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

Karl Schwede

paper

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

arXiv:0711.3380

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 $, an d$ t > 0 $i s a r e a l n u mb er . I n t hi ss i t u a t i o n w e ha v e t h e i d e a l$ τ_R(\ba^t) $, t h e g e n er a l i z e d t es t i d e a l a ssoc ia t e d t o$ (R, a^t) $a s d e f in e d b y H a r aan d Y os hi d a . W es h o w t ha t$ τ_R(a^t) \cap R^{\circ} $i s ma d e u p o f a pp r o p r ia t e l y d e f in e d g e n er a l i z e d t es t e l e m e n t s w hi c h w ec a l l sharp test elements . W e a l so d e f in e a v a r ian t o f$ F$-purity for pairs, \emph{sharp $F$-purity}, which interacts well with sharp test elements and agrees with previously defined notions of $F $- p u r i t y inman y co mm o n s i t u a t i o n s . W es h o w t ha t i f$ (R, \ba^t) $i ss ha r pl y F - p u r e, t h e n$ τ_R(\ba^t) $i s a r a d i c a l i d e a l . F u r t h er m or e, b y f o l l o w in g ana r g u m e n t o f V a ss i l e v, w es h o w t ha t i f$ R $i s a q u o t i e n t o f an$ F $- f ini t er e g u l a r l oc a l r in g an d$ (R, \ba^t) $i ss ha r pl y$ F $- p u r e, t h e n$ R/{τ_R(\ba^t)} $i t se l f i s$ F $- p u r e . W eco n c l u d e b y s h o w in g t ha t s ha r p$ F $- p u r i t y c anb e u se d t o d e f in e t h e$ F $- p u r e t h r es h o l d . A s ana ppl i c a t i o n w es h o w t ha tt h e$ F$-pure threshold must be a rational number under certain hypotheses.

Theorem 2.9 added. Several typos corrected and proofs expanded. To appear in Mathematical Research Letters

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