Cofiniteness of Generalized Local Cohomology Modules

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 Ext j R ( M /𝔞 M , H i 𝔞 ( N )) is finite for all i < s and all j ≥0 . We show that Hom R ( R /𝔞, H s 𝔞 ( M , N )) is finite provided Ext s R ( M /𝔞 M , N ) is a finite R -module. In addition, for finite R -modules M and N , we prove that if H i 𝔞 ( M , N ) is minimax for all i < s , then Hom R ( R /𝔞, H s 𝔞 ( 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.