A new version of Local-Global Principle for annihilations of local cohomology modules
Volume 100 / 2004
Abstract
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.
