Symbolic powers of monomial ideals

TL;DR

This paper studies the asymptotic behavior of the lengths of certain symbolic powers of monomial ideals, showing that specific coefficients in their quasi-polynomial expressions become constant under particular conditions.

Contribution

It proves that the second leading coefficient in the quasi-polynomial describing lengths of symbolic powers stabilizes to a constant when the ideal is generated by elements of the same degree and has height at least 2.

Findings

The function describing lengths is of quasi-polynomial type with periodic coefficients.

The leading coefficient is constant for large n, as previously shown.

The second coefficient also becomes constant under specified conditions.

Abstract

Let $A = K[X_1,\ldots, X_d]$ and let $I$, $J$ be monomial ideals in $A$. Let $I_n(J) = (I^n \colon J^\infty)$ be the $n^{th}$ symbolic power of $I$ \wrt \ $J$. It is easy to see that the function $f^I_J(n) = e_0(I_n(J)/I^n)$ is of quasi-polynomial type, say of period $g$ and degree $c$. For $n \gg 0$ say \[ f^I_J(n) = a_c(n)n^c + a_{c-1}(n)n^{c-1} + \text{lower terms}, \] where for $i = 0, \ldots, c$, $a_i \colon \mathbb{N} \rt \mathbb{Z}$ are periodic functions of period $g$ and $a_c \neq 0$. In an earlier paper we (together with Herzog and Verma) proved that $\dim I_n(J)/I^n$ is constant for $n \gg 0$ and $a_c(-)$ is a constant. In this paper we prove that if $I$ is generated by some elements of the same degree and height $I \geq 2$ then $a_{c-1}(-)$ is also a constant.

Equations

$$f_J^I(n)=a_c(n)n^c+a_{c-1}(n)n^{c-1}+\text{lower terms},$$

$$f(n)=a_c(n)n^c+a_{c-1}(n)n^{c-1}+\text{lower terms}$$

$$f_J^I(n)=a_c(n)n^c+a_{c-1}(n)n^{c-1}+\text{lower terms}.$$

Full text

[TABLE]

where for $i=0,\ldots,c$, $a_{i}\colon\mathbb{N}\rightarrow\mathbb{Z}$ are periodic functions of period $g$ and $a_{c}\neq 0$. In [HPV, 2.4] we (together with Herzog and Verma) proved that $\dim I_{n}(J)/I^{n}$ is constant for $n\gg 0$ and $a_{c}(-)$ is a constant. In this paper we prove that if $I$ is generated by some elements of the same degree and $0ptI\geq 2$ then $a_{c-1}(-)$ is also a constant.

1. introduction

Let us recall that a function $f\colon\mathbb{N}\rightarrow\mathbb{Z}$ is called of quasi-polynomial of period $g$ and degree $c$ if

[TABLE]

where for $i=0,\ldots,c$, $a_{i}\colon\mathbb{N}\rightarrow\mathbb{Z}$ are periodic functions of period $g$ and $a_{c}\neq 0$. Many interesting functions are of quasi-polynomial type (i.e., it coincides with a quasi-polynomial for $n\gg 0$). For instance the Hilbert function of a graded module over a not-necessarily standard graded $K$-algebra is of quasi-polynomial type; see [BH, 4.4.3].

Let $f$ be of quasi-polynomial type. Following Erhart we let the grade of $f$ denote the smallest integer $\delta\geq-1$ such that $a_{j}(-)$ is constant for all $j>\delta$. Erhart had conjectured that if $P$ is a $d$-dimensional rational polytope in $\mathbb{R}^{m}$ such that for some $\delta$ the affine span of every $\delta$-dimensional face of $P$ contains a point with integer coordinates then the grade of the Erhart quasi-polynomial of $P$ is $<\delta$. This conjecture was proved by independently by McMullen [M] and Stanley [S, Theorem 2.8]. For a purely algebraic proof see [BI, Theorem 5].

1.1**.**

Our interest in quasi-polynomials arise in the following context (see [HPV]): Let $A=K[X_{1},\ldots,X_{d}]$ be a standard graded polynomial ring over a field $K$. If $M$ is a graded $A$-module then let $e_{0}(M)$ denote its multiplicity. Let $I,J$ be monomial ideals. Let $I_{n}(J)=I^{n}\colon J^{\infty}$ be the $n^{th}$ symbolic power of $I$ with respect to $J$. Then by [HHT, 3.2] we have that $\bigoplus_{n\geq 0}I_{n}(J)$ is a finitely generated $A$-algebra. From this it is easy to see that $f^{I}_{J}(n)=e_{0}(I_{n}(J)/I^{n})$ is of quasi-polynomial type, say of period $g$ and degree $c$. We write

[TABLE]

In [HPV, 2.4] we proved that $\dim I_{n}(J)/I^{n}$ is constant for $n\gg 0$ and $a_{c}(-)$ is a positive constant. In this short paper we prove

Theorem 1.2**.**

[with hypotheses as in 1.1] Further assume that $I$ is generated by some elements of the same degree and $0ptI\geq 2$. Then $a_{c-1}(-)$ is also a constant.

2. Preliminaries

Let $K$ be a field and let $R=\bigoplus_{m\geq 0}R_{m}$ be a standard graded, finitely generated $K=R_{0}$-algebra. Let $M=\bigoplus_{m\geq 0}M_{m}$ be a graded $R$-module. Its Hilbert series is $H_{M}(z)=\sum_{m\geq 0}\dim_{K}M_{m}$. It is well-known that $H_{M}(z)=h_{M}(z)/(1-z)^{\dim M}$ where $h_{M}(z)\in\mathbb{Z}[z]$. The number $e_{0}(M)=h_{M}(1)$ is called the multiplicity of $M$.