The algebraic fundamental group of a reductive group scheme over an arbitrary base scheme

Mikhail Borovoi 1  and Cristian González-Avilés 2
  • 1 Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978, Tel Aviv, Israel
  • 2 Departamento de Matemáticas, Universidad de La Serena, Cisternas 1200, La Serena, 1700000, Chile

Abstract

We define the algebraic fundamental group π 1(G) of a reductive group scheme G over an arbitrary non-empty base scheme and show that the resulting functor G↦ π1(G) is exact.

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  • [1] Borovoi M., Abelian Galois Cohomology of Reductive Groups, Mem. Amer. Math. Soc., 132(626), American Mathematical Society, Providence, 1998

  • [2] Borovoi M., Demarche C., Le groupe fondamental d’un espace homogène d’un groupe algébrique linéaire, preprint available at http://arxiv.org/abs/1301.1046

  • [3] Borovoi M., Kunyavskiĭ B., Gille P., Arithmetical birational invariants of linear algebraic groups over two-dimensional geometric fields, J. Algebra, 2004, 276(1), 292–339 http://dx.doi.org/10.1016/j.jalgebra.2003.10.024

  • [4] Colliot-Thélène J.-L., Résolutions flasques des groupes linéaires connexes, J. Reine Angew. Math., 2008, 618, 77–133

  • [5] Conrad B., Reductive group schemes (SGA3 Summer School, 2011), available at http://math.stanford.edu/~conrad/papers/luminysga3.pdf

  • [6] Demazure M., Grothendieck A. (Eds.), Schémas en Groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962–64 (SGA 3), re-edition available at http://www.math.jussieu.fr/~polo/SGA3; volumes 1 and 3 have been published: Documents Mathématiques, 7–8, Société Mathématique de France, Paris, 2011

  • [7] Gelfand S.I., Manin Yu.I., Methods of Homological Algebra, 2nd ed., Springer Monogr. Math., Springer, Berlin, 2003 http://dx.doi.org/10.1007/978-3-662-12492-5

  • [8] González-Avilés C.D., Quasi-abelian crossed modules and nonabelian cohomology, J. Algebra, 2012, 369, 235–255 http://dx.doi.org/10.1016/j.jalgebra.2012.07.031

  • [9] González-Avilés C.D., Abelian class groups of reductive group schemes, Israel J. Math., 2013, 196(1), 175–214 http://dx.doi.org/10.1007/s11856-012-0147-4

  • [10] González-Avilés C.D., Flasque resolutions of reductive group schemes, Cent. Eur. J. Math., 2013, 11(7), 1159–1176 http://dx.doi.org/10.2478/s11533-013-0235-7

  • [11] Kottwitz R.E., Stable trace formula: cuspidal tempered terms, Duke Math. J., 1984, 51(3), 611–650 http://dx.doi.org/10.1215/S0012-7094-84-05129-9

  • [12] Merkurjev A.S., K-theory and algebraic groups, In: European Congress of Mathematics, II, Budapest, July 22–26, 1996, Progr. Math., 169, Birkhäuser, Basel, 1998, 43–72

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