On a category of cofinite modules which is abelian
Let \(I\) be an ideal of a commutative noetherian local ring \((R,\mathfrak{m})\) and \(M\) a finitely generated \(R\)-module. In 1968, \textit{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 \(\Hom_R(R/I,H_{I}^i(M))\) is finitely generated for all \(i\geq 0\). \textit{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.
scientific article; zbMATH DE number 5963464
scientific article