# Depth of an initial ideal

## TL;DR

This paper constructs specific ideals in polynomial rings demonstrating that the depth of initial ideals can vary widely, even when the quotient ring is Cohen–Macaulay, highlighting the complexity of initial ideal behavior.

## Contribution

It provides a construction of ideals with prescribed initial ideal depths across all possible values, revealing new insights into the depth behavior of initial ideals.

## Key findings

- Initial ideals can have any depth between 0 and d.
- Constructed ideals are Cohen–Macaulay with variable initial ideal depths.
- Depth of initial ideals is highly flexible for certain Cohen–Macaulay rings.

## Abstract

Given an arbitrary integer $d>0$, we construct a homogeneous ideal $I$ of the polynomial ring $S = K[x_1, \ldots, x_{3d}]$ in $3d$ variables over a filed $K$ for which $S/I$ is a Cohen--Macaulay ring of dimension $d$ with the property that, for each of the integers $0 \leq r \leq d$, there exists a monomial order $<_r$ on $S$ with ${\rm depth} (S/{\rm in}_{<_r}(I)) = r$, where ${\rm in}_{<_r}(I)$ is the initial ideal of $I$ with respect to $<_r$.