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- W2023845061 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." @default.
- W2023845061 created "2016-06-24" @default.
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- W2023845061 date "2007-11-21" @default.
- W2023845061 modified "2023-09-27" @default.
- W2023845061 title "Generalized test ideals, sharp F-purity, and sharp test elements" @default.
- W2023845061 hasPublicationYear "2007" @default.
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