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
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
H. Lenzing (Paderborn) Michael Butler 3 / 1
Infinite Abelian groups
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.
H. Lenzing (Paderborn) Michael Butler 4 / 1
Infinite 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
· · ·
H. Lenzing (Paderborn) Michael Butler 5 / 1
Infinite Abelian groups
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.
H. Lenzing (Paderborn) Michael Butler 6 / 1
Infinite Abelian 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
H. Lenzing (Paderborn) Michael Butler 7 / 1
Infinite Abelian groups
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)
H. Lenzing (Paderborn) Michael Butler 8 / 1
Infinite Abelian groups
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.
H. Lenzing (Paderborn) Michael Butler 9 / 1
Infinite Abelian groups
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?
H. Lenzing (Paderborn) Michael Butler 10 / 1
Infinite Abelian groups
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.
H. Lenzing (Paderborn) Michael Butler 11 / 1
Infinite Abelian groups
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.
H. Lenzing (Paderborn) Michael Butler 12 / 1
Infinite Abelian groups
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.
H. Lenzing (Paderborn) Michael Butler 13 / 1
Infinite Abelian groups
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.
H. Lenzing (Paderborn) Michael Butler 14 / 1
Infinite Abelian groups
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.
H. Lenzing (Paderborn) Michael Butler 15 / 1
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.
H. Lenzing (Paderborn) Michael Butler 17 / 1
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.
H. Lenzing (Paderborn) Michael Butler 18 / 1
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.
H. Lenzing (Paderborn) Michael Butler 19 / 1
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.
H. Lenzing (Paderborn) Michael Butler 20 / 1