On the cofiniteness of generalized local cohomology modules. (English) Zbl 1316.13024

Cuong, Nguyen Tu; Goto, Shiro; van Hoang, Nguyen

Kyoto J. Math. 55, No. 1, 169-185 (2015).

Let \(R\) be a commutative Noetherian ring, let \(I\) be an ideal of \(R\), and let \(M\), \(N\) be two finitely generated \(R\)-modules. In the paper under review, the authors show that if \(I\) is a principal ideal, then the generalized local cohomology module \(H^j_I(M,N)\) is \(I\)-cofinite for all \(M\), \(N\) and all \(j<t\). In addition, they show that for a non-negative integer \(t\) if \(\dim(M)\leq 2\) or \(\dim(N)\leq 2\), then \(H^j_I(M,N)\) is \(I\)-cofinite for all \(j\).

Reviewer: Siamak Yassemi (Tehran)

MSC:

13D45	Local cohomology and commutative rings
13E99	Chain conditions, finiteness conditions in commutative ring theory
18G60	Other (co)homology theories (MSC2010)

Keywords:

generalized local cohomology; \(I\)-cofiniteness

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