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BERNSTEIN–SATO ROOTS FOR MONOMIAL IDEALS IN POSITIVE CHARACTERISTIC

Part of: (Co)homology theory General commutative ring theory

Published online by Cambridge University Press:  20 March 2020

EAMON QUINLAN-GALLEGO Open the ORCID record for EAMON QUINLAN-GALLEGO [Opens in a new window]
EAMON QUINLAN-GALLEGO*
Affiliation:
Department of Mathematics, East Hall, 530 Church Street, 48109Ann Arbor, MI, USA email equinlan@umich.edu
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Abstract

Following the work of Mustaţă and Bitoun, we recently developed a notion of Bernstein–Sato roots for arbitrary ideals, which is a prime characteristic analogue for the roots of the Bernstein–Sato polynomial. Here, we prove that for monomial ideals the roots of the Bernstein–Sato polynomial (over $\mathbb{C}$) agree with the Bernstein–Sato roots of the mod $p$ reductions of the ideal for $p$ large enough. We regard this as evidence that the characteristic-$p$ notion of Bernstein–Sato root is reasonable.

MSC classification

Primary: 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure
Secondary: 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
Type
Article
Information
Nagoya Mathematical Journal , Volume 244 , December 2021 , pp. 25 - 34
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Copyright
© 2020 Foundation Nagoya Mathematical Journal

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Footnotes

Partially supported by the National Science Foundation grant DMS-1801697 and by the Ito Foundation for International Education Exchange.

References

Bernstein, J. N., Analytic continuation of generalized functions with respect to a parameter , Funct. Anal. Appl. 6 (1972), 26–40.Google Scholar
Bitoun, T., On a theory of the b-function in positive characteristic , Selecta Math. 24 (2018), 3501–3528.10.1007/s00029-017-0383-xCrossRefGoogle Scholar
Blickle, M., Mustata, M. and Smith, K. E., Discreteness and rationality of F-thresholds , Michigan Math. J. 57 (2008), 43–61.Google Scholar
Blickle, M. and Stäbler, A., Bernstein–Sato polynomials and test modules in positive characteristic , Nagoya Math. J. 222 (2016), 74–99.Google Scholar
Budur, N., Mustaţă, M. and Saito, M., Bernstein–Sato polynomials of arbitrary varieties , Compos. Math. 142 (2006), 779–797.10.1112/S0010437X06002193CrossRefGoogle Scholar
Budur, N., Mustaţă, M. and Saito, M., Roots of Bernstein–Sato polynomials for monomial ideals: a positive characteristic approach , Math. Res. Lett. 13 (2006), 125–142.Google Scholar
Hara, N. and Yoshida, K.-I., A generalization of tight closure and multiplier ideals , Trans. Amer. Math. Soc. 355 (2003), 3143–3174.CrossRefGoogle Scholar
Hochster, M. and Huneke, C., Tight closure, invariant theory, and the Briançon–Skoda theorem , J. Amer. Math. Soc. 3 (1990), 31–116.Google Scholar
Kashiwara, M., B-functions and holonomic systems , Invent. Math. 38 (1976/77), 33–53.10.1007/BF01390168CrossRefGoogle Scholar
Kashiwara, M., “ Vanishing cycle sheaves and holonomic systems of differential equations ”, in Algebraic Geometry (Tokyo/Kyoto, 1982), Lecture Notes in Mathematics 1016 , Springer, Berlin, 1983, 134–142.CrossRefGoogle Scholar
Leykin, A. and Tsai, H., Dmodules: functions for computations with $D$ -modules. Version 1.4.0.1.Google Scholar
Malgrange, B., Sur les polynômes de I. N. Bernstein, Séminaire Équations aux dérivées partielles (Polytechnique), (1973–1974). talk:20.Google Scholar
Malgrange, B., “ Polynômes de Bernstein–Sato et cohomologie évanescente ”, in Analysis and Topology on Singular Spaces, II, III (Luminy, 1981), Astérisque 101 , Soc. Math. France, Paris, 1983, 243–267.Google Scholar
Mustaţă, M., Bernstein–Sato polynomials in positive characteristic , J. Algebra 321 (2009), 128–151.10.1016/j.jalgebra.2008.08.014CrossRefGoogle Scholar
Mustaţă, M., Takagi, S. and Watanabe, K.-I., “ F-thresholds and Bernstein–Sato polynomials ”, in Proceedings of the Fourth European Congress of Mathematics, European Mathematical Society, Zürich, 2005, 341–364.Google Scholar
Quinlan-Gallego, E., Bernstein–Sato theory for arbitrary ideals in positive characteristic, preprint, 2019, arXiv:1907.07297.Google Scholar
Sato, M., Theory of prehomogeneous vector spaces (algebraic part)—the English translation of Sato’s lecture from Shintani’s note , Nagoya Math. J. 120 (1990), 1–34; Notes by Takuro Shintani, Translated from the Japanese by Masakazu Muro.10.1017/S0027763000003214CrossRefGoogle Scholar
Stadnik, T., The Lemma on $b$ -functions in Positive Characteristic, preprint, 2012, arXiv:1206.4039.Google Scholar