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Comptes Rendus
Mathématique
Marcin Lara, Vasudevan Srinivas and Jakob Stix
Fundamental groups of proper varieties are finitely presented
Volume 362 (2024), p. 51-54
Online since: 2 February 2024
https://doi.org/10.5802/crmath.518
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Comptes Rendus. Mathématique
2024, Vol. 362, p. 51-54
https://doi.org/10.5802/crmath.518
Research article / Article de recherche
Algebraic geometry / Géométrie algébrique
Fundamental groups of proper varieties are
finitely presented
Le groupe fondamental d’une variété propre est de
présentation finie
Marcin Lara ,∗,a, Vasudevan Srinivas b and Jakob Stix c
a Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian
University, Łojasiewicza 6, 30-348 Kraków, Poland
b TIFR, School of Mathematics, Homi Bhabha Road, Colaba, 400005 Mumbai, India
c Institut für Mathematik, Goethe–Universität Frankfurt, Robert-Mayer-Straße 6–8,
60325 Frankfurt am Main, Germany
E-mails: lara@math.uni-frankfurt.de, srinivas@math.tifr.res.in, stix@math.uni-frankfurt.de
Abstract. It was proven in [1], that the étale fundamental group of a connected smooth projective variety
over an algebraically closed field k is topologically finitely presented. In this note, we extend this result to all
connected proper schemes over k.
Résumé. Il a été prouvé dans [1], que le groupe fondamental étale d’une variété projective lisse connexe sur
un corps algébriquement clos k est topologiquement de présentation finie. Dans cette note, nous étendons
ce résultat à tous les schémas propres connexes sur k.
2020 Mathematics Subject Classification. 14F35, 14F20.
Funding. The authors acknowledge support by Deutsche Forschungsgemeinschaft (DFG) through the Col-
laborative Research Centre TRR 326 “Geometry and Arithmetic of Uniformized Structures”, project number
444845124. The second author (VS) was supported during part of the preparation of the article by a J. C. Bose
Fellowship of the Department of Science and Technology, India. He also acknowledges support of the De-
partment of Atomic Energy, India under project number RTI4001. This material is partly based upon work
supported by the NSF Grant No. DMS-1928930 while the third named author (JS) was in residence at MSRI in
Berkeley (Spring 2023).
Manuscript received 4 April 2023, revised and accepted 25 May 2023.
1. Introduction
For a connected algebraic variety X over an algebraically closed field k of characteristic 0, the
étale fundamental group πét1 (X , x) of X is a topologically finitely presented profinite group.
This is proven by first reducing to the case of k = C and then applying the Riemann Existence
Theorem [3, Exp. XII, Thm. 5.1] together with the fact that the topological fundamental group
π
top
1 (X (C), x) is of finite presentation as a discrete group (see e.g. [7] or [4]).
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
52 Marcin Lara, Vasudevan Srinivas and Jakob Stix
In characteristic p > 0, the picture is much more subtle due to the existence of Artin–Schreier
covers of affine schemes, which makes πét1 (Spec(A), x) typically not even topologically finitely
generated. Remark 5.7 of [3, Exp. IX] raised doubts whether πét1 (X , x) is topologically finitely pre-
sented for proper varieties, even for proper smooth curves. In recent work, however, Shusterman
(in the case of curves [9]), and Esnault, Shusterman and the second named author [1] have shown
that for smooth projective varieties πét1 (X , x) is still topologically finitely presented. Both results
are based on a criterion for finite presentation of profinite groups due to Lubotzky [8].
From now on, we will omit the base points.
Theorem 1 (part of Thm. 1.1 of [1]). Let X be a connected smooth projective variety over an
algebraically closed field k. Then the étale fundamental group πét1 (X ) is topologically finitely
presented.
Our goal is to generalize Theorem 1 to all connected schemes that are proper over Spec(k)
(which is new only if k has characteristic p > 0). Such a generalization responds affirmatively to
a question raised by Esnault.
Theorem 2. Let X be a connected scheme that is proper over Spec(k) for an algebraically closed
field k. Then πét1 (X ) is topologically finitely presented.
To prove the theorem, we use descent along an alteration map to X and the van Kampen
presentation of πét1 (X ) arising in this way. More precisely, we use this trick twice.
2. The proof
For a scheme T , let FÉtT denote the category of finite étale covers of T . This gives rise to
a category fibred over schemes. Recall that a morphism g : T ′ → T of schemes is said to be
of effective descent for FÉt, if g∗ induces an equivalence of categories between FÉtT and the
category of descent data in FÉt along g .
Proposition 3 ([3, Exp. IX, Thm. 4.12]). Let f : X ′ → X be a proper surjective morphism of finite
presentation. Then f is of effective descent for FÉt.
Morphisms of effective descent f : X ′ → X for FÉt give rise to a van Kampen-like presentation
of πét1 (X ) as the profinite completion of a quotient of the free topological product of the étale
fundamental groups of the connected components of X ′ and the usual topological fundamental
group of a suitably defined “dual graph”. This goes back to [3, Exp. IX, §5] and has been worked
out in detail in [10, Cor. 5.3].
The existence of such a presentation allows one to “descend” finite generation/presentation of
the fundamental groups involved, as made precise in the following proposition. Every statement
below is about topological finite generation/presentation.
Proposition 4 ([3, Exp. IX, Cor. 5.2 + Cor. 5.3]). Let f : X ′ → X be a morphism of effective descent
for FÉt. We denote X ′×X X ′ by X ′′ and X ′×X X ′×X X ′ by X ′′′.
