Models of Torsors and the Fundamental Group Scheme

Antei, Marco; Emsalem, Michel

doi:10.1017/nmj.2016.67

Cited by 5 publications

(3 citation statements)

References 28 publications

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“…If H is the full subcategory of Coh(X R ) formed by vector bundles trivialized by the H-torsor h, then we can consider the functor γ * : T R → H, that is faithful, because γ is faithfully flat. Using [4,Lemma 2.11] we obtain an equivalence H Rep 0 R (H), and therefore we obtain a functor Rep 0 R (M R ) → Rep 0 R (H), and thus an affine group morphism H → M R whose generic fibre H K → (M R ) K is an isomorphism. This induces in turn an immersion R [M R ] → R [H], as both are R -flat modules.…”

Section: Annales De L'institut Fouriermentioning

confidence: 97%

Extension of torsors and prime to p fundamental group scheme

Antei¹,

Calvo-Monge²

2022

Annales De l'Institut Fourier

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Let R be a discrete valuation ring. Let X be a proper and faithfully flat R-scheme, endowed with a section x ∈ X(R), with integral and normal fibres. Let f : Y → Xη be a finite Nori-reduced G-torsor. In this paper we provide a useful criterion to extend f : Y → Xη to a torsor over X. Furthermore in the particular situation where R is a complete discrete valuation ring of residue characteristic p > 0 and X → Spec(R) is smooth we apply our criterion to prove that the natural morphismbetween the prime-to-p fundamental group scheme of Xη and the generic fibre of the prime-to-p fundamental group scheme of X is an isomorphism. This generalizes a well known result for the étale fundamental group. The methods used are purely tannakian.Résumé. -Soit R un anneau de valuation discrète. Soit X un R-schéma propre et fidèlement plat, muni d'une section x ∈ X(R), ayant des fibres intègres et normales. Soit f : Y → Xη un G-torseur fini et Nori-réduit. Dans cet article on introduit une condition suffisante pour étendre f : Y → Xη en un torseur au dessus de X. De plus, lorsque R est complet de caractéristique résiduelle p > 0 et X → Spec(R) est lisse, nous utilisons notre méthode pour démontrer que le morphisme naturel ψ (p ) : π(Xη, xη) (p ) → π(X, x) (p ) η entre la partie première à p du schéma en groupe fondamental de Xη et la fibre générique de la partie première à p du schéma en groupes fondamental de X est un isomorphisme. Cela généralise un résultat bien connu pour le groupe fondamental étale. Les méthodes utilisées sont purement tannakiennes.

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Section: Annales De L'institut Fouriermentioning

confidence: 97%

Extension of torsors and prime to p fundamental group scheme

Antei¹,

Calvo-Monge²

2022

Annales De l'Institut Fourier

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“…Antei and Emsalem approached the issue with a different point of view in [5]. Since a model of G that is finite and flat does not always exist, they choose to work with a more general model of G that is flat but only quasi-finite, and then, extend the torsor over some scheme X , which is obtained by modifying the special fiber of X .…”

Section: Introductionmentioning

confidence: 99%

Extending torsors over regular models of curves

Mehidi¹

2020

Preprint

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Let R be a discrete valuation ring with field of fractions K and perfect residue field k of characteristic p > 0. Given a finite flat and commutative group scheme G over K and a smooth projective curve C over K with a rational point, we study the extension of pointed fppf G-torsors over C to pointed torsors over a regular model of C over R. We first study this problem in the category of log schemes: we prove that extending a G-torsor into a log flat torsor amounts to finding a finite flat model of G over R for which a certain group scheme morphism to the Jacobian J of C extends to the Néron model of J. Assuming this condition is satisfied, we give a necessary and sufficient condition for the torsor to have an fppf-extension. Finally, we give two detailed examples of extension of torsors when C is a hyperelliptic curve defined over Q.In a second part, generalizing a result of Chiodo [9], we give a criterion for the r-torsion subgroup of the Néron model of J to be a finite flat group scheme, and we combine it with the results of the first part.

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“…In this paper we study the problem of extending torsors under the action of very general G, that is when G is only affine and of finite type. This approach has been already used in [5] where it has been proved that, (at least when X has dimension 2, the case dim(X) > 2 having a different formulation for which we refer the reader to [5]) every torsor over X η under the action of an affine and flat group scheme can be extended to a torsor over X up to a finite number of Néron blow up of X at a closed subscheme of its special fiber. In this paper we focus essentially, but not only, on a precise example, the case when X is the affine space A n R , i.e.…”

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Models of Torsors Over Affine Spaces

Antei

Araya

2019

Mathematika

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Let X := A n R be the n-dimensional affine space over a discrete valuation ring R with fraction field K. We prove that any pointed torsor Y over A n K under the action of an affine finite type group scheme can be extended to a torsor over A n R possibly after pulling Y back over an automorphism of A n K . The proof is effective. Other cases, including X = α p,R , will also be discussed.Mathematics Subject Classification. Primary: 14L30, 14L15. Secondary: 11G99.

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Models of Torsors and the Fundamental Group Scheme

Cited by 5 publications

References 28 publications

Extension of torsors and prime to p fundamental group scheme

Extension of torsors and prime to p fundamental group scheme

Extending torsors over regular models of curves

Models of Torsors Over Affine Spaces

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