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

AbstractLet k⊂K be fields, let k0 be the maximal separable extension of k in K, and let x1,…,xn be analytically independent indeterminates over K, where n≥1. If K has finite exponent over k0 and [k0:k]<∞, then K〚x1,…,xn〛 is integral over k〚x1,…,xn〛, but if K has infinite exponent over k0 or [k0:k]=∞, then the generic fibre of the extension k〚x1,…,xn〛↪K〚x1,…,xn〛 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, dimR〚x1,…,xn〛=mn+1 if K has finite exponent over k0 and [k0:k]<∞ but equals mn+n otherwise. More generally, it is also shown that, if R is an m-dimensional SFT globalized pseudo-valuation domain, then dimR〚x1,…,xn〛=mn+1 or mn+n