Regularity functions of powers of graded ideals

Abstract

This paper studies the problem of which sequences of non-negative integers arise as the functions $\mathrm{reg}{I}^{n-1}/I^n$, $\mathrm{reg}R/I^n$, $\mathrm{reg}{I}^n$ 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 $\mathrm{reg}{I}^{n-1}/I^n$, $\mathrm{reg}R/I^n$ and show that $\mathrm{reg}{I}^n$ 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 $\mathrm{reg}{I}^{n-1}/I^n$ can be any numerical asymptotically linear function $f(n)\ge dn-1$ with slope $d$ and $\mathrm{reg}R/I^n$ 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.

24 pages