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

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.

Minor changes. To appear in the Transactions of the AMS

References in corpus (3)

• Test ideals via algebras of $p^{-e}$-linear maps
• On the behavior of Castelnuovo-Mumford regularity with respect to some functors
• A note on discreteness of $F$-jumping numbers