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Cofinite modules and local cohomology. (English) Zbl 0893.13005
Summary: We show that if \(M\) is a finitely generated module over a commutative Noetherian local ring \(R\) and \(I\) is a dimension one ideal of \(R\) (i.e., \(\dim R/I=1)\), then the local cohomology modules \(H^i_I(M)\) are \(I\)-cofinite; that is, \(\text{Ext}^j_R (R/I, H^i_I (M))\) is finitely generated for all \(i,j\). We also show that if \(R\) is a complete local ring and \(P\) is a dimension one prime ideal of \(R\), then the set of \(P\)-cofinite modules form an abelian subcategory of the category of all \(R\)-modules. Finally, we prove that if \(M\) is an \(n\)-dimensional finitely generated module over a Noetherian local ring \(R\) and \(I\) is any ideal of \(R\), then \(H^n_I(M)\) is \(I\)-cofinite.

MSC:
13D45 Local cohomology and commutative rings
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