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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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