Quillen’s stratification for fusion systems

Abstract

The purpose of this note is to provide a reference for the fact that the proof of Quillen's stratification for finite group cohomology carries over to fusion system. As in the case of Quillen's stratification for block varieties, the proof is similar to the usual proof for group cohomology except for the use of fusion stable bisets, whose existence is due to Broto, Levi, and Oliver.

Stratification and duality for homotopical groups

“…The Quillen stratification of this theorem generalizes previous work of Linckelmann [Lin17], and in fact gives an alternative proof that does not rely on the existence of certain bisets for p-local finite groups.…”

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

“…To prove Theorem 1.2 we will use Quillen stratification for cohomology of saturated fusion systems given by Markus Linckelmann in [13], for which the proof is the same as for block algebras [11], with some minor adjustments. Since it has not appeared in this form in the literature we state it here for completeness.…”

“…We will not give the proof of Theorem 1.3 from [13], since the way to obtain it is to copy the proof from [11] in the context of saturated fusion systems. This proof follows in turn very closely Bensons presentation in [3] of parts of Quillens original work in [14], with only additional ingredient the fusion stable bisets whose existence was proved by Broto, Levi, and Oliver in [5].…”

“…Acknowledgments. We are grateful to Professor Markus Linckelmann for some email discussions and for the reference [13].…”

On the Brauer Indecomposability of Scott Modules

“…, then the saturation of F P (G) is equivalent to requiring that N G (P )/P C G (P ) has order prime to p. For any remaining notation and terminology, see the books of [14] and [16], and also [2] and [11] for fusion systems.…”