Essentially Finite Vector Bundles on Normal Pseudo-Proper Algebraic Stacks

Tonini, Fabio; Zhang, Lei

doi:10.4171/dm/742

Cited by 3 publications

References 6 publications

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Finite torsors on projective schemes defined over a discrete valuation ring

Hái¹,

Santos²

2023

Alg. Geom.

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Given a Henselian and Japanese discrete valuation ring A and a flat and projective A-scheme X, we follow the approach of Biswas and dos Santos [J. Inst. Math. Jussieu 10 (2011), no. 2, 225-234] to introduce a full subcategory of coherent modules on X which is then shown to be Tannakian. We then prove that, under normality of the generic fibre, the associated affine and flat group is pro-finite in a strong sense (so that its ring of functions is a Mittag-Leffler A-module) and that it classifies finite torsors Q → X. This establishes an analogy to Nori's theory of the essentially finite fundamental group. In addition, we compare our theory with the ones recently developed by Mehta-Subramanian and Antei-Emsalem-Gasbarri. Using the comparison with the former, we show that any quasi-finite torsor Q → X has a reduction of the structure group to a finite one.

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Finite torsors on projective schemes defined over a discrete valuation ring

Hái¹,

Santos²

2023

Alg. Geom.

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Nori fundamental gerbe of essentially finite covers and Galois closure of towers of torsors

Antei¹,

Biswas²,

Emsalem³

et al. 2017

Preprint

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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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Essentially Finite Vector Bundles on Normal Pseudo-Proper Algebraic Stacks

Cited by 3 publications

References 6 publications

Finite torsors on projective schemes defined over a discrete valuation ring

Finite torsors on projective schemes defined over a discrete valuation ring

Nori fundamental gerbe of essentially finite covers and Galois closure of towers of torsors

Extension of torsors and prime to p fundamental group scheme

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