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The Hochschild cohomology ring of a group algebra. (English) Zbl 1044.16005

The paper studies the Hochschild cohomology ring of a group ring. In order to do this the paper describes a natural way of describing the cup product in general using the fact that any two projective resolutions are quasi-isomorphic. The authors are interested in describing this with more detail in the group ring case. For this they use a decomposition of the Hochschild cohomology as the direct sum of the cohomology rings of the centralizers of representatives of the conjugacy classes of \(G\). They describe the cup product in terms of this decomposition. As applications, they determine presentations for the Hochschild cohomology rings of (1) the mod-\(3\) group algebra of the symmetric group \(S_3\), (2) the mod-\(2\) group algebra of the alternating group \(A_4\), and (3) the mod-\(2\) group algebras of the dihedral \(2\)-groups.

MSC:

16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16S34 Group rings
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
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