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Universal covering spaces and fundamental groups in algebraic geometry as schemes. (English. French summary) Zbl 1228.14019

The authors construct, for any connected quasi-compact and quasi-separated scheme \(X\), a group scheme \(\pi_1(X)\) over \(X\) whose fibre over a geometric point \(\bar x\) identifies with Grothendieck’s étale fundamental group \(\pi_1(X, \bar x)\). They also work out the existence of an algebraic universal cover which is a scheme and not just a pro-object as in SGA1 [A. Grothendieck (ed.), Seminar on algebraic geometry at Bois Marie 1960-61. Documents Mathématiques (Paris) 3. Paris: Société Mathématique de France. (2003; Zbl 1039.14001)]. The possibility of such constructions was certainly known to Grothendieck, and an exposition can be found in §10 of [P. Deligne, “Le groupe fondamental de la droite projective moins trois points”, Galois groups over \(\mathbb{Q}\), Proc. Workshop, Berkeley/CA (USA) 1987, Publ., Math. Sci. Res. Inst. 16, 79–297 (1989; Zbl 0742.14022)], under somewhat more restrictive assumptions. There is also an obvious connection to Nori’s fundamental group scheme as constructed in [M. Nori, “The fundamental group-scheme”, Proc. Indian Acad. Sci., Math. Sci. 91, 73–122 (1982; Zbl 0586.14006)] which the authors mention but do not work out. On the other hand, their exposition is very reader-friendly, emphasizes points that are not always stressed in other treatments and contains a number of worked-out examples which are helpful for the novice.
A somewhat puzzling feature of the text is the unusually large number of facts and examples that the authors claim to have heard from other colleagues (though many of them can be found in the literature). At points the paper resembles those justly popular websites that contain a wealth of information reflecting the joint effort of many excellent mathematicians but which often lack precise references. As such it is warmly recommended for beginners in the subject but some may wonder whether the place of such a text is in a refereed journal.

MSC:

