Auto-equivalences of derived categories acting on group cohomology
by
Alexander Zimmermann
Universite de Picardie Jules Verne

The derived category of a group ring attained quite some interest in recent years. Let G be a finite group, let k be a field of characteristic p, let P be a p-Sylow subgroup of G, let H:=N_G(P) be the normalizer of P in G and let C_G(P) be the centralizer of P in G.

Broué conjectures that if P is abelian and k large enough, then the derived category D^b(B₀(kG)) of the principal block of kG and the derived category D^b(B₀(kH)) of the principal block of kH are equivalent. Equivalences which are induced by tensoring with a complex of bimodules are called standard. A stronger conjecture of Broué and Rickard predicts that there is even an equivalence of a special kind, a splendid equivalence. A splendid equivalence F induces compatible equivalences F_Q between the the derived categories of the group rings over the centralizers C_G(Q) and C_H(Q).

In a joint work with Raphaël Rouquier we defined and studied the group of auto-equivalences TrPic_k(\Lambda) of standard type of the derived category D^b(\Lambda) of the k-algebra \Lambda. This group is the natural automorphism group for several homological constructions.

Theorem If a splendid auto-equivalence F of the derived category D^b(B₀(kG)) of the principal block of the group ring kG fixes the trivial kG-module, then it induces an automorphism F^* of the cohomology ring H^*(G, k). This action is functorial with respect to k. Suppose F_Q fixes the trivial module as well. In this case F commutes with restriction to centralizers C_G(Q) of p-subgroups Q and F commutes with transfer from C_G(Q) to G.

Define

HDk(G):={F in TrPic(B0(kG))|  F(k) =~ k}

and

HSplenk(G):={F in HDk(G)|  F is splendid }  .

It is clear that the group of augmented automorphisms of kG modulo inner automorphisms is a subgroup of HD_k(G). The group automorphisms modulo inner automorphisms of kG is a subgroup of HSplen_k(G).

Theorem Let P be the partially ordered set of p-subgroups of G furnished with inclusions. HSplen_k(C_G(-)) is a group sheaf on this poset, inclusions being mapped to the Brauer construction.

From the above two results it is possible to derive a group sheaf action on H^*(G, k) as group sheaf of Mackey functors.

References

[1] R. Rouquier and A. Zimmermann, A Picard group for derived categories, preprint 1998.

[2] A. Zimmermann, Auto-equivalences of derived categories acting on group cohomology, preprint 1999.

[2] A. Zimmermann, Cohomology of groups and splendid equivalences of derived categories, preprint 2000.

Date received: May 15, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cafe-02.