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≠ P 2. We show that V〚X 1,…,X n〛 V−(0) is an n-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 dimV〚X 1,…,X n〛 V−(0)=∞ and t- dimV〚X 1,…,X n〛 V-(0)=∞ . In the process of obtaining the above result, we introduce a product of infinitely many power series.