(a) Assume that X ′, X ′′ have finite π0’s and that πét1 ’s of the connected components of X
′ are
finitely generated. Then πét1 (X ) is finitely generated.
(b) Assume that X ′, X ′′, X ′′′ have finite π0’s, that πét1 ’s of the connected components of X
′ are of
finite presentation and that πét1 ’s of the connected components of X
′′ are finitely generated.
Then πét1 (X ) is of finite presentation.
Let us now recall a result of de Jong specialized to our setting.
Proposition 5 (see [6, Thm. 4.1]). Let X be a scheme that is proper over Spec(k) for an alge-
braically closed field k. Then there exists a proper surjective morphism (of finite presentation)
f : X ′ → X from a smooth projective variety X ′ over Spec(k).
Marcin Lara, Vasudevan Srinivas and Jakob Stix 53
Proof. Let ν : X ν → X be the normalization of X . The map ν is finite, and thus the scheme X ν is
still proper over Spec(k).
Let then f1 : X ′ → X ν be the alteration map of [6, Thm. 4.1] applied to each connected
component of X ν. The map f1 is proper, dominant, and thus surjective. Moreover, loc. cit.
guarantees that X ′ is regular and projective (and not merely proper!) over Spec(k). Now, as k
is algebraically closed, X ′ is smooth over Spec(k). The composition f = ν◦ f1 : X ′ → X has all the
requested properties. □
We are now ready to finish the proof of the main result.
Proof of Theorem 2. Take f : X ′ → X as in Proposition 5. By Theorem 1 and Proposition 3 the
map f satisfies the assumptions for Proposition 4(a). This shows that πét1 (X ) is finitely generated,
for any scheme X that is proper over Spec(k). In fact, finite generation was already proven in [3,
Exp. X, Thm. 2.9], and we included the argument here for the convenience of the reader.
We are going to apply finite generation to the connected components of X ′′ = X ′×X X ′, which
are connected proper schemes over Spec(k). Indeed, using Theorem 1 (this time crucially!) and
Proposition 3 again, the map f now satisfies the assumptions of Proposition 4(b). This shows
that πét1 (X ) is finitely presented. □
3. More general base fields
Similarly to [1, §5], our main result extends to more arithmetic settings. We thank Peter Haine for
essentially suggesting the following corollary.
Corollary 6. Let X be a connected scheme that is proper over Spec(k) for a field k. Then πét1 (X ) is
finitely presented if and only if the absolute Galois group Galk is finitely presented.
Proof. We may assume X is reduced and thus k ′ = H0(X ,OX ) is a finite field extension of k. Let
k be an algebraic closure of k containing k ′. Then X → Spec(k ′) being the Stein factorization of
X → Spec(k) implies that X = X ×k ′ k is connected. By Theorem 2, the group π1(X ) is finitely
presented. The fundamental exact sequence [3, Exp. IX, Thm. 6.1]
1 →π1(X ) →π1(X ) → Galk ′ → 1
shows that π1(X ) is finitely presented if and only if Galk ′ is finitely presented. The latter is
equivalent to Galk being finitely presented, see for example [2, Prop. 2.3]. □
Remark 7. Examples of fields k with finitely presented Galk include: fields algebraic over a finite
field, local p-adic fields, R and more generally real closed fields, K ((t )) for a field of characteristic
0 with GalK of finite presentation, and by [5, Thm. 5.1] for a hilbertian field k, probabilistically
almost always (for the Haar measure on Galk ) the fixed field k
Σ in the separable closure k of a
finite subset Σ⊂ Galk .
References
[1] H. Esnault, M. Shusterman, V. Srinivas, “Finite presentation of the tame fundamental group”, Sel. Math., New Ser. 28
(2022), no. 2, article no. 37 (19 pages).
[2] H. Esnault, V. Srinivas, J. Stix, “An obstruction to lifting to characteristic 0”, Algebr. Geom. 10 (2023), no. 3, p. 327-347.
[3] A. Grothendieck (ed.), Revêtements étales et groupe fondamental (SGA1), Documents Mathématiques, vol. 3, So-
ciété Mathématique de France, 2003, Séminaire de géométrie algébrique du Bois Marie 1960–61., Directed by A.
Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of the 1971 original, xviii+327 pages.
[4] H. Hironaka, “Triangulations of algebraic sets”, in Algebraic geometry (Arcata, Calif., 1974), Proc. Sympos. Pure Math.,
vol. 29, American Mathematical Society, 1975, p. 165-185.
[5] M. Jarden, “Algebraic extensions of finite corank of Hilbertian fields”, Isr. J. Math. 18 (1974), p. 279-307.
54 Marcin Lara, Vasudevan Srinivas and Jakob Stix
[6] A. J. de Jong, “Smoothness, semi-stability and alterations”, IHES Publ. Math. 83 (1996), p. 51-93.
[7] S. Łojasiewicz, “Triangulation of semi-analytic sets”, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 18 (1964), p. 449-474.
[8] A. Lubotzky, “Pro-finite presentations”, J. Algebra 242 (2001), no. 2, p. 672-690.
[9] M. Shusterman, “Balanced presentations for fundamental groups of curves over finite fields”, Math. Res. Lett. 29
(2022), no. 4, p. 1251-1259.
[10] J. Stix, “A general Seifert-Van Kampen theorem for algebraic fundamental groups”, Publ. Res. Inst. Math. Sci. 42
(2006), no. 3, p. 763-786.


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