Faltings’ local–global principle and annihilator theorem for the finiteness dimensions

Abstract

Let R be a commutative Noetherian ring, M be a finitely generated R-module and n be a non-negative integer. In this article, it is shown that for a positive integer t, there is a finitely generated submodule N_i of $H_{\mathfrak{a}}^i(M)$ such that $\dim \text{Supp}{H}_{\mathfrak{a}}^i(M)/N_i<n$ for all i < t if and only if there is a finitely generated submodule $N_{i,\mathfrak{p}}$ of $H_{\mathfrak{a}{R}_{\mathfrak{p}}}^i(M_{\mathfrak{p}})$ such that $\dim \text{Supp}{H}_{\mathfrak{a}{R}_{\mathfrak{p}}}^i(M_{\mathfrak{p}})/N_{i,\mathfrak{p}}<n$ for all i < t and all p ∈ Spec(R). This generalizes Faltings’ Local–global Principle for the finiteness of local cohomology modules (Faltings’ in Math. Ann. 255:45–56, 1981). Also, it is shown that whenever R is a homomorphic image of a Gorenstein local ring, then the invariants $\inf \{i\in {\mathbb{N}}_0|\dim \text{Supp}({\mathfrak{b}}^t{H}_{\mathfrak{a}}^i(M))⩾n\text{for}\text{all}t\in {\mathbb{N}}_0\}$ and $\inf \{\text{depth}{M}_{\mathfrak{p}}+\text{ht}(\mathfrak{a}+\mathfrak{p})/\mathfrak{p}|\mathfrak{p}\in \text{Spec}(R)\setminus \mathrm{V}(\mathfrak{b}),\dim R/(\mathfrak{a}+\mathfrak{p})⩾n\}$ are equal, for every finitely generated R-module M and for all ideals $\mathfrak{a},\mathfrak{b}$ of R with $\mathfrak{b}\subseteq \mathfrak{a}$. As a consequence, we determine the least integer i where the local cohomology module $H_{\mathfrak{a}}^i(M)$ is not minimax (resp. weakly Laskerian).

Keywords:

• Faltings’ annihilator theorem
• Faltings’ local–global principle
• in dimension < n modules
• local cohomology
• minimax modules

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D45
• 14B15
• 13E05

Additional information

Funding

This research was in part supported by a grant from IPM (No. 96130027).