Quintasymptotic primes, local cohomology and ideal topologies

A. A. Mehrvarz ¹   ; R. Naghipour ¹   ; M. Sedghi ²

¹ Department of Mathematics University of Tabriz Tabriz, Iran
² Department of Mathematics Azarbaidjan University of Tarbiat Moallem Azarshahr, Tabriz, Iran

Colloquium Mathematicum, Tome 106 (2006) no. 1, pp. 25-37

Résumé

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.

Détail

DOI : 10.4064/cm106-1-3

Keywords: mit phi system ideals commutative noetherian ring multiplicatively closed subset first result shows topologies defined mit phi mit phi equivalent only disjoint quintasymptotic primes mit phi using generalized lichtenbaum hartshorne vanishing theorem mathfrak d dimensional local quasi unmixed ring mit phi dth local cohomology module respect mit phi vanishes only there exists multiplicatively closed subset cap mathfrak neq emptyset mit phi topology finer mit phi a topology