Faltings’ local–global principle for finiteness dimension of cofinite modules

Abstract

Let R denote a commutative Noetherian ring, \(\mathfrak {a}\) an ideal of R, and M an \(\mathfrak {a}\)-cofinite R-module. The purpose of this article is to show that for a positive integer t, the R-module \(H_\mathfrak {a}^i(M)\) is finitely generated for all \(i<t\) if and only if the \(R_{\mathfrak {p}}\)-module \(H_{\mathfrak {a}R_{\mathfrak {p}}}^i(M_{{\mathfrak {p}}})\) is finitely generated for all \(i<t\) and all \(\mathfrak {p}\in \mathrm {Spec}(R)\). As a consequence, we provide a generalization and short proof of Faltings’ local–global principle for finiteness dimensions; i.e., \(f_\mathfrak {a}(M)=\mathrm{{inf}}\{f_{\mathfrak {a}R_{\mathfrak {p}}}(M_{{\mathfrak {p}}}) | \ \mathfrak {p}\in \mathrm {Spec}(R)\}.\)