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COFINITENESS OF GENERALIZED LOCAL COHOMOLOGY MODULES
Published online by Cambridge University Press: 29 March 2011
Abstract
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.
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- Research Article
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- 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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