Cofiniteness of generalized local cohomology modules for ideals of small dimension

TL;DR

This paper investigates conditions under which generalized local cohomology modules are cofinite, focusing on rings with small dimension and using spectral sequences to establish cofinateness.

Contribution

It provides new criteria for the cofiniteness of generalized local cohomology modules in rings of small dimension using spectral sequence techniques.

Findings

Cofiniteness holds if $ ext{dim}_R M extless= 2$ and certain local cohomology modules are cofinite.

Cofiniteness is established when local cohomology modules have dimension $ extless= 1$ for all degrees.

Results apply when the cohomological dimension $q(rak{a}, R) extless= 1$.

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 $\mathrm{dim}_RM\leq2$ and $\mathrm{H}^{i}_\mathfrak{a}(N)$ are $\mathfrak{a}$-cofinite for all $i\geq0$, or $\mathrm{H}^{i}_\mathfrak{a}(N)$ is an $\mathfrak{a}$-cofinite module of dimension $\leq1$ for each $i\geq1$, or $q(\mathfrak{a},R)\leq1$, then the $R$-modules $\mathrm{H}^{t}_\mathfrak{a}(M,N)$ are $\mathfrak{a}$-cofinite for all $t\geq0$.