Deligne–Illusie Classes as Arithmetic Kodaira–Spencer Classes
Taylor Dupuy; David Zureick-Brown
Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 2, p. 371-383

Faltings showed that “arithmetic Kodaira–Spencer classes” satisfying a certain compatibility axiom cannot exist. By modifying his definitions slightly, we show that the Deligne–Illusie classes satisfy what could be considered an “arithmetic Kodaira–Spencer” compatibility condition.

Faltings a montré qu’il n’y a pas de « classes de Kodaira–Spencer arithmétiques » satisfaisant à un certain axiome de compatibilité. En modifiant légèrement ses définitions, nous montrons que les classes de Deligne–Illusie satisfont à ce que l’on pourrait considérer comme « condition de compatibilité de Kodaira–Spencer arithmétique ».

Received : 2018-09-14
Accepted : 2019-04-08
Published online : 2019-10-29
DOI : https://doi.org/10.5802/jtnb.1086
Classification:  12H05,  11G99
Keywords: p-derivations, Frobenius lifts, semi-linear
@article{JTNB_2019__31_2_371_0,
     author = {Taylor Dupuy and David Zureick-Brown},
     title = {Deligne--Illusie Classes as Arithmetic Kodaira--Spencer Classes},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {2},
     year = {2019},
     pages = {371-383},
     doi = {10.5802/jtnb.1086},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2019__31_2_371_0}
}
Dupuy, Taylor; Zureick-Brown, David. Deligne–Illusie Classes as Arithmetic Kodaira–Spencer Classes. Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 371-383. doi : 10.5802/jtnb.1086. https://jtnb.centre-mersenne.org/item/JTNB_2019__31_2_371_0/

[1] Piotr Achinger; Jakub Witaszek; Maciej Zdanowicz Liftability of the Frobenius morphism and images of toric varieties (2017) (https://arxiv.org/abs/1708.03777)

[2] James Borger Lambda-rings and the field with one element (2009) (https://arxiv.org/abs/0906.3146)

[3] Alexandru Buium Differential function fields and moduli of algebraic varieties, Springer, Lecture Notes in Mathematics, Tome 1226 (1986) | MR 874111 | Zbl 0613.12018

[4] Alexandru Buium Differential characters of abelian varieties over p-adic fields, Invent. Math., Tome 122 (1995) no. 1, pp. 309-340 | Article | MR 1358979 | Zbl 0841.14037

[5] Alexandru Buium Arithmetic differential equations, American Mathematical Society, Mathematical Surveys and Monographs, Tome 118 (2005) | MR 2166202 | Zbl 1088.14001

[6] Pierre Deligne; Luc Illusie Relèvements modulo p 2 et décomposition du complexe de de Rham, Invent. Math., Tome 89 (1987), pp. 247-270 | Article | Zbl 0632.14017

[7] Taylor Dupuy Deligne-Illusie classes I: Lifted torsors of lifts of the Frobenius for curves (2014) (https://arxiv.org/abs/1403.2025)

[8] Taylor Dupuy; Eric Katz; Joseph Rabinoff; David Zureick-Brown Total p-differentials on schemes over /p 2 , J. Algebra, Tome 524 (2019), pp. 110-123 | Article | MR 3903661 | Zbl 1408.13054

[9] Gerd Faltings Does there exist an arithmetic Kodaira–Spencer class?, Algebraic geometry: Hirzebruch 70, American Mathematical Society (Contemporary Mathematics) Tome 241 (1999), pp. 141-146 | MR 1718142 | Zbl 0944.14009

[10] Shinichi Mochizuki A survey of the Hodge–Arakelov theory of elliptic curves. I, Arithmetic fundamental groups and noncommutative algebra, American Mathematical Society (Proceedings of Symposia in Pure Mathematics) Tome 70 (2002), pp. 533-569 | Article | MR 1935421 | Zbl 1058.14039

[11] Stacks Project Authors Stacks Project (2014) (http://stacks.math.columbia.edu)