On the complete relative homology and cohomology of Frobenius extensions

Abstract

Let G be a finite group, K a subgroup of G and M a left G-module. Then for reZ the complete relative homology group Hr(G, K, M) and cohomology group Hr(G, K, M) are defined in [6]. Let 1 be the unit element of G. For the case of K={1 Hr(G, K, M)^H~r'\G, K, M) holds. But itis not true that for any G, K, M and r there exists an isomorphism from Hr(G, K, M) into H~r-\G, K, M). In fact, in [6, p. 262] there are G, K and M such that Hr(G, K, M)=Z/2Z and Hr{G, K, Af)=0 for all reZ. And if we set M=Q/Z in [6, p. 262], Hr(G, K, M)=0 and Hr(G, K, M)^Z/2Z hold for all reZ. Let A be an algebra over a commutative ring K and F a subalgebra such that the ring extension A/F is a Frobenius extension. In section 1 we shall introduce the complete relative cohomology group Hr(A, F, ―) and homology group Hr(A, F, ―) for r£Z. When the ring extension F/K is also a Frobenius extension, We can define a if-homomorphism W%r'. Hr{A, F, (―)A)―> H~r~\A,F, ―) for reZ, where A is the Nakayama automorphism. The main purpose of this paper is to show necessary and sufficientconditions on which WrA/r is an isomorphism. Theorems 6.3, 7.1 and 7.2 provide the necessary and sufficientconditions. In section 8 we apply our results to extensions defined by a finitegroup G and a subgroup K. In generalization of the well-known duality for Tate cohomology we show that Hr{G, K, ―)=H~r~\G,K, ―) if and only if K is a Hall subgroup of G.