A new proof of Faltings’ local-global principle for the finiteness of local cohomology modules

Abstract

Let R be a commutative Noetherian ring, and let ${\mathfrak{b} \subseteq \mathfrak{a}}$ be ideals of R. The goal of this paper is to show that, for a finitely generated R-module M, if the set ${{\rm Ass}_R (H_{\mathfrak{a}}^{f_{\mathfrak{a}}^{\mathfrak{b}}(M)}(M))}$ is finite or ${f_{\mathfrak{a}}(M) \neq c_{\mathfrak{a}}^{\mathfrak{b}}(M)}$, then ${f_{\mathfrak{a}}^{\mathfrak{b}}(M) = {\rm inf} \{f_{\mathfrak{a} R_{\mathfrak{p}}}^{\mathfrak{b} R_{\mathfrak{p}}}(M_{\mathfrak{p}})|\,\,\,\mathfrak{p} \in {\rm Spec}(R)\}}$, where ${c_{\mathfrak{a}}^{\mathfrak{b}}(M)}$ denotes the first non ${\mathfrak{b}}$-cofiniteness of the local cohomology module ${H^i_{\mathfrak{a}}(M)}$. As a consequence of this, we provide a new and short proof of the Faltings’ local-global principle for finiteness dimensions. Also, several new results concerning the finiteness dimensions are given.