Detecting completeness from ext-vanishing

Abstract

Motivated by work of C. U. Jensen, R.-O. Buchweitz, and H. Flenner, we prove the following result. Let $R$ be a commutative Noetherian ring and $a$ an ideal in the Jacobson radical of $R$. Let ${\hat{R}}^a$ be the $a$-adic completion of $R$. If $M$ is a finitely generated $R$-module such that $\ext^i_R(\hat{R}^a,M)=0$ for all $i\ne 0$, then $M$ is $a$-adically complete.

10 pages; proof of Theorem A revised; Theorem B added; to appear in Proceedings of the American Mathematical Society