Test ideals in non-Q-Gorenstein rings

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 $τ_b(R) = \tld τ(R)$ 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.

18 pages, minor changes, to appear in Transactions of the American Mathematical Society

References in corpus (1)

• Globally $F$-regular and log Fano varieties

Cited by in corpus (2)

• F-signature of pairs and the asymptotic behavior of Frobenius splittings
• A note on discreteness of $F$-jumping numbers