14F35 Homotopy theory and fundamental groups in algebraic geometry
14H30 Coverings of curves, fundamental group
14A15 Schemes and morphisms
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[1] M. Artin, Grothendieck Topologies. Lecture Notes, Harvard University Math. Dept., Cambridge, Mass 1962. · Zbl 0208.48701
[2] M. Artin and B. Mazur, Etale Homotopy. Lecture Notes in Math. 100, Springer-Verlag, Berlin-New York 1969. · Zbl 0182.26001
[3] E. Artin and J. Tate, Class Field Theory. W. A. Benjamin, Inc., New York-Amsterdam, 1968. · Zbl 0176.33504
[4] D. K. Biss, A generalized approach to the fundamental group. Amer. Math. Monthly 107 (2000), no. 8, 711-720. · Zbl 1016.57003
[5] D. K. Biss, The topological fundamental group and generalized covering spaces. Topology Appl. 124 (2002), no. 3, 355-371. · Zbl 1016.57002
[6] M. Boggi, Profinite Teichmüller theory. Math. Nachr. 279 (2006), no. 9-10, 953-987. · Zbl 1105.14031
[7] S. Bosch, W. Lutkebohmert, M. Raynaud, Néron Models. Springer, 1990. · Zbl 0705.14001
[8] F. Bogomolov and Y. Tschinkel, Unramified correspondences. In Algebraic number theory and algebraic geometry, 17-25, Contemp. Math., 300, Amer. Math. Soc., Providence, RI, 2002. · Zbl 1082.14511
[9] P. Deligne, Le groupe fondamental de la droite projective moins trois points. In Galois groups over \(\mathbb{Q} \) (Berkeley, CA, 1987), 79-297, Math. Sci. Res. Inst. Publ., 16, Springer, New York, 1989. · Zbl 0742.14022
[10] M. Demazure and A. Grothendieck, Séminaire de Géométrie Algébrique du Bois Marie 1962/64, Schémas en Groupes II. Lecture Notes in Mathematics 152, Springer-Verlag, 1970. · Zbl 0209.24201
[11] L. Ehrenpreis, Cohomology with bounds. In Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) pp. 389-395, Academic Press, London. · Zbl 0239.55006
[12] H. Esnault and P. H. Hai, The fundamental groupoid scheme and applications. Preprint 2006, arXiv:math/0611115v2. · Zbl 1167.14011
[13] W.-D. Geyer, Unendliche algebraische Zahlkörper, über denen jede Gleichung auflösbar von beschränkter Stufe ist. J. Number Theory 1 (1969), 346-374. · Zbl 0192.40101
[14] P. Gille, Le groupe fondamental sauvage d’une courbe affine en caractétristique \(p>0\). In Courbes semi-stables et groupe fondamental en géométrie algébrique (Luminy, 1998), 217-231, Progr. Math. 187, Birkhäuser, Basel, 2000. · Zbl 0978.14034
[15] A. Grothendieck, Éléments de géométrie algébrique II. Étude globale élémentaire de quelques classes de morphismes. IHES Publ. Math. No. 11, 1961.
[16] A. Grothendieck, Éléments de géométrie algébrique III, Première Partie. IHES Publ. Math. No. 11, 1961. · Zbl 0235.14007
[17] A. Grothendieck, Éléments de géométrie algébrique IV: Étude locale des schémas et des morphismes de schémas, Seconde partie. IHES Publ. Math. No. 24, 1965. · Zbl 0135.39701
[18] A. Grothendieck, Éléments de géométrie algébrique IV: Étude locale des schémas et des morphismes de schémas, Troisième partie. IHES Publ. Math. No. 28, 1967. · Zbl 0144.19904
[19] A. Grothendieck, Éléments de géométrie algébrique IV: Étude locale des schémas et des morphismes de schémas, Quatrième partie. IHES Publ. Math. No. 32, 1967. · Zbl 0153.22301
[20] A. Grothendieck, Brief an G. Faltings. (German) [Letter to G. Faltings] With an English translation on pp. 285-293. London Math. Soc. Lecture Note Ser., 242, Geometric Galois actions, 1, 49-58, Cambridge Univ. Press, Cambridge, 1997. · Zbl 0901.14002
[21] A. Grothendieck, Esquisse d’un programme. (French. French summary) [Sketch of a program] With an English translation on pp. 243-283. London Math. Soc. Lecture Note Ser., 242, Geometric Galois actions, 1, 5-48, Cambridge Univ. Press, Cambridge, 1997. · Zbl 0901.14001
[22] A. Grothendieck (dir.), Revêtements étales et groupe fondamental (SGA1). Documents Mathématiques 3, Soc. Math. Fr., Paris, 2003.
[23] J. Kahn and V. Markovic, Random ideal triangulations and the Weil-Petersson distance between finite degree covers of punctured Riemann surfaces. Preprint 2008, arXiv:0806.2304v1.
[24] H. W. Lenstra, Galois Theory for Schemes. Course notes available from the server of the Universiteit Leiden Mathematics Department, http://websites.math.leidenuniv.nl/algebra/GSchemes.pdf, Electronic third edition: 2008.
[25] A. Magid, Covering spaces of algebraic groups. Amer. Math. Montly 83 (1976), 614-621. · Zbl 0346.14008
[26] V. Markovic and D. Šarić, Teichmüller mapping class group of the universal hyperbolic solenoid. Trans. Amer. Math. Soc. 358 (2006), no. 6, 2637-2650. · Zbl 1095.30040
[27] C. McMullen, Thermodynamics, dimension, and the Weil-Petersson metric. Invent. Math. 173 (2008), no. 2, 365-425. · Zbl 1156.30035
[28] J. S. Milne, Étale cohomology. Princeton Math. Ser., 33, Princeton U. P., Princeton, N.J., 1980. · Zbl 0433.14012
[29] M. Miyanishi, On the algebraic fundamental group of an algebraic group. J. Math. Kyoto Universit. 12 (1972), 351-367. · Zbl 0241.14012
[30] S. Mochizuki, Correspondences on hyperbolic curves. Preprint, available at http://www.kurims.kyoto-u.ac.jp/ motizuki/papers-english.html · Zbl 0965.14013
[31] F. Morel, \( \mathbb{A}^1\)-algebraic topology over a field. Preprint, available at http://www.mathematik.uni-muenchen.de/ morel/preprint.html · Zbl 1097.14014
[32] J. W. Morgan and I. Morrison, A van Kampen theorem for weak joins. Proc. London Math. Soc. (3) 53 (1986), no. 3, 562-576. · Zbl 0609.57002
[33] J. P. Murre, Lectures on an Introduction to Grothendieck’s theory of the Fundamental Group. Notes by S. Anantharaman, TIFR Lect. on Math., no. 40, TIFR, Bombay, 1967. · Zbl 0198.26202
[34] J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of number fields. Second edition. Grundlehren der Mathematischen Wissenschaften, 323. Springer-Verlag, Berlin, 2008. · Zbl 1136.11001
[35] M. Nori, On the representations of the fundamental group. Compositio Math., 33, (1976), no. 1, 29-41. · Zbl 0337.14016
[36] M. Nori, The fundamental group-scheme. Proc. Indian Acad. Sci. (Math. Sci.) 91, no. 2, 1982, 73-122. · Zbl 0586.14006
[37] F. Oort, The algebraic fundamental group. In Geometric Galois Actions, 1, 67-83, London Math. Soc. Lecture Note Ser., 242, Cambridge UP, Cambridge, 1997. · Zbl 0911.14006
[38] R. Pardini, Abelian covers of algebraic varieties. J. Reine Angew. Math. 417 (1991), 191-213. · Zbl 0721.14009
[39] F. Pop, Anabelian Phenomena in Geometry and Arithmetic. 2005 notes, available at http://modular.math.washington.edu/swc/aws/notes/files/05PopNotes.pdf · Zbl 1303.11123
[40] I. I. Piatetski-Shapiro and I. R. Shafarevich, Galois theory of transcendental extensions and uniformization. In Igor R. Shafarevich: Collected Mathematical Papers, Springer-Verlag, New York, 1980, 387-421.
[41] L. Ribes and P. Zalesskii, Profinite Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, 40, Springer, Berlin, 2000. · Zbl 0949.20017
[42] J.-P. Serre, Construction de revêtements étales de la droite affine en caractéristique \(p\). C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 6, 341-346. · Zbl 0726.14021
[43] J.-P. Serre, Groupes proalgébriques. IHES Publ. Math. No. 7, 1960. · Zbl 0097.35901
[44] J.-P. Serre, Galois cohomology. P. Ion trans., Springer-Verlag, Berlin, 2002. · Zbl 1004.12003
[45] E. Spanier, Algebraic Topology. McGraw-Hill Book Co., New York, 1966. · Zbl 0477.55001
[46] D. Sullivan, Linking the universalities of Milnor-Thurston, Feigenbaum, and Ahlfors-Bers. In Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), 543-564, Publish or Perish, Houston TX, 1993. · Zbl 0803.58018
[47] T. Szamuely, Le théorème de Tamagawa I. In Courbes semi-stables et groupe fondamental en géométrie algébrique (Luminy, 1998), 185-201, Progr. Math. 187, Birkhäuser, Basel, 2000. · Zbl 0978.14014
[48] T. Szamuely, Galois Groups and Fundamental Groups. Cambridge Studies in Advanced Mathematics, vol. 117, to be published by Cambridge University Press in 2009. · Zbl 1189.14002
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