Markus Linckelmann

Transfer in Hochschild Cohomology of Blocks of Finite Groups

We develop the notion of a cohomology ring of blocks of finite groups and study its basic properties by means of transfer maps between the Hochschild cohomology rings of symmetric algebras associated with bounded complexes of finitely generated bimodules which are projective on either side.

Splendid Derived Equivalences for Blocks of Finite p-Solvable Groups

Since the remarkable discovery of the relevance of derived equivalences in the theory of p -blocks of finite groups, where p is a prime, by J. Rickard in [ 15 , 16 ], various attempts have been made to understand this phenomenon. In particular, J. Rickard defines in [ 18 ] a certain class of derived equivalences between the derived module categories of p -blocks of finite groups that he calls splendid equivalences (and that we are going to call splendid derived equivalences in this paper) which take into account the local structure , that is, which under suitable hypotheses induce a family of derived equivalences at all ‘local levels’ of the considered p -blocks (see [ 18 ] for a more detailed motivation). The main condition for a derived equivalence to be splendid is that it is given by a two-sided tilting complex consisting of p -permutation bimodules (see Definitions 1.3 and 1.4 below for the precise terminology).

ZJ-THEOREMS FOR FUSION SYSTEMS

For p an odd prime, we generalise the Glauberman-Thompson p-nilpotency theorem [5, Ch. 8, Theorem 3.1] to arbitrary fusion systems. We define a notion of Qd(p)- free fusion systems and show that if F is a Qd(p)-free fusion system on some finite p-group P then F is controlled by W(P) for any Glauberman functor W, generalising Glauberman’s ZJ-theorem [3] to arbitrary fusion systems.

Local control in fusion systems of p-blocks of finite groups

Abstract If p is an odd prime, b a p -block of a finite group G such that SL (2, p ) is not involved in N G ( Q , e )/ C G ( Q ) for any b -subpair ( Q , e ), then N G ( Z ( J ( P ))) controls b -fusion, where P is a defect group of b . This is a block theoretic analogue of Glaubermans ZJ -Theorem. Several results of general interest about fusion and blocks are also proved.

Conjectures of Alperin and Broué for 2-blocks with elementary abelian defect groups of order 8

Abstract Using the classification of finite simple groups, we prove Alperins weight conjecture and the character theoretic version of Broués abelian defect conjecture for 2-blocks of finite groups with an elementary abelian defect group of order 8.

TRIVIAL SOURCE BIMODULE RINGS FOR BLOCKS AND p-PERMUTATION EQUIVALENCES

We associate with any p-block of a finite group a Grothendieck ring of certain p-permutation bimodules. We extend the notion of p-permutation equivalences introduced by Boltje and Xu [4] to source algebras of p-blocks of finite groups. We show that a p-permutation equivalence between two source algebras A, B of blocks with a common defect group and same local structure induces an isotypy.

On the Glauberman and Watanabe correspondences for blocks of finite -solvable groups

If G is a finite p-solvable group for some prime p, A a solvable subgroup of the automorphism group of G of order prime to |G| such that A stabilises a p-block b of G and acts trivially on a defect group P of b, then there is a Morita equivalence between the block b and its Watanabe correspondent w(b) of C G (A), given by a bimodule M with vertex ΔP and an endo-permutation module as source, which on the character level induces the Glauberman correspondence (and which is an isotypy by Watanabes results).

On simple modules over twisted finite category algebras

The purpose of this note is to show that the recent proof, by Ganyushkin, Mazorchuk and Steinberg, of the parametrisation of simple modules over finite semigroup algebras due to Clifford, Munn and Ponizovski˘i carries over to twisted finite category algebras. We observe that the parametrisations of simple modules over Brauer algebras, Temperley-Lieb algebras, and Jones algebras due to Graham and Lehrer, can be obtained as special cases of our main result. We further note that the notion of weights in the context of Alperin’s weight conjecture extends to twisted finite category algebras.

Quillen stratification for block varieties

Abstract The classical results on stratifications for cohomology varieties of finite groups and their modules due to Quillen (Ann. Math. 94 (1971) 549–572; 573–602) and Avrunin–Scott (Invent. Math. 66 (1982) 277–286) carry over to the varieties associated with finitely-generated modules over p -blocks of finite groups, introduced in Linckelmann (J. Algebra 215 (1999) 460–480).

On graded centres and block cohomology

We extend the group theoretic notions of transfer and stable elements to graded centers of triangulated categories. When applied to the center H∗Db(B)) of the derived bounded category of a block algebra B we show that the block cohomology H∗(B) is isomorphic to a quotient of a certain subalgebra of stable elements of H∗(Db(B)) by some nilpotent ideal, and that a quotient of H∗(Db(B)) by some nilpotent ideal is Noetherian over H∗(B).