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On the finiteness properties of extension and torsion functors of local cohomology modules


Author: Kazem Khashyarmanesh
Journal: Proc. Amer. Math. Soc. 135 (2007), 1319-1327
MSC (2000): Primary 13D45, 13D07
DOI: https://doi.org/10.1090/S0002-9939-06-08664-3
Published electronically: December 27, 2006
MathSciNet review: 2276640
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ R$ be a commutative Noetherian ring with non-zero identity, $ \mathfrak{a}$ and $ \mathfrak{b}$ ideals of $ R$ with $ \mathfrak{a} \subseteq \mathfrak{b}$, and $ M$ a finitely generated $ R$-module. In this paper, for fixed integers $ j$ and $ n$, we study the finiteness of $ \operatorname{Ext}^j_R(R/\mathfrak{b},H^n_{\mathfrak{a}}(M))$ and $ Tor_j^R(R/\mathfrak{b},H^n_{\mathfrak{a}}(M))$ in several cases.


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  • [1] ASADOLLAHI, J.; SCHENZEL, P., Some results on associated primes of local cohomology modules, Japan. J. Math. (N.S.) 29 (2003), no. 2, 285-296.MR 2035541 (2004m:13043)
  • [2] BRODMANN, M. P.; LASHGARI F. A., A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc. 128 (2000), no. 10, 2851-2853. MR 1664309 (2000m:13028)
  • [3] BRODMANN, M.; SHARP, R.Y., `Local cohomology - an algebraic introduction with geometric applications', Cambridge Studies in Advanced Mathematics No. 60, Cambridge University Press (1998). MR 1613627 (99h:13020)
  • [4] DELFINO, D., On the cofiniteness of local cohomology modules, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 1, 79-84.MR 1253283 (94m:13023)
  • [5] DELFINO, D; MARLEY, T., Cofinite modules and local cohomology, J. Pure Appl. Algebra 121 (1997), no. 1, 45-52. MR 1471123 (98g:13015)
  • [6] DIBAEI, M. T.; YASSEMI, S., Associated primes and cofiniteness of local cohomology modules, Manuscripta Math. 117 (2005), no. 2, 199-205.MR 2150481
  • [7] DIBAEI, M. T.; YASSEMI, S., Finiteness of extension functors of local cohomology modules, preprint (arXiv: math. AC/0509340 V1 15 Sep. 2005).
  • [8] GROTHENDIECK, A., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux $ (SGA$ $ 2)$, North-Holland Publishing Co., Amsterdam (1968). MR 0476737 (57:16294)
  • [9] HARTSHORNE, R., Affine duality and cofiniteness, Invent. Math. 9 (1969/1970), 145-164. MR 0257096 (41:1750)
  • [10] HUNEKE, C; KOH, J., Cofiniteness and vanishing of local cohomology modules, Math. Proc. Cambridge Philos. Soc. 110 (1991), no. 3, 421-429. MR 1120477 (92g:13021)
  • [11] KHASHYARMANESH, K.; SALARIAN, SH., Filter regular sequences and the finiteness of local cohomology modules, Comm. in Algebra 26(8) (1998), 2483-2490. MR 1627876 (99h:13021)
  • [12] KHASHYARMANESH, K.; SALARIAN, SH., On the associated primes of local cohomology modules, Comm. in Algebra 27 (1999), no. 12, 6191-6198. MR 1726302 (2000m:13029)
  • [13] MARLEY, T.; VASSILEV, J. C., Cofiniteness and associated primes of local cohomology modules, J. Algebra 256 (2002), no. 1, 180-193.MR 1936885 (2003j:13025)
  • [14] MELKERSSON, L., Properties of cofinite modules and applications to local cohomology, Math. Proc. Cambridge Philos. Soc. 125 (1999), no. 3, 417-423. MR 1656785 (99k:13024)
  • [15] SCHENZEL, P.; TRUNG, N. V.; CUONG, N. T., Verallgemeinerte Cohen-Macaulay-Moduln, Math. Nachr. 85 (1978), 57-73. MR 0517641 (80i:13008)
  • [16] ST¨UCKRAD, J.; VOGEL, W., `Buchsbaum rings and applications', VEB Deutscher Verlag der Wissenschaftan, Berlin (1986).MR 0881220 (88h:13011a)
  • [17] VASCONCELOS, W. V., `Divisor theory in module categories', North-Holland Mathematics Studies, No. 14, North-Holland, Amsterdam (1974). MR 0498530 (58:16637)
  • [18] YASSEMI, S., Coassociated primes, Comm. in Algebra 23 (1995), no. 4, 1473-1498.MR 1317409 (96e:13003)
  • [19] YOSHIDA, K.-I., Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J. 147 (1997), 179-191. MR 1475172 (98j:13014)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-06-08664-3
Keywords: Local cohomology modules, extension functor, torsion functor, cofinite modules, associated primes, cohomological dimension, coassociated primes, filter regular sequences.
Received by editor(s): November 2, 2005
Received by editor(s) in revised form: February 2, 2006
Published electronically: December 27, 2006
Additional Notes: The author was partially supported by a grant from the Institute for Studies in Theoretical Physics and Mathematics (IPM) Iran (No. 84130025).
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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