Canadian Mathematical Society www.cms.math.ca
Abstract view

# On the Endomorphism Rings of Local Cohomology Modules

Published:2010-07-26
Printed: Dec 2010
• Kazem Khashyarmanesh,
Ferdowsi University of Mashhad, Department of Mathematics, Mashhad, Iran
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
 Format: HTML LaTeX MathJax PDF

## Abstract

Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal of $R$. We show that if $n:=\operatorname{grade}_R\mathfrak{a}$, then $\operatorname{End}_R(H^n_\mathfrak{a}(R))\cong \operatorname{Ext}_R^n(H^n_\mathfrak{a}(R),R)$. We also prove that, for a nonnegative integer $n$ such that $H^i_\mathfrak{a}(R)=0$ for every $i\neq n$, if $\operatorname{Ext}_R^i(R_z,R)=0$ for all $i >0$ and $z \in \mathfrak{a}$, then $\operatorname{End}_R(H^n_\mathfrak{a}(R))$ is a homomorphic image of $R$, where $R_z$ is the ring of fractions of $R$ with respect to a multiplicatively closed subset $\{z^j \mid j \geqslant 0 \}$ of $R$. Moreover, if $\operatorname{Hom}_R(R_z,R)=0$ for all $z \in \mathfrak{a}$, then $\mu_{H^n_\mathfrak{a}(R)}$ is an isomorphism, where $\mu_{H^n_\mathfrak{a}(R)}$ is the canonical ring homomorphism $R \rightarrow \operatorname{End}_R(H^n_\mathfrak{a}(R))$.
 Keywords: local cohomology module, endomorphism ring, Matlis dual functor, filter regular sequence
 MSC Classifications: 13D45 - Local cohomology [See also 14B15] 13D07 - Homological functors on modules (Tor, Ext, etc.) 13D25 - Complexes

© Canadian Mathematical Society, 2013 : http://www.cms.math.ca/