A LOCALIZATION OF A POWER SERIES RING OVER A VALUATION DOMAIN

Abstract

Let V be a valuation domain. It is known that V[X-1,...,X-n]v-(0), is an n-dimensional Noetherian UFD if V has a height 1 prime ideal P and P not equal P-2. We show that V[X-1,...,X-n]v-(0) is an it-dimensional Noetherian regular local ring if V does not have a height 1 prime ideal. If V has a height 1 prime ideal that is idempotent, then dim V[X-1,...,X-n]v-(0) = infinity and t-dim V[X-1,..., X-n]v-(0) = infinity. In the process of obtaining the above result, we introduce a product of infinitely many power series. (C) 1999 Elsevier Science B.V. All rights reserved.