Basic Morita equivalences and a conjecture of Sasaki for cohomology of block algebras of finite groups. (Q2794440)

The article under review confirms a question due to Sasaki in certain quite specific cases. This conjecture expresses the group cohomology of a block of a group algebra, as defined by Linckelmann, as image of the ordinary group cohomology of the defect group under certain trace maps. It is known that the conjecture is true if the inertia group \(N_G(P_\gamma)\) of the block controls the fusion of Bauer pairs.NEWLINENEWLINE Puig defined a fairly technical property between two blocks of two groups \(G\) and \(H\), called basic Morita equivalence. If this happens, and another technical hypothesis concerning source algebras of the two blocks happens, then in a first result the Sasaki trace map has image in the fixed point space of the action of the normaliser of the defect group on the group cohomology of the defect group. As a consequence, the author shows that nilpotent covered blocks satisfy Sasaki's conjecture. Here a block \(b\) of \(G\) is nilpotent covered if the initial group \(G\) is a normal subgroup of a finite group \(\widetilde G\) and there is a nilpotent block of \(\widetilde G\) covering \(b\). -- In a second result the author gives a compatibility, under certain conditions, of Sasaki transfer maps and transfer maps of Hochschild cohomology of block algebras.