∞71γ1fin8∞

j . We have E1N = 7∞ (the ∞-adic mod-

ule over A1) and neighbourhoods of this module consist of cofinite subsets ofP ∪ T∞ (together with 7∞ itself, of course). Pulling back such a neighbourhoodto RZg and discarding finite-dimensional points, we obtain a set of modules ofthe forms 71,∞, ∞71γ1&, fin′γ1& (where & denotes any finite or infinite string, asappropriate and fin′ denotes any finite string). Also E2N = fin8∞

j which has a

neighbourhood consisting of modules of the form fin · m(αjβ−1j )αj& (where & can

be finite or infinite) as well as fin8∞j itself. Pulling back to RZg and discarding

finite-dimensional modules we obtain, as well as ∞71γ1fin8∞j , modules of the

form ∞71γ1fin · fin′ 8∞k with k > j , ∞71γ1fin · fin′, fin · fin′ 8∞

k with k > j

(fin′ denoting an arbitrary finite string). On intersecting these neighbourhoods weobtain, apart from ∞71γ1fin8∞

j itself, modules of the form ∞71γ1fin · fin′ 8∞k

with k > j , ∞71γ1fin · fin′, fin · fin′ 8∞k with k > j . By induction on (n+ 1)− j ,

CB(∞71γ1fin · fin′ 8∞k ) = 1 + (n + 1) − k for k < j and, by what has been

shown earlier in the proof, the Cantor–Bendixson ranks of the other points are nomore than (n + 1) − k. Furthermore, we can obtain ∞71γ1fin8∞

j as a direct limit

of the modules ∞71γ1fin · m(αjβ−1j )αj8

∞j+1 which have rank 1 + n − j . Hence

CB(∞71γ1fin8∞j ) = 1 + (n + 1) − j , as required.

The computation for ∞7iγifin8∞j (i > 1) is similar (see also the computation

for ∞7iγifin).N = 71,λ: First consider λ �= ∞. One neighbourhood of E1N = 71,λ is

{7λ}∪P∪Tλ. Pulling back to RZg we obtain an open set containing, among infinite-dimensional modules, just those of the form 71,λ, m(β−1

1 α1)&8∞j for j > 1. Hence

CB(71,λ) � n − 1 (= CB(m(β−11 α1) & 8

∞2 )). On the other hand, just as for 2,

71,λ lies in the Ziegler-closure of the set of modules of the form m(β−11 α1) & 8

∞2

and so CB(71,λ) = n for λ �= ∞. In the case of 71,∞ the neighbourhood {7∞} ∪P ∪ T∞ of E1(71,∞) pulls back to one whose infinite-dimensional members havethe form 71,∞, m(β−1

1 α1) & 8∞j , m(α1β

−11 )α1 & 8

∞j . Intersecting this with the open

neighbourhood (ker γ1 > ker γ1 ∩ imβ1) of 71,∞ gives an open set whose onlymember is 71,∞ and so CB(71,∞) = 1.

The arguments for 7i,λ are of the same sort.


Elementary duality then gives us the Cantor–Bendixson ranks for the modules8i,λ.

Therefore the Cantor–Bendixson ranks of the points of RZg, apart from thegeneric modules, are bounded by n, the bound being achieved, for each i, by amodule which has the generic Gi in its Ziegler-closure. Hence CB(Gi) = n + 1,as required.

Note that, using the arguments above, one obtains sufficient information on thetopology to allow the induction to proceed. ✷

Finally, one may check that, as for 2, the dimension of nZar(1) is 1.

Acknowledgements

This work was done while the first author was supported by EPSRC grant numberGR/K19686. Both authors thank the EPSRC for this financial support.

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The Ziegler and Zariski Spectra of Some Domestic String Algebras