# 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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Preprints, Working Papers, ...
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Cited literature [11 references]

https://hal.univ-cotedazur.fr/hal-01342091
Contributor : Marco Antei <>
Submitted on : Tuesday, July 5, 2016 - 2:01:47 PM
Last modification on : Tuesday, May 26, 2020 - 6:50:52 PM
Long-term archiving on: : Thursday, October 6, 2016 - 11:41:42 AM

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1602.04644v1.pdf
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• HAL Id : hal-01342091, version 1

### Citation

Marco Antei, Arijit Dey. FURTHER REMARKS ON THE FUNDAMENTAL GROUP SCHEME. 2016. ⟨hal-01342091⟩

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