FINITE GENERATIVITY OF HOMOLOGY AND COHOMOLOGY MODULES

Abstract

In this paper, we consider the following question: if all homology groups of a space X are finitely generated, and if R is a commutative ring with identity, is it true that the homology and cohomology R-modules H<inf>i</inf> (X; R) and Hⁱ (X; R) are also finitely generated? We show that the answer to this question is negative in general, but affirmative if R is an integral domain. In the case when R is a principal ideal domain, and H<inf>i</inf> (X; R) is finitely generated for all i, we also discuss computing H<inf>i</inf> (X; M) and Hⁱ (X; M) for a finitely generated R-module M.