On the discreteness and rationality of F-jumping coefficients

TL;DR

This paper extends the understanding of F-jumping coefficients, proving their discreteness and rationality in a broader class of regular local rings using novel methods that do not rely on D-modules.

Contribution

It introduces a new approach to analyze F-jumping coefficients in excellent regular local rings, broadening applicability beyond F-finite cases.

Findings

F-jumping coefficients are discrete and rational in the studied rings.

New method avoids D-modules, analyzing nilpotent modules under Frobenius actions.

Results extend previous theorems to more general regular local rings.

Abstract

This paper studies the jumping coefficients of principal ideals of regular local rings. Recently M. Blickle, M. Mustata and K. Smith showed that, when $R$ is of essentially finite type over a field and $F$-finite, bounded intervals contain finitely many jumping coefficients and that those are rational. In a later paper they extended these results to principal ideals of $F$-finite complete regular local rings. The aim of this paper is to extend these results on the discreteness and rationality of jumping coefficients to principal ideals of arbitrary (i.e. not necessarily $F$-finite) excellent regular local rings containing fields of positive characteristic. Our proof uses a very different method: we do not use $D$-modules and instead we analyze the modules of nilpotents elements in the injective hull or $R$ under some non-standard Frobenius actions. This new method undoubtedly holds a potential for more applications.