The pseudo-fundamental group scheme

Let X be any scheme defined over a Dedekind scheme S with a given section x∈X(S). We prove the existence of a pro-finite S-group scheme ℵ(X,x) and a universal ℵ(X,x)-torsor dominating all the pro-finite pointed torsors over X. Though ℵ(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 ℵalg(X,x) and a ℵalg(X,x)-torsor dominating all the pro-algebraic and affine pointed torsors over X. The case where X→S has no sections is also considered.