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Nori fundamental gerbe of essentially finite covers and Galois closure of towers of torsors

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Abstract

We prove the existence of a Galois closure for towers of torsors under finite group schemes over a proper, geometrically connected and geometrically reduced algebraic stack X over a field k. This is done by describing the Nori fundamental gerbe of an essentially finite cover of X. A similar result is also obtained for the \(S \)-fundamental gerbe.

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Acknowledgements

We would like to thank Hélène Esnault and Angelo Vistoli for helpful conversations and suggestions received. We would also like to thank the referee for many helpful remarks.

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Authors and Affiliations

  1. Universidad de Costa Rica, Ciudad universitaria Rodrigo Facio Brenes, San Pedro, Costa Rica

    Marco Antei

  2. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay, 400005, India

    Indranil Biswas

  3. Laboratoire Paul Painlevé, U.F.R. de Mathématiques, Université des Sciences et des Technologies de Lille 1, 59 655, Villeneuve d’Ascq, France

    Michel Emsalem

  4. Dipartimento di Matematica e Informatica ’Ulisse Dini’, Universitá degli Studi di Firenze, Viale Morgagni, 67/a, Florence, 50134, Italy

    Fabio Tonini

  5. Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

    Lei Zhang

  6. National Tsing Hua University, No. 101, Section 2, Guangfu Road, East District, Hsinchu City, Taiwan

    Lei Zhang

Authors
  1. Marco Antei
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  2. Indranil Biswas
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  3. Michel Emsalem
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  4. Fabio Tonini
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  5. Lei Zhang
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Corresponding author

Correspondence to Lei Zhang.

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This work is supported by the departmental fund and the Direct Grant for Research (4053343) of the Chinese University of Hong Kong and the Labex CEMPI (ANR-11-LABX-01). The second author is supported by the J. C. Bose Fellowship.

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Antei, M., Biswas, I., Emsalem, M. et al. Nori fundamental gerbe of essentially finite covers and Galois closure of towers of torsors. Sel. Math. New Ser. 25, 18 (2019). https://doi.org/10.1007/s00029-019-0449-z

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  • DOI: https://doi.org/10.1007/s00029-019-0449-z

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