Generalized test ideals, sharp F-purity, and sharp test elements

Abstract

Consider a pair $(R, \ba^t)$ where $R$ is a ring of positive characteristic, $\ba$ is an ideal such that $a \cap $R^{\circ} \neq \emptyset$, and $t > 0$ is a real number. In this situation we have the ideal $\tau_R(\ba^t)$, the generalized test ideal associated to $(R, a^t)$ as defined by Hara and Yoshida. We show that $\tau_R(a^t) \cap R^{\circ}$ is made up of appropriately defined generalized test elements which we call \emph{sharp test elements}. We also define a variant of $F$-purity for pairs, \emph{sharp $F$-purity}, which interacts well with sharp test elements and agrees with previously defined notions of $F$-purity in many common situations. We show that if $(R, \ba^t)$ is sharply F-pure, then $\tau_R(\ba^t)$ is a radical ideal. Furthermore, by following an argument of Vassilev, we show that if $R$ is a quotient of an $F$-finite regular local ring and $(R, \ba^t)$ is sharply $F$-pure, then $R/{\tau_R(\ba^t)}$ itself is $F$-pure. We conclude by showing that sharp $F$-purity can be used to define the $F$-pure threshold. As an application we show that the $F$-pure threshold must be a rational number under certain hypotheses.