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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
##### Keywords:
Cofinite modules; Local Cohomology module
Zbl 0196.24301
Full Text:
##### References:
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