A new version of Local-Global Principle for annihilations of local cohomology modules

K. Khashyarmanesh ¹   ; M. Yassi ²   ; A. Abbasi ²

¹ Institute for Studies in Theoretical Physics and Mathematics P.O. Box 19395-5746 Tehran, Iran
² Department of Mathematics Mashhad University P.O. Box 1159-91775 Mashhad, Iran

Colloquium Mathematicum, Tome 100 (2004) no. 2, pp. 213-219

Résumé

Let $R$ be a commutative Noetherian ring. Let $\mathfrak a$ and $\mathfrak b$ be ideals of $R$ and let $N$ be a finitely generated $R$-module. We introduce a generalization of the $\mathfrak b$-finiteness dimension of $f^{\mathfrak b}_{\mathfrak a}(N)$ relative to $\mathfrak a$ in the context of generalized local cohomology modules as $$f^{\mathfrak b}_{\mathfrak a}(M,N):= \hbox{inf} \{ i\geq 0\mid {\mathfrak b} \subseteq \sqrt{(0:_R H^i_{\mathfrak a}(M,N))}\,\}, $$ where $M$ is an $R$-module. We also show that $f^{\mathfrak b}_{\mathfrak a}(N)\leq f^{\mathfrak b}_{\mathfrak a}(M,N)$ for any $R$-module $M$. This yields a new version of the Local-Global Principle for annihilation of local cohomology modules. Moreover, we obtain a generalization of the Faltings Lemma.

Détail

DOI : 10.4064/cm100-2-5

Keywords: commutative noetherian ring mathfrak mathfrak ideals finitely generated r module introduce generalization mathfrak b finiteness dimension mathfrak mathfrak relative mathfrak context generalized local cohomology modules mathfrak mathfrak hbox inf geq mid mathfrak subseteq sqrt mathfrak where r module mathfrak mathfrak leq mathfrak mathfrak r module yields version local global principle annihilation local cohomology modules moreover obtain generalization faltings lemma