Noetherian Domains Inside a Homomorphic Image of a Completion

Over the past 60 years, important examples of Noetherian domains have been constructed using power series, homomorphic images, and intersections. In these constructions it is often crucial that the resulting domain is computable as a directed union. In this article we analyse this construction and show that the Noetherian property for the associated directed union is equivalent to a flatness condition. LetRbe a Notherian integral domain with fraction fieldL. Letxbe a nonzero nonunit ofRand letR* denote the (x)-adic completion ofR. SupposeIis an ideal ofR* with the propertyp∩R=(0) for eachp∈Ass(R*/I).Theorem. The embeddingR↪(R*/I)xis flat if and only ifA≔L∩(R*/I) is Noetherian and is realizable as a localization of a subring ofRx=R[1/x]. We present several examples where this flatness condition holds; one of these examples is a local Noetherian domain that is not universally catenary, but has geometrically regular formal fibers.