Ext-Tor Vanishing and Completeness Criteria

Abstract

As a classical result, Jensen proved that, if $\mathfrak a=\mathfrak m$ is the maximal ideal of a Noetherian local ring R, then a finitely generated R-module M is $\mathfrak m$-complete if and only if $\text {Ext}_R^1(F, M)=0$ for all countably generated flat modules F if and only if $\text {Ext}_R^i(F, M)=0$ for all $i\ge 1$ and any flat module F (see Jensen in J. Algebra 15:151–166, 1970 [1] and Les foncteurs dérivés de lim et leurs applications en théorie des modules, Springer, Berlin, Heidelberg, New York [2] [Proposition 8.2]).