Cup products on the complete relative cohomologies of finite groups and group algebras (Q1962714)

A cup product on the complete relative cohomology \(H^*(\mathbb{Z} G,\mathbb{Z} K;A)\) of the group algebra \(\mathbb{Z} G\) of a finite group \(G\) has been defined by the first author [Tsukuba J. Math. 17, No. 1, 99-113 (1993; Zbl 0793.13005)]. The authors define a cup product on the complete relative cohomology \(H^*(G,K;M)\) of a finite group \(G\) and show an isomorphism between these cohomology groups preserving cup products. In particular, it follows that the ring \(H^*(G,K;\mathbb{Z})\) is a direct summand of the ring \(H^*(\mathbb{Z} G,\mathbb{Z} K;\mathbb{Z} G)\).