A criterion for I-adic completeness

Abstract

Let I denote an ideal in a commutative Noetherian ring R. Let M be an R-module. The I-adic completion is defined by ${\hat{M}^I = \varprojlim{}_{\alpha} M/I^{\alpha}M}$. Then M is called I-adic complete whenever the natural homomorphism ${M \to \hat{M}^I}$ is an isomorphism. Let M be I-separated, i.e. ${\cap_{\alpha} I^{\alpha}M = 0}$. In the main result of the paper, it is shown that M is I-adic complete if and only if ${{\rm Ext}_R^1(F,M) = 0}$ for the flat test module ${F = \oplus_{i = 1}^r R_{x_i}}$, where ${\{x_1,\ldots,x_r\}}$ is a system of elements such that ${{\rm Rad} I = {\rm Rad}\, \underline{{\it x}} R}$. This result extends several known statements starting with Jensen’s result [9, Proposition 3] that a finitely generated R-module M over a local ring R is complete if and only if ${{\rm Ext}^1_R(F,M) = 0}$ for any flat R-module F.