Test ideals vs. multiplier ideals

TL;DR

This paper explores the relationship between test ideals and multiplier ideals, highlighting their differences through examples and proving that every ideal in an F-finite regular local ring can be expressed as a generalized test ideal.

Contribution

It demonstrates that all ideals in F-finite regular local rings can be represented as generalized test ideals and examines the semicontinuity of F-pure thresholds.

Findings

Examples showing different behaviors of test and multiplier ideals

Every ideal in an F-finite regular local ring can be a generalized test ideal

Semicontinuity of F-pure thresholds established

Abstract

The generalized test ideals introduced in [HY] are related to multiplier ideals via reduction to characteristic p. In addition, they satisfy many of the subtle properties of the multiplier ideals, which in characteristic zero follow via vanishing theorems. In this note we give several examples to emphasize the different behavior of test ideals and multiplier ideals. Our main result is that every ideal in an F-finite regular local ring can be written as a generalized test ideal. We also prove the semicontinuity of $F$-pure thresholds (though the analogue of the Generic Restriction Theorem for multiplier ideals does not hold).