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Cofiniteness with respect to ideals of dimension one. (English) Zbl 1273.13029
Let \(A\) denote a commutative Noetherian ring and let \(\mathfrak{a}\) denote an ideal of \(A\). An \(A\)-module \(M\) is called \(\mathfrak{a}\)-cofinite, if \(M\) is an \(\mathfrak{a}\)-torsion module and the modules \(\text{Ext}^i_A(A/\mathfrak{a}, M)\) are finitely generated for all \(i \in \mathbb{Z}\). The main result of the paper is the following: If \(\mathfrak{a}\) is an ideal with \(\dim A/\mathfrak{a} = 1\), then \(M\) is \(\mathfrak{a}\)-cofinite if and only if \(M\) is an \(\mathfrak{a}\)-torsion module and the modules \(\text{Ext}^i_A(A/\mathfrak{a}, M)\) are finitely generated for all \(i \in \{0,1\}\). As an application it follows that for \(\dim A/\mathfrak{a} = 1\) the category of \(\mathfrak{a}\)-cofinite modules is a full subcategory of the category of \(A\)-modules. Moreover, if \(\dim A/\mathfrak{a} = 1\), then all the local cohomology modules \(H^i_{\mathfrak{a}}(N)\) are \(\mathfrak{a}\)-cofinite for all \(i \in \mathbb{Z}\) and all finitely generated \(A\)-modules \(N\). The author’s clever arguments extends, simplify and generalizes several results on the subject starting with R. Hartshorne’s investigations in [Invent. Math. 9, 145–164 (1970; Zbl 0196.24301)].

MSC:
13D45 Local cohomology and commutative rings
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