Top local cohomology modules with respect to a pair of ideals

Author:
Lizhong Chu

Journal:
Proc. Amer. Math. Soc. **139** (2011), 777-782

MSC (2010):
Primary 13D45

Published electronically:
October 28, 2010

MathSciNet review:
2745630

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a commutative Noetherian local ring, and let and be two proper ideals of . Let be a non-zero finitely generated module. We investigate the top local cohomology module . We get some results about attached prime ideals of the local cohomology module . As a consequence, we find that there exists a quotient of such that . Also, we give the generalized version of the Lichtenbaum-Hartshorne Vanishing Theorem for local cohomology modules of a finitely generated module with respect to a pair of ideals.

**1.**Nicolas Bourbaki,*Elements of mathematics. Commutative algebra*, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972. Translated from the French. MR**0360549****2.**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****3.**Lizhong Chu and Wing Wang,*Some results on local cohomology modules defined by a pair of ideals*, J. Math. Kyoto Univ.**49**(2009), no. 1, 193–200. MR**2531134****4.**Mohammad T. Dibaei and Siamak Yassemi,*Attached primes of the top local cohomology modules with respect to an ideal*, Arch. Math. (Basel)**84**(2005), no. 4, 292–297. MR**2135038**, 10.1007/s00013-004-1156-2**5.**K. Divaani-Aazar and P. Schenzel,*Ideal topologies, local cohomology and connectedness*, Math. Proc. Cambridge Philos. Soc.**131**(2001), no. 2, 211–226. MR**1857116**, 10.1017/S0305004101005229**6.**Robin Hartshorne,*Cohomological dimension of algebraic varieties*, Ann. of Math. (2)**88**(1968), 403–450. MR**0232780****7.**I. G. Macdonald,*Secondary representation of modules over a commutative ring*, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971) Academic Press, London, 1973, pp. 23–43. MR**0342506****8.**I. G. MacDonald and Rodney Y. Sharp,*An elementary proof of the non-vanishing of certain local cohomology modules*, Quart. J. Math. Oxford Ser. (2)**23**(1972), 197–204. MR**0299598****9.**Ryo Takahashi, Yuji Yoshino, and Takeshi Yoshizawa,*Local cohomology based on a nonclosed support defined by a pair of ideals*, J. Pure Appl. Algebra**213**(2009), no. 4, 582–600. MR**2483839**, 10.1016/j.jpaa.2008.09.008

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Additional Information

**Lizhong Chu**

Affiliation:
Department of Mathematics, Soochow University, 215006, Jiangsu, People’s Republic of China

Email:
Chulizhong@suda.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-2010-10471-9

Keywords:
Local cohomology modules,
vanishing,
Artinian modules.

Received by editor(s):
April 12, 2009

Received by editor(s) in revised form:
March 2, 2010

Published electronically:
October 28, 2010

Additional Notes:
This work was supported by NSF (10771152, 10926094) of China, by the NSF (09KJB110006) for Colleges and Universities in Jiangsu Province, by the Research Foundation (Q4107805) of Suzhou University and by the Research Foundation of Pre-research Project (Q3107852) of Suzhou University.

Communicated by:
Bernd Ulrich

Article copyright:
© Copyright 2010
American Mathematical Society