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COUNTING GEOMETRIC BRANCHES VIA THE FROBENIUS MAP AND F-NILPOTENT SINGULARITIES

Part of: Ring extensions and related topics Commutative algebra: Homological methods General commutative ring theory

Published online by Cambridge University Press:  27 February 2024

HAILONG DAO Open the ORCID record for HAILONG DAO [Opens in a new window] ,
KYLE MADDOX Open the ORCID record for KYLE MADDOX [Opens in a new window]  and
VAIBHAV PANDEY Open the ORCID record for VAIBHAV PANDEY [Opens in a new window]
HAILONG DAO
Affiliation:
Department of Mathematics University of Kansas 1460 Jayhawk Boulevard Lawrence, Kansas 66045 United States [email protected]
KYLE MADDOX*
Affiliation:
Department of Mathematical Sciences University of Arkansas 850 West Dickson Street Fayetteville, Arkansas 72701 United States
VAIBHAV PANDEY
Affiliation:
Department of Mathematics Purdue University 150 North University Street West Lafayette, Indiana 47907 United States [email protected]
*
[email protected]
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Abstract

We give an explicit formula to count the number of geometric branches of a curve in positive characteristic using the theory of tight closure. This formula readily shows that the property of having a single geometric branch characterizes F-nilpotent curves. Further, we show that a reduced, local F-nilpotent ring has a single geometric branch; in particular, it is a domain. Finally, we study inequalities of Frobenius test exponents along purely inseparable ring extensions with applications to F-nilpotent affine semigroup rings.

Keywords

F-nilpotent ringsgeometric branchesintegral closureweak normalization

MSC classification

Primary: 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure
Secondary: 13D45: Local cohomology 13B40: Étale and flat extensions; Henselization; Artin approximation
Type
Article
Information
Nagoya Mathematical Journal , Volume 255 , September 2024 , pp. 724 - 741
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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References

Blickle, M., The intersection homology D-module in finite characteristic, Math. Ann. 328 (2004), no. 3, 425–450.CrossRefGoogle Scholar
Blickle, M. and Bondu, R., Local cohomology multiplicities in terms of étale cohomology , Ann. Inst. Fourier. 55 (2005), no. 7, 2239–2256.CrossRefGoogle Scholar
Brenner, H., Bounds for test exponents , Compos. Math. 142 (2006), 451–463.CrossRefGoogle Scholar
Dieudonne, J., Lie groups and Lie hyperalgebras over a field of characteristic p $>$ 0 (II) , Amer. J. Math. 77 (1955), no. 2, 218–244.CrossRefGoogle Scholar
Fedder, R., F-purity and rational singularity , Trans. Amer. Math. Soc. 278 (1983), no. 2, 461–480.Google Scholar
Hartshorne, R. and Speiser, R., Local cohomological dimension in characteristic p , Ann. of Math. 105 (1977), no. 1, 45–79.CrossRefGoogle Scholar
Hochster, M. and Huneke, C., Tight closure, invariant theory, and the Briançon–Skoda theorem , J. Amer. Math. Soc. 3 (1990), no. 1, 31–116.Google Scholar
Katzman, M. and Sharp, R. Y., Uniform behaviour of the Frobenius closures of ideals generated by regular sequences , J. Algebra. 295 (2005), 231–246.CrossRefGoogle Scholar
Kawasaki, T., On Macaulayfication of Noetherian schemes , Trans. Amer. Math. Soc. 352 (2000), no. 6, 2517–2552.CrossRefGoogle Scholar
Kenkel, J., Maddox, K., Polstra, T. and Simpson, A., $F$ -nilpotent rings and permanence properties , J. Comm. Algebra. 15 (2023), no. 4, 559–575.Google Scholar
Leahy, J. V. and Vitulli, M. A., Seminormal rings and weakly normal varieties , Nagoya Math. J. 82 (1981), 27–56.CrossRefGoogle Scholar
Maddox, K., A sufficient condition for the finiteness of Frobenius test exponents , Proc. Amer. Math. Soc. 147 (2019), no. 12, 5083–5092.CrossRefGoogle Scholar
Maddox, K. and Miller, L. E., Generalized F-depth and graded nilpotent singularities, preprint, arXiv:2101.00365, 2021.Google Scholar
Maddox, K. and Pandey, V., Homological properties of pinched Veronese rings , J. Algebra. 614 (2023), 307–329.CrossRefGoogle Scholar
Polstra, T. and Quy, P. H., Nilpotence of Frobenius actions on local cohomology and Frobenius closure of ideals , J. Algebra. 529 (2019), 196–225.CrossRefGoogle Scholar
Quy, P. H., On the uniform bound of Frobenius test exponents , J. Algebra. 518 (2019), 119–128.CrossRefGoogle Scholar
Quy, P. H. and Shimomoto, K., F-injectivity and Frobenius closure of ideals in Noetherian rings of characteristic $p>0$ , Adv. Math. 313 (2017), 127–166.CrossRefGoogle Scholar
Schwede, K., F-injective singularities are Du bois , Amer. J. Math. 131 (2009), no. 2, 445–473.CrossRefGoogle Scholar
Sharp, R. Y., Tight closure test exponents for certain parameter ideals , Mich. Math. J. 54 (2006), no. 2, 307–317.CrossRefGoogle Scholar
Shimomoto, K., F-coherent rings with applications to tight closure theory , J. Algebra. 338 (2011), no. 1, 24–34.CrossRefGoogle Scholar
Singh, A. K. and Walther, U., A connectedness result in positive characteristic , Trans. Amer. Math. Soc. 360 (2008), no. 6, 3107–3120.CrossRefGoogle Scholar
Smith, K. E., F-rational rings have rational singularities , Amer. J. Math. 119 (1997), no. 1, 159–180.CrossRefGoogle Scholar
Smith, K. E., Tight closure in graded rings , J. Math. Kyoto Univ. 37 (1997), no. 1, 35–53.Google Scholar
Srinivas, V. and Takagi, S., Nilpotence of Frobenius action and the Hodge filtration on local cohomology , Adv. Math. 305 (2017), 456–478.CrossRefGoogle Scholar
Swanson, I. and Huneke, C., Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, Vol. 336, Cambridge University Press, Cambridge, 2006.Google Scholar
The Stacks Project Authors, Stacks project, 2018. https://stacks.math.columbia.edu.Google Scholar