ON THE FINITENESS OF LOCAL HOMOLOGY MODULES | ||
| Journal of Algebraic Systems | ||
| دوره 13، شماره 1، تیر 2025، صفحه 37-52 اصل مقاله (155.22 K) | ||
| نوع مقاله: Original Manuscript | ||
| شناسه دیجیتال (DOI): 10.22044/jas.2023.12118.1631 | ||
| نویسندگان | ||
| Ali Fathi* 1؛ Alireza Hajikarimi2 | ||
| 1Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran. | ||
| 2Department of Mathematics, Mobarakeh Branch, Islamic Azad University, Isfahan, Iran. | ||
| چکیده | ||
| Let $R$ 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. | ||
| کلیدواژهها | ||
| Local homology؛ Tor functor؛ finiteness؛ filter coregular sequence | ||
| مراجع | ||
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