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On the cofiniteness of generalized local cohomology modules (1207.0703v1)

Published 28 Jun 2012 in math.AC

Abstract: Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$, $N$ two finitely generated $R$-modules. The aim of this paper is to investigate the $I$-cofiniteness of generalized local cohomology modules $\displaystyle Hj_I(M,N)=\dlim\Extj_R(M/InM,N)$ of $M$ and $N$ with respect to $I$. We first prove that if $I$ is a principal ideal then $Hj_I(M,N)$ is $I$-cofinite for all $M, N$ and all $j$. Secondly, let $t$ be a non-negative integer such that $\dim\Supp(Hj_I(M,N))\le 1 \text{for all} j<t.$ Then $Hj_I(M,N)$ is $I$-cofinite for all $j<t$ and $\Hom(R/I,Ht_I(M,N))$ is finitely generated. Finally, we show that if $\dim(M)\le 2$ or $\dim(N)\le 2$ then $Hj_I(M,N)$ is $I$-cofinite for all $j$.

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