A theorem of Mislin for cohomology of fusion systems and applications to block algebras of finite groups

Abstract

The aim of this short expository article is to give an algebraic proof for a theorem of Mislin in the case of cohomology of saturated fusion systems defined on p-groups when p is odd. Some applications of this theorem, which includes different proofs of known results regarding block algebras of finite groups, are also given.

Quillen’s stratification for fusion systems

“…It is pointed out in work of Benson,Grodal,and Henke [3,Remark 3.7] that there seems to be no reference for Quillen's stratification for fusion systems and that such a reference would be the only missing link to generalise [3, Theorem A] to arbitrary fusion systems. See Todea [8,Theorem 1.1] for an explicit statement and proof of [3, Theorem A] for fusion systems along these lines. Quillen's stratification for arbitrary fusion systems is a straightforward adaptation of the block cohomology version in [5], which is in turn an adaptation of Benson's presentation in [2] of some of Quillen's results in [6], [7].…”

Mislin’s theorem for fusion systems through Mackey functors

“…We prove the above theorem for all primes p. Consequently Todea's results in [11] on block algebras now hold for all primes p. Our approach follows closely Symonds's proof [10] of Mislin's theorem for finite groups, which uses Mackey functors. Throughout this paper, p is a fixed prime and all modules are finitely generated left modules.…”

“…Recently Todea [11] proved the above theorem when p is odd following Benson, Grodal and Henke's new algebraic proof [2] of Mislin's theorem for finite groups for p odd. We prove the above theorem for all primes p. Consequently Todea's results in [11] on block algebras now hold for all primes p. Our approach follows closely Symonds's proof [10] of Mislin's theorem for finite groups, which uses Mackey functors.…”

Control of fusion by Abelian subgroups of the hyperfocal subgroup

“…In particular, if A ≤ S is elementary abelian, one can define V + F ,A = res * F ,A V + A . In an unpublished preprint, Markus Linckelmann [15, Theorem 1] proves a version of the Quillen stratification theorem; see also Theorem 1.3 and Remark 1.1 in [19]. Using this, one can similarly prove the following version of Theorem C for fusion systems:…”