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Cofiniteness and associated primes of local cohomology modules. (English) Zbl 1042.13010
Summary: Let \(R\) be a \(d\)-dimensional regular local ring, \(I\) an ideal of \(R\), and \(M\) a finitely generated \(R\)-module of dimension \(n\). We prove that the set of associated primes of \(\text{Ext}_R^i(R/I, H^j_I(M))\) is finite for all \(i\) and \(j\) in the following cases:
(a) \(\dim M\leq 3\);
(b) \(\dim R\leq 4\);
(c) \(\dim M/IM\leq 2\) and \(M\) satisfies Serre’s condition \(S_{n-3}\);
(d) \(\dim M/IM\leq 3\), \(\text{ann}_RM=0\), \(R\) is unramified, and \(M\) satisfies \(S_{n-3}\).
In these cases we also prove that \(H^i_I(M)_p\) is \(I_p\)-cofinite for all but finitely many primes \(p\) of \(R\).
Additionally, we show that if \(\dim R/I\geq 2\) and \(\text{Spec}\,R/I-\{m/I\}\) is disconnected then \(H_I^{d-1}(R)\) is not \(I\)-cofinite, generalizing a result due to C. Huneke and I. Koh [Math. Proc. Camb. Philos. Soc. 110, No. 3, 421–429 (1991; Zbl 0749.13007)].

MSC:
13D45 Local cohomology and commutative rings
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
13H05 Regular local rings
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