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.

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 $<_r$.

Equations

$$I=(x_1^2-x_2{x}_3,x_1{x}_2-x_3^2,x_1{x}_3-x_2^2)\subset S$$

$$\{x_1^2-x_2{x}_3,x_1{x}_2-x_3^2,x_1{x}_3-x_2^2,x_2^3-x_3^3\}$$

$$\{x_2{x}_3-x_1^2,x_3^2-x_1{x}_2,x_2^2-x_1{x}_3\}$$

$$I=(\{x_{3i-2}^2-x_{3i-1}{x}_{3i},x_{3i-2}{x}_{3i-1}-x_{3i}^2,x_{3i-2}{x}_{3i}-x_{3i-1}^2:1\le i\le d\})\subset S.$$

$$I_i=(x_{3i-2}^2-x_{3i-1}{x}_{3i},x_{3i-2}{x}_{3i-1}-x_{3i}^2,x_{3i-2}{x}_{3i}-x_{3i-1}^2)\subset S_i,$$

$$S/I\cong S_1/I_1{\otimes }_K\cdots {\otimes }_K{S}_d/I_d$$

$$x_{3d}\prec x_{3d-1}\prec \cdots \prec x_1$$

$$w_r=(\underset{3r}{\underbrace{1,2,2,\dots ,1,2,2}},\underset{3(d-r)}{\underbrace{1,1,1,\dots ,1,1,1}})\in {\mathbb{Z}}^{3d},0\le r\le d.$$

$$A=\{x_{3i-1}{x}_{3i}-x_{3i-2}^2,x_{3i-1}^2-x_{3i-2}{x}_{3i},x_{3i}^2-x_{3i-2}{x}_{3i-1}:1\le i\le r\}$$

$$B=\{x_{3i-2}^2-x_{3i-1}{x}_{3i},x_{3i-2}{x}_{3i-1}-x_{3i}^2,x_{3i-1}{x}_{3i}-x_{3i}^2,x_{3i-1}^3-x_{3i}^3:r+1\le i\le d\},$$

$$S/{\mathrm{in}}_{{<}_r}(I)$$

$$\mathrm{depth}(S/{\mathrm{in}}_{{<}_r}(I))+\mathrm{reg}(S/{\mathrm{in}}_{{<}_r}(I))=2d$$

$$I_D=(\{x_{\alpha }{x}_{\beta }-x_{\alpha \wedge \beta }{x}_{\alpha \vee \beta }:\alpha \nleqq \beta ,\beta \nleqq \alpha \})\subset S.$$

Full text

The first author was partially supported by JSPS KAKENHI 19H00637. The second author was partially supported by JSPS KAKENHI 19K14505.

1. Background

In order to answer a question suggested in [2, p. 38], the first author [6, p. 285] discovered a graded Gorenstein Hodge algebra whose corresponding discrete Hodge algebra is not a Cohen–Macaulay ring. In the modern language of Gröbner bases and initial ideals, the work guarantees the existence of a homogeneous ideal $I$ of the polynomial ring $S=K[x_{1},\ldots,x_{n}]$ over a field $K$ for which $S/I$ is Cohen–Macaulay with the property that there is an initial ideal ${\rm in}_{<}(I)$ of $I$ for which $S/({\rm in}_{<}(I))$ is not Cohen–Macaulay. On the other hand, in [1, Corollary 3.9], it is shown that if $A$ is an ASL (algebra with straightening laws [3]) and $A_{0}$ is its discrete ASL, then $\operatorname{depth}A=\operatorname{depth}A_{0}$. In particular the discrete ASL of a Cohen–Macaulay ASL is again Cohen–Macaulay.

Take the above background into consideration, one cannot escape the temptation to study the question as follows:

Question 1.1**.**

Given an arbitrary integer $d>0$, does there exist a homogeneous ideal $I$ of the polynomial ring $S$ 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 is 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 $<_{r}$? **

The purpose of the present paper is to solve Question 1.1 and, in addition, to supply related questions.

2. Result

We refer the reader to [4, Chapter 2] for fundamental materials and standard notation on Gröbner bases.

Let $S=K[x_{1},\ldots,x_{n}]$ denote the polynomial ring in $n$ variables over a field $K$. Given a vector ${\mathbb{w}}=(w_{1},\ldots,w_{d})\in{\mathbb{Z}}^{n}$ and a monomial $u=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}\in S$, the weight of $u$ with respect to ${\mathbb{w}}$ is defined to be $a_{1}w_{1}+\cdots+a_{n}w_{n}$.

Theorem 2.1**.**

Given an arbitrary integer $d>0$, there exists 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 is a monomial order $<_{r}$ on $S$ with ${\rm depth}(S/{\rm in}_{<_{r}}(I))=r$.

Proof.

