Vol. 13, No. 3, 2019

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 7, 1239–1357
Issue 6, 1127–1237
Issue 5, 981–1126
Issue 4, 805–980
Issue 3, 541–804
Issue 2, 267–539
Issue 1, 1–266

Volume 16, 10 issues

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
Other MSP Journals
Fundamental gerbes

Niels Borne and Angelo Vistoli

Vol. 13 (2019), No. 3, 531–576

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.

fundamental group scheme, Tannaka theory, algebraic stacks
Mathematical Subject Classification 2010
Primary: 14A20
Secondary: 14H30
Received: 18 September 2017
Revised: 28 August 2018
Accepted: 21 January 2019
Published: 23 March 2019
Niels Borne
Laboratoire Paul Painlevé
Université de Lille
U.M.R. CNRS 8524
U.F.R. de Mathématiques
Angelo Vistoli
Department of Mathematics
Scuola Normale Superiore