In memoriam of Michael Butler (19282012) Helmut Lenzing Universit - - PowerPoint PPT Presentation
In memoriam of Michael Butler (19282012) Helmut Lenzing Universit at Paderborn Auslander Conference 2013, Woods Hole, 19. April H. Lenzing (Paderborn) Michael Butler 1 / 1 ICRA Beijing (2000). M. Butler with W. Crawley-Boevey H.
Helmut Lenzing
Universit¨ at Paderborn
Auslander Conference 2013, Woods Hole, 19. April
Michael Butler 1 / 1
ICRA Beijing (2000). M. Butler with W. Crawley-Boevey
Michael Butler 2 / 1
Scope
We restrict to two aspects of the work of M.C.R. Butler having a major influence on the course of mathematics: Infinite Abelian groups Tilting theory
Michael Butler 3 / 1
Infinite Abelian groups
The header of this slide refers to Butler’s paper ”A class of torsion-free abelian groups of finite rank”, Proc. London Math.
It is Butler’s first paper on Abelian Groups.
Michael Butler 4 / 1
Infinite Abelian groups
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 · · ·
Michael Butler 5 / 1
Infinite Abelian groups
The extension group Ext(X, Y ) formed by classes of extensions of short exact sequences. (R. Baer) Existence of enough injective modules (R. Baer, 1940). Ringel’s study of infinite dimensional modules over tame hereditary algebras, exploiting the sophisticated structure theory of infinite abelian groups.
Michael Butler 6 / 1
Infinite Abelian groups
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 of rank one The factor group Q/Z is torsion. It decomposes Q/Z =
Z(p∞) into Pr¨ ufer groups Q = E(Z), Z(p∞) = E(Z/(p)), p prime. Here, E=injective hull
Michael Butler 7 / 1
Infinite Abelian groups
Each injective abelian group is a direct sum of copies of the rationals Q and of Pr¨ ufer groups Z(p∞), hence of injective hulls of Z/p, where p 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)
Michael Butler 8 / 1
Infinite Abelian groups
Torsion-free groups are an almost hopeless case. Indeed, I. Kaplansky (1954, 1969) in his nice little book states about torsion-free groups: ”In this strange part of the subject anything that can conceivably happen actually does happen.” It is here, where Michael Butler changed the direction of abelian group theory.
Michael Butler 9 / 1
Infinite Abelian groups
Kaplansky coined the nice concept of ”Test Problems”, not being so much important in itself, but suitable to testing the maturity of a mathematical theory. Problem 1. Assume G is isomorphic to a direct summand of H, and H is isomorphic to a direct summand of G. Does it follow that G ∼ = H? Problem 2. Assume G ⊕ G ∼ = H ⊕ H. Does it follow G ∼ = H?
Michael Butler 10 / 1
Infinite Abelian groups
A finite direct sum of torsion-free rank-one abelian groups = completely decomposable Butler (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 completely
decomposable group These groups, now called Butler groups, allow a classification by types/typesets.
Michael Butler 11 / 1
Infinite Abelian groups
Such a group H is just a non-zero subgroup of Q. We may assume Z ⊆ H ⊆ Q. Then H/Z ⊆ Q/Z =
Z(p∞). thus H/Z =
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 is called the type τ(H) of H.
Michael Butler 12 / 1
Infinite Abelian groups
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 finite
dimensional 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 ordered
Michael Butler 13 / 1
Infinite Abelian groups
Motto: Abelian groups form the spearhead of module theory. We discuss a shaking instance: the Whitehead problem. An abelian group W is called a Whitehead group if Ext1(W, Z) = 0.If W is projective, then the condition is satisfied. What about the converse?
free.
Michael Butler 14 / 1
Infinite Abelian groups
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 group
Michael Butler 15 / 1
Tilting theory
Shela Brenner and Michael Butler. Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors. 1980 ICRA Ottawa 1992, S. Brenner and Lutz Hille
Michael Butler 16 / 1
Tilting theory
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 by Bernstein-Gelfand-Ponomarev: Coxeter functors and Gabriel’s theorem An important rephrasing is Auslander-Platzeck-Reiten: Coxeter groups without diagrams from 1979 The concept of — nowadays called classical — tilting, in full generality, is developed by Brenner-Butler in 1980 In 1981/82 Happel-Ringel and Bongartz substantial deepened the conceptual understanding of tilting It is, however, fair to say, that the real break-through was initiated by the 1980-paper from Brenner-Butler.
Michael Butler 17 / 1
Tilting theory
Brenner-Butler, on page 1 of their paper, give the following explanation of the name ”tilting”. ”It turns out that . . . we like to think [of our functors] as a change of basis for a fixed root-system — a tilting of the axes relative to the roots which results in a different subset of roots lying in the positive cone” ”For this reason . . . we call our functors tilting functors or simply tilts.
Michael Butler 18 / 1
Tilting theory
The proper understanding of tilting is through D. Happel’s habilitation thesis from the mid-1980s: Finite dimensional algebras related by (sequences of) tilts are derived-equivalent. From now on Representation Theory enjoys a competition between the abelian and the triangulated point of view.
Michael Butler 19 / 1
Tilting theory
Happel’s interpretation of tilting as derived equivalence puts another group
Beilinson with ”Coherent sheaves on P n and problems of linear algebra”, (1978) Bernstein-Gelfand-Gelfand with ”Algebraic bundles over P n and problems 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 of their role for the development of tilting — had a decisive role in the development for cluster theory, through the concept of mutations.
Michael Butler 20 / 1
Tilting theory
Cluster Conference, Mexico City, 2008
Michael Butler 21 / 1