Test ideals in non-Q-Gorenstein rings

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 $\sum_Δ τ(R; Δ)$ where the sum is over $Δ$ such that $K_X + Δ$ 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.

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