On the Endomorphism Rings of Local Cohomology Modules

Khashyarmanesh, Kazem

Canadian mathematical bulletin, Tome 53 (2010) no. 4, pp. 667-673

Résumé

Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal of $R$ . We show that if $n\,:=\,\text{grad}{{\text{e}}_{R}}\,\mathfrak{a}$ , then $\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))\,\cong \,\text{Ext}_{R}^{n}(H_{\mathfrak{a}}^{n}(R),\,R)$ . We also prove that, for a nonnegative integer $n$ such that $H_{\mathfrak{a}}^{i}(R)\,=\,0$ for every $i\,\ne \,n$ , if $\text{Ext}_{R}^{i}({{R}_{z}},\,R)\,=\,0$ for all $i\,>\,0$ and $z\,\in \,\mathfrak{a}$ , then $\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))$ is a homomorphic image of $R$ , where ${{R}_{z}}$ is the ring of fractions of $R$ with respect to a multiplicatively closed subset $\{{{z}^{j}}|j\,\ge \text{0}\}$ of $R$ . Moreover, if $\text{Ho}{{\text{m}}_{R}}({{R}_{z}},R)=0$ for all $z\,\in \,\mathfrak{a}$ , then ${{\mu }_{H_{\mathfrak{a}}^{n}(R)}}$ is an isomorphism, where ${{\mu }_{H_{\mathfrak{a}}^{n}(R)}}$ is the canonical ring homomorphism $R\,\to \,\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))$ .

Détail

DOI : 10.4153/CMB-2010-072-1

Mots-clés : 13D45, 13D07, 13D25, local cohomology module, endomorphism ring, Matlis dual functor, filter regular sequence