Auto-equivalences of derived categories acting on cohomology (Q1849547)

Let \(R\) be a commutative ring, \(A\) an \(R\)-algebra projective as an \(R\)-module, and let \(M\) be an \(A\)-module. Denote by \(\text{TrPic}_R(A)\) the group of standard self-equivalences of the derived category of bounded complexes of \(A\)-modules, and by \(HD_M(A)\) the stabilizer of \(M\) in \(\text{TrPic}_R(A)\). The author proves that if any automorphism of \(M\) is induced by multiplication by an invertible element of \(Z(A)\), then \(\text{Ext}_A^n(M,M)\) is a module for the group algebra \(RHD_M(A)\). This generalizes the action of the automorphism group of a group \(G\) on the cohomology group \(H^n(G,R)\).