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COFINITENESS OF GENERALIZED LOCAL COHOMOLOGY MODULES

Part of: Commutative algebra: Homological methods

Published online by Cambridge University Press:  29 March 2011

KEIVAN BORNA ,
PARVIZ SAHANDI  and
SIAMAK YASSEMI
KEIVAN BORNA*
Affiliation:
Faculty of Mathematical Sciences and Computer, Tarbiat Moallem University, Tehran, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran (email: borna@ipm.ir)
PARVIZ SAHANDI
Affiliation:
Department of Mathematics, University of Tabriz, Tabriz, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran (email: sahandi@ipm.ir)
SIAMAK YASSEMI
Affiliation:
Department of Mathematics, University of Tehran, Tehran, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran (email: yassemi@ipm.ir)
*
✉For correspondence; e-mail: keivan.borna@gmail.com
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Abstract

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Let 𝔞 be an ideal of a Noetherian ring R. Let s be a nonnegative integer and let M and N be two R-modules such that ExtjR(M/𝔞M,Hi𝔞(N)) is finite for all i<s and all j≥0 . We show that HomR (R/𝔞,Hs𝔞(M,N)) is finite provided ExtsR(M/𝔞M,N) is a finite R-module. In addition, for finite R-modules M and N, we prove that if Hi𝔞(M,N) is minimax for all i<s, then HomR (R/𝔞,Hs𝔞(M,N)) is finite. These are two generalizations of the result of Brodmann and Lashgari [‘A finiteness result for associated primes of local cohomology modules’, Proc. Amer. Math. Soc. 128 (2000), 2851–2853] and a recent result due to Chu [‘Cofiniteness and finiteness of generalized local cohomology modules’, Bull. Aust. Math. Soc. 80 (2009), 244–250]. We also introduce a generalization of the concept of cofiniteness and recover some results for it.

Keywords

cofinitenessgeneralized local cohomologylocal cohomologyminimaxness

MSC classification

Secondary: 13D45: Local cohomology
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 3 , June 2011 , pp. 382 - 388
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The research of K. Borna was in part supported by a grant from IPM (No. 89130049). The research of P. Sahandi was in part supported by a grant from IPM (No. 89130051). The research of S. Yassemi was in part supported by a grant from IPM (No. 89130213).

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