Vol. 13, No. 3, 2019

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Fundamental gerbes

Niels Borne and Angelo Vistoli

Vol. 13 (2019), No. 3, 531–576
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.

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