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COFINITENESS OF GENERALIZED LOCAL COHOMOLOGY MODULES

AbstractLet ???? 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/????M,Hi????(N)) is finite for all i<s and all j≥0 . We show that HomR (R/????,Hs????(M,N)) is finite provided ExtsR(M/????M,N) is a finite R-module. In addition, for finite R-modules M and N, we prove that if Hi????(M,N) is minimax for all i<s, then HomR (R/????,Hs????(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.

Cambridge University Press (CUP)

KEIVAN BORNA PARVIZ SAHANDI SIAMAK YASSEMI

Bulletin of the Australian Mathematical Society

2011

Title: COFINITENESS OF GENERALIZED LOCAL COHOMOLOGY MODULES

Description:

AbstractLet ???? 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/????M,Hi????(N)) is finite for all i<s and all j≥0 .

We show that HomR (R/????,Hs????(M,N)) is finite provided ExtsR(M/????M,N) is a finite R-module.

In addition, for finite R-modules M and N, we prove that if Hi????(M,N) is minimax for all i<s, then HomR (R/????,Hs????(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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