Growth of multiplicities of graded families of ideals

TL;DR

This paper investigates the asymptotic behavior of the length function of graded families of ideals in Noetherian local rings, establishing bounds on the difference of lengths that grow polynomially with degree related to the ring's dimension.

Contribution

It provides a bound on the difference of lengths of successive ideals in graded families, showing the growth is controlled by a polynomial of degree d-1.

Findings

Existence of a constant b3 controlling the growth

Difference of lengths is bounded by b3 n^{d-1}

Growth rate is polynomial in n

Abstract

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d > 0$. Let $I_\bullet = \{I_n\}_{n \in \mathbb{N}}$ be a graded family of $\mathfrak{m}$-primary ideals in $R$. We examine how far off from a polynomial can the length function $\ell_R(R/I_n)$ be asymptotically. More specifically, we show that there exists a constant $\gamma > 0$ such that for all $n \ge 0$, $$\ell_R(R/I_{n+1}) - \ell_R(R/I_n) < \gamma n^{d-1}.$$