Auto-equivalences of derived categories acting on cohomology

Abstract

Let A be a k-algebra which is projective as a k-module, let M be an A-module whose endomorphisms are given by multiplication by central elements of A, and let TrPic k ( A) be the group of standard self-equivalences of the derived category of bounded complexes of A-modules. Then we define an action of the stabilizer of M in TrPic k ( A) on the Ext-algebra of M. In case M is the trivial module for the group algebra kG = A, this defines an action on the cohomology ring of G which extends the well-known action of the automorphism group of G on the cohomology group. Introduction. Let A and B be R-algebras over the commutative ring R so that A is projective as an R-module. If there is an equivalence between the derived categories of bounded complexes of A-modules D b ( A) and the derived category D b (B), Rickard and Kellerequivalences of this type form a group which, in an earlier work with R. Rouquier [8], is called TrPic R ( A). Let M be an A-module. Then, it is reasonable to expect that the set HD M ( A) of elements in TrPic R ( A), which fix M up to isomorphism, acts on the Ext-algebra Ext * A (M, M) of M as ring automorphisms since Ext n A (M, M) = Hom D b (A) (M, M[n]) for any integer n. To get an actual action one has to be a bit more careful. We prove the above statement if any automorphism of M is induced by multiplication by an invertible element of the centre of A. For other modules with more complicated automorphism groups an extension HD M ( A) of HD M ( A) by some quotient of the automorphism group of M acts on Ext * A (M, M). The above defined action is well behaved with respect to change of base rings. In case A is the group ring RG of a group G, the action of HD R (RG) extends the well known action of the outer automorphism group of G on the cohomology ring H * (G, R). This action of HD R (RG) is functorial with respect to the second variable. In further work [10,11] we study the functoriality with respect to the first variable. There, the situation is more complicate and we only have partial answers.

Cohomology of groups and splendid equivalences of derived categories

“…(S p , R) is trivial if the action of HSplen R (C p ) on H * (C p , R) is trivial and one may prove that the trivial module is fixed also when one takes the Brauer functor of the auto-equivalence in question. This can be checked individually for the auto-equivalences considered in [16]. Moreover, One would like to get the same result for R being the p-adic integers.…”

“…Hence, the fact that the restriction to the centralizer of C p is compatible with the action of HSplen k (S p ) proves that the action of HSplen R (G) on H * (S p , R) is trivial if the action of HSplen R (C p ) on H * (C p , R) is trivial and one may prove that the trivial module is fixed also when one takes the Brauer functor of the auto-equivalence in question. This can be checked individually for the auto-equivalences considered in [16]. Moreover,…”

“…Let HD R (G) be the subgroup of those auto-equivalences which fix the trivial module up to isomorphism. In [16] an action of HD R (G) on H * (G, R) is defined so that the cohomology ring is an R (HD R (G))-module. We study in the present paper the connection between the functor 'cohomology of a finite group G' and, for fixed G, the action of HD R (G).…”

Self-tilting complexes yield unstable modules

“…If any automorphism of M is given by multiplication with a unit in the centre of A, then in [11] it is shown that Ext *…”

“…For any A-module M let HD M (A) be the subgroup of T rP ic R (A) which is formed by the self-equivalences mapping M to an isomorphic copy. Then, in an earlier paper [11] I showed that, under a certain hypothesis on M , the group HD M (A) acts in a natural way on the Extalgebra Ext * A (M, M ). When A is a group algebra RG, with R being a field of characteristic p and G being a finite group, J. Rickard defined in [8] a splendid equivalence by some technical conditions, basically by asking that the homogeneous components of a tilting complex be p-permutation modules induced from diagonal p-subgroups, and by an invertibility condition in the homotopy category.…”