Sobre el umbral F-puro del ideal homogéneo máximo de un anillo de Stanley-ReisnerResumen
En característica prima, el umbral F-puro es un invariante numérico que mide singularidades. Se conocen pocas estimaciones de este número. En esta nota, calculamos explícitamente el umbral F-puro del ideal homogéneo máximo en un anillo de Stanley-Reisner y demostramos que este número y la dimensión de escisión son iguales.
Palabras clave: Umbral F-puro; Anillos de Stanley-Reisner; Anillos de característica prima.
Abstract
In prime characteristic, the F-pure threshold is a numerical invariant measuring singularities. Few estimates of this number are known. In this note, we explicitly compute the F-pure threshold of the homogeneous maximal ideal in a Stanley-Reisner ring and prove that this number and the splitting dimension are same.
Keywords: F-pure threshold; Stanley-Reisner rings; Prime characteristic rings.
Mathematics Subject Classification: Primary: 13A35, 13F55; Secondary: 14B05.
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Agradecimientos
El autor agradece a Luis Núñez-Betancourt por sus útiles comentarios y sugerencias. También agradece al árbitro anónimo por sus útiles comentarios. El autor hizo este trabajo durante una estancia posdoctoral realizada gracias al Programa de Becas Posdoctorales en la UNAM (POSDOC).
Referencias
-
I,M, Aberbach; F, Enescu. The structure of F-pure rings. Math. Z. 250(2005), no. 4, 791-806. doi: 10.1007/s00209-005-0776-y
» https://doi.org/10.1007/s00209-005-0776-y -
W, Badilla-Céspedes. F-invariants of Stanley-Reisner rings. J. Pure Appl. Algebra 225(2021), no. 9, Paper No. 106671, 19. doi: 10.1016/j.jpaa.2021.106671
» https://doi.org/10.1016/j.jpaa.2021.106671 -
A, Benito; E, Faber; K,E, Smith. Measuring singularities with Frobenius: the basics. Commutative algebra. Springer, New York, 2013, 57-97. doi: 10.1007/978-1-4614-5292-8_3
» https://doi.org/10.1007/978-1-4614-5292-8_3 -
M, Blickle; M, Mustata; KE, Smith. Discreteness and rationality of Fthresholds. Michigan Math. J. 57(2008). Special volume in honor of Melvin Hochster, 43-61. doi: 10.1307/mmj/1220879396
» https://doi.org/10.1307/mmj/1220879396 -
A, De Stefani; L, Núñez-Betancourt. F-thresholds of graded rings. Nagoya Math. J.229(2018), 141-168. doi: 10.1017/nmj.2016.65
» https://doi.org/10.1017/nmj.2016.65 -
M, González Villa; D, Jaramillo-Velez; L, Núñez-Betancourt. F-thresholds and test ideals of Thom-Sebastiani type polynomials. Proc. Amer. Math. Soc. 150(2022), no. 9, 3739-3755. doi: 10.1090/proc/16025
» https://doi.org/10.1090/proc/16025 -
N, Hará; K,I, Yoshida. A generalization of tight closure and multiplier ideals. Trans. Amer. Math. Soc. 355(2003), no. 8, 3143-3174. doi: 10.1090/S0002-9947-03-03285-9
» https://doi.org/10.1090/S0002-9947-03-03285-9 -
D,J, Hernández. F-pure thresholds of binomial hypersurfaces. Proc. Amer. Math. Soc. 142(2014), no. 7, 2227-2242. doi: 10.1090/S0002- 9939- 2014-11941-1
» https://doi.org/10.1090/S0002- 9939- 2014-11941-1 -
D,J, Hernández. F-purity versus log canonicity for polynomials. Nagoya Math. J. 224(2016), no. 1, 10-36. doi: 10.1017/nmj.2016.14
» https://doi.org/10.1017/nmj.2016.14 -
D,J, Hernández; K, Schwede; P, Teixeira; E,E, Witt. The FrobeniusThresholds package for Macaulay2. J. Softw. Algebra Geom. 11(2021), no. 1, 25-39. doi: 10.2140/jsag.2021.11.25
» https://doi.org/10.2140/jsag.2021.11.25 -
M, Hochster; C, Huneke. Tight closure, invariant theory, and the Briancon-Skoda theorem. J. Amer. Math. Soc. 3(1990), no. 1, 31-116. doi: 10.2307/1990984
» https://doi.org/10.2307/1990984 -
M, Hochster; J,L, Roberts. The purity of the Frobenius and local cohomology. Advances in Math. 21(1976), no. 2, 117-172. doi: 10.1016/0001-8708(76)90073-6
» https://doi.org/10.1016/0001-8708(76)90073-6 - C, Huneke. Tight closure and its applications. Vol. 88. Regional Conference Series in Mathematics. Conference Board of the Mathematical Sciences, 1996.
-
C, Huneke; G,J, Leuschke. Two theorems about maximal Cohen-Macaulay modules. Math. Ann. 324(2002), no. 2, 391-404. doi: 10.1007/s00208-002-0343-3
» https://doi.org/10.1007/s00208-002-0343-3 -
Z, Kadyrsizova; et al. Lower bounds on the F-pure threshold and extremal singularities. Trans. Amer. Math. Soc. Ser. B 9(2022), 977-1005. doi: 10.1090/btran/106
» https://doi.org/10.1090/btran/106 -
E, Kunz. Characterizations of regular local rings for characteristic p. Amer. J. Math. 91(1969), 772-784. doi: 10.2307/2373351
» https://doi.org/10.2307/2373351 -
S, Müller. The F-pure threshold of quasi-homogeneous polynomials. J. Pure Appl. Algebra 222(2018), no. 1, 75-96. doi: 10.1016/j.jpaa.2017.03.005
» https://doi.org/10.1016/j.jpaa.2017.03.005 -
M, Mustata; S, Takagi; Ki, Watanabe. F-thresholds and Bernstein-Sato polynomials. European Congress of Mathematics. Eur. Math. Soc., Zürich, 2005, 341-364. doi: 10.48550/arXiv.math/0411170
» https://doi.org/10.48550/arXiv.math/0411170 -
T, Shibuta; S, Takagi. Log canonical thresholds of binomial ideals. Manuscripta Math. 130(2009), no. 1, 45-61. doi: 10.1007/s00229-009-0270-7
» https://doi.org/10.1007/s00229-009-0270-7 -
K,E, Smith; M, Van den Bergh. Simplicity of rings of differential operators in prime characteristic. Proc. London Math. Soc. (3) 75(1997), no. 1, 32-62. doi: 10.1112/S0024611597000257
» https://doi.org/10.1112/S0024611597000257 -
S, Takagi; Ki, Watanabe. On F-pure thresholds. J. Algebra 282(2004), no. 1, 278-297. doi: 10.1016/j.jalgebra.2004.07.011
» https://doi.org/10.1016/j.jalgebra.2004.07.011 -
K, Tucker. F-signature exists. Invent. Math. 190(2012), no. 3, 743-765. doi: 10.1007/s00222-012-0389-0
» https://doi.org/10.1007/s00222-012-0389-0 -
Y, Yao. Observations on the F-signature of local rings of characteristic p. J. Algebra 299(2006), no. 1, 198-218. doi: 10.1016/j.jalgebra.2005.08.013
» https://doi.org/10.1016/j.jalgebra.2005.08.013
Fechas de Publicación
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Fecha del número
Jul-Dec 2024
Histórico
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Recibido
03 Ago 2023 -
Acepto
16 Mayo 2024
