On equivalences for cohomological Mackey functors

Abstract

Abstract. By results of Rognerud, a source algebra equivalence between two p-blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories of cohomological Mackey functors. We prove this by giving an intrinsic description of cohomological Mackey functors of a block in terms of the source algebras of the block, and then using this description to construct explicit two-sided tilting complexes realising the above mentioned derived equivalence. We show further that a splendid stable equivalence of Morita type between two blocks induces an equivalence between the categories of cohomological Mackey functors which vanish at the trivial group. We observe that the module categories of a block, the category of cohomological Mackey functors, and the category of cohomological Mackey functors which vanish at the trivial group arise in an idempotent recollement. Finally, we extend a result of Tambara on the finitistic dimension of cohomological Mackey functors to blocks.

The dominant dimension of cohomological Mackey functors

“…See [2] for details and more general versions of this notion. We will show that Theorem 1.1 is an immediate consequence of results from [3], [4], [6], and the following. Theorem 1.2.…”

“…Set V = ⊕ Q kP ⊗ kQ k, where Q runs as before over the subgroups of P , and set F = End kP (V ). As in the proof of [3,Theorem 1.6], the functor A⊗ kP − sends V to add(U ) and the functor kP A⊗ A − sends U to add(V ), because A has a P × P -stable k-basis, hence preserves the classes of p-permutation modules. By definition, the dominant dimension of coMack(B) is equal to ddim(E).…”

“…With the notation above, set U = ⊕ Q A ⊗ kQ k, where Q runs over the subgroups of P , and set E = End A (U ). By [3,Theorem 1.1] we have coMack(B) ∼ = mod(E op ). Set V = ⊕ Q kP ⊗ kQ k, where Q runs as before over the subgroups of P , and set F = End kP (V ).…”

“…Proof of Theorem 1.2. The following argument from the proof of [3,Theorem 3.1] shows that add(M ⊗ B V ) = add(U ) and add(M * ⊗ A U ) = add(V ). By the assumptions, we have add…”

On Morita and derived equivalences for cohomological Mackey algebras

“…In particular, this proof does not show that eM f and f N e are Rickard complexes, and it seems unclear whether the induced derived equivalence preserves the subcategories of chain complexes over add(X) and add(Y ). The proof does show that the derived equivalence induced by M and N restricts to an equivalence The proof of Theorem 1.1 is played back to the theorems 1.3 and 1.4, together with description of cohomological Mackey functors in terms of source algebras of blocks in [5], extending ideas going back to Yoshida [15].…”