Cofiniteness of generalized local cohomology modules for ideals of small dimension

Abstract

Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$ and $M,N$ two finitely generated $R$-modules. By using a spectral sequence argument, it is shown that if either ${\dim }_{R}M\le 2$ and ${\mathrm{H}}_{\mathfrak{a}}^i(N)$ are $\mathfrak{a}$-cofinite for all $i\ge 0$, or ${\mathrm{H}}_{\mathfrak{a}}^i(N)$ is an $\mathfrak{a}$-cofinite module of dimension $\le 1$ for each $i\ge 1$, or $q(\mathfrak{a},R)\le 1$, then the $R$-modules ${\mathrm{H}}_{\mathfrak{a}}^t(M,N)$ are $\mathfrak{a}$-cofinite for all $t\ge 0$.

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