Formal fibers and complete homomorphic images

Abstract:

Let $(R,{\mathbf {m}})$ be an excellent normal local Henselian domain, and suppose that ${\mathbf {q}}$ is a prime ideal in $R$ of height $> 1$. We show that, if $R/{\mathbf {q}}$ is not complete, then there are infinitely many height one prime ideals ${\mathbf {p}} \subseteq {\mathbf {q}}\hat R$ of $\hat R$ with ${\mathbf {p}} \cap R = 0$; in particular, the dimension of the generic formal fiber of $R$ is at least one. This result may in fact indicate that a much stronger relationship between maximal ideals in the formal fibers of an excellent Henselian local ring and its complete homomorphic images is possibly satisfied. The second half of the paper is concerned with a property of excellent normal local Henselian domains $R$ with zero-dimensional formal fibers. We show that for such an $R$ one has the following good property with respect to intersection: for any field $L$ such that $\mathcal {Q}(R) \subseteq L \subseteq \mathcal {Q}(\hat R)$, the ring $L \cap \hat R$ is a local Noetherian domain which has completion $\hat R$.