On the proalgebraic fundamental group of topological spaces and amalgamated products of affine group schemes

Author:

Christopher Deninger, Michael Wibmer

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Number Theory (math.NT)

journal:

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date:

2023-06-04 16:00:00

Abstract

The proalgebraic fundamental group of a connected topological space $X$, recently introduced by the first author, is an affine group scheme whose representations classify local systems of finite-dimensional vector spaces on $X$. In this article, we further develop the theory of the proalgebraic fundamental group, in particular, we establish homotopy invariance and a Seifert-van Kampen theorem. To facilitate the latter, we study amalgamated free product of affine group schemes. We also compute the proalgebraic fundamental group of the arithmetically relevant Kucharcyzk-Scholze spaces and compare it to the motivic Galois group.