In memoriam of Michael Butler (1928–2012) 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. Lenzing (Paderborn) Michael Butler 2 / 1
Scope 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 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. Three examples R. Baer (1902–1979) 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. 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 / ( p n ) , 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 ∞ ) p prime into Pr¨ ufer 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 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) 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 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. 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 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 ? H. Lenzing (Paderborn) Michael Butler 10 / 1
Infinite Abelian groups Butler 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. 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 assume Z ⊆ H ⊆ Q . Then � H/ Z ⊆ Q / Z = Z ( p ∞ ) . p prime thus � H/ Z = U p p prime where U p is a subgroup of length n p with p ∈ { 0 , 1 , . . . , ∞} . Up to change of finitely many entries, each by a finite value, the sequence n = ( n 2 , n 3 , n 5 , . . . ) characterizes the isoclass of H . The resulting equivalence class of n is called 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 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 ordered 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 group W is called a Whitehead group if Ext 1 ( 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 is free. 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 group of cardinality ℵ 1 that is not free. H. Lenzing (Paderborn) Michael Butler 15 / 1
Tilting theory Michael Butler, the co-creator of tilting theory Shela Brenner and Michael Butler. Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors. 1980 ICRA Ottawa 1992, S. Brenner and Lutz Hille H. 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 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. H. Lenzing (Paderborn) Michael Butler 17 / 1
Tilting theory The name ”tilting” 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. H. Lenzing (Paderborn) Michael Butler 18 / 1
Tilting theory Dieter Happel: Tilting as derived equivalence 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. H. Lenzing (Paderborn) Michael Butler 19 / 1
Tilting theory Further ancestors of tilting Happel’s interpretation of tilting as derived equivalence puts another group of mathematicians in the list of ancestors of tilting: 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. H. Lenzing (Paderborn) Michael Butler 20 / 1
Tilting theory Cluster Conference, Mexico City, 2008 H. Lenzing (Paderborn) Michael Butler 21 / 1
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