Siegel, Stephen F.; Witherspoon, Sarah J. The Hochschild cohomology ring of a group algebra. (English) Zbl 1044.16005 Proc. Lond. Math. Soc., III. Ser. 79, No. 1, 131-157 (1999). 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. Reviewer: Eduardo Marcos (São Paulo) Cited in 5 ReviewsCited in 51 Documents 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) Keywords:Hochschild cohomology rings; group rings PDFBibTeX XMLCite \textit{S. F. Siegel} and \textit{S. J. Witherspoon}, Proc. Lond. Math. Soc. (3) 79, No. 1, 131--157 (1999; Zbl 1044.16005) Full Text: DOI