The Nori fundamental gerbe of a fibered category
Authors:
Niels Borne and Angelo Vistoli
Journal:
J. Algebraic Geom. 24 (2015), 311-353
DOI:
https://doi.org/10.1090/S1056-3911-2014-00638-X
Published electronically:
September 18, 2014
MathSciNet review:
3311586
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Abstract |
References |
Additional Information
Abstract: We extend Nori’s theory of the fundamental group scheme to a theory of the fundamental gerbe, which applies to schemes, algebraic stacks, and more general fibered categories, even in the absence of a rational point. We give a Tannakian interpretation of the fundamental gerbe in terms of essentially finite bundles, extending Nori’s correspondence for complete varieties with a rational point. We also show how our formalism allows a natural formulation of Grothendieck’s Section Conjecture in arbitrary characteristic.
References
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- Schémas en groupes. I: Propriétés générales des schémas en groupes, Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274458
- Neantro Saavedra Rivano, Catégories Tannakiennes, Lecture Notes in Mathematics, Vol. 265, Springer-Verlag, Berlin-New York, 1972 (French). MR 0338002
- Jakob Stix, On cuspidal sections of algebraic fundamental groups, arXiv:0809.0017 (2008).
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References
- Dan Abramovich, Martin Olsson, and Angelo Vistoli, Tame stacks in positive characteristic, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 4, 1057–1091 (English, with English and French summaries). MR 2427954 (2009c:14002)
- M. Artin, Brauer-Severi varieties, Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), Lecture Notes in Math., vol. 917, Springer, Berlin, 1982, pp. 194–210. MR 0657430 (83j:14015)
- M. Atiyah, On the Krull-Schmidt theorem with application to sheaves, Bull. Soc. Math. France 84 (1956), 307–317. MR 0086358 (19,172b)
- Sylvain Brochard, Duality for commutative group stacks. arXiv:1404.0285 (2012)
- Brian Conrad, Chow’s $K/k$-image and $K/k$-trace, and the Lang-Néron theorem, Enseign. Math. (2) 52 (2006), no. 1-2, 37–108. MR 2255529 (2007e:14068)
- P. Deligne, Le groupe fondamental de la droite projective moins trois points, Galois groups over $\textbf {Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 79–297 (French). MR 1012168 (90m:14016), DOI https://doi.org/10.1007/978-1-4613-9649-9_3
- P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 111–195 (French). MR 1106898 (92d:14002)
- Hélène Esnault and Phùng Hô Hai, The fundamental groupoid scheme and applications, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 7, 2381–2412 (English, with English and French summaries). MR 2498355 (2010b:14046)
- Jean Giraud, Cohomologie non abélienne, Die Grundlehren der mathematischen Wissenschaften, Band 179, Springer-Verlag, Berlin, 1971 (French). MR 0344253 (49 \#8992)
- Alexandre Grothendieck (ed.), Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1); Augmenté de deux exposés de M. Raynaud, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin, 1971 (French). MR 0354651 (50 \#7129)
- Alexander Grothendieck, Brief an G. Faltings, with an English translation on pp. 285–293. Geometric Galois actions, 1, London Math. Soc. Lecture Note Ser., vol. 242, Cambridge Univ. Press, Cambridge, 1997, pp. 49–58 (German). MR 1483108 (99c:14023)
- Philippe Gille and Tamás Szamuely, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, Cambridge, 2006. MR 2266528 (2007k:16033)
- Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927 (2001f:14006)
- S. Lang and A. Néron, Rational points of abelian varieties over function fields, Amer. J. Math. 81 (1959), 95–118. MR 0102520 (21 \#1311)
- J. S. Milne, Quotients of Tannakian categories, Theory Appl. Categ. 18 (2007), No. 21, 654–664. MR 2369113 (2009a:18004)
- V. B. Mehta and S. Subramanian, On the fundamental group scheme, Invent. Math. 148 (2002), no. 1, 143–150. MR 1892846 (2004c:14089), DOI https://doi.org/10.1007/s002220100191
- Madhav V. Nori, The fundamental group-scheme, Proc. Indian Acad. Sci. Math. Sci. 91 (1982), no. 2, 73–122. MR 682517 (85g:14019), DOI https://doi.org/10.1007/BF02967978
- Madhav V. Nori, The fundamental group-scheme of an abelian variety, Mathematische Annalen 263 (1983), no. 3, 263–266. MR 0704291 (85b:14063)
- Martin C. Olsson, $\underline \textrm {Hom}$-stacks and restriction of scalars, Duke Math. J. 134 (2006), no. 1, 139–164. MR 2239345 (2007f:14002), DOI https://doi.org/10.1215/S0012-7094-06-13414-2
- Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser Boston, Mass., 1980. MR 561910 (81b:14001)
- Christian Pauly, A smooth counterexample to Nori’s conjecture on the fundamental group scheme, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2707–2711 (electronic). MR 2317943 (2008e:14039), DOI https://doi.org/10.1090/S0002-9939-07-08805-3
- Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589 (87f:16027)
- Matthieu Romagny, Group actions on stacks and applications, Michigan Math. J. 53 (2005), no. 1, 209–236. MR 2125542 (2005m:14005), DOI https://doi.org/10.1307/mmj/1114021093
- M. Demazure and A. Grothendieck, editors, Schémas en groupes. I: Propriétés générales des schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin, 1970. MR 0274458 (43 \#223a)
- Neantro Saavedra Rivano, Catégories Tannakiennes, Lecture Notes in Mathematics, Vol. 265, Springer-Verlag, Berlin, 1972 (French). MR 0338002 (49 \#2769)
- Jakob Stix, On cuspidal sections of algebraic fundamental groups, arXiv:0809.0017 (2008).
- Jakob Stix, Rational points and arithmetic of fundamental groups. Evidence for the section conjecture. Lecture Notes in Mathematics, vol. 2054, Springer-Verlag, Heidelberg, 2013. MR 2977471
Additional Information
Niels Borne
Affiliation:
Laboratoire Paul Painlevé, Université de Lille, U.M.R. CNRS 8524, U.F.R. de Mathématiques, 59 655 Villeneuve d’Ascq Cedex, France
Email:
Niels.Borne@math.univ-lille1.fr
Angelo Vistoli
Affiliation:
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
MR Author ID:
194370
ORCID:
0000-0003-3857-3755
Email:
angelo.vistoli@sns.it
Received by editor(s):
June 14, 2012
Received by editor(s) in revised form:
January 29, 2013
Published electronically:
September 18, 2014
Additional Notes:
The first author was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01) and Anr ARIVAF (ANR-10-JCJC 0107). The second author was supported in part by the PRIN project “Geometria delle varietà algebriche e dei loro spazi di moduli” from MIUR
Article copyright:
© Copyright 2014
University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.
