Castelnuovo-Mumford regularity and the discreteness of $F$-jumping coefficients in graded rings

TL;DR

This paper proves that the sets of $F$-jumping coefficients in certain graded rings are discrete by establishing linear bounds on Castelnuovo-Mumford regularity growth, with applications to specific determinantal rings.

Contribution

It introduces a criterion based on linear bounds for regularity growth to prove discreteness of $F$-jumping coefficients in graded rings.

Findings

Discreteness of $F$-jumping coefficients in certain graded $F$-finite rings.

Existence of linear bounds for Castelnuovo-Mumford regularity in one- and two-dimensional rings.

Discreteness of $F$-jumping coefficients in determinantal rings from $2\times 2$ minors.

Abstract

In this paper we show that the sets of $F$-jumping coefficients of ideals form discrete sets in certain graded $F$-finite rings. We do so by giving a criterion based on linear bounds for the growth of the Castelnuovo-Mumford regularity of certain ideals. We further show that these linear bounds exists for one-dimensional rings and for ideals of (most) two-dimensional domains. We conclude by applying our technique to prove that all sets of $F$-jumping coefficients of all ideals in the determinantal ring given as the quotient by $2\times 2$ minors in a $2\times 3$ matrix of indeterminates form discrete sets.