A localization of a power series ring over a valuation domain Original Research Article

Let V be a valuation domain. It is known that Vleft double bracket delimiterX1,…,Xnright double bracket delimiterV−(0)is an n-dimensional Noetherian UFD if V has a height 1 prime ideal P and P≠P2. We show that Vleft double bracket delimiterX1,…,Xnright double bracket delimiterV−(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 dimVleft double bracket delimiterX1,…,Xnright double bracket delimiterV−(0)=∞ and t-dimVleft double bracket delimiterX1,…,Xnright double bracket delimiterV-(0)=∞. In the process of obtaining the above result, we introduce a product of infinitely many power series.