On the fundamental groups of commutative algebraic groups

Michel Brion

doi:10.5802/ahl.25

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 ${\rm Pro}({\mathcal C})$ (resp. ${\rm Pro}({\mathcal F})$) of pro-algebraic (resp. profinite) group schemes. When $k$ is perfect, we show that the profinite fundamental group $\varpi_1 : {\rm Pro}({\mathcal C}) \to {\rm Pro}({\mathcal F})$ is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors $\varpi_i$ vanish for $i \geq 2$. Along the way, we describe the indecomposable projective objects of ${\rm Pro}({\mathcal C})$ over an arbitrary field $k$.

Highlights

Every real Lie group G gives rise to two exact sequences0 → G0 → G → π0(G) → 0, 0 → π1(G) → G → G0 → 0, where G0 denotes the identity component, G its universal cover, and π0(G), π1(G) are discrete groups; the second homotopy group π2(G) vanishes
This classical result has a remarkable analogue for commutative algebraic groups over an algebraically closed field k, as shown by Serre and Oort via a categorical approach
We describe the projective objects of Pro(C) (Proposition 3.11); for this, we use results of Demazure and Gabriel on the projectives of Pro(L) over a perfect field, combined with properties of the isogeny category C/F

Summary

Introduction

0 → G0 → G → π0(G) → 0, 0 → π1(G) → G → G0 → 0, where G0 denotes the identity component, G its universal cover, and π0(G), π1(G) are discrete groups; the second homotopy group π2(G) vanishes. — When k is perfect, the functor 1 : Pro(C) → Pro(F) is left exact and commutes with base change under algebraic field extensions. We show that the profinite homotopy functors commute with base change under separable algebraic field extensions (Proposition 3.15), thereby completing the proof of the main result. Over an imperfect field k, the functors 0, 1 do not commute with purely inseparable field extensions, nor does the pro-étale p-primary part of 1 (see Remarks 3.19, 3.20 and 3.21) In this setting, it seems very likely that 2 is nontrivial, but we have no explicit example for this; the profinite fundamental group scheme 1 deserves further investigation, already for smooth connected unipotent groups. It would be interesting to relate the above (affine, profinite or pro-étale) fundamental groups with further notions of fundamental group schemes considered in the literature. When k is algebraically closed, the affine fundamental group of A coincides with its S-fundamental group scheme introduced by Langer in [Lan11], as follows from [Lan, Thm. 6.1]

Pro-artinian categories and colocalizing subcategories

Homotopy groups

Structure of projective objects

Compatibility properties

The affine fundamental group

The profinite fundamental group

Projective covers of abelian varieties

Structure of indecomposable projectives

Field extensions

The Milne spectral sequence