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Cofiniteness and vanishing of local cohomology modules. (English) Zbl 0749.13007
For an ideal $$I$$ of a local Noetherian ring $$(R,m)$$ let $$H^i_I(M)$$ denote the $$i$$-th local cohomology module of $$M$$, a finitely generated $$R$$-module. By virtue of the situation $$I=m$$, A. Grothendieck [see Sémin. Géométrie Algébrique, SGA 2 (1962; Zbl 0159.50402), Exposé 13] asked whether $$\operatorname{Hom}_R(R/I,H^i_I(M))$$ is a finitely generated $$R$$-module. R. Hartshorne [Invent. Math. 9, 145–164 (1970; Zbl 0196.24301)] has shown that this is not true in general for $$R$$ a hypersurface ring. In particular, $$H^i_I(R)$$ is not $$I$$-cofinite, where an $$R$$-module $$N$$ is $$I$$-cofinite provided $$\text{Supp}_R(N)\subseteq V(I)$$ and $$\text{Ext}_R(R/I,N)$$ is finitely generated for all $$i$$. Extending these results in the case of $$R$$ a regular local ring the authors prove – among others – the following vanishing results:
1. If $$\operatorname{Hom}_R(R/I,H^i_I(R))$$ is a finitely generated $$R$$-module for all $$i>r$$ for some $$r\geq\text{bight}(I)$$, then $$H^i_I(R)=0$$ for all $$i>r$$.
2. If $$\text{char}(R)=p>0$$, and $$\operatorname{Hom}_R(R/I,H^i_I(R))$$ is finitely generated for any $$i>\text{bight}(I)$$, then $$H^i_I(R)=0$$.
Here $$\text{bight}(I)$$ denotes the maximum of the heights of minimal prime ideals of $$I$$. Furthermore, for a complete local Gorenstein domain $$(R,m)$$, $$\dim R/I=1$$, and $$M$$ a finitely generated $$R$$-module it turns out that $$H^i_I(M)$$ is $$I$$-cofinite for all $$i$$. This extends one of R. Hartshorne’s results, see the paper cited above.

##### MSC:
 13D45 Local cohomology and commutative rings 14B15 Local cohomology and algebraic geometry
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##### References:
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