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

Keywords: Homology; cohomology; finitely generated module.

DOI: 10.57016/TM-NSXY8680

Pages:  112-118     

Volume  XXVII ,  Issue  2 ,  2024