The pseudo-fundamental group-scheme

TL;DR

This paper constructs universal pro-finite and pro-algebraic group schemes associated with a scheme over a Dedekind base, generalizing fundamental groups to a broader algebraic context.

Contribution

It introduces the pseudo-fundamental group-scheme concepts, establishing existence of universal torsors that dominate all pro-finite and pro-algebraic pointed torsors over a scheme.

Findings

Existence of a pro-finite $S$-group scheme $ ap(X,x)$ with a universal torsor.

Existence of a pro-algebraic $S$-group scheme $ ap^{ m alg}(X,x)$ with a universal torsor.

Applicable even when $X o S$ has no sections.

Abstract

Let $X$ be any scheme defined over a Dedekind scheme $S$ with a given section $x\in X(S)$. We prove the existence of a pro-finite $S$-group scheme $\aleph(X,x)$ and a universal $\aleph(X,x)$-torsor dominating all the pro-finite pointed torsors over $X$. Though $\aleph(X,x)$ may not be unique in general it still can provide useful information in order to better understand $X$. In a similar way we prove the existence of a pro-algebraic $S$-group scheme $\aleph^{\rm alg}(X,x)$ and a $\aleph^{\rm alg}(X,x)$-torsor dominating all the pro-algebraic and affine pointed torsors over $X$. The case where $X\to S$ has no sections is also considered.