On the vanishing and finiteness properties of generalized local cohomology modules

TL;DR

This paper investigates the properties of generalized local cohomology modules over noetherian rings, establishing equivalences between finiteness, coassociated primes, and coatomicity, and exploring conditions for vanishing and artinian properties.

Contribution

It provides new equivalences characterizing finiteness, coatomicity, and coassociated primes of generalized local cohomology modules, and identifies conditions for their vanishing and artinian behavior.

Findings

Equivalence of finiteness, coatomicity, and prime support for modules.

Finiteness or minimax conditions imply vanishing or artinian properties beyond a certain degree.

Results extend understanding of the structure and behavior of generalized local cohomology modules.

Abstract

Let $R$ be a commutative noetherian ring, $\fa$ an ideal of $R$ and $M,N$ finite $R$--modules. We prove that the following statements are equivalent. \begin{enumerate} \item[(i)] $\lc^{i}_{\fa}(M,N)$ is finite for all $i< n$. \item[(ii)] $\Coass_R(\lc^{i}_{\fa}(M,N)) \subset \V{(\fa)}$ for all $i< n$. \item[(iii)] $\lc^{i}_{\fa}(M,N)$ is coatomic for all $i< n$. \end{enumerate} If $\pd M$ is finite and $r$ be a non-negative integer such that $r>\pd M$ and $\lc^{i}_{\fa}(M,N)$ is finite (resp. minimax) for all $i\geq r$, then $\lc^{i}_{\fa}(M,N)$ is zero (resp. artinian) for all $i\geq r$.