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Associated primes of local cohomology modules

Authors: Kamran Divaani-Aazar and Amir Mafi
Journal: Proc. Amer. Math. Soc. 133 (2005), 655-660
MSC (2000): Primary 13D45, 13E99
Published electronically: October 7, 2004
MathSciNet review: 2113911
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Abstract: Let $\mathfrak{a}$ be an ideal of a commutative Noetherian ring $R$ and $M$ a finitely generated $R$-module. Let $t$ be a natural integer. It is shown that there is a finite subset $X$ of $\operatorname{Spec}R$, such that $\operatorname{Ass}_R(H_{\mathfrak{a}}^t(M))$ is contained in $X$ union with the union of the sets $\operatorname{Ass}_R(\operatorname{Ext} _R^j(R/\mathfrak{a},H_{\mathfrak{a}}^i(M)))$, where $0\leq i<t$ and $0\leq j\leq t^2+1$. As an immediate consequence, we deduce that the first non- $\mathfrak{a}$-cofinite local cohomology module of $M$ with respect to $\mathfrak{a}$ has only finitely many associated prime ideals.

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  • 1. R. Belshoff, E. Enochs and J. R. Garcia Rozas, Generalized Matlis duality, Proc. Amer. Math. Soc., 128(5) (2000), 1307-1312. MR 1641645 (2000j:13015)
  • 2. M. P. Brodmann and F. A. Lashgari, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc., 128(10) (2000), 2851-2853. MR 1664309 (2000m:13028)
  • 3. M. P. Brodmann and R. Y. Sharp: Local cohomology-An algebraic introduction with geometric applications, Cambr. Univ. Press, 1998.MR 1613627 (99h:13020)
  • 4. E. Enochs, Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc., 92(2) (1984), 179-184. MR 0754698 (85j:13016)
  • 5. C. Faith and D. Herbera, Endomorphim rings and tensor products of linearly compact modules, Comm. Algebra, 25(4) (1997),1215-1255. MR 1437670 (98b:16026)
  • 6. R. Hartshorne, Affine duality and cofiniteness, Invent. Math., 9 (1970), 145-164. MR 0257096 (41:1750)
  • 7. C. Huneke, Problems on local cohomology, Free resolutions in commutative algebra and algebraic geometry, Res. Notes Math., 2 (1992), 93-108.MR 1165320 (93f:13010)
  • 8. K. I. Kawasaki, Cofiniteness of local cohomology modules for principle ideals, Bull. London Math. Soc., 30 (1998), 241-246. MR 1608094 (98m:13025)
  • 9. K. Khashyarmanesh and Sh. Salarian, On the associated primes of local cohomology modules, Comm. Algebra, 27(12) (1999), 6191-6198. MR 1726302 (2000m:13029)
  • 10. G. Lyubeznik, A partial survey of local cohomology, Local cohomology and its applications, Lecture Notes in Pure and Appl. Math., 226 (2002), 121-154.MR 1888197 (2003b:14006)
  • 11. J. Rotman, Introduction to homological algebra, Academic Press, 1979.MR 0538169 (80k:18001)
  • 12. A. K. Singh, $p$-torsion elements in local cohomology modules, Math. Res. Lett., 7(2-3) (2000), 165-176. MR 1764314 (2001g:13039)
  • 13. W. Xue, Rings with Morita duality, Lecture Notes in Mathematics, 1523, Springer-Verlag, Berlin, 1992. MR 1184837 (94b:16002)
  • 14. K. I. Yoshida, 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

Kamran Divaani-Aazar
Affiliation: Department of Mathematics, Az-Zahra University, Vanak, Post Code 19834, Tehran, Iran — and — Institute for Studies in Theoretical Physics and Mathematics, P. O. Box 19395-5746, Tehran, Iran

Amir Mafi
Affiliation: Institute of Mathematics, University for Teacher Education, 599 Taleghani Avenue, Tehran 15614, Iran

Keywords: Local cohomology, associated prime ideals, cofiniteness, weakly Laskerian modules, spectral sequences
Received by editor(s): October 16, 2003
Published electronically: October 7, 2004
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2004 American Mathematical Society

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