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

Metadata

Keywordscommutative algebraic groups, fundamental groups

Abstract

Consider the abelian category 𝒞 of commutative group schemes of finite type over a field k, its full subcategory of finite group schemes, and the associated pro-category Pro(𝒞) (resp. Pro()) of pro-algebraic (resp. profinite) group schemes. When k is perfect, we show that the profinite fundamental group ϖ 1 :Pro(𝒞)Pro() is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors ϖ i vanish for i2. 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, Springer, Lecture Notes in Mathematics, Volume 100 (1969) | Zbl 0182.26001

[BR07] Beligiannis, Apostolos; Reiten, Idun Homological and homotopical aspects of torsion theories, American Mathematical Society, Memoirs of the American Mathematical Society, Volume 883 (2007) | Zbl 1124.18005

[Bri09] Brion, Michel Anti-affine algebraic groups, J. Algebra, Volume 321 (2009) no. 3, pp. 934-952 | Article | MR 2488561 | Zbl 1166.14029

[Bri15] Brion, Michel On extensions of algebraic groups with finite quotient, Pac. J. Math., Volume 279 (2015) no. 1-2, pp. 135-153 | Article | MR 3437773 | Zbl 1332.14057

[Bri17] Brion, Michel Commutative algebraic groups up to isogeny, Doc. Math., Volume 22 (2017), pp. 679-725 | MR 3650225 | Zbl 1378.14040

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

[Bri19] Brion, Michel Homological dimension of isogeny categories of commutative algebraic groups, Eur. J. Math., Volume 5 (2019) no. 4, pp. 1107-1138 | Article | MR 4015448 | Zbl 07137464

[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 | Article | Zbl 1272.14036

[CGP15] Conrad, Brian; Gabber, Ofer; Prasad, Gopal Pseudo-reductive groups, Cambridge University Press, New Mathematical Monographs, Volume 26 (2015) | MR 3362817 | Zbl 1314.20037

[DG70] Demazure, Michel; Gabriel, Pierre Groupes algébriques, Masson (1970) | Zbl 0203.23401

[Fri82] Friedlander, Eric Étale homotopy of simplicial schemes, Princeton University Press, Annals of Mathematics Studies, Volume 104 (1982) | Zbl 0538.55001

[Gab62] Gabriel, Pierre Des catégories abéliennes, Bull. Soc. Math. Fr., Volume 90 (1962), pp. 323-448 | Article | Numdam | Zbl 0201.35602

[GL91] Geigle, Werner; Lenzing, Helmut Perpendicular categories with applications to representations and sheaves, J. Algebra, Volume 144 (1991) no. 2, pp. 273-343 | Article | MR 1140607 | Zbl 0748.18007

[Gro57] Grothendieck, Alexander Sur quelques points d’algèbre homologique, Tôhoku Math. J., Volume 9 (1957), pp. 119-221 | Zbl 0118.26104

[KS06] Kashiwara, Masaki; Schapira, Pierre Categories and sheaves, Springer, Grundlehren der Mathematischen Wissenschaften, Volume 332 (2006) | MR 2182076 | Zbl 1118.18001

[Lan11] Langer, Adrian On the S-fundamental group scheme, Ann. Inst. Fourier, Volume 26 (2011) no. 5, pp. 2077-2119 | Article | Numdam | MR 2961849 | Zbl 1247.14019

[Lan12] Langer, Adrian On the S-fundamental group scheme. II, J. Inst. Math. Jussieu, Volume 11 (2012) no. 4, pp. 835-854 | Article | MR 2979824 | Zbl 1252.14028

[Mil70] Milne, James S. The homological dimension of commutative group schemes over a perfect field, J. Algebra, Volume 16 (1970), pp. 436-441 | Article | MR 266929 | Zbl 0212.53102

[Nor76] Nori, Madhav V. On the representations of the fundamental group, Compos. Math., Volume 33 (1976), pp. 29-41 | Numdam | MR 417179 | Zbl 0337.14016

[Nor82] Nori, Madhav V. The fundamental group-scheme, Proc. Indian Acad. Sci., Math. Sci., Volume 91 (1982), pp. 73-122 | Article | MR 682517 | Zbl 0586.14006

[Nor83] Nori, Madhav V. The fundamental group-scheme of an Abelian variety, Math. Ann., Volume 263 (1983), pp. 263-266 | Article | MR 704291 | Zbl 0497.14018

[Oor66] Oort, Frans Commutative group schemes, Springer, Lecture Notes in Mathematics, Volume 15 (1966) | MR 213365 | Zbl 0216.05603

[Rus70] Russell, Peter Forms of the affine line and its affine group, Pac. J. Math., Volume 32 (1970) no. 2, pp. 527-539 | Article | Zbl 0199.24502

[Sai17] Saito, Takeshi Wild ramification and the cotangent bundle, J. Algebr. Geom., Volume 26 (2017), pp. 399-473 | Article | MR 3647790 | Zbl 1401.14107

[Ser60] Serre, Jean-Pierre Groupes proalgébriques, Publ. Math., Inst. Hautes Étud. Sci., Volume 7 (1960), pp. 341-403 | Numdam | Zbl 0097.35901

[Ser97] Serre, Jean-Pierre Galois cohomology, Springer (1997) | Zbl 0902.12004

[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 | Article | Numdam | MR 3185350 | Zbl 1286.14061