On the fundamental groups of commutative algebraic groups

Brion, Michel

Annales Henri Lebesgue, Tome 3 (2020), pp. 1-34.

Considérons la catégorie abélienne $𝒞$ des schémas en groupes de type fini sur un corps $k$, la sous-catégorie pleine $ℱ$ des schémas en groupes finis, et la catégorie correspondante $\mathrm{Pro}\left(𝒞\right)$ (resp. $\mathrm{Pro}\left(ℱ\right)$) des groupes proalgébriques (resp. profinis). Lorsque $k$ est parfait, nous montrons que le groupe fondamental profini ${\varpi }_1:\mathrm{Pro}\left(𝒞\right)\to \mathrm{Pro}\left(ℱ\right)$ est exact à gauche et commute aux extensions algébriques de corps ; il en résulte que les groupes d’homotopie profinis supérieurs ${\varpi }_i$ sont nuls pour $i\ge 2$. Au passsage, nous décrivons les objects projectifs indécomposables de $\mathrm{Pro}\left(𝒞\right)$ sur un corps $k$ arbitraire.

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 $\mathrm{Pro}\left(𝒞\right)$ (resp. $\mathrm{Pro}\left(ℱ\right)$) of pro-algebraic (resp. profinite) group schemes. When $k$ is perfect, we show that the profinite fundamental group ${\varpi }_1:\mathrm{Pro}\left(𝒞\right)\to \mathrm{Pro}\left(ℱ\right)$ is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors ${\varpi }_i$ vanish for $i\ge 2$. Along the way, we describe the indecomposable projective objects of $\mathrm{Pro}\left(𝒞\right)$ over an arbitrary field $k$.

Classification : 14K05, 14L15, 18E15, 20G07
Mots clés : commutative algebraic groups, fundamental groups

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