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A characterization of cofinite complexes over complete Gorenstein domains. (English) Zbl 1252.13010
Let $$R$$ be a Noetherian ring and $$J$$ and ideal of $$R$$. An $$R$$-module is said to be $$J$$-cofinite if $$\text{Supp} N \subset V(J)$$ and $$\text{Ext}^i(R/J, N)$$ is finitely generated for each $$j \geq 0$$. On the other hand, the definition of $$J$$-cofiniteness of complexes is quite different. That is, a complex $$N^\bullet$$ of $$R$$-modules is said to be $$J$$-cofinite if there is a complex $$M^\bullet$$ with finitely generated cohomologies such that $N^\bullet \cong \text{RHom}(M^\bullet, {R \Gamma_J}(D^\bullet)),$ where $$D^\bullet$$ is a dualizing complex of $$R$$.
In [Math. Z. 269, No. 1-2, 587–608 (2011; Zbl 1228.13020)], the second author proved that if $$R$$ is a regular local ring, it is $$J$$-adic compete and $$\dim R/J = 1$$, then a bounded-below complex $$N^\bullet$$ is $$J$$-cofinite if and only if all the cohomology modules of $$N^\bullet$$ are $$J$$-cofinite. It is a partial answer to R. Hartshorne’s question [Invent. Math. 9, 145–164 (1970; Zbl 0196.24301)]. In the present paper, the authors replace the assumption “$$R$$ is a regular local ring and $$J$$-adic complete” with “$$R$$ is a Gorenstein local domain.”

##### MSC:
 13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) 14B15 Local cohomology and algebraic geometry 18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
##### Keywords:
local cohomology; cofinite modules; abelian category
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##### References:
 [1] S.I. Gelfant and Yu.I. Manin, Methods of Homological Algebra , Springer-Verlag, Berlin, 1996. [2] A. Grothendieck, Cohomologie locale des faisceaux cohérants et théorèmes de Lefschetz locaux et globaux (SGA 2), North-Holland, Amsterdam, 1968. [3] A. Grothendieck, Local cohomology , with notes by R. Hartshorne, Springer Lecture Notes Math. 41 , Springer-Verlag, Berlin, 1967. · Zbl 0185.49202 · doi:10.1007/BFb0073971 [4] R. Hartshorne, Affine duality and cofiniteness , Invent. Math. 9 (1969/1970), 145-164. · Zbl 0196.24301 · doi:10.1007/BF01404554 · eudml:142002 [5] —, Residue and duality , Springer Lecture Notes Math. 20 , Springer-Verlag, New York, 1966. [6] —, Algebraic geometry , Grad. Texts Math. 52 , Springer-Verlag, New York, 1977. [7] C. Huneke and J. Koh, Cofiniteness and vanishing of local cohomology modules , Math. Proc. Cambridge Philos. Soc. 110 , (1991), 421-429. · Zbl 0749.13007 · doi:10.1017/S0305004100070493 [8] K.-I. Kawasaki, On the finiteness of Bass numbers of local cohomology modules , Proc. Amer. Math. Soc. 124 (1996), 3275-3279. · Zbl 0860.13011 · doi:10.1090/S0002-9939-96-03399-0 [9] —, On a category of cofinite modules which is Abelian , Math. Z., · Zbl 1228.13020 · doi:10.1007/s00209-010-0751-0 [10] J. Lipman, Lectures on local cohomology and duality , in Local cohomology and its applications , Lect. Notes Pure Appl. Math. 226 , Marcel Dekker, Inc., New York, 2002%, 39-89. · Zbl 1011.13010 [11] H. Matsumura, Commutative algebra , 2nd ed., Benjamin Cummings, Reading, Massachusetts, 1980. · Zbl 0441.13001 [12] —, Commutative ring theory , Cambridge Stud. Adv. Math. 8 , Cambridge University Press, Cambridge, 1986. [13] L. Melkersson, Properties of cofinite modules and applications to local cohomology , Math. Proc. Cambridge Philos. Soc. 125 , (1999), 417-423. · Zbl 0921.13009 · doi:10.1017/S0305004198003041
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