Algebra & Number Theory
- Algebra Number Theory
- Volume 13, Number 3 (2019), 531-576.
For a class of affine algebraic groups over a field , we define the notion of -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 implying that a fibered category over satisfying mild hypotheses admits a Nori -fundamental gerbe. We also give a tannakian interpretation of the gerbe that results by taking as the class of virtually unipotent group schemes, under a properness condition on .
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.
Algebra Number Theory, Volume 13, Number 3 (2019), 531-576.
Received: 18 September 2017
Revised: 28 August 2018
Accepted: 21 January 2019
First available in Project Euclid: 9 April 2019
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 14A20: Generalizations (algebraic spaces, stacks)
Secondary: 14H30: Coverings, fundamental group [See also 14E20, 14F35]
Borne, Niels; Vistoli, Angelo. Fundamental gerbes. Algebra Number Theory 13 (2019), no. 3, 531--576. doi:10.2140/ant.2019.13.531. https://projecteuclid.org/euclid.ant/1554775222