On the fundamental groups of commutative algebraic groups

2018/05/24 by Brion, Michel · 1 citation
#14K05 #14L15 #18E15 #20G07 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.09525

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 \varpi₁ : \rm Pro(\mathcal C) → \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 ≥ 2. Along the way, we describe the indecomposable projective objects of \rm Pro(\mathcal C) over an arbitrary field k.

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• Class field theory for function fields and finite abelian torsors

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