Nonnoetherian coordinate rings with unique maximal depictions

TL;DR

This paper investigates the structure of nonnoetherian coordinate rings, establishing conditions under which they have unique maximal depictions and how their geometric properties can be derived from these depictions.

Contribution

It proves the existence and uniqueness of maximal depictions for certain nonnoetherian rings and relates their geometric dimensions to these depictions.

Findings

Unique maximal depiction exists if R is noetherian in codimension 1.

Normal depiction S is the unique maximal depiction when it exists.

Geometric dimensions can be computed directly from the maximal depiction.

Abstract

A depiction of a nonnoetherian integral domain $R$ is a special coordinate ring that provides a framework for describing the geometry of $R$. We show that if $R$ is noetherian in codimension 1, then $R$ has a unique maximal depiction $T$. In this case, the geometric dimensions of the points of $\operatorname{Spec}R$ may be computed directly from $T$. If in addition $R$ has a normal depiction $S$, then $S$ is the unique maximal depiction of $R$.