Generalized test ideals, sharp F-purity, and sharp test elements
| dc.creator | Schwede, Karl | |
| dc.date | 2007-11-21 | |
| dc.date | 2008-01-26 | |
| dc.date.accessioned | 2026-07-07T13:08:35Z | |
| dc.date.available | 2026-07-07T13:08:35Z | |
| dc.description | 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 $τ_R(\ba^t)$, the generalized test ideal associated to $(R, a^t)$ as defined by Hara and Yoshida. We show that $τ_R(a^t) \cap R^{\circ}$ 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 $τ_R(\ba^t)$ 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 $R/{τ_R(\ba^t)}$ 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. | |
| dc.description | Theorem 2.9 added. Several typos corrected and proofs expanded. To appear in Mathematical Research Letters | |
| dc.identifier | https://arxiv.org/abs/0711.3380 | |
| dc.identifier | http://arxiv.org/abs/0711.3380 | |
| dc.identifier | Math. Res. Lett. 15 (2008), no. 6, 1251--1261 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/228461 | |
| dc.subject | Commutative Algebra | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 13A35, 14B05 | |
| dc.title | Generalized test ideals, sharp F-purity, and sharp test elements | |
| dc.type | text |