On the cofiniteness of generalized local cohomology modules

Abstract

Let $R$ be a commutative Noetherian ring with non-zero identity and $I$ be an ideal of $R$. Let $M$ and $N$ be two finitely generated $R$-modules. In this paper it is shown that, if ${\rm cd}(I,R)\leq 1$ and ${\rm pd_{R}}(M)<\infty$, then $H_{I}^{i}(M,N)$ are $I$-cofinite for each $i\geq 0$. Moreover, it is shown that if $\mbox{q}(I,R) \leq 1$ and ${\rm pd_{R}}(M)<\infty$, then the Bass numbers of generalized local cohomology modules $H_{I}^{i}(M,N)$ are finite for each $i\geq 0$. We also characterize the greatest integer $i$ such that $H_{I}^{i}(M,N)$ is not Artinian and $I$-cofinite.