Fundamental gerbes

Borne, Niels; Vistoli, Angelo

doi:10.2140/ant.2019.13.531

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Fundamental gerbes

Niels Borne and Angelo Vistoli

Vol. 13 (2019), No. 3, 531–576

DOI: 10.2140/ant.2019.13.531

Abstract

For a class of affine algebraic groups $C$ over a field $κ$ , we define the notion of $C$ -fundamental gerbe of a fibered category, generalizing what we did for finite group schemes in a 2015 paper.

We give necessary and sufficient conditions on $C$ implying that a fibered category $X$ over $κ$ satisfying mild hypotheses admits a Nori $C$ -fundamental gerbe. We also give a tannakian interpretation of the gerbe that results by taking as $C$ the class of virtually unipotent group schemes, under a properness condition on $X$ .

Finally, we prove a general duality result, generalizing the duality between group schemes of multiplicative type and Galois modules, that yields a construction of the multiplicative gerbe of multiplicative type which is independent of the previous theory, and requires weaker hypotheses. This gives a conceptual interpretation of the universal torsor of Colliot-Thélène and Sansuc.

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Keywords

fundamental group scheme, Tannaka theory, algebraic stacks

Mathematical Subject Classification 2010

Primary: 14A20

Secondary: 14H30

Milestones

Received: 18 September 2017

Revised: 28 August 2018

Accepted: 21 January 2019

Published: 23 March 2019

Authors

Niels Borne
Laboratoire Paul Painlevé Université de Lille U.M.R. CNRS 8524 U.F.R. de Mathématiques Villeneuve-d’Ascq France

Angelo Vistoli
Department of Mathematics Scuola Normale Superiore Pisa Italy

Vol. 13, No. 3, 2019