FURTHER REMARKS ON THE FUNDAMENTAL GROUP SCHEME

Abstract : 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. When ℵ(X, x) and ℵ(X, x) ′ are two distinct group schemes with the same universal property then there exist two faithfully flat morphisms ℵ(X, x) → ℵ(X, x) ′ and ℵ(X, x) ′ → ℵ(X, x) whose compositions may not be isomorphisms. In a similar way we prove the existence of a pro-algebraic S-group scheme ℵ alg (X, x) and a ℵ alg (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.
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Submitted on : Tuesday, July 5, 2016 - 2:01:47 PM
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Marco Antei, Arijit Dey. FURTHER REMARKS ON THE FUNDAMENTAL GROUP SCHEME. 2016. ⟨hal-01342091⟩

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