Quintasymptotic primes, local cohomology and ideal topologies
Volume 106 / 2006
Abstract
Let ${\mit\Phi}$ be a system of ideals on a commutative Noetherian ring $R$, and let $S$ be a multiplicatively closed subset of $R$. The first result shows that the topologies defined by $\{I_a\}_{I\in{\mit\Phi}}$ and $\{S(I_a)\}_{I\in{\mit\Phi}}$ are equivalent if and only if $S$ is disjoint from the quintasymptotic primes of ${\mit\Phi}$. Also, by using the generalized Lichtenbaum–Hartshorne vanishing theorem we show that, if $(R,{\mathfrak m})$ is a $d$-dimensional local quasi-unmixed ring, then $H^d_{{\mit\Phi}} (R)$, the $d$th local cohomology module of $R$ with respect to ${\mit\Phi}$, vanishes if and only if there exists a multiplicatively closed subset $S$ of $R$ such that $S\cap {\mathfrak m}\neq \emptyset$ and the $S({\mit\Phi})$-topology is finer than the ${\mit\Phi}_a$-topology.
