×

zbMATH — the first resource for mathematics

On the characterizations of cofinite complexes over affine curves and hypersurfaces. (English) Zbl 1386.13042
In [Invent. Math. 9, 145–164 (1970; Zbl 0196.24301)] R. Hartshorne asked about four questions. The fourth one was as follows:
Question. Let \(R\) be a regular ring of finite Krull dimension and \(I\) an ideal of \(R\). Suppose that \(R\) is complete with respect to an \(I\)-adic topology. Then does there exist an abelian subcategory \(\mathcal{M}_{cof}\) consisting of \(R\)-modules, such that \(I\)-cofinite complexes \(N^*\) are characterized by the property \(\mathrm{H}^i (N^*) \in \mathcal{M}_{cof}\) for all \(i\)?
Let \(A\) be a homomorphic image of a Gorenstein ring of finite Krull dimension, \(J\) an ideal of \(A\) of dimension one, and \(N^*\) a bounded below complex of \(A\)-modules. suppose that \(A\) is complete with respect to a \(J\)-adic topology. In this paper, it is proved that \(N^*\) is a \(J\)-cofinite complex if and only if \(\mathrm{H}^i(N^*)\) is a \(J\)-cofinite module for all \(i\). The same result is also proved for principal ideals \(J\). Consequently, for the above question, one obtain an answer over the ring, on affine curves and hypersurfaces.
Note: The section 4 of the reviewer’s paper [J. Pure Appl. Algebra 95, No. 1, 103–119 (1994; Zbl 0843.13005)] gives some information about cofinite complexes.
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)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abazari, R.; Bahmanpour, K., Cofiniteness of extension functors of cofinite modules, J. Algebra, 330, 507-516, (2011) · Zbl 1227.13010
[2] Bahmanpour, K.; Naghipour, R., Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra, 321, 1997-2011, (2009) · Zbl 1168.13016
[3] Cartan, H.; Eilenberg, S., Homological algebra, Princeton Mathematical Series, vol. 19, (1956), Princeton University Press New Jersey/London · Zbl 0075.24305
[4] Eto, K.; Kawasaki, K.-i., A characterization of cofinite complexes over complete Gorenstein domains, J. Commut. Algebra, 3, 4 Winter, 537-550, (2011) · Zbl 1252.13010
[5] Gelfand, S. I.; Manin, Yu. I., Methods of homological algebra, (1996), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0855.18001
[6] Hartshorne, R., Affine duality and cofiniteness, Invent. Math., 9, 145-164, (1969-1970) · Zbl 0196.24301
[7] Hartshorne, R., Residue and duality, Springer Lecture Note in Mathematics, vol. 20, (1966), Springer-Verlag New York/Berlin/Heidelberg
[8] Huneke, C.; Koh, J., Cofiniteness and vanishing of local cohomology modules, Math. Proc. Cambridge Philos. Soc., 110, 3, 421-429, (1991) · Zbl 0749.13007
[9] Kawasaki, K.-i., On a category of cofinite modules which is abelian, Math. Z., 269, 1, 587-608, (2011) · Zbl 1228.13020
[10] Kawasaki, K.-i., On the finiteness of bass numbers of local cohomology modules, Proc. Amer. Math. Soc., 124, 3275-3279, (1996) · Zbl 0860.13011
[11] Kawasaki, K.-i., On a characterization of cofinite complexes, addendum to ‘on a category of cofinite modules which is abelian’, Math. Z., 275, 641-646, (2013) · Zbl 1278.13016
[12] Kawasaki, K.-i., Cofiniteness of local cohomology modules for principal ideals, Bull. Lond. Math. Soc., 30, 3, 241-246, (1998) · Zbl 0930.13013
[13] Kawasaki, K.-i., On the category of cofinite modules for principal ideals, Nihonkai Math. J., 22, 2, 67-71, (2011) · Zbl 1247.14003
[14] Kawasaki, T., On arithmetic macaulayfication of Noetherian rings, Trans. Amer. Math. Soc., 354, 1, 123-149, (2002) · Zbl 1087.13502
[15] Lipman, J., Lectures on local cohomology and duality, local cohomology and its applications, Lecture Notes in Pure and Applied Mathematics, vol. 226, 39-89, (2002), Marcel Dekker, Inc. New York/Basel · Zbl 1011.13010
[16] Matsumura, H., Commutative algebra, (1980), Benjamin-Cummings Reading, Massachusetts · Zbl 0211.06501
[17] Matsumura, H., Commutative ring theory, Cambridge Studies in Advance Mathematics, vol. 8, (1986), Cambridge University Press Cambridge
[18] Melkersson, L., Modules cofinite with respect to an ideal, J. Algebra, 285, 649-668, (2005) · Zbl 1093.13012
[19] Melkersson, L., Cofiniteness with respect to ideals of dimension one, J. Algebra, 372, 459-462, (2012) · Zbl 1273.13029
[20] Sharp, R. Y., Necessary conditions for the existence of dualizing complexes in commutative algebra, (Sem. Algebre P. Dubreil 1977/78, Lecture Notes in Mathematics, vol. 740, (1979), Springer-Verlag), 213-229
[21] Spaltenstein, N., Resolution of unbounded complexes, Compos. Math., 65, 2, 121-154, (1988) · Zbl 0636.18006
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.