The converse of a theorem by Bayer and Stillman

TL;DR

This paper characterizes the reverse lexicographic order as the unique monomial order that preserves the regularity of ideals through generic initial ideals, and establishes the uniqueness of monomial orders based on their generic initial ideals.

Contribution

It proves the reverse lexicographic order is uniquely characterized by the property of preserving regularity for all ideals and shows that identical generic initial ideals imply identical monomial orders.

Findings

Reverse lexicographic order uniquely preserves regularity.

Identical generic initial ideals imply identical monomial orders.

Characterization of monomial orders based on generic initial ideals.

Abstract

Bayer-Stillman showed that $reg(I) = reg(gin_\tau(I))$ when $\tau$ is the graded reverse lexicographic order. We show that the reverse lexicographic order is the unique monomial order $\tau$ satisfying $reg(I) = reg(gin_\tau(I))$ for all ideals $I$. We also show that if $gin_{\tau_1}(I) = gin_{\tau_2}(I)$ for all $I$, then $\tau_1 = \tau_2$.