The pseudo-fundamental group-scheme

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 ${\aleph }^{\mathrm{alg}}(X,x)$ and a ${\aleph }^{\mathrm{alg}}(X,x)$-torsor dominating all the pro-algebraic and affine pointed torsors over $X$. The case where $X\to S$ has no sections is also considered.

Final accepted version