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 Hi(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 Hi(X; R) is finitely generated for all i, we also discuss computing Hi(X; M ) and Hi(X; M ) for a finitely generated R-module M .