Hochschild cohomology of tame blocks.
In the paper under review the author determines completely the additive structure of the Hochschild cohomology \(HH^*(B)\) of blocks of group rings \(B\) with dihedral, semidihedral or quaternion defect group and one or three simple modules. Using Erdmann's classification of these algebras up to Morita equivalence the author gave in an earlier work a complete classification of these algebras \(B\) up to derived equivalence. Since the Hochschild cohomology is an invariant of the derived category, this can be used. Further, the author discovers in each of the classes a \(B\) which is a principal block of a group ring of a finite group. For one simple module this is the dihedral, semidihedral or quaternion group. For three simple modules, this is the alternating group of degree \(4\), or a \(\text{PSL}_2(q)\) for some choice of \(q\) in the case of a dihedral defect, a \(\text{PSL}_3(q)\) for semidihedral defect, and an \(\text{SL}_2(q)\) for quaternion defect group. Then, the author uses the well known decomposition of the Hochschild cohomology of a group ring into a direct sum group cohomology of centralizers of elements, as graded vector spaces. Finally, the group ring decomposes into various blocks, and the author manages to identify the non principal block contribution in the Hochschild cohomology in each case. Quite sophisticated results are used at this place, such as a result of Bonnafé and Rouquier on Deligne-Lusztig induction.
scientific article; zbMATH DE number 2037112
scientific article