ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

2. Algebra

1991 MS Classification: 13 Wiegand, Sylvia, University of Nebraska, Lincoln, NE 68502; William Heinzer, Purdue University, West Lafayette, IN 47907; Christel Rotthaus, Michigan State University, East Lansing MI 48824 Building Noetherian domains using completions Over the past sixty years many important examples of Noetherian domains have been constructed using a technique developed by Akizuki in the 1930s and Nagata in the fifites. The rings constructed are intersections of the form L ∩ R̂/I; that is, an intersection of a homomorphic image of a completion of a well-understood and well-behaved Noetherian domain R with an extension field of the fraction field of R. (Often R is taken to be a polynomial ring in two variables over a field and the completion can be considered as a power series ring.) We analyze this technique in a general way and show how flatness is crucial to whether an intersection A := L ∩ R̂/I is Noetherian and can be computed easily. We also give some results concerning excellence of such an intersection.