In memoriam of Michael Butler (1928â€“2012)

In memoriam of Michael Butler (1928–2012)

Helmut Lenzing

Universitat Paderborn

Auslander Conference 2013, Woods Hole, 19. April

H. Lenzing (Paderborn) Michael Butler 1 / 1

ICRA Beijing (2000). M. Butler with W. Crawley-BoeveyH. Lenzing (Paderborn) Michael Butler 2 / 1

Scope

Scope

We restrict to two aspects of the work of M.C.R. Butler having a majorinfluence on the course of mathematics:

Infinite Abelian groups

Tilting theory



Michael Butler, the man changing the direction of”Abelian group theory”.

The header of this slide refers to Butler’s paper

”A class of torsion-free abelian groups of finite rank”, Proc. London Math.Soc. 15 (1965), 680–98.

It is Butler’s first paper on Abelian Groups.



Infinite Abelian Groups, a popular misconception

Isn’t it that everything is already known?Is there anything interesting left?

Abelian group theory often suffers from these misconceptions.

Nothing could be more wrong.

Instead, Abelian Group Theory is the grandmother of

Ring- and Module Theory

Homological Algebra

Abelian Categories

· · ·



The pioneering role of Abelian Group Theory. Threeexamples

R. Baer (1902–1979)

The extension group Ext(X,Y ) formed by classes of extensions ofshort exact sequences. (R. Baer)

Existence of enough injective modules (R. Baer, 1940).

Ringel’s study of infinite dimensional modules over tame hereditaryalgebras, exploiting the sophisticated structure theory of infiniteabelian groups.



Infinite Abelian Groups, Examples

Each finitely generated abelian group is the direct sum of(indecomposable) cyclic groups

The indecomposable ones are Z and Z/(pn), p prime

The additive group Q of rational numbers. It is infinitely generated ofrank one

The factor group Q/Z is torsion. It decomposes

Q/Z =⊕

p prime

Z(p∞)

into Prufer groups

Q = E(Z), Z(p∞) = E(Z/(p)), p prime. Here, E=injective hull



Infinite Abelian Groups, Classification

Each injective abelian group is a direct sum of copies of the rationalsQ and of Prufer groups Z(p∞), hence of injective hulls of Z/p, wherep is a prime ideal in Z.

This result foreshadows a corresponding theorem of Eben Matlis(1958) for injective modules over commutative noetherian rings.

Torsion groups admit a satisfactory classification by invariants (Ulm)



What about torsion-free abelian groups?

Torsion-free groups are an almost hopeless case.Indeed, I. Kaplansky (1954, 1969) in his nice little book states abouttorsion-free groups:

”In this strange part of the subject anything that can conceivably happenactually does happen.”

It is here, where Michael Butler changed the direction of abelian grouptheory.



Kaplansky’s Test Problems

Kaplansky coined the nice concept of ”Test Problems”, not being so muchimportant in itself, but suitable to testing the maturity of a mathematicaltheory.

Problem 1. Assume G is isomorphic to a direct summand of H, and H isisomorphic to a direct summand of G. Does it follow that G ∼= H?

Problem 2. Assume G⊕G ∼= H ⊕H. Does it follow G ∼= H?



Butler groups

A finite direct sum of torsion-free rank-one abelian groups = completelydecomposableButler (1986): For an abelian group H the following are equivalent:

1 H is a pure subgroup of a completely decomposable group

2 H is a pure quotient (torsion-free image) of a completelydecomposable group

These groups, now called Butler groups, allow a classification bytypes/typesets.



The type of a rank-one group

Such a group H is just a non-zero subgroup of Q. We may assumeZ ⊆ H ⊆ Q. Then

H/Z ⊆ Q/Z =⊕

p prime

Z(p∞).

thusH/Z =

⊕p prime

Up

where Up is a subgroup of length np with p ∈ {0, 1, . . . ,∞}.Up to change of finitely many entries, each by a finite value, the sequence

n = (n2, n3, n5, . . .)

characterizes the isoclass of H. The resulting equivalence class of n iscalled the type τ(H) of H.



