Krull dimension and generic fibres for mixed polynomial and power series integral domains
This article has two main results. For a ring \(R\) and variables \(x_1,\dots,x_n\), let \(R[x_1]],\dots,[x_n]]\) denote a mixed extension ring of \(R\), where each \(x_i\) is fixed as either \([x_i] \) for polynomials in the variable \(x_i\) or \(x_i\) for power series in \([x_i]\). It is well known that if \(R\) is a Noetherian ring of Krull dimension \(m\), then \(R[x_1]],\dots,[x_n]]\) has Krull dimension \(m+n\). In on result by assuming that at least one \([x_i]]\) is \([x_i]]\) author prove that for a certain class of integral domains \(R\) of dimension \(m\), that the dimension of \(R[x_1]],\dots,[x_n]]\) is \(mn+1\) or \(mn+n\). They observe that each integral domains \(R\) is close to being Noetherian and has a canonically associated Prüfer overring \(T\) such that the contraction map \(\mathrm{Spec}(T)\rightarrow \mathrm{Spec}(R)\) is a homeomorphism. In the second result of this article, used in the proof of the above result, appears not to have been noticed before: For an extension \(k\subset K\) of fields, the extension \(k[x_1]],\dots,[x_n]] \hookrightarrow K[x_1]],\dots,[x_n]]\) is integral and the dimension of the generic fibre is \(0\), if the maximal separable extension \(k_0\) of \(k \in K\) is a finite extension of \(k\) and \(K\) has finite exponent over \(k_0\). Otherwise the generic fibre has dimension \(n-1\).
scientific article; zbMATH DE number 6104613
scientific article