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- W2972881844 abstract "Let $R$ be a commutative Noetherian ring with non-zero identity, $n$ a non-negative integer, $mathfrak{a}$ an ideal of $R$ with $dim(R/mathfrak{a}) leq n+1$, and $X$ an arbitrary $R$-module. In this paper, we prove the following results: (i) If $X$ is an $mathfrak{a}$-torsion $R$-module such that $operatorname{Hom}_{R}(R/mathfrak{a},X)$ and $operatorname{Ext}_{R}^{1}(R/mathfrak{a},X)$ are $operatorname{FD}_{lt n}$ $R$-modules, then $X$ is an $(operatorname{FD}_{lt n},mathfrak{a})$-cofinite $R$-module; (ii) The category of $(operatorname{FD}_{lt n},mathfrak{a})$-cofinite $R$-modules is an Abelian category; (iii) $operatorname{H}^{i}_{mathfrak{a}}(X)$ is an $(operatorname{FD}_{lt n},mathfrak{a})$-cofinite $R$-module and ${ mathfrak{p} in operatorname{Ass}_R(operatorname{H}^{i}_{mathfrak{a}}(X)) : dim(R/mathfrak{p}) geq n }$ is a finite set for all $i$ when $operatorname{Ext}^{i}_{R}(R/mathfrak{a},X)$ is an $operatorname{FD}_{lt n}$ $R$-module for all $i$. We observe that, among other things, $operatorname{Ass}_R(operatorname{H}^{i}_{mathfrak{a}}(X))$ is a finite set for all $i$ whenever $R$ is a semi-local ring with $dim(R/mathfrak{a}) leq 2$ and $operatorname{Ext}^{i}_{R}(R/mathfrak{a},X)$ is an $operatorname{FD}_{lt 1}$ $R$-module for all $i$." @default.
- W2972881844 created "2019-09-19" @default.
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- W2972881844 date "2020-08-01" @default.
- W2972881844 modified "2023-09-26" @default.
- W2972881844 title "Cofiniteness with Respect to the Class of Modules in Dimension less than a Fixed Integer" @default.
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- W2972881844 doi "https://doi.org/10.11650/tjm/190902" @default.
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