BCM-thresholds of non-principal ideals
Sandra Rodríguez-Villalobos, Karl Schwede
TL;DR
The paper defines BCM-thresholds as a characteristic-free analogue of $F$-thresholds for non-principal ideals using fractional integral closures in $R^+$ and big Cohen-Macaulay algebras. It shows that these thresholds recover the classical $F$-threshold in weakly $F$-regular rings and correspond to $F$-jumping numbers in regular rings; it also yields a valuative/integral-closure interpretation via $(\mathfrak{a}R^+)_{>t}$ and a tight-closure variant $c_*^J(\mathfrak{a})$. The work derives parameter-ideal bounds analogous to Huneke–Mustata–Takagi–Watanabe and relates thresholds to BCM-test ideals, including explicit formulas for jumping numbers in positive and mixed characteristic using perfectoid BCM algebras. It provides a coherent framework tying thresholds, integral closure, and test ideals across equal, positive, and mixed characteristic, and ends with open questions about extending these results further and understanding functoriality and finiteness in this characteristic-free setting.
Abstract
Generalizing previous work of the first author, we introduce and study a characteristic free analog of the $F$-threshold for non-principal ideals. We show that this coincides with the classical $F$-threshold for weakly $F$-regular rings and that the set of $F$-thresholds coincides with the set of $F$-jumping numbers in a regular ring. We obtain results on $F$-thresholds of parameter ideals analogous to results of Huneke-Musta{ţ}{ă}-Takagi-Watanabe. Instead of taking ordinary powers of ideal, our definition uses fractional integral closure in an absolute integral closure of our ambient ring.