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

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.

Considérons la catégorie abélienne 𝒞 des schémas en groupes de type fini sur un corps k, la sous-catégorie pleine des schémas en groupes finis, et la catégorie correspondante Pro(𝒞) (resp. Pro()) des groupes proalgébriques (resp. profinis). Lorsque k est parfait, nous montrons que le groupe fondamental profini ϖ 1 :Pro(𝒞)Pro() est exact à gauche et commute aux extensions algébriques de corps ; il en résulte que les groupes d’homotopie profinis supérieurs ϖ i sont nuls pour i2. Au passsage, nous décrivons les objects projectifs indécomposables de Pro(𝒞) sur un corps k arbitraire.

Received:
Revised:
Accepted:
Online First:
Published online:
DOI: 10.5802/ahl.25
Classification: 14K05, 14L15, 18E15, 20G07
Keywords: commutative algebraic groups, fundamental groups
Brion, Michel 1

1 Institut Fourier 100 rue des Mathématiques 38610 Gières (France)
@article{AHL_2020__3__1_0,
     author = {Brion, Michel},
     title = {On the fundamental groups of commutative algebraic groups},
     journal = {Annales Henri Lebesgue},
     pages = {1--34},
     publisher = {\'ENS Rennes},
     volume = {3},
     year = {2020},
     doi = {10.5802/ahl.25},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/ahl.25/}
}
TY  - JOUR
AU  - Brion, Michel
TI  - On the fundamental groups of commutative algebraic groups
JO  - Annales Henri Lebesgue
PY  - 2020
SP  - 1
EP  - 34
VL  - 3
PB  - ÉNS Rennes
UR  - http://www.numdam.org/articles/10.5802/ahl.25/
DO  - 10.5802/ahl.25
LA  - en
ID  - AHL_2020__3__1_0
ER  - 
%0 Journal Article
%A Brion, Michel
%T On the fundamental groups of commutative algebraic groups
%J Annales Henri Lebesgue
%D 2020
%P 1-34
%V 3
%I ÉNS Rennes
%U http://www.numdam.org/articles/10.5802/ahl.25/
%R 10.5802/ahl.25
%G en
%F AHL_2020__3__1_0
Brion, Michel. On the fundamental groups of commutative algebraic groups. Annales Henri Lebesgue, Volume 3 (2020), pp. 1-34. doi : 10.5802/ahl.25. http://www.numdam.org/articles/10.5802/ahl.25/

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

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

Cited by Sources: