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On the category of weakly Laskerian cofinite modules. (English) Zbl 1306.13010
Let \(R\) be a Noetherian commutative ring with identity and \(I\) an ideal of \(R\). Recall that an \(R\)-module \(M\) is said to be \(I\)-cofinite if \(\text{Supp}_RM\subseteq V(I)\) and \(\text{Ext}_R^i(R/I,M)\) is finitely generated for all \(i\geq 0\). Also, \(M\) is said to be weakly Laskerian if any quotient module of \(M\) has finitely many associated prime ideals.
The main achievement of this paper is to show that the category of \(I\)-cofinite weakly Laskerian \(R\)-modules forms an abelian subcategory of the category of all \(R\)-modules. Also, the author provides a nice characterization for weakly Laskerian modules. He proves that an \(R\)-module \(M\) is weakly Laskerian if and only if it possesses a submodule \(N\) such that \(\text{Supp}_RM/N\) is finite.

MSC:
13D45 Local cohomology and commutative rings
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