Moduli of formal torsors
Authors:
Fabio Tonini and Takehiko Yasuda
Journal:
J. Algebraic Geom. 29 (2020), 753-801
DOI:
https://doi.org/10.1090/jag/771
Published electronically:
June 12, 2020
MathSciNet review:
4158465
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We construct the moduli stack of torsors over the formal punctured disk in characteristic $p>0$ for a finite group isomorphic to the semidirect product of a $p$-group and a tame cyclic group. We prove that the stack is a limit of separated Deligne-Mumford stacks with finite and universally injective transition maps.
References
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- Michael D. Fried and Ariane Mézard, Configuration spaces for wildly ramified covers, Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999) Proc. Sympos. Pure Math., vol. 70, Amer. Math. Soc., Providence, RI, 2002, pp. 353–376. MR 1935413, DOI https://doi.org/10.1090/pspum/070/1935413
- David Harbater, Moduli of $p$-covers of curves, Comm. Algebra 8 (1980), no. 12, 1095–1122. MR 579791, DOI https://doi.org/10.1080/00927878008822511
- Nicholas M. Katz, Local-to-global extensions of representations of fundamental groups, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 4, 69–106 (English, with French summary). MR 867916
- Andrew Obus and Rachel Pries, Wild tame-by-cyclic extensions, J. Pure Appl. Algebra 214 (2010), no. 5, 565–573. MR 2577662, DOI https://doi.org/10.1016/j.jpaa.2009.06.017
- Rachel J. Pries, Families of wildly ramified covers of curves, Amer. J. Math. 124 (2002), no. 4, 737–768. MR 1914457
- Matthieu Romagny, Group actions on stacks and applications, Michigan Math. J. 53 (2005), no. 1, 209–236. MR 2125542, DOI https://doi.org/10.1307/mmj/1114021093
- The Stacks Project Authors. Stacks Project. http://stacks.math.columbia.edu, 2017.
- Richard G. Swan, On seminormality, J. Algebra 67 (1980), no. 1, 210–229. MR 595029, DOI https://doi.org/10.1016/0021-8693%2880%2990318-X
- Fabio Tonini, Stacks of ramified covers under diagonalizable group schemes, Int. Math. Res. Not. IMRN 8 (2014), 2165–2244. MR 3194016
- Fabio Tonini, Ramified Galois covers via monoidal functors, Transform. Groups 22 (2017), no. 3, 845–868. MR 3682838, DOI https://doi.org/10.1007/s00031-016-9395-4
- Fabio Tonini and Takehiko Yasuda, Notes on the motivic McKay correspondence for the group scheme $\alpha _p$, Proceedings of FJV2017 Kagoshima, (to appear), arXiv:1803.09558, 2018.
- Fabio Tonini and Takehiko Yasuda, Moduli of formal torsors II, arXiv:1909.09276, 2019.
- Fabio Tonini and Lei Zhang, Algebraic and Nori fundamental gerbes, J. Inst. Math. Jussieu 18 (2019), no. 4, 855–897. MR 3963521, DOI https://doi.org/10.1017/s147474801700024x
- Melanie Machett Wood and Takehiko Yasuda, Mass formulas for local Galois representations and quotient singularities. I: a comparison of counting functions, Int. Math. Res. Not. IMRN 23 (2015), 12590–12619. MR 3431631, DOI https://doi.org/10.1093/imrn/rnv074
- Melanie Wood and Takehiko Yasuda, Mass formulas for local Galois representations and quotient singularities II: Dualities and resolution of singularities, Algebra Number Theory 11 (2017), no. 4, 817–840. MR 3665638, DOI https://doi.org/10.2140/ant.2017.11.817
- Takehiko Yasuda, Motivic integration over wild Deligne-Mumford stacks, arXiv:1908.02932.
- Takehiko Yasuda, The $p$-cyclic McKay correspondence via motivic integration, Compos. Math. 150 (2014), no. 7, 1125–1168. MR 3230848, DOI https://doi.org/10.1112/S0010437X13007781
- Takehiko Yasuda, Toward motivic integration over wild Deligne-Mumford stacks, Higher dimensional algebraic geometry—in honour of Professor Yujiro Kawamata’s sixtieth birthday, Adv. Stud. Pure Math., vol. 74, Math. Soc. Japan, Tokyo, 2017, pp. 407–437. MR 3791224, DOI https://doi.org/10.2969/aspm/07410407
References
- Michael Artin, Alexander Grothendieck, and Jean-Louis Verdier, SGA4-2 - Théorie des topos et cohomologie étale des schémas - Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - vol. 2 (Lecture notes in mathematics “‘270”’). 1964.
- Dan Abramovich, Martin Olsson, and Angelo Vistoli, Tame stacks in positive characteristic, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 4, 1057–1091 (English, with English and French summaries). MR 2427954
- Victor V. Batyrev, Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 1, 5–33. MR 1677693, DOI https://doi.org/10.1007/PL00011158
- Alexis Bouthier and Kestutis Česnavičius, Torsors on loop groups and the Hitchin fibration, arXiv:1908.07480v1, 2019.
