Regularity functions of powers of graded ideals

TL;DR

This paper characterizes the possible growth patterns of regularity functions of powers of graded ideals, revealing their asymptotic linearity and answering longstanding questions about their behavior.

Contribution

It provides a complete characterization of regularity functions for ideals with zero-dimensional quotients and explores their asymptotic linearity in higher dimensions.

Findings

Regularity functions are asymptotically linear with slope equal to the degree of generators.

Any numerical function above certain bounds can be realized as a regularity function.

The saturation degree of powers of an ideal is asymptotically linear.

Abstract

This paper studies the problem of which sequences of non-negative integers arise as the functions $\operatorname{reg} I^{n-1}/I^n$, $\operatorname{reg} R/I^n$, $\operatorname{reg} I^n$ for an ideal $I$ generated by forms of degree $d$ in a standard graded algebra $R$. These functions are asymptotically linear with slope $d$. If $\dim R/I = 0$, we give a complete characterization of all numerical functions which arise as the functions $\operatorname{reg} I^{n-1}/I^n$, $\operatorname{reg} R/I^n$ and show that $\operatorname{reg} I^n$ can be any numerical function $f(n) \ge dn$ that weakly decreases until it becomes a linear function with slope $d$. The latter result gives a negative answer to a question of Eisenbud and Ulrich. If $\dim R/I \ge 1$, we show that $\operatorname{reg} I^{n-1}/I^n$ can be any numerical asymptotically linear function $f(n) \ge dn-1$ with slope $d$ and $\operatorname{reg} R/I^n$ can be any numerical asymptotically linear function $f(n) \ge dn-1$ with slope $d$ that is weakly increasing. Inspired of a recent work of Ein, Ha and Lazarsfeld on non-singular complex projective schemes, we also prove that the function of the saturation degree of $I^n$ is asymptotically linear for an arbitrary graded ideal $I$ and study the behavior of this function.