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

Let k⊂Kk⊂K be fields, let k0k0 be the maximal separable extension of kk in KK, and let x1,…,xnx1,…,xn be analytically independent indeterminates over KK, where n≥1n≥1. If KK has finite exponent over k0k0 and [k0:k]<∞[k0:k]<∞, then K〚x1,…,xn〛K〚x1,…,xn〛 is integral over k〚x1,…,xn〛k〚x1,…,xn〛, but if KK has infinite exponent over k0k0 or [k0:k]=∞[k0:k]=∞, then the generic fibre of the extension k〚x1,…,xn〛↪K〚x1,…,xn〛k〚x1,…,xn〛↪K〚x1,…,xn〛 is (n−1)(n−1)-dimensional. As an application, it is shown that, for an mm-dimensional SFT pseudo-valuation domain RR with residue field kk and the associated valuation domain VV with residue field KK, dimR〚x1,…,xn〛=mn+1 if KK has finite exponent over k0k0 and [k0:k]<∞[k0:k]<∞ but equals mn+nmn+n otherwise. More generally, it is also shown that, if RR is an mm-dimensional SFT globalized pseudo-valuation domain, then dimR〚x1,…,xn〛=mn+1 or mn+nmn+n.