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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
##### Citations:
Zbl 0196.243; Zbl 0159.504; Zbl 0749.13007
Full Text:
##### 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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