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Cohomological dimension, cofiniteness and abelian categories of cofinite modules. (English) Zbl 1451.13060
Let \(R\) be a commutative Noetherian ring, \(I\) be an ideal of \(R\), and \(M\) be an \(R\)-module. Recall that, the notion of the cohomological dimension of \(M\) relative to \(I\), denoted by \(\mathrm{cd}(I,M)\), is defined to be the greatest integer \(i\) such that \(H_I^i(M)\neq 0\) if there exist such \(i\)’s and \(-infty\) otherwise. Also, \(q(I,M)\) denotes the greatest integer \(i\) such that \(H_I^i(M)\) is not Artian if there exist such \(i\)’s and \(-\infty\) otherwise. In the paper under review, it is proved that, if \(J\) is an ideal of \(R\) with \(I\subseteq J\), and \(M\) is finitely generated, then \(q(J,M)\leq q(I,M)+\mathrm{cd}(J,M/IM)\).
Recall that an \(R\)-module \(M\) is called \(I\)-cofinite, if \(\mathrm{Supp}_RM\subseteq\mathrm{V}(I)\) and \(\mathrm{Ext}^i_R(R/I,M)\) is a finitely generated module for all \(i\). In the paper under review, the author defined \(\tilde{q}(I,M)\) as the greatest integer \(i\) such that \(H_I^i(M)\) is not Artian \(I\)-cofinite if there exist such \(i\)’s and \(-infty\) otherwise. It is shown that if \(M\) is finitely generated and \(q(I,M)\leq 1\), then \(\tilde{q}(I,M)=q(I,M)\), and \(H_I^i(M)\) is \(I\)-cofinite for any \(i\geq 0\). As a consequence, it is proved that, if \(M\) is finitely generated and \(q(I,R)\leq 1\), then \(H_I^i(M)\) is \(I\)-cofinite for any \(i\geq 0\).
Let \(R,\mathfrak{m}\) be a complete Noetherian local ring with \(q(I,R)\leq 1\). It is shown that the category of all \(I\)-cofinite \(R\)-modules is an abelian subcategory of the category of all \(R\)-modules; that is, if \(f:M\rightarrow N\) is an \(R\)-homomorphism of \(I\)-cofinite modules, then \(\ker f\) and \(\mathrm{coker }f\) are \(I\)-cofinite \(R\)-modules.

MSC:
13D45 Local cohomology and commutative rings
14B15 Local cohomology and algebraic geometry
13E05 Commutative Noetherian rings and modules
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