On the finiteness properties of extension and torsion functors of local cohomology modules
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- by Kazem Khashyarmanesh PDF
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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.References
- J. Asadollahi and P. Schenzel, Some results on associated primes of local cohomology modules, Japan. J. Math. (N.S.) 29 (2003), no. 2, 285–296. MR 2035541, DOI 10.4099/math1924.29.285
- M. P. Brodmann and A. Lashgari Faghani, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc. 128 (2000), no. 10, 2851–2853. MR 1664309, DOI 10.1090/S0002-9939-00-05328-4
- M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics, vol. 60, Cambridge University Press, Cambridge, 1998. MR 1613627, DOI 10.1017/CBO9780511629204
- Donatella Delfino, On the cofiniteness of local cohomology modules, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 1, 79–84. MR 1253283, DOI 10.1017/S0305004100071929
- Donatella Delfino and Thomas Marley, Cofinite modules and local cohomology, J. Pure Appl. Algebra 121 (1997), no. 1, 45–52. MR 1471123, DOI 10.1016/S0022-4049(96)00044-8
- Mohammad T. Dibaei and Siamak Yassemi, Associated primes and cofiniteness of local cohomology modules, Manuscripta Math. 117 (2005), no. 2, 199–205. MR 2150481, DOI 10.1007/s00229-005-0538-5
- Dibaei, M. T.; Yassemi, S., Finiteness of extension functors of local cohomology modules, preprint (arXiv: math. AC/0509340 V1 15 Sep. 2005).
- Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux $(SGA$ $2)$, North-Holland Publishing Co., Amsterdam; Masson & Cie, Editeur, Paris, 1968 (French). Augmenté d’un exposé par Michèle Raynaud; Séminaire de Géométrie Algébrique du Bois-Marie, 1962; Advanced Studies in Pure Mathematics, Vol. 2. MR 0476737
- Robin Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1969/70), 145–164. MR 257096, DOI 10.1007/BF01404554
- Craig Huneke and Jee Koh, Cofiniteness and vanishing of local cohomology modules, Math. Proc. Cambridge Philos. Soc. 110 (1991), no. 3, 421–429. MR 1120477, DOI 10.1017/S0305004100070493
- K. Khashyarmanesh and Sh. Salarian, Filter regular sequences and the finiteness of local cohomology modules, Comm. Algebra 26 (1998), no. 8, 2483–2490. MR 1627876, DOI 10.1080/00927879808826293
- K. Khashyarmanesh and Sh. Salarian, On the associated primes of local cohomology modules, Comm. Algebra 27 (1999), no. 12, 6191–6198. MR 1726302, DOI 10.1080/00927879908826816
- Thomas Marley and Janet C. Vassilev, Cofiniteness and associated primes of local cohomology modules, J. Algebra 256 (2002), no. 1, 180–193. MR 1936885, DOI 10.1016/S0021-8693(02)00151-5
- Leif Melkersson, Properties of cofinite modules and applications to local cohomology, Math. Proc. Cambridge Philos. Soc. 125 (1999), no. 3, 417–423. MR 1656785, DOI 10.1017/S0305004198003041
- Peter Schenzel, Ngô Viêt Trung, and Nguyễn Tụ’ Cu’ò’ng, Verallgemeinerte Cohen-Macaulay-Moduln, Math. Nachr. 85 (1978), 57–73 (German). MR 517641, DOI 10.1002/mana.19780850106
- Jürgen Stückrad and Wolfgang Vogel, Buchsbaum rings and applications, Springer-Verlag, Berlin, 1986. An interaction between algebra, geometry and topology. MR 881220, DOI 10.1007/978-3-662-02500-0
- Wolmer V. Vasconcelos, Divisor theory in module categories, North-Holland Mathematics Studies, No. 14, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1974. MR 0498530
- Siamak Yassemi, Coassociated primes, Comm. Algebra 23 (1995), no. 4, 1473–1498. MR 1317409, DOI 10.1080/00927879508825288
- Ken-Ichi Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J. 147 (1997), 179–191. MR 1475172, DOI 10.1017/S0027763000006371
Additional Information
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2276640