On the Vanishing and the Finiteness of Supports of Generalized Local Cohomology Modules

TL;DR

This paper proves a vanishing theorem for generalized local cohomology modules over Noetherian local rings when the first module has finite projective dimension, and characterizes when their supports are infinite.

Contribution

It establishes a vanishing result for generalized local cohomology modules and characterizes the support finiteness conditions, advancing understanding of their structure.

Findings

Vanishing of $H^j_I(M,N)$ for $j > ext{dim}(R)$ when $M$ has finite projective dimension.

Characterizations of the least and last integers with infinite support of $H^r_I(M,N)$.

Abstract

Let $(R,\fr m)$ be a Noetherian local ring, $I$ an ideal of $R$ and $M, N$ two finitely generated $R$-modules. The first result of this paper is to prove a vanishing theorem for generalized local cohomology modules which says that $H^j_I(M,N)=0$ for all $j>\dim(R)$, provided $M$ is of finite projective dimension. Next, we study and give characterizations for the least and the last integer $r$ such that $\Supp(H^r_I(M,N))$ is infinite.