BCM-thresholds of non-principal ideals

TL;DR

This paper introduces BCM-thresholds, a characteristic-free analog of F-thresholds for non-principal ideals, establishing their properties and relations to classical thresholds and jumping numbers.

Contribution

It generalizes the concept of F-thresholds to non-principal ideals using BCM-thresholds and explores their properties in various ring settings.

Findings

BCM-thresholds coincide with BCM-jumping numbers in complete local regular rings.

The new thresholds match classical F-thresholds for weakly F-regular rings.

Results on F-thresholds of parameter ideals and a mixed characteristic version of multiplicity are obtained.

Abstract

Generalizing previous work of the first author, we introduce and study a characteristic free analog of the $F$-threshold for non-principal ideals, BCM-thresholds. We show that this coincides with the classical $F$-threshold for weakly $F$-regular rings and that the set of BCM-thresholds coincides with the set of BCM-jumping numbers in a complete local regular ring. We obtain results on $F$-thresholds of parameter ideals analogous to results of Huneke-\mustata-Takagi-Watanabe as well as a mixed characteristic version of one of their results on multiplicity. Instead of taking ordinary powers of an ideal, our definition uses fractional integral closure in an absolute integral closure of our ambient ring.