Cohomology of groups, abelian Sylow subgroups and splendid equivalences

Let G be a finite group and let R be a complete discrete valuation domain of characteristic 0 with residue field k of characteristic p and let S be R or k. The cohomology rings H∗(K,S) for subgroups K of G together with restriction to subgroups of G, transfer from subgroups of G and conjugation by elements of G gives H∗(−, S) the structure of a Mackey functor. Moreover, the group HSplenS(K) of splendid auto-equivalences of the bounded derived category of finitely generated SG-modules fixing the trivial module acts S-linearly on H∗(K,S). In this note we study the compatibility of these structures and get some consequences when G has an abelian Sylow p subgroup. In particular we see that in case G has an abelian Sylow p subgroup, then HSplenR(G) acts by automorphisms of the Sylow subgroup on the cohomology. 2000 AMS Subject Classification : 20C05, 20J06, 16G30, 18E30, 14L30 Let G be a finite group and let R be a commutative ring, considered as a trivial RG-module. The bounded derived category D(RG) of finitely generated RG-modules is used in modular representation theory of finite groups to provide a geometric framework for the classical conjectures like Dade’s conjecture or Alperin’s conjecture [3, 4]. These two are consequences of Broué’s conjecture [1, 4] which states that in case R = k is a field of characteristic p and G is a finite group with abelian Sylow p subgroup P , the derived categories of the principal block B0(kG) of kG and the principal block B0(kNG(P )) of kNG(P ) are equivalent. Besides the above conjectures of Alperin and Dade a positive answer to Broué’s conjecture implies for example that the K-theory, the cyclic and the Hochschild (co-)homology of the principal blocks of kG and of kNG(P ) coincide. For an account of other consequences and most known results see [4]. If there is an equivalence between two derived categories, an immediate question is, how many equivalences there are. This way, one is lead to the definition of the group TrP icR(B0(RG)) of autoequivalences of the derived category D(B0(RG)) (see [8]). This group comes into the play from a very different approach as well. The Mirror symmetry conjecture of Kontsevitch imply that symplectic automorphisms of a symplectic manifold with vanishing first Chern class induce auto-equivalences of the derived category of sheaves of the mirror Calabi-Yau manifold. From there as well one is lead to the group of auto-equivalences of the derived category (see [9]). Studying a group is done most naturally by studying its modules. So, one should look for a natural module on which these groups act on. From many points of view the derived category is the right object to consider homology. Since the derived categories, we are interested in, are derived categories of group rings, in the context of auto-equivalences of group rings the natural module we asked for is the cohomology of groups. In the present note we construct a module structure on H∗(G,R) coming from interpreting this object in the derived category. Let A be an R-algebra. Bernhard Keller proved [2] that D(B0(RG)) ' D(A) as triangulated categories if and only if there is an X ∈ D(A⊗RB0(RG)) so that X⊗B0(RG)− is an equivalence. Such an X is called a two-sided tilting complex. In [11] it is proved that if R is hereditary, then A is R-projective again and by [2] the inverse equivalence is again a derived tensor product by a complex of bimodules. For a complex X note by [X] its isomorphism class in the derived category. Set [8] TrP icR(B0(RG)) := {[X]| X ∈ D(B0(RG)⊗B0(RG)) is a 2-sided tilting complex} and HDR(G) := {[X] ∈ TrP icR(B0(RG))| X ⊗B0(RG) R ' R} . It is shown (cf. [12]) that the group cohomology H∗(G,R) is an R HDR(G)-module by composing the following morphisms: Take X with