Test ideals in non-Q-Gorenstein rings

TL;DR

This paper proves that the big test ideal in certain positive characteristic rings can be expressed as a sum over divisors with $Q$-Cartier canonical class, extending to non-normal rings.

Contribution

It establishes a new characterization of the big test ideal in non-Q-Gorenstein rings, answering a question posed by several researchers.

Findings

The big test ideal equals the sum over $ au(R; riangle)$ for $ riangle$ with $K_X + riangle$ $Q$-Cartier.

The result holds even when the ring $R$ is not necessarily normal.

Abstract

Suppose that $X = \Spec R$ is an $F$-finite normal variety in characteristic $p > 0$. In this paper we show that the big test ideal $\tau_b(R) = \tld \tau(R)$ is equal to $\sum_{\Delta} \tau(R; \Delta)$ where the sum is over $\Delta$ such that $K_X + \Delta$ is $\bQ$-Cartier. This affirmatively answers a question asked by various people, including Blickle, Lazarsfeld, K. Lee and K. Smith. Furthermore, we have a version of this result in the case that $R$ is not even necessarily normal.