- Jan Denef and François Loeser, Motivic integration, quotient singularities and the McKay correspondence, Compositio Math. 131 (2002), no. 3, 267–290. MR 1905024, DOI https://doi.org/10.1023/A%3A1015565912485
- Michael D. Fried and Ariane Mézard, Configuration spaces for wildly ramified covers, Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999) Proc. Sympos. Pure Math., vol. 70, Amer. Math. Soc., Providence, RI, 2002, pp. 353–376. MR 1935413, DOI https://doi.org/10.1090/pspum/070/1935413
- David Harbater, Moduli of $p$-covers of curves, Comm. Algebra 8 (1980), no. 12, 1095–1122. MR 579791, DOI https://doi.org/10.1080/00927878008822511
- Nicholas M. Katz, Local-to-global extensions of representations of fundamental groups, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 4, 69–106 (English, with French summary). MR 867916
- Andrew Obus and Rachel Pries, Wild tame-by-cyclic extensions, J. Pure Appl. Algebra 214 (2010), no. 5, 565–573. MR 2577662, DOI https://doi.org/10.1016/j.jpaa.2009.06.017
- Rachel J. Pries, Families of wildly ramified covers of curves, Amer. J. Math. 124 (2002), no. 4, 737–768. MR 1914457
- Matthieu Romagny, Group actions on stacks and applications, Michigan Math. J. 53 (2005), no. 1, 209–236. MR 2125542, DOI https://doi.org/10.1307/mmj/1114021093
- The Stacks Project Authors. Stacks Project. http://stacks.math.columbia.edu, 2017.
- Richard G. Swan, On seminormality, J. Algebra 67 (1980), no. 1, 210–229. MR 595029, DOI https://doi.org/10.1016/0021-8693%2880%2990318-X
- Fabio Tonini, Stacks of ramified covers under diagonalizable group schemes, Int. Math. Res. Not. IMRN 8 (2014), 2165–2244. MR 3194016
- Fabio Tonini, Ramified Galois covers via monoidal functors, Transform. Groups 22 (2017), no. 3, 845–868. MR 3682838, DOI https://doi.org/10.1007/s00031-016-9395-4
- Fabio Tonini and Takehiko Yasuda, Notes on the motivic McKay correspondence for the group scheme $\alpha _p$, Proceedings of FJV2017 Kagoshima, (to appear), arXiv:1803.09558, 2018.
- Fabio Tonini and Takehiko Yasuda, Moduli of formal torsors II, arXiv:1909.09276, 2019.
- Fabio Tonini and Lei Zhang, Algebraic and Nori fundamental gerbes, J. Inst. Math. Jussieu 18 (2019), no. 4, 855–897. MR 3963521, DOI https://doi.org/10.1017/s147474801700024x
- Melanie Machett Wood and Takehiko Yasuda, Mass formulas for local Galois representations and quotient singularities. I: a comparison of counting functions, Int. Math. Res. Not. IMRN 23 (2015), 12590–12619. MR 3431631, DOI https://doi.org/10.1093/imrn/rnv074
- Melanie Wood and Takehiko Yasuda, Mass formulas for local Galois representations and quotient singularities II: Dualities and resolution of singularities, Algebra Number Theory 11 (2017), no. 4, 817–840. MR 3665638, DOI https://doi.org/10.2140/ant.2017.11.817
- Takehiko Yasuda, Motivic integration over wild Deligne-Mumford stacks, arXiv:1908.02932.
- Takehiko Yasuda, The $p$-cyclic McKay correspondence via motivic integration, Compos. Math. 150 (2014), no. 7, 1125–1168. MR 3230848, DOI https://doi.org/10.1112/S0010437X13007781
- Takehiko Yasuda, Toward motivic integration over wild Deligne-Mumford stacks, Higher dimensional algebraic geometry—in honour of Professor Yujiro Kawamata’s sixtieth birthday, Adv. Stud. Pure Math., vol. 74, Math. Soc. Japan, Tokyo, 2017, pp. 407–437. MR 3791224, DOI https://doi.org/10.2969/aspm/07410407
Additional Information
Fabio Tonini
Affiliation:
Universitá degli Studi di Firenze, Dipartimento di Matematica e Informatica “Ulisse Dini”, Viale Giovanni Battista Morgagni, 67/A, 50134 Firenze, Italy
MR Author ID:
931746
Email:
fabio.tonini@unifi.it
Takehiko Yasuda
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
MR Author ID:
683747
ORCID:
0000-0001-8875-4533
Email:
takehikoyasuda@math.sci.osaka-u.ac.jp
Received by editor(s):
October 25, 2017
Received by editor(s) in revised form:
February 3, 2020
Published electronically:
June 12, 2020
Additional Notes:
The first author was supported by GNSAGA of INdAM. The second author was supported by JSPS KAKENHI Grant Numbers JP15K17510 and JP16H06337
Article copyright:
© Copyright 2020
University Press, Inc.
