Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules

TL;DR

This paper extends the Lichtenbaum-Hartshorne vanishing theorem to generalized local cohomology modules over Noetherian rings, providing conditions for their vanishing and describing their associated primes.

Contribution

It generalizes the vanishing theorem for local cohomology to a broader class of modules and characterizes the associated primes of specific generalized local cohomology modules.

Findings

Characterization of coassociated prime ideals of H^{p+c}_a(M,N)

Necessary and sufficient conditions for vanishing of generalized local cohomology in local rings

Extension of the Lichtenbaum-Hartshorne vanishing theorem to generalized modules

Abstract

Let $R$ be a commutative Noetherian ring, and let $\mathfrak a$ be a proper ideal of $R$. Let $M$ be a non-zero finitely generated $R$-module with the finite projective dimension $p$. Also, let $N$ be a non-zero finitely generated $R$-module with $N\neq\mathfrak{a} N$, and assume that $c$ is the greatest non-negative integer with the property that $\operatorname{H}^i_{\mathfrak a}(N)$, the $i$-th local cohomology module of $N$ with respect to $\mathfrak a$, is non-zero. It is known that $\operatorname{H}^i_{\mathfrak a}(M, N)$, the $i$-th generalized local cohomology module of $M$ and $N$ with respect to $\mathfrak a$, is zero for all $i>p+c$. In this paper, we obtain the coassociated prime ideals of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$. Using this, in the case when $R$ is a local ring and $c$ is equal to the dimension of $N$, we give a necessary and sufficient condition for the vanishing of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$ which extends the Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules.