Frobenius Categories versus Brauer Blocks

On the reduction of Alperin's Conjecture to the quasi-simple groups

“…16] our proof also depends on checking, in the list of quasi-simple groups, the technical condition [10, 16.22.1] -which, as a matter of fact, is not always fulfilled: it is not difficult to exhibit a counter-example from Example 4.2 in [12]! -and on the other hand, in the second half of [10,Chap. 16] some arguments have been scratched.…”

“…Unfortunately, on the one hand in [10,Chap. 16] our proof also depends on checking, in the list of quasi-simple groups, the technical condition [10, 16.22.1] -which, as a matter of fact, is not always fulfilled: it is not difficult to exhibit a counter-example from Example 4.2 in [12]!…”

“…To carry out this purpose, in [10,Chap. 15] we develop reduction results relating both members of isomorphism (1.4.1) with the Grothendieck groups coming from suitable proper normal subblocks; recall that a normal sub-block of (b,Ĝ) is a pair (c,Ĥ) formed by a normal k * -subgroupĤ ofĜ and a block c ofĤ fulfilling cb = 0.…”

“…Let p be a prime number, k an algebraically closed field of characteristic p, O a complete discrete valuation ring of characteristic zero admitting k as the residue field,Ĝ a k * -group of finite k * -quotient G [10, 1.23], b a block ofĜ [10, 1.25] and G k (Ĝ, b) the scalar extension from Z to O of the Grothendieck group of the category of finitely generated k * Ĝ bmodules [10, 14.3]. In [10,Chap. 14], choosing a maximal Brauer (b,Ĝ)-pair (P , e), the existence of a suitable k * -Gr-valued functor aut (F (b,Ĝ) ) nc over some full subcategory (F (b,Ĝ) ) nc of the Frobenius P -category F (b,Ĝ) [10, 3.2] allows us to consider an inverse limit of Grothendieck groups -noted G k (F (b,Ĝ) , aut (F (b,Ĝ) ) nc ) and called the Grothendieck group of F (b,Ĝ) -such that Alperin's Conjecture is actually equivalent to the existence of an O-module isomorphism [10,I 32 and Corollary 14.32] G k (Ĝ, b) ∼ = G k (F (b,Ĝ) , aut (F (b,Ĝ) ) nc ).…”

“…In [10,Chap. 16] we propose a refinement of Alperin's Conjecture whose possible proof can be reduced to check that this refinement holds on the so-called quasi-simple groups.…”