Krull dimension and generic fibres for mixed polynomial and power series integral domains

For a ring R and variables x1,…,xn, we let R[x1〛⋯[xn〛 denote a mixed extension ring of R, where each [xi〛 is fixed as either [xi] for polynomials in the variable xi or 〚xi〛 for power series in xi. It is well known that if R is a Noetherian ring of Krull dimension m, then R[x1〛⋯[xn〛 has Krull dimension m+n. We assume throughout that at least one [xi〛 is 〚xi〛 and we prove, for a certain class of integral domains R of dimension m, that the dimension of R[x1〛⋯[xn〛 is mn+1 or mn+n. Each of our integral domains R is close to being Noetherian and has a canonically associated Prüfer overring T such that the contraction map Spec(T)→Spec(R) is a homeomorphism. A second result in this paper, used in the proof of the above result, appears not to have been noticed before: For an extension k⊂K of fields, the extension k[x1〛⋯[xn〛↪K[x1〛⋯[xn〛 is integral and the dimension of the generic fibre is 0, if the maximal separable extension k0 of k in K is a finite extension of k and K has finite exponent over k0. Otherwise the generic fibre has dimension n−1.