Descent of projectivity for locally free modules

Abstract:

Let $R \to \hat R$ be the natural homomorphism from the commutative ring $R$ into its associated von Neumann regular ring, and let $M$ be a locally free $R$-module such that $\hat R \otimes M$ is a projective $\hat R$-module. We show that if $M$ is either countably generated or locally finitely generated, then $M$ is projective, and we deduce that the trace of any projective ideal is projective. These results are a consequence of a more general theorem on the descent of the Mittag-Leffler condition. The “locally free” hypothesis may be weakened to “flat” if and only if $R$ is locally perfect.