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On a category of cofinite modules which is abelian. (English) Zbl 1228.13020
Math. Z. 269, No. 1-2, 587-608 (2011); addendum ibid. 275, No. 1-2, 641-646 (2013).
Let \(I\) be an ideal of a commutative noetherian local ring \((R,\mathfrak{m})\) and \(M\) a finitely generated \(R\)-module. In 1968, A. Grothendieck [Séminaire de géométrie algébrique: Cohomologie locale des faisceaux cohérents et théoremes de Lefschetz locaux et globaux (1962; Zbl 0159.50402)] has conjectured that \(\operatorname{Hom}_R(R/I,H_{I}^i(M))\) is finitely generated for all \(i\geq 0\). R. Hartshorne gave a counter-example to this conjecture [Invent. Math. 9, 145–164 (1970; Zbl 0196.24301)]. In the same paper, Hartshorne introduced the notion of \(I\)-cofinite modules and asked the following question: For which local rings \(R\) and ideals \(I\) is \(H_{I}^i(N)\) \(I\)-cofinite for all \(i\geq 0\) and any finitely generated \(R\)-module \(N\)? An \(R\)-module \(X\) is said to be \(I\)-cofinite if \(\mathrm{Supp}_RX\subseteq V(I)\) and \(\mathrm{Ext}^i_R(R/I,X)\) is finitely generated for all \(i\geq 0\). Now, it is known that if either \(I\) is principal or \(\dim R/I=1\), then \(H_{I}^i(N)\) is \(I\)-cofinite for all \(i\geq 0\) and any finitely generated \(R\)-module \(N\).
Let \(\mathcal{M}(R,I)_{cof}\) denote the full subcategory of \(I\)-cofinite \(R\)-modules. When \(I\) is principal, Hartshorne showed that \(\mathcal{M}(R,I)_{cof}\) is abelian. He asked whether \(\mathcal{M}(R,I)_{cof}\) is abelian when \(\dim R/I=1\). In the paper under review, the author provides an affirmative answer to this questions.

13D45 Local cohomology and commutative rings
13D09 Derived categories and commutative rings
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