Local-global principle for annihilation of general local cohomology
Volume 87 / 2001
Colloquium Mathematicum 87 (2001), 129-136
MSC: Primary 13D45.
DOI: 10.4064/cm87-1-8
Abstract
Let $A$ be a Noetherian ring, let $M$ be a finitely generated $A$-module and let ${\mit \Phi } $ be a system of ideals of $A$. We prove that, for any ideal ${\mathfrak a}$ in ${\mit \Phi } $, if, for every prime ideal ${\mathfrak p}$ of $A$, there exists an integer $k({\mathfrak p})$, depending on ${\mathfrak p}$, such that ${\mathfrak a}^{k({ \mathfrak p})}$ kills the general local cohomology module $H_{{\mit \Phi } _{{\mathfrak p}}}^j(M_{{ \mathfrak p}})$ for every integer $j$ less than a fixed integer $n$, where ${\mit \Phi } _{{ \mathfrak p}}:=\{ {\mathfrak a}_{{\mathfrak p}}:{\mathfrak a}\in {\mit \Phi } \} $, then there exists an integer $k$ such that ${\mathfrak a}^kH_{{\mit \Phi } }^j(M)=0$ for every $j< n$.
