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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
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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 local cohomology module, endomorphism ring, Matlis dual functor, filter regular sequence
MSC Classifications: 13D45, 13D07, 13D25 show english descriptions Local cohomology [See also 14B15]
Homological functors on modules (Tor, Ext, etc.)
Complexes
13D45 - Local cohomology [See also 14B15]
13D07 - Homological functors on modules (Tor, Ext, etc.)
13D25 - Complexes
 

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