Geodesic
Finite generativity of homology and cohomology modules
The Teaching of Mathematics, XXVII (2024) no. 2, p. 112 .

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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 $M$.
DOI : 10.57016/TM-NSXY8680
Classification : 97H99, H75
Keywords: homology, cohomology, finitely generated module. 
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Milica Jovanović; Petar Stojčić. Finite generativity of homology and cohomology modules. The Teaching of Mathematics, XXVII (2024) no. 2, p. 112 . doi : 10.57016/TM-NSXY8680. http://geodesic.mathdoc.fr/articles/10.57016/TM-NSXY8680/

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