Test ideals can be computed by boundaries

Abstract

Let $R$ be a normal $F$-finite ring of characteristic $p>0$. We show that one effective $\mathbb{Q}$-divisor $\Delta$, with $K_R+\Delta$ $\mathbb{Q}$-Cartier of index prime to $p$, computes the big test ideal. This answers a question of Schwede and Smith. More generally, we prove their conjecture that test module of a Cartier algebra acting on an ideal is the test module of a single map $\psi$ in the algebra and even the image of $\psi$. We prove a global version for schemes quasi-projective over an $F$-finite local ring as well. In particular, every such normal, locally strongly $F$-regular scheme admits an effective boundary $\Delta$ for which $(X,\Delta )$ is locally strongly $F$-regular and $K_X+\Delta$ is $\mathbb{Q}$-Cartier of index prime to $p$.

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