Equivalences between blocks of cohomological Mackey algebras

Abstract

Let G be a finite group and (K, O, k) be a p-modular system "large enough". Let R = O or k. There is a bijection between the blocks of the group algebra RG and the central primitive idempotents (the blocks) of the so-called cohomological Mackey algebra coµ R (G). Here, we prove that a so-called permeable derived equivalence between two blocks of group algebras implies the existence of a derived equivalence between the corresponding blocks of cohomological Mackey algebras. In particular, in the context of Broué's abelian defect group conjecture, if two blocks are splendidly derived equivalent, then the corresponding blocks of cohomological Mackey algebras are derived equivalent.The main result of this paper is the following theorem which settles the question for the cohomological Mackey algebra in the case of a splendid equivalence (see [14]): Theorem 1.2. Let G and H be two finite groups, let b be a block of RG and c be a block of RH. If RGb and RHc are splendidly derived equivalent, then D b (coµ R (G)ι(b)) ∼ = D b (coµ R (H)ι(c)).

Equivalences between blocks of p-local Mackey algebras

“…Since the p-local Mackey algebra and the cohomological Mackey algebra share a lot of properties, for example, they have the same number of simple modules in each block and the projective cohomological Mackey functors are the biggest cohomological quotients of the p-local projective Mackey functors, one may ask if an equivalence between blocks of p-local Mackey algebras induces in some sense, an equivalence between the corresponding blocks of the cohomological Mackey algebras. Such a result would allow us to use [21]. The following example shows that the situation is unfortunately not that simple.…”

“…If D is abelian, it is conjectured by Broué that the block algebras RGb and RN G (D)b ′ are derived equivalent. In [21], the Author proved that if two blocks of group algebras are splendidly derived equivalent, then the corresponding blocks of the so-called cohomological Mackey algebras are derived equivalent. The cohomological Mackey algebra is a quotient of the p-local Mackey algebra.…”

On equivalences for cohomological Mackey functors

“…The algebra A = iOGi with its canonical image P i of P in A × is called a source algebra of b. By a result of Rognerud [10,Proposition 4.5], the last statement holds more generally for permeable Morita equivalences; that is, Morita equivalences preserving p-permutation modules (and the proof presented below shows this as well). It is not necessary to take the direct sum over all subgroups of P in the statement of 1.1.…”

“…By a result of Yoshida in [13], the category coMack(OG) of cohomological Mackey functors of G with coefficients in mod(O) is equivalent to the category of finitely generated modules over the algebra End OG (⊕ H OG/H) op , where H runs over the subgroups of G. This description implies that the category coMack(OG) decomposes as a direct sum of the categories coMack(G, b), with b running over the blocks of OG, such that each coMack(G, b) is equivalent to the category of finitely generated modules over the algebra End OG (⊕ H b · OG/H) op , where H runs as before over the subgroups of G. By results of Thévenaz and Webb in [12, §17], this is the block decomposition of coMack(OG). We use Yoshida's result to describe coMack(G, b) in an analogous way at the source algebra level, obtaining as a consequence an alternative proof of a result of Rognerud in [10], stating that blocks with isomorphic source algebras have equivalent categories of cohomological Mackey functors. A defect group of a block b of OG is a maximal p-subgroup P of G such that OP is isomorphic to a direct summand of OGb as an OP -OP -bimodule.…”

On Morita and derived equivalences for cohomological Mackey algebras

“…Set X = ⊕ R A ⊗ OR O, where R runs over the subgroups of P , and set E = End A (X). Similarly, A permeable Morita equivalence between block algebras over k need not be splendid; see [11,Remark 4.7]. In characteristic zero, however, permeable Morita equivalences are splendid.…”