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

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.