Cohomological dimension filtration and annihilators of top local cohomology modules

Ali Atazadeh¹ ; Monireh Sedghi¹ ; Reza Naghipour²

¹ Department of Mathematics Azarbaijan Shahid Madani University Tabriz, Iran
² Department of Mathematics University of Tabriz Tabriz, Iran

Colloquium Mathematicum, Tome 139 (2015) no. 1, pp. 25-35

Résumé

Let $\mathfrak a$ denote an ideal in a Noetherian ring $R$, and $M$ a finitely generated $R$-module. We introduce the concept of the cohomological dimension filtration $\mathscr {M} =\{M_i\}_{i=0}^c$, where $ c=\mathop {\rm cd}\nolimits (\mathfrak a,M)$ and $M_i$ denotes the largest submodule of $M$ such that $\mathop {\rm cd}\nolimits (\mathfrak a,M_i)\leq i.$ Some properties of this filtration are investigated. In particular, if $(R, \mathfrak m)$ is local and $c= \dim M$, we are able to determine the annihilator of the top local cohomology module $H_{\mathfrak a}^c(M)$, namely ${\rm Ann}_R(H_{\mathfrak a}^c(M))= {\rm Ann}_R(M/M_{c-1}).$ As a consequence, there exists an ideal $\mathfrak b$ of $R$ such that ${\rm Ann}_R(H_{\mathfrak a}^{c}(M))={\rm Ann}_R(M/H_{\mathfrak b}^{0}(M))$. This generalizes the main results of Bahmanpour et al. (2012) and Lynch (2012).

Détail

DOI : 10.4064/cm139-1-2

Keywords: mathfrak denote ideal noetherian ring finitely generated r module introduce concept cohomological dimension filtration mathscr where mathop nolimits mathfrak denotes largest submodule mathop nolimits mathfrak leq properties filtration investigated particular mathfrak local dim able determine annihilator top local cohomology module mathfrak namely ann mathfrak ann consequence there exists ideal mathfrak ann mathfrak ann mathfrak generalizes main results bahmanpour lynch