A Local Flatness Criterion for Complete Modules

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 Torn (Ω, k) = 0 for all n = 1, . . . , depth(R). In low dimensions, we have the following criteria. If R is onedimensional and reduced, then Ω is flat if and only if Tor1 (Ω, k) = 0. If R is twodimensional, then in order for Ω to be flat, it suffices that it is separated, that its projective dimension is finite and that Tor1 (Ω, k) = 0. Many of these criteria have global counterparts and in particular, it is shown that the aadic completion of a flat module of finite projective dimension over an arbitrary Noetherian ring is again flat.