Link to representation theory

For a torsion-free group H, each non-zero element h sits in its pure hull〈h〉∗ which is a rank-one group.Facts

1 For a Butler group, the typeset T (H) consisting of all types τ(〈h〉∗),is always finite.

2 The typeset T = T (H) is a poset. Butler groups relate to finitedimensional Q-linear representations of posets derived from T .

The subject of Butler groups is still alive today, see the book of David M.Arnold: Abelian groups and representations of finite partially orderedordered sets from 2000.



The Whitehead problem

Motto: Abelian groups form the spearhead of module theory.

We discuss a shaking instance: the Whitehead problem. An abelian groupW is called a Whitehead group if Ext1(W,Z) = 0.If W is projective, then

the condition is satisfied.

What about the converse?K. Stein (1951): If W is countable (=countably generated), then W isfree.



Shela’s discovery

Shelah, 1978

Theorem (Shelah 1974)

On the basis of ZFC-set theory, the Whitehead problem is undecidable.More precisely:

1 V = L implies that every Whitehead group is free.

2 Martin’s axiom and ¬CH implies the existence of a Whitehead groupof cardinality ℵ1 that is not free.


Tilting theory

Michael Butler, the co-creator oftilting theoryShela Brenner and Michael Butler. Generalizations of theBernstein-Gelfand-Ponomarev reflection functors. 1980

ICRA Ottawa 1992, S. Brenner and Lutz HilleH. Lenzing (Paderborn) Michael Butler 16 / 1

Tilting theory

The first steps of tilting

Non-experts of Representation Theory usually will have heard about

Quivers

Auslander-Reiten theory

Tilting

As many important concepts, tilting theory has many fathers and mothers.

The first instance of tilting, appears in the 1973-paper byBernstein-Gelfand-Ponomarev: Coxeter functors and Gabriel’s theorem

An important rephrasing is Auslander-Platzeck-Reiten: Coxetergroups without diagrams from 1979

The concept of — nowadays called classical — tilting, in fullgenerality, is developed by Brenner-Butler in 1980

In 1981/82 Happel-Ringel and Bongartz substantial deepened theconceptual understanding of tilting

It is, however, fair to say, that the real break-through was initiated by the1980-paper from Brenner-Butler.


Tilting theory

The name ”tilting”

Brenner-Butler, on page 1 of their paper, give the following explanation ofthe name ”tilting”.

”It turns out that . . . we like to think [of our functors] as a change of basisfor a fixed root-system — a tilting of the axes relative to the roots whichresults in a different subset of roots lying in the positive cone”

”For this reason . . . we call our functors tilting functors or simply tilts.


Tilting theory

Dieter Happel: Tilting as derived equivalence

The proper understanding of tilting is through D. Happel’shabilitation thesis from the mid-1980s:Finite dimensional algebras related by (sequences of) tilts arederived-equivalent.From now on Representation Theory enjoys a competition betweenthe abelian and the triangulated point of view.


Tilting theory

Further ancestors of tilting

Happel’s interpretation of tilting as derived equivalence puts another groupof mathematicians in the list of ancestors of tilting:

Beilinson with ”Coherent sheaves on Pn and problems of linearalgebra”, (1978)

Bernstein-Gelfand-Gelfand with ”Algebraic bundles over Pn andproblems of linear algebra” (1978)

Rudakov and his school of algebraic geometers (end of the 1980s).

Remark

It is worth to note that the BGP-reflection functors — independent oftheir role for the development of tilting — had a decisive role in thedevelopment for cluster theory, through the concept of mutations.


Tilting theory

Cluster Conference, Mexico City, 2008H. Lenzing (Paderborn) Michael Butler 21 / 1

Documents

In memoriam of Michael Butler (1928â€“2012)