Generic formal fibers and analytically ramified stable rings

TL;DR

This paper investigates the relationship between the structure of generic formal fibers of local Noetherian domains and the properties of analytically ramified stable rings, especially focusing on stable rings and their embedding dimensions.

Contribution

It extends the understanding of the correspondence between prime ideals in generic formal fibers and analytically ramified stable rings, providing new characterizations and insights into their embedding dimensions.

Findings

Embedding dimension of stable rings reflects the deviation from regularity.

Characterizations of analytically ramified local stable domains are established.

The work deepens the link between formal fiber properties and stable ring structures.

Abstract

Let $A$ be a local Noetherian domain of Krull dimension $d$. Heinzer, Rotthaus and Sally have shown that if the generic formal fiber of $A$ has dimension $d-1$, then $A$ is birationally dominated by a one-dimensional analytically ramified local Noetherian ring having residue field finite over the residue field of $A$. We explore further this correspondence between prime ideals in the generic formal fiber and one-dimensional analytically ramified local rings. Our main focus is on the case where the analytically ramified local rings are stable, and we show that in this case the embedding dimension of the stable ring reflects the embedding dimension of a prime ideal maximal in the generic formal fiber, thus providing a measure of how far the generic formal fiber deviates from regularity. A number of characterizations of analytically ramified local stable domains are also given.