Krull dimension of power series rings over a globalized pseudo-valuation domain

Abstract

Let k ⊂ K be fields, let k 0 be the maximal separable extension of k in K , and let x 1 , … , x n be analytically independent indeterminates over K , where n ≥ 1 . If K has finite exponent over k 0 and [ k 0 : k ] < ∞ , then K 〚 x 1 , … , x n 〛 is integral over k 〚 x 1 , … , x n 〛 , but if K has infinite exponent over k 0 or [ k 0 : k ] = ∞ , then the generic fibre of the extension k 〚 x 1 , … , x n 〛 ↪ K 〚 x 1 , … , x n 〛 is ( n − 1 ) -dimensional. As an application, it is shown that, for an m -dimensional SFT pseudo-valuation domain R with residue field k and the associated valuation domain V with residue field K , dim R 〚 x 1 , … , x n 〛 = m n + 1 if K has finite exponent over k 0 and [ k 0 : k ] < ∞ but equals m n + n otherwise. More generally, it is also shown that, if R is an m -dimensional SFT globalized pseudo-valuation domain, then dim R 〚 x 1 , … , x n 〛 = m n + 1 or m n + n .