Extension of torsors and prime to p fundamental group scheme
Annales de l'Institut Fourier, Volume 72 (2022) no. 1, pp. 367-386.

Let R be a discrete valuation ring. Let X be a proper and faithfully flat R-scheme, endowed with a section xX(R), with integral and normal fibres. Let f:YX η be a finite Nori-reduced G-torsor. In this paper we provide a useful criterion to extend f:YX η to a torsor over X. Furthermore in the particular situation where R is a complete discrete valuation ring of residue characteristic p>0 and XSpec(R) is smooth we apply our criterion to prove that the natural morphism ψ (p ) :π(X η ,x η ) (p ) π(X,x) η (p ) between the prime-to-p fundamental group scheme of X η 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.

Soit R un anneau de valuation discrète. Soit X un R-schéma propre et fidèlement plat, muni d’une section xX(R), ayant des fibres intègres et normales. Soit f:YX η un G-torseur fini et Nori-réduit. Dans cet article on introduit une condition suffisante pour étendre f:YX η en un torseur au dessus de X. De plus, lorsque R est complet de caractéristique résiduelle p>0 et XSpec(R) est lisse, nous utilisons notre méthode pour démontrer que le morphisme naturel ψ (p ) :π(X η ,x η ) (p ) π(X,x) η (p ) entre la partie première à p du schéma en groupe fondamental de X η 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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3475
Classification: 14L30, 14L15, 11G99
Keywords: torsors, affine group schemes, models, prime to $p$ torsors.
Mot clés : torseurs, schémas en groupes affines, modèles, torseurs premiers à $p$
Antei, Marco 1; Calvo-Monge, Jimmy 1

1 Universidad de Costa Rica, Ciudad Universitaria Rodrigo Facio Brenes, San Pedro de Montes de Oca, San José (Costa Rica)
@article{AIF_2022__72_1_367_0,
     author = {Antei, Marco and Calvo-Monge, Jimmy},
     title = {Extension of torsors and prime to $p$ fundamental group scheme},
     journal = {Annales de l'Institut Fourier},
     pages = {367--386},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {72},
     number = {1},
     year = {2022},
     doi = {10.5802/aif.3475},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.3475/}
}
TY  - JOUR
AU  - Antei, Marco
AU  - Calvo-Monge, Jimmy
TI  - Extension of torsors and prime to $p$ fundamental group scheme
JO  - Annales de l'Institut Fourier
PY  - 2022
SP  - 367
EP  - 386
VL  - 72
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.3475/
DO  - 10.5802/aif.3475
LA  - en
ID  - AIF_2022__72_1_367_0
ER  - 
%0 Journal Article
%A Antei, Marco
%A Calvo-Monge, Jimmy
%T Extension of torsors and prime to $p$ fundamental group scheme
%J Annales de l'Institut Fourier
%D 2022
%P 367-386
%V 72
%N 1
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.3475/
%R 10.5802/aif.3475
%G en
%F AIF_2022__72_1_367_0
Antei, Marco; Calvo-Monge, Jimmy. Extension of torsors and prime to $p$ fundamental group scheme. Annales de l'Institut Fourier, Volume 72 (2022) no. 1, pp. 367-386. doi : 10.5802/aif.3475. http://www.numdam.org/articles/10.5802/aif.3475/

[1] Antei, Marco Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber, J. Théor. Nombres Bordeaux, Volume 22 (2008) no. 3, pp. 537-555 | DOI | MR | Zbl

[2] Antei, Marco On the abelian fundamental group scheme of a family of varieties, Isr. J. Math., Volume 186 (2011), pp. 427-446 | DOI | MR | Zbl

[3] Antei, Marco; Biswas, Indranil; Emsalem, Michel; Tonini, Fabio; Zhang, Lei Nori fundamental gerbe of essentially finite covers and Galois closure of towers of torsors, Sel. Math., New Ser., Volume 25 (2019) no. 2, 18, 37 pages | MR | Zbl

[4] Antei, Marco; Emsalem, Michel Models of torsors and the fundamental group scheme, Nagoya Math. J., Volume 230 (2018), pp. 18-34 | DOI | MR | Zbl

[5] Antei, Marco; Emsalem, Michel; Gasbarri, Carlo Sur l’existence du schéma en groupes fondametal, Épijournal de Géom. Algébr., EPIGA, Volume 4 (2020), 5, 15 pages | Zbl

[6] Biswas, Indranil; dos Santos, João Pedro Vector bundles trivialized by proper morphisms and the fundamental group scheme, J. Inst. Math. Jussieu, Volume 10 (2011) no. 2, pp. 225-234 | DOI | MR | Zbl

[7] Duong, Nguyen Dai; Hai, Phùng Hô Tannakian duality over Dedekind rings and applications, Math. Z., Volume 288 (2018) no. 3-4, pp. 1103-1142 | DOI | MR | Zbl

[8] Grothendieck, Alexander Éléments de géométrie algébrique: IV. Étude locale de schémas et de morphismes de schémas, Seconde partie, Publ. Math., Inst. Hautes Étud. Sci., Volume 24 (1965), pp. 5-231 | Numdam | Zbl

[9] Revêtements étales et groupe fondamental. Séminaire de géométrie algébrique du Bois Marie 1960-61 (SGA 1) (Grothendieck, Alexander, ed.), Documents Mathématiques, 3, Société Mathématique de France, 2003

[10] Hai, Phùng Hô; dos Santos, João Pedro Finite torsors on projective schemes defined over a discrete valuation ring (2019) (https://arxiv.org/abs/1904.10659)

[11] Hartshorne, Robin Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer, 1977 | DOI

[12] Nori, Madhav On the representations of the Fundamental Group, Compos. Math., Volume 33 (1976) no. 1, pp. 29-41 | Numdam | MR | Zbl

[13] Nori, Madhav The fundamental group scheme, Proc. Indian Acad. Sci., Math. Sci., Volume 91 (1982) no. 2, pp. 73-122 | DOI | MR | Zbl

[14] Serre, Jean-Pierre Corps Locaux, Publications de l’Institut de Mathématique de l’Université de Nancago, 8, Hermann, 1968

[15] Stack Project Authors Stack project, 2015 (http://stacks.math.columbia.edu)

[16] Szamuely, Tamás Galois Groups and Fundamental Groups, Cambridge Studies in Advanced Mathematics, 117, Cambridge University Press, 2009 | DOI

[17] Tonini, Fabio; Zhang, Lei Essentially finite vector bundles on normal pseudo-proper algebraic stacks, Doc. Math., Volume 25 (2020), pp. 159-169 | MR | Zbl

[18] Vasconcelos, Wolmer On finitely generated flat modules, Trans. Am. Math. Soc., Volume 138 (1969), pp. 505-512 | DOI | MR

[19] Wedhorn, Torsten On Tannakian duality over valuation rings, J. Algebra, Volume 282 (2004) no. 2, pp. 575-609 | DOI | MR | Zbl

[20] Zhang, Lei The homotopy sequence of Nori’s fundamental group, J. Algebra, Volume 393 (2013) no. 1, pp. 79-91 | DOI | MR | Zbl

Cited by Sources: