Growth of Hilbert coefficients of Syzygy modules

TL;DR

This paper investigates the growth patterns of Hilbert coefficients of syzygy modules over complete intersection rings, revealing quasi-polynomial behaviors and stability properties of associated graded modules' depths.

Contribution

It establishes new quasi-polynomial formulas for Hilbert coefficients of syzygy modules and demonstrates stability of depth functions under certain Cohen-Macaulay conditions.

Findings

Hilbert coefficients of syzygy modules follow quasi-polynomial patterns with period 2.

Depth functions of associated graded modules stabilize for large syzygy indices.

Under Cohen-Macaulay conditions, the limits of depth functions are eventually constant.

Abstract

Let $(A,\mathfrak{m})$ be a complete intersection ring of dimension $d$ and let $I$ be an $\mathfrak{m}$-primary ideal. Let $M$ be a maximal \CM \ $A$-module. For $i = 0,1,\cdots,d$, let $e_i^I(M)$ denote the $i^{th}$ Hilbert -coefficient of $M$ with respect to $I$. We prove that for $i = 0, 1, 2$, the function $j \mapsto e_i^I(Syz_j^A(M))$ is of quasi-polynomial type with period $2$. Let $G_I(M)$ be the associated graded module of $M$ with respect to $I$. If $G_I(A)$ is Cohen-Macaulay and $\dim A \leq 2$ we also prove that the functions $j \mapsto depth \ G_I(Syz^A_{2j+i}(M))$ are eventually constant for $i = 0, 1$. Let $\xi_I(M) = \lim_{l \rightarrow \infty} depth \ G_{I^l}(M)$. Finally we prove that if $\dim A = 2$ and $G_I(A)$ is Cohen-Macaulay then the functions $j \mapsto \xi_I(Syz^A_{2j + i}(M))$ are eventually constant for $i = 0, 1$.