On a cohomology theory for pairs of groups

Abstract

of crossed homomorphisms from G to A vanishing on H. Clearly this is a left-exact functor in the category GMJF of left G-modules. The nth rightderived functor of X(G, H, -) in Ga9Z isdenoted byHn+'(G, H, -). The group Hn(G, H, A), A C GM, is called the nth cohomology group of the pair (G, H) with coefficients in A. These groups were first described and studied by M. Auslander in [1], who also found the sequence of Proposition 1.2. In this note we prove an excision property for the functors Hn(G, H, -), Theorem 2.2, and we find a direct sum decomposition of them under suitable conditions, Propositions 2.3 and 2.5. From this one deduces by standard methods a Mayer-Vietoris type sequence for the cohomology of groups, Proposition 2.6. The results in this paper are part of the author's doctoral dissertation at the University of Rochester. The author wishes to thank C. E. Watts for his advise and encouragement.