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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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##### References:
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