Algebra & Number Theory

Fundamental gerbes

Niels Borne and Angelo Vistoli

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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.

Article information

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

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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]

fundamental group scheme Tannaka theory algebraic stacks


Borne, Niels; Vistoli, Angelo. Fundamental gerbes. Algebra Number Theory 13 (2019), no. 3, 531--576. doi:10.2140/ant.2019.13.531.

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  • D. Abramovich, M. Olsson, and A. Vistoli, “Tame stacks in positive characteristic”, Ann. Inst. Fourier $($Grenoble$)$ 58:4 (2008), 1057–1091.
  • D. Bergh, “Functorial destackification of tame stacks with abelian stabilisers”, Compos. Math. 153:6 (2017), 1257–1315.
  • B. Bhatt and P. Scholze, “The pro-étale topology for schemes”, pp. 99–201 in Astérisque 369, Soc. Math. France, Paris, 2015.
  • N. Borne and A. Vistoli, “The Nori fundamental gerbe of a fibered category”, J. Algebraic Geom. 24:2 (2015), 311–353.
  • A. Braverman and R. Bezrukavnikov, “Geometric Langlands correspondence for $\mathscr D$-modules in prime characteristic: the ${\rm GL}(n)$ case”, Pure Appl. Math. Q. 3:1 (2007), 153–179.
  • S. Brochard, “Duality for commutative group stacks”, preprint, 2014.
  • J.-L. Colliot-Thélène and J.-J. Sansuc, “Torseurs sous des groupes de type multiplicatif; applications à l'étude des points rationnels de certaines variétés algébriques”, C. R. Acad. Sci. Paris Sér. A-B 282:18 (1976), A1113–A1116.
  • J.-L. Colliot-Thélène and J.-J. Sansuc, “La descente sur les variétés rationnelles, II”, Duke Math. J. 54:2 (1987), 375–492.
  • P. Deligne, “Catégories tannakiennes”, pp. 111–195 in The Grothendieck Festschrift, II, edited by P. Cartier et al., Progr. Math. 87, Birkhäuser, Boston, 1990.
  • J. Giraud, Cohomologie non abélienne, Grundlehren der Math. Wissenschaften 179, Springer, 1971.
  • G. Laumon and L. Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik $($3$)$ 39, Springer, 2000.
  • M. V. Nori, “The fundamental group-scheme”, Proc. Indian Acad. Sci. Math. Sci. 91:2 (1982), 73–122.
  • S. Otabe, “An extension of Nori fundamental group”, Comm. Algebra 45:8 (2017), 3422–3448.
  • N. Saavedra Rivano, Catégories tannakiennes, Lecture Notes in Math. 265, Springer, 1972.
  • P. Gabriel, “Construction de préschémas quotient”, pp. 251–286 in Schémas en groupes, Tome I: Propriétés générales des schémas en groupes, Exposés I–VII (Séminaire de Géométrie Algébrique du Bois Marie 1962–1964), edited by M. Demazure and A. Grothendieck, Lecture Notes in Math. 151, Springer, 1970.
  • M. Artin, A. Grothendieck, and J. L. Verdier, Théorie des topos et cohomologie étale des schémas, Tome 3: Exposés IX–XIX (Séminaire de Géométrie Algébrique du Bois Marie 1963–1964), Lecture Notes in Math. 305, Springer, 1973.
  • P. Belmans, A. J. de Jong, et al., “The Stacks project”, electronic reference, 2005–,
  • F. Tonini and L. Zhang, “Algebraic and Nori fundamental gerbes”, J. Inst. Math. Jussieu (online publication July 2017).
  • F. Tonini and L. Zhang, “$F$-divided sheaves trivialized by dominant maps are essentially finite”, Trans. Amer. Math. Soc. (online publication August 2018).