Faltings' local-global principle for finiteness dimension of cofinite modules
Let \(R\) be a commutative Noetherian ring with non-zero identity, \(\mathfrak{a}\) an ideal of \(R\), \(M\) an arbitrary \(R\)--module, and \(t\) a non-negative integer. In this paper, the authors generalize Faltings' local-global principle for the finiteness of local cohomology modules with respect to \(\mathfrak{a}\) to the class of all \(R\)-modules that are \(\mathfrak{a}\)-cofinite. Recall that \(M\) is said to be \(\mathfrak{a}\)-cofinite if \(\text{Supp}_R(M)\subseteq \{\mathfrak{p}\in \text{Spec}(R) : \mathfrak{p}\supseteq \mathfrak{a}\}\) and \(\text{Ext}^i_R(R/\mathfrak{a}, M)\) is finitely generated for all \(i\). They prove that if \(\text{Ext}^i_R(R/\mathfrak{a}, M)\) is finitely generated for all \(i< t\), then the following conditions are equivalent: {\parindent=0.7cm\begin{itemize}\item[(i)] \(\text{H}^{i}_{\mathfrak{a}}(M)\) is a finitely generated \(R\)-module for all \(i< t\); \item[(ii)] \(\text{H}^{i}_{\mathfrak a}R_{\mathfrak p}(M_{\mathfrak p})\) is a finitely generated \(R_{\mathfrak p}\)-module for all \(i< t\) and all \(\mathfrak{p}\in \text{Spec}(R)\); \item[(iii)] \(\mathfrak{a}\subseteq \text{Rad}(0:_R\text{H}^{i}_{\mathfrak{a}}(M))\) for all \(i< t\). \end{itemize}} Here, \(\text{Rad}(\mathfrak{a})= \{x\in R : x^n\in \mathfrak{a}\;\text{for some}\;n\in \mathbb{N}\}\) is the radical of \(\mathfrak{a}\). As a consequence, the authors provide a generalization and short proof of the Faltings' local-global principle for the finiteness dimensions by showing that if \(\text{Ext}^i_R(R/\mathfrak{a}, M)\) is finitely generated for all \(i\) (e.g., \(M\) is an \(\mathfrak{a}\)-cofinite \(R\)-module), then \[ f_{\mathfrak a}(M)= \inf\{f_{\mathfrak a}R_{\mathfrak p}(M_{\mathfrak p}) : \mathfrak{p}\in \text{Spec}(R)\}= \inf\{i\in \mathbb{N}_0 : \mathfrak{a}\nsubseteq \text{Rad}(0:_R\text{H}^{i}_{\mathfrak{a}}(M))\}, \] where \[ f_{\mathfrak a}(M)= \inf\{i\in \mathbb{N}_0 : \text{H}^{i}_{\mathfrak{a}}(M)\;\text{is not a finitely generated}\;R\text{-module}\} \] is the finiteness dimension of \(M\) with respect to \(\mathfrak{a}\).
scientific article; zbMATH DE number 7005657
scientific article