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On the cofiniteness of local cohomology modules. (English) Zbl 0806.13005
Let \(I\) denote an ideal in a local Noetherian ring \((R,{\mathfrak m})\). In his paper in Invent. Math. 9, 145-164 (1970; Zbl 0196.243), R. Hartshorne defined an \(R\)-module to be \(I\)-cofinite provided \(\text{Supp } N \subseteq V(I)\) and Ext\(_ R^ i(R/I,N)\) is a finitely generated \(R\)-module for all \(i\). While \(H^ i_{\mathfrak m} (M)\) is an Artinian \(R\)-module for \(M\) a finitely generated \(R\)-module it is \({\mathfrak m}\)- cofinite. Disproving a conjecture of A. Grothendieck [“Cohomologie locale des faisceaux cohérents et théorémes de Lefschetz locaux et globaux”, SGA 2 (1962; Zbl 0159.504)], R. Hartshorne has shown that \(H^ i_ I(M)\) is in general not \(I\)-cofinite. In extending another result of R. Hartshorne [loc. cit.] it was shown by C. Huneke and J. Koh [Math. Proc. Camb. Philos. Soc. 110, No. 3, 421-429 (1991; Zbl 0749.13007)] that Ext\(^ i_ R (N,H^ j_ I(M)\) is finitely generated for all \(i,j\) provided
(a) \(M,N\) are finitely generated and Supp \(N \subseteq V(I)\),
(b) \(\dim R/I=1\), and
(c) \(R\) is a local complete Gorenstein domain.
In the present paper the author weakened the assumptions for the finiteness by replacing \(R\) in (c) by a complete local domain satisfying one of the following conditions
(1) \((R,{\mathfrak m})\) is equicharacteristic,
(2) if \(k\) is a coefficient ring of \(R\) with uniformizing parameter \(q\) then \(q \in \text{Rad} I\),
(3) if \(k\) and \(q\) are as in (2) then \(q\) does not belong to the union of the minimal prime ideals of \(I\).
The proof uses a variant of the Cohen structure theorem which enables to construct a Gorenstein ring \(S\) inside \(R\) [see M. Brodmann and C. Huneke, “A quick proof of the Hartshorne-Lichtenbaum vanishing theorem”, in: Algebraic geometry and its applications, Coll. pap. Abhyankar’s 60th birthd., Conf. Purdue Univ. 1990, 305-308 (1994)]. Then the author is able to reduce the more general situation to the case of a Gorenstein domain handled by C. Huneke and J. Koh [loc. cit.].

MSC:
13D45 Local cohomology and commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14B15 Local cohomology and algebraic geometry
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References:
[1] Rotman, An Introduction to Homological Algebra (1979) · Zbl 0441.18018
[2] Milne, Etale Cohomology (1980)
[3] Matsumura, Commutative Algebra (1980)
[4] Grothendieck, Cohomologie locale des faisceaux coh?rents et th?or?m.es de Lefschelz locaux et globaux (1968)
[5] Huneke, Math. Proc. Cambridge Philos. Soc 110 pp 421– (1991)
[6] DOI: 10.1007/BF01404554 · Zbl 0196.24301
[7] DOI: 10.1007/BF01233420 · Zbl 0717.13011
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