Extensions of local domains with trivial generic fiber

Abstract

We consider injective local maps from a local domain $R$ to a local domain $S$ such that the generic fiber of the inclusion map $R \hookrightarrow S$ is trivial, that is, $P \cap R \ne (0)$ for every nonzero prime ideal $P$ of $S$. We present several examples of injective local maps involving power series that have or fail to have this property. For an extension $R \hookrightarrow S$ having this property, we give some results on the dimension of $S$; in some cases we show $\dim S = 2$ and in some cases $\dim S = 1$.