On F-pure thresholds

Takagi, Shunsuke; Watanabe, Kei-ichi

doi:10.1016/j.jalgebra.2004.07.011

Cited by 97 publications

(79 citation statements)

References 19 publications

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“…Then R/τ R (a t ) is F -pure. Finally, we also show that sharp F -purity can be used to define c(a), the F -pure threshold of (R, a * ); see [TW04]. As an easy corollary we are able to show that if (R, a c(a) ) is sharply F -pure, then c(a) is a rational number that can be written as a quotient whose denominator is not a multiple of p = char R; see Corollary 5.4.…”

Section: Corollary 28 the Set Of A T -Sharp Test Elements Is Equal Tmentioning

confidence: 80%

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

Schwede

2008

Mathematical Research Letters

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Abstract. Consider a pair (R, a t ) where R is a ring of positive characteristic, a is an ideal such that a ∩ R • = ∅, and t > 0 is a real number. In this situation we have the ideal τ R (a t ), the generalized test ideal associated to (R, a t ) as defined by Hara and Yoshida. We show that τ R (a t ) ∩ R • is made up of appropriately defined generalized test elements which we call sharp test elements. We also define a variant of F -purity for pairs, 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, a t ) is sharply F -pure, then τ R (a 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, a t ) is sharply F -pure, then R/τ R (a 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.

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Section: Corollary 28 the Set Of A T -Sharp Test Elements Is Equal Tmentioning

confidence: 80%

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

Schwede

2008

Mathematical Research Letters

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“…When I is generated in a single degree, the above lower bound has been shown in [TW,Proprosition 4.2]. A (more powerful) variant for the log-canonical threshold is in [dEM,Theorem 3.4].…”

Section: Theorem 27 For a Subset σ ⊆ H N The Integral Closure Of mentioning

confidence: 99%

Test, multiplier and invariant ideals

Henriques

Varbaro

2016

Advances in Mathematics

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ABSTRACT. This paper gives an explicit formula for the multiplier ideals, and consequently for the log canonical thresholds, of any GL(V ) × GL(W )-invariant ideal in S = Sym(V ⊗ W * ), where V and W are vector spaces over a field of characteristic 0. This characterization is done in terms of a polytope constructed from the set of Young diagrams corresponding to the Schur modules generating the ideal.Our approach consists in computing the test ideals of some invariant ideals of S in positive characteristic: Namely, we will compute the test ideals (and so the F-pure thresholds) of any sum of products of determinantal ideals. Even in characteristic 0, not all the invariant ideals are as the latter, but they are up to integral closure, and this is enough to reach our goals.The results concerning the test ideals are obtained as a consequence of general results holding true in a special situation. Within such framework fall determinantal objects of a generic matrix, as well as of a symmetric matrix and of a skew-symmetric one. Similar results are thus deduced for the GL(V )-invariant ideals in Sym(Sym 2 V ) and in Sym( 2 V ). (Also monomial ideals fall in this framework, thus we recover Howald's formula for their multiplier ideals and, more generally, we get the formula for their test ideals). During the proof, we introduce the notion of "floating test ideals", a property that in a sense is satisfied by ideals defining schemes with singularities as nice as possible. As we will see, products of determinantal ideals, and by passing to characteristic 0 ideals generated by a single Schur module, have this property.

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“…Particular attention has been given to F -pure [HR76,Fed83] and Fregular singularities [HH90,HH94a,HH94b,FW89]. Attached to these singularity types, there are numerical invariants that measure how good or bad a singular point is; for instance, the F -thresholds [MTW05,HMTW08], the F -pure thresholds [TW04], and the F -signature [SVdB97,HL02,Tuc12]. In this manuscript, we study these numbers, compare them, and obtain consequences regarding the singularities of the ring.…”

Section: Introductionmentioning

confidence: 99%

“…The F -pure threshold of an ideal a ⊆ R, denoted fpt(a), was defined by Takagi and Watanabe [TW04]. Roughly speaking, the F -pure threshold of an ideal measures its splitting order.…”

Section: Introductionmentioning

confidence: 99%

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On the existence of 𝐹-thresholds and related limits

Stefani

Núñez-Betancourt²,

Pérez

2018

Trans. Amer. Math. Soc.

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Dedicated to Professor Craig Huneke on the occasion of his sixty-fifth birthday.Abstract. We show the existence of F -thresholds in full generality. In addition, we study properties of standard graded algebras over a field for which F -pure threshold and F -threshold at the irrelevant maximal ideal agree. We also exhibit explicit bounds for the a-invariants and Castelnuovo-Mumford regularity of Frobenius powers of ideals in terms of F -thresholds and F -pure thresholds, obtaining the existence of related limits in certain cases.

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On F-pure thresholds

Cited by 97 publications

References 19 publications

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

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

Test, multiplier and invariant ideals

On the existence of 𝐹-thresholds and related limits

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