Consider the abelian category of commutative group schemes of finite type over a field , its full subcategory of finite group schemes, and the associated pro-category (resp. ) of pro-algebraic (resp. profinite) group schemes. When is perfect, we show that the profinite fundamental group is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors vanish for . Along the way, we describe the indecomposable projective objects of over an arbitrary field .
Considérons la catégorie abélienne des schémas en groupes de type fini sur un corps , la sous-catégorie pleine des schémas en groupes finis, et la catégorie correspondante (resp. ) des groupes proalgébriques (resp. profinis). Lorsque est parfait, nous montrons que le groupe fondamental profini est exact à gauche et commute aux extensions algébriques de corps ; il en résulte que les groupes d’homotopie profinis supérieurs sont nuls pour . Au passsage, nous décrivons les objects projectifs indécomposables de sur un corps arbitraire.
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Keywords: commutative algebraic groups, fundamental groups
@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/} }
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/
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