(First Step)   Let $d=1$. Let $S=K[x_{1},x_{2},x_{3}]$ and

[TABLE]

([4, Example 3.3.6]). Then $S/I$ is a one-dimensional Cohen–Macaulay ring. Let $\prec$ be the lexicographic order on $S$ with $x_{3}\prec x_{2}\prec x_{1}$. Let, in addition, ${\mathbb{w}}_{0}=(1,1,1)$ and ${\mathbb{w}}_{1}=(1,2,2)$. For each $r\in\{0,1\}$, we introduce the monomial order $<_{r}$ on $S$ as follows: One has $u<_{r}v$ if and only if one of the following holds:

• •

The weight of $u$ is less than that of $v$ with respect to ${\mathbb{w}}_{r}$;

• •

The weight of $u$ is equal to that of $v$ with respect to ${\mathbb{w}}_{r}$ and $u\prec v$.

Then

[TABLE]

is a Gröbner basis of $I$ with respect to $<_{0}$ and ${\rm depth}(S/{\rm in}_{<_{0}}(I))=0$. On the other hand,

[TABLE]

is a Gröbner basis of $I$ with respect to $<_{1}$ and ${\rm depth}(S/{\rm in}_{<_{1}}(I))=1$.

(Second Step)   Let $d>1$. Let $S=K[x_{1},\ldots,x_{3d}]$ and

[TABLE]

Let $S_{i}=K[x_{3_{i}-2},x_{3i-1},x_{3i}]$ and

[TABLE]

where $1\leq i\leq d$. Thus

[TABLE]

and $S/I$ is a Cohen–Macaulay ring of dimension $d$.

Now, we employ the lexicographic order $\prec$ on $S$ with

[TABLE]

on $S$ and the vectors

[TABLE]

For each $0\leq r\leq d$, we introduce the monomial order $<_{r}$ on $S$ as follows: One has $u<_{r}v$ if and only if one of the following holds:

• •

The weight of $u$ is less than that of $v$ with respect to ${\mathbb{w}}_{r}$;

• •

The weight of $u$ is equal to that of $v$ with respect to ${\mathbb{w}}_{r}$ and $u\prec v$.

It then follows that the set $A\cup B$, where

[TABLE]

and

[TABLE]

is a Gröbner basis of $I$ with respect to $<_{r}$. Since

[TABLE]

one has ${\rm depth}(S/{\rm in}_{<_{r}}(I))=r$, as desired.

Remark 2.2**.**

Let $S/I$ be the quotient ring studied in the proof of Theorem 2.1 and $\operatorname{reg}(S/I)$ its regularity. On has $\operatorname{reg}(S/I)=d$. Furthermore, it follows that

[TABLE]

for each of $0\leq r\leq d$. **

3. Questions

We conclude the present paper with related questions.

Question 3.1**.**

In Question 1.1, one may ask if $S/I$ can be a Gorenstein ring. **

Question 3.2**.**

In Question 1.1, one may ask if $I$ can be a prime ideal. **

Question 3.3**.**

Let $I\subset S=K[x_{1},\ldots,x_{n}]$ be a homogeneous ideal with $\operatorname{reg}(S/I)=r$ and ${\rm depth}(S/I)=e$. Suppose that there is a monomial order $\prec$ on $S$ with $\operatorname{reg}(S/{\rm in}_{\prec}(I))=r^{\prime}$ and ${\rm depth}(S/{\rm in}_{\prec}(I))=e^{\prime}$. Then, for each $r\leq r^{\prime\prime}\leq r^{\prime}$ and for each $e^{\prime}\leq e^{\prime\prime}\leq e$, does there exist a monomial order $\prec^{\prime}$ on $S$ with $\operatorname{reg}(S/{\rm in}_{\prec^{\prime}}(I))=r^{\prime\prime}$ and ${\rm depth}(S/{\rm in}_{\prec^{\prime}}(I))=e^{\prime\prime}$? **

Let $I\subset S=K[x_{1},\ldots,x_{n}]$ be the toric ideal of a unimodular convex polytope [5, p. 107]. Thus, for any monomial order $\prec$ on $S$, its initial ideal ${\rm in}_{\prec}(I)$ is generated by squarefree monomials. It then follows from [1] that $\operatorname{reg}(S/{\rm in}_{\prec}(I))=\operatorname{reg}(S/I)$ and ${\rm depth}(S/I)={\rm depth}(S/{\rm in}_{\prec}(I))$.

Question 3.4**.**

Is there a nice class of toric ideals $I\subset S$ for which $\operatorname{reg}(S/{\rm in}_{\prec}(I))=\operatorname{reg}(S/I)$ for any monomial order $\prec$ on $S$ and for which there is a monomial order $\prec$ on $S$ with ${\rm depth}(S/{\rm in}_{\prec}(I))<{\rm depth}(S/I)$? **

Let $D$ be a finite distributive lattice and ${\mathcal{R}}_{K}[D]$ the ASL on $D$ over a field $K$ ([7, p. 98]). It is known that ${\mathcal{R}}_{K}[D]$ is normal and Cohen–Macaulay. Its discrete ASL is Cohen–Macaulay. Let $S=K[\{x_{\alpha}:\alpha\in D\}]$ the polynomial ring in $|D|$ variables over $K$. The defining ideal of ${\mathcal{R}}_{K}[D]$ is

[TABLE]

Question 3.5**.**

For which finite distributive lattices $D$, does there exist a monomial order $\prec$ on $S=K[\{x_{\alpha}:\alpha\in D\}]$ with $\min{\rm depth}(S/{\rm in}_{\prec}(I_{D}))=0$?**