Extensions of simple cohomological Mackey functors

This is a report on some recent joint work with Radu Stancu, to appear in [4]. It is an expanded version of a talk given at the RIMS workshop Cohomology of finite groups and related topics, February 18-20, 2015. 1. Cohomological Mackey functors 1.1. Let G be a finite group, and k be a commutative ring. There are many equivalent definitions of Mackey functors for G over k. For the “naive” one, this is an assignment H 7→ M(H) of a k-module M(H) to any subgroup H of G, together with k-linear maps M(H) tH −→M(K) rK H −→M(H), M(H) cx,H −→M(H) whenever H ≤ K ≤ G and x ∈ G, subject to a list of compatibility conditions, e.g. transitivity of transfers and restrictions, or the Mackey formula (see [6] for details). A Mackey functor M is called cohomological if ∀H ≤ K ≤ G, tH ◦ r H = |K : H|IdM(K) . The cohomological Mackey functors for G over k form a category Mk(G). 1.2. Examples : • Let V be a kG-module. The fixed points functor FPV is defined by M(H) = V H , for any H ≤ G, and by ∀H ≤ K ≤ G, r H : V K ↪→ V H , tH = TrH : V H → V K , and by cx,H(v) = x · v, for x ∈ G. More generally, for n ∈ N, the cohomology functor H(−, V ) is a cohomological Mackey functor. • Let k be a field of characteristic p, let G be a finite p-group. The simple cohomological Mackey functors for G over k are the functors SX = S G X , where X ≤ G (up to G-conjugation), defined by ∀H ≤ G, SX(H) = { k if H =G X, {0} otherwise.