Volume 3 (2020)
Brion, Michel
On the fundamental groups of commutative algebraic groups
Annales Henri Lebesgue, Volume 3 (2020), pp. 1-34.

Metadata

DOI10.5802/ahl.25
Keywords commutative algebraic groups, fundamental groups

Abstract

Consider the abelian category 𝒞 of commutative group schemes of finite type over a field k, its full subcategory ℱ of finite group schemes, and the associated pro-category Pro(𝒞) (resp. Pro(ℱ)) of pro-algebraic (resp. profinite) group schemes. When k is perfect, we show that the profinite fundamental group ϖ 1 :Pro(𝒞)→Pro(ℱ) is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors ϖ i vanish for i≥2. Along the way, we describe the indecomposable projective objects of Pro(𝒞) over an arbitrary field k.


References

[AM69] Artin, Michael; Mazur, Barry Étale homotopy, Lecture Notes in Mathematics, Volume 100, Springer, 1969 | Zbl

[BR07] Beligiannis, Apostolos; Reiten, Idun Homological and homotopical aspects of torsion theories, Memoirs of the American Mathematical Society, Volume 883, American Mathematical Society, 2007 | Zbl

[Bri09] Brion, Michel Anti-affine algebraic groups, J. Algebra, Volume 321 (2009) no. 3, pp. 934-952 | DOI | MR | Zbl

[Bri15] Brion, Michel On extensions of algebraic groups with finite quotient, Pac. J. Math., Volume 279 (2015) no. 1-2, pp. 135-153 | DOI | MR | Zbl

[Bri17] Brion, Michel Commutative algebraic groups up to isogeny, Doc. Math., Volume 22 (2017), pp. 679-725 | MR | Zbl

[Bri18] Brion, Michel Homogeneous vector bundles over abelian varieties via representation theory (2018) (preprint) | Zbl

[Bri19] Brion, Michel Homological dimension of isogeny categories of commutative algebraic groups, Eur. J. Math., Volume 5 (2019) no. 4, pp. 1107-1138 | DOI | MR | Zbl

[BS13] Brion, Michel; Szamuely, Tamás Prime-to-p étale covers of algebraic groups, Bull. Lond. Math. Soc., Volume 45 (2013) no. 3, pp. 602-612 | DOI | Zbl

[CGP15] Conrad, Brian; Gabber, Ofer; Prasad, Gopal Pseudo-reductive groups, New Mathematical Monographs, Volume 26, Cambridge University Press, 2015 | MR | Zbl

[DG70] Demazure, Michel; Gabriel, Pierre Groupes algébriques, Masson, 1970 | Zbl

[Fri82] Friedlander, Eric Étale homotopy of simplicial schemes, Annals of Mathematics Studies, Volume 104, Princeton University Press, 1982 | Zbl

[Gab62] Gabriel, Pierre Des catégories abéliennes, Bull. Soc. Math. Fr., Volume 90 (1962), pp. 323-448 | DOI | Numdam | Zbl

[GL91] Geigle, Werner; Lenzing, Helmut Perpendicular categories with applications to representations and sheaves, J. Algebra, Volume 144 (1991) no. 2, pp. 273-343 | DOI | MR | Zbl

[Gro57] Grothendieck, Alexander Sur quelques points d’algèbre homologique, Tôhoku Math. J., Volume 9 (1957), pp. 119-221 | Zbl

[KS06] Kashiwara, Masaki; Schapira, Pierre Categories and sheaves, Grundlehren der Mathematischen Wissenschaften, Volume 332, Springer, 2006 | MR | Zbl

[Lan11] Langer, Adrian On the S-fundamental group scheme, Ann. Inst. Fourier, Volume 26 (2011) no. 5, pp. 2077-2119 | DOI | Numdam | MR | Zbl

[Lan12] Langer, Adrian On the S-fundamental group scheme. II, J. Inst. Math. Jussieu, Volume 11 (2012) no. 4, pp. 835-854 | DOI | MR | Zbl

[Mil70] Milne, James S. The homological dimension of commutative group schemes over a perfect field, J. Algebra, Volume 16 (1970), pp. 436-441 | DOI | MR | Zbl

[Nor76] Nori, Madhav V. On the representations of the fundamental group, Compos. Math., Volume 33 (1976), pp. 29-41 | Numdam | MR | Zbl

[Nor82] Nori, Madhav V. The fundamental group-scheme, Proc. Indian Acad. Sci., Math. Sci., Volume 91 (1982), pp. 73-122 | DOI | MR | Zbl

[Nor83] Nori, Madhav V. The fundamental group-scheme of an Abelian variety, Math. Ann., Volume 263 (1983), pp. 263-266 | DOI | MR | Zbl

[Oor66] Oort, Frans Commutative group schemes, Lecture Notes in Mathematics, Volume 15, Springer, 1966 | MR | Zbl

[Rus70] Russell, Peter Forms of the affine line and its affine group, Pac. J. Math., Volume 32 (1970) no. 2, pp. 527-539 | DOI | Zbl

[Sai17] Saito, Takeshi Wild ramification and the cotangent bundle, J. Algebr. Geom., Volume 26 (2017), pp. 399-473 | DOI | MR | Zbl

[Ser60] Serre, Jean-Pierre Groupes proalgébriques, Publ. Math., Inst. Hautes Étud. Sci., Volume 7 (1960), pp. 341-403 | Numdam | Zbl

[Ser97] Serre, Jean-Pierre Galois cohomology, Springer, 1997 | Zbl

[Sta18] Stacks Project Authors Stacks Project, 2018 (http://stacks.math.columbia.edu)

[Tot13] Totaro, Burt Pseudo-abelian varieties, Ann. Sci. Éc. Norm. Supér., Volume 46 (2013) no. 5, pp. 693-721 | DOI | Numdam | MR | Zbl