On the discreteness and rationality of F-jumping coefficients

Katzman, Mordechai; Lyubeznik, Gennady; Zhang, Wenliang

doi:10.1016/j.jalgebra.2008.11.032

Cited by 23 publications

(26 citation statements)

References 10 publications

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“…This viewpoint also recovers (and slightly generalizes, with new proofs) the main results on the rationality and discreteness of jumping numbers of Blickle et al [2008] and the results on generators of certain D-modules of Alvarez-Montaner et al [2005]. A similar generalization, however using slightly different (but related, see Remark 2.12) methods, was recently obtained independently by Katzman et al [2007].…”

Section: Introductionsupporting

confidence: 80%

“…If one applies this to the Matlis dual U = M ∨ of M and the union of its σ -nilpotent submodules (M/N i ) ∨ in the above statement, an alternative proof is obtained. This approach via the Hartshorne-Speiser result is used in [Katzman et al 2007] to study F-thresholds and hence appears to be directly related to the observations we make in Section 3.3 below.…”

Section: Minimal γ -Sheavesmentioning

confidence: 94%

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Minimal γ-sheaves

Blickle

2008

ANT

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In this note we show that finitely generated unit O X [σ]-modules for X regular and F -finite have a minimal root (in the sense of [Lyu97] Definition 3.6). This question was asked by Lyubeznik and answered by himself in the complete case. In fact, we construct a minimal subcategory of the category of coherent γsheaves (in the sense of [BB06]) which is equivalent to the category of γ-crystals. Some applications to tight closure are included at the end of the paper.

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Section: Introductionsupporting

confidence: 80%

Section: Minimal γ -Sheavesmentioning

confidence: 94%

Minimal γ-sheaves

Blickle

2008

ANT

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“…The technical core is Lemma 6.2 which shows that if t = a/b ∈ Q where b is relatively prime to p, then τ (R, a s ) is constant for s ∈ (t − ε, t) and 1 ≫ ε > 0 (compare with [BMS09,KLZ09]). We already know the result in the principal case, which gives that τ (ω Y , (t − ε)G) is constant for 1 ≫ ε > 0 where π : Y − → X is the normalized blowup and a · O Y = O Y (−G).…”

Section: Sketch Of Main Ideas In Simple Casesmentioning

confidence: 99%

“…Indeed, this has been shown to be the case for arbitrary a when R is finite type over a field (for regular R in [BMS08] and in general by [BSTZ10,STZ12,Bli13]). However, for more general rings, all previous proofs (again for regular R in [BMS09,KLZ09] and in general by [BSTZ10,STZ12]) have required a to be principal. This was particularly frustrating as the discreteness and rationality of F -jumping numbers for non-principal ideals in formal power series rings has remained elusive.…”

Section: Introductionmentioning

confidence: 99%

Test ideals of non-principal ideals: Computations, jumping numbers, alterations and division theorems

Schwede

Tucker

2014

Journal de Mathématiques Pures et Appliquées

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Abstract. Given an ideal a ⊆ R in a (log) Q-Gorenstein F -finite ring of characteristic p > 0, we study and provide a new perspective on the test ideal τ (R, a t ) for a real number t > 0. Generalizing a number of known results from the principal case, we show how to effectively compute the test ideal and also describe τ (R, a t ) using (regular) alterations with a formula analogous to that of multiplier ideals in characteristic zero. We further prove that the F -jumping numbers of τ (R, a t ) as t varies are rational and have no limit points, including the important case where R is a formal power series ring. Additionally, we obtain a global division theorem for test ideals related to results of Ein and Lazarsfeld from characteristic zero, and also recover a new proof of Skoda's theorem for test ideals which directly mimics the proof for multiplier ideals.

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“…The F-pure threshold is known to be rational in a number of cases, see, for example, [BMS1,BMS2,BSTZ,Ha,KLZ]. The theory of F-pure thresholds is motivated by connections to log canonical thresholds; for simplicity, and to conform to the above context, let I be a homogeneous ideal in a polynomial ring over the field of rational numbers.…”

Section: Introductionmentioning

confidence: 99%