Formal Fibers of Prime Ideals in Polynomial Rings

TL;DR

This paper proves that in certain Noetherian local domains, the maximal prime ideals in the generic formal fiber have height n-1, extending Matsumura's previous results and exploring related extensions.

Contribution

It establishes that all maximal prime ideals in the generic formal fiber of specific local domains have height n-1, confirming and extending Matsumura's findings.

Findings

Maximal prime ideals in the generic formal fiber have height n-1.

Results apply to extensions of mixed polynomial-power series rings.

Confirms the maximal height in the generic formal fiber for these domains.

Abstract

Let (R,m) be a Noetherian local domain of dimension n that is essentially finitely generated over a field and let R^ denote the m-adic completion of R. Matsumura has shown that n-1 is the maximal height possible for prime ideals of R^ in the generic formal fiber of R. In this article we prove that every prime ideal of R^ that is maximal in the generic formal fiber of R has height n-1. We also present a related result concerning the generic formal fibers of certain extensions of mixed polynomial-power series rings.