Shahrood University of TechnologyJournal of Algebraic Systems2345-5128Articles in Press20240409ON THE FINITENESS OF LOCAL HOMOLOGY MODULES309610.22044/jas.2023.12118.1631ENAli FathiDepartment of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran.0000-0003-4739-9576Alireza HajikarimiDepartment of Mathematics, Mobarakeh Branch, Islamic Azad University, Isfahan, Iran.Journal Article20220721Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ be an ideal of $R$. Suppose $M$ is a finitely generated $R$-module and $N$ is an Artinian $R$-module. We define the concept of filter coregular sequence to determine the infimum of integers $i$ such that the generalized local homology $\textrm{H}^{\mathfrak{a}}_i(M, N)$ is not finitely generated as an $\widehat{R}^{\mathfrak{a}}$-module, where $\widehat{R}^{\mathfrak{a}}$ denotes the $\mathfrak{a}$-adic completion of $R$. In particular, if $R$ is a complete semi-local ring, then $\textrm{H}^{\mathfrak{a}}_i(M, N)$ is a finitely generated $\widehat{R}^{\mathfrak{a}}$-module for all non-negative integers $i$ if and only if $(0:_N\mathfrak{a}+\textrm{Ann}(M))$ has finite length.https://jas.shahroodut.ac.ir/article_3096_ebdecf06945b4a8979309c94013e9701.pdf