inR. We then have a canonical immediate extensionν of ν to QF(R/P (R) ∞ ) which dominatesR/P (R) ∞ .The following lemma appears in [11].Lemma 1.2. Suppose that ν is a rank 1 valuation of a field K and R is a Noetherian local domain which is dominated by ν. Letν be the canonical extension of. There exists n 0 such that f n − h ∈ m m RR for n ≥ n 0 . Then in ν (f n ) = inν(h) for n ≥ n 0 .From Lemma 1.2, we obtain a positive answer to Question 1.1 for local domains R and rank 1 valuations ν which admit local unformization. A positive answer to Question 1.1 for rank 1 valuations with the additional conclusion that R ′ /p is a regular local ring is given in [4] and in [7, Theorem 7.2] for R which are essentially of finite type over a field of characteristic zero. This is generalized somewhat in [6] and [9].Related to Question 1.1 is the following question, which we will also answer.