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.


13D45 Local cohomology and commutative rings
14B15 Local cohomology and algebraic geometry
Full Text: DOI


[1] DOI: 10.1007/BF01404554 · Zbl 0196.24301
[2] Grothendieck, Cohomologie Locale des Faisceaux et Theoremes de Lefshetz Locaux et Globaux (1969)
[3] Grothendieck, Local Cohomology, notes by R. Hart shorne (1966)
[4] Brodmann, Einige Ergebnisse aus der Lokalen Kohomologietheorie und Ihre Anwendung (1983)
[5] Serre, Algebra Locale; Multiplicities 11 (1965)
[6] DOI: 10.2307/1970720 · Zbl 0169.23302
[7] DOI: 10.2307/1970785 · Zbl 0308.14003
[8] Hartshorne, Inst. Hautes ?tudes Sci. Publ. Math. 42 pp 323– (1973)
[9] Matlis, Pacific J. Math. 8 pp 511– (1958) · Zbl 0084.26601
[10] DOI: 10.1007/BF01233420 · Zbl 0717.13011
[11] DOI: 10.2307/1971025 · Zbl 0362.14002
[12] Rotman, An Introduction to Homological Algebra (1979) · Zbl 0441.18018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.