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Cofiniteness of generalized local cohomology modules. (English) Zbl 1072.13011

Let \(R\) be a commutative Noetherian ring. The notion of local cohomology has been extended by J. Herzog. He considered functors \(H^i_I(M,M')=\varinjlim_k{\text{Ext}}^i_R(M/I^k\cdot M,M')\) for \(R\)-modules \(M,M'\) and an \(R\)-ideal \(I\). In the present paper the following is shown.
Let \(M\) have finite projective dimension over \(R\), and assume that both \(M\) and \(M'\) are finitely generated. If either \(I\) is principal, or if \(R\) is complete local and \(I\) is a prime ideal with \(\dim(R/I)=1\), then \(H^i_I(M,M')\) is \(I\)-cofinite. This gives an affirmative answer to a question proposed by the present reviewer [S. Yassemi, Commun. Algebra 29, No. 6, 2333–2340 (2001; Zbl 1023.13013)].

MSC:

13D45 Local cohomology and commutative rings

Citations:

Zbl 1023.13013
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