A Local Flatness Criterion for Complete Modules

Abstract

We prove various extensions of the Local Flatness Criterion over a Noetherian local ring R with residue field k. For instance, if Ω is a complete R-module of finite projective dimension, then Ω is flat if and only if for all n = 1,…, depth (R). In low dimensions, we have the following criteria. If R is one-dimensional and reduced, then Ω is flat if and only if . If R is two-dimensional, then in order for Ω to be flat, it suffices that it is separated, that its projective dimension is finite and that . Many of these criteria have global counterparts and in particular, it is shown that the 𝔞-adic completion of a flat module of finite projective dimension over an arbitrary Noetherian ring is again flat.