On the fundamental groups of commutative algebraic groups

Abstract

Consider the abelian category $\mathcal{C}$ of commutative group schemes of finite type over a field $k$, its full subcategory $\mathcal{F}$ of finite group schemes, and the associated pro category $\mathrm{Pro}(\mathcal{C})$ (resp. $\mathrm{Pro}(\mathcal{F})$) of pro-algebraic (resp. profinite) group schemes. When $k$ is perfect, we show that the profinite fundamental group ${\varpi }_1:\mathrm{Pro}(\mathcal{C})\to \mathrm{Pro}(\mathcal{F})$ 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 $\mathrm{Pro}(\mathcal{C})$ over an arbitrary field $k$.

Final version, accepted for publication at Annales Henri Lebesgue