On the complete relative homology and cohomology of Frobenius extensions (Q1904460)

Let \(\Lambda\) be an algebra over a commutative ring \(k\) and \(\Gamma\) a subalgebra such that the ring extension \(\Lambda/\Gamma\) is a Frobenius extension. The author introduces the complete relative homology \(H_r(\Lambda,\Gamma,-)\) and cohomology \(H^r(\Lambda,\Gamma,-)\) groups for all integers \(r\) [cf. \textit{G. Hochschild}, Trans. Am. Math. Soc. 82, 246-269 (1956; Zbl 0070.269)]\ and defines a map \(\Psi^r_{\Lambda/\Gamma}:H_r(\Lambda,\Gamma,(-)^\Delta)\to H^{-r-1}(\Lambda,\Gamma,-)\), where \(\Delta\) is the Nakayama automorphism. Then necessary and sufficient conditions under which \(\Psi^r_{\Lambda/\Gamma}\) is an isomorphism are presented. In particular, if extensions are defined by a finite group \(G\) and a subgroup \(K\) a generalization of the well-known duality for the Tate cohomology is shown.