Generic tropical initial ideals of Cohen-Macaulay algebras

TL;DR

This paper investigates the structure of generic tropical initial ideals of Cohen-Macaulay algebras, providing formulas and properties that enhance understanding of their algebraic and geometric features.

Contribution

It offers a new formula for initial ideals of Cohen-Macaulay algebras and characterizes when these ideals are prime, especially in the domain case.

Findings

Formulas for generic tropical initial ideals.

Initial ideals from codimension-1 skeletons are prime in domains.

Enhanced understanding of algebraic structure of Cohen-Macaulay algebras.

Abstract

We study the generic tropical initial ideals of a positively graded Cohen-Macaulay algebra $R$ over an algebraically closed field $\mathbf{k}$. Building on work of R\"omer and Schmitz, we give a formula for each initial ideal, and we express the associated quasivaluations in terms of certain $I$-adic filtrations. As a corollary, we show that in the case that $R$ is a domain, every initial ideal coming from the codimension-$1$ skeleton of the tropical variety is prime, so "generic presentations of Cohen-Macaulay domains are well-poised in codimension-$1$."