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 $\Ext_R^1(F,M) = 0$ 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 $\Rad I = \Rad \xx R$. This result extends several known statements starting with C. U. Jensen's result (see \cite[Proposition 3]{J}) that a finitely generated $R$-module $M$ over a local ring $R$ is complete if and only if $\Ext^1_R(F,M) = 0$ for any flat $R$-module $F$.