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 t hese constructions it is often crucial that the resulting domain is computa ble as a directed union. In this article we analyse this construction and s how that the Noetherian property for the associated directed union is equiv alent to a flatness condition. Let R be a Notherian integral domain with fr action field L. Let x be a nonzero nonunit of R and let R* denote the (x)-a dic completion of R. Suppose I is an ideal of R* with the property p boolea n AND R = (0) for each p is an element of Ass(R*/I). Theorem. The embedding R --> (R*/I)(x) is flat if and only if A := L boolean AND (R*/I) is Noethe rian and is realizable as a localization of a subring of R-x = A[1/x]. We p resent several examples where this flatness condition holds; one of these e xamples is a local Noetherian domain that is not universally catenary, but has geometrically regular formal fibers. (C) 1999 Academic Press.