The unfinished specialization theory of the fundamental group after SGA 1, Exposé X: higher étale homotopy exact sequences in positive characteristic, purely algebraic presentation of curve fundamental groups, and the continuous component of the full fundamental group

Exposé X of SGA 1 (Grothendieck) proved the homotopy exact sequence of étale fundamental groups for proper separable morphisms and the surjectivity of the specialization homomorphism for smooth proper families, and then listed what a complete, purely algebraic theory of specialization still lacked. Two of the listed gaps are now closed: finite presentability of the fundamental group of every proper scheme over an algebraically closed field — which the source expected to fail — was proved between 2018 and 2024; and the missing foundations for purity of the branch locus were supplied by SGA 2. A precise conjecture recorded in the source, that the 'true' fundamental groups of the fibers of a smooth proper family form a local system of finite flat group schemes, was disproved by an example of M. Artin recorded there. What remains is the problem posed here: (1) a theory of higher étale homotopy groups carrying the full homotopy exact sequence for smooth proper morphisms, valid in positive characteristic; (2) a purely algebraic proof that the prime-to-characteristic fundamental group of a smooth projective curve of genus $g\ge 2$ is the maximal prime-to-$p$ quotient of the profinite completion of the genus-$g$ surface group; (3) a structural description of the 'continuous component' of an enlarged fundamental group, with a specialization theory for it.

Background

In 1960–61 the Séminaire de Géométrie Algébrique du Bois-Marie (SGA 1), directed by Alexander Grothendieck, founded the étale fundamental group, the algebraic-geometric replacement of the topological fundamental group. For a connected scheme $X$ and a geometric point $x$ of $X$ (a morphism from the spectrum of an algebraically closed field), the étale fundamental group $\pi_1(X,x)$ classifies the finite étale covers of $X$ — finite surjective morphisms that are étale, the algebraic-geometric analogue of local isomorphisms. It is a profinite group (an inverse limit of finite groups with the natural topology), and for a complex variety it recovers the profinite completion of the topological fundamental group. Exposé X of SGA 1 studies $\pi_1$ in families. If $f\colon X\to Y$ is a proper separable morphism (separable means flat with geometrically reduced fibers; smooth morphisms qualify), $y$ a geometric point of $Y$, and $\overline X_y$ the corresponding geometric fiber, then the sequence of profinite groups $\pi_1(\overline X_y)\to\pi_1(X)\to\pi_1(Y)\to 1$ is exact (Corollary X.1.4, the homotopy exact sequence); and when $Y$ is the spectrum of a strictly henselian discrete valuation ring (a local ring with separably closed residue field and no nontrivial finite étale extensions) with generic point $\eta$ and closed point $s$, there is a specialization homomorphism $sp\colon\pi_1(\overline X_\eta)\to\pi_1(\overline X_s)$ between the fundamental groups of the geometric generic and special fibers, surjective when $f$ is smooth and proper (Theorem X.2.3). Over the complex numbers these statements mirror the topology of a fibration; but topology provides more, namely a long exact homotopy sequence involving the higher homotopy groups of base and total space, and there the analogy stopped. The exposé lists five gaps.

(i) Higher homotopy. In the topological situation the kernel of $\pi_1(\overline X_y)\to\pi_1(X)$ is the image of $\pi_2(Y)$; the source asks for homotopy groups in all dimensions in algebraic geometry and the complete homotopy exact sequence for a proper morphism satisfying suitable hypotheses (for instance smoothness), and records that no result of this kind existed at the time (X, §I.6). (ii) Presentations. For a smooth projective connected curve $C$ of genus $g$ over an algebraically closed field (genus $g$ means that over $\mathbb C$ the associated compact surface has $g$ handles; equivalently $\dim H^1(C,\mathcal O_C)=g$), the group $\pi_1(C)$ admits $2g$ topological generators (elements generating a dense subgroup for the profinite topology) with the single relation $\prod_{i=1}^{g}[\gamma_i,\sigma_i]=1$ (Theorem X.2.6); the proof is explicitly by a 'transcendental route' (reduction to $\mathbb C$, the topological classification of compact surfaces, and specialization via the ring of Witt vectors, which lifts perfect fields of characteristic $p$ to characteristic $0$), and no purely algebraic proof was known except in genus $0$ and $1$ (X.2.7). (iii) The full fundamental group. A satisfactory theory of specialization should also account for the 'continuous component' of the 'true' fundamental group, meaning covers beyond the finite étale ones, in particular principal covers under infinitesimal group schemes (finite group schemes whose underlying scheme is a point, visible only in positive characteristic); the expected precise statement — that the true fundamental groups of the geometric fibers of a smooth proper family form a local system (a family that varies locally constantly over the base), as projective limits of finite flat group schemes — was disproved by an unpublished example of M. Artin, already for families of curves of genus $g\ge 2$ (X.2.5). (iv) Finite presentation. It was not known whether $\pi_1$ of a proper scheme over an algebraically closed field is topologically finitely presented, that is, a quotient of a free profinite group on finitely many generators by the closed normal subgroup generated by finitely many relations; the source recorded the expectation that this is 'quite improbable' for curves of genus $g\ge 2$ in positive characteristic (Remark IX.5.7, repeated at X.2.8), while finite generation holds (Theorem X.2.9). (v) Foundations. The purity of the branch locus then used in the theory rested on results of Nagata and Chow that, as the source itself records, no participant of the seminar had verified (X.3.2).

Three further notions are needed for the modern status. The tame fundamental group $\pi_1^t$ of a smooth variety with a fixed boundary divisor is the quotient of $\pi_1$ classifying covers tamely ramified along the boundary (all ramification indices prime to the residue characteristic), and the prime-to-$p$ fundamental group $\pi_1'$ is the maximal quotient of $\pi_1$ whose finite quotients have orders prime to $p$; in characteristic $p>0$ the full $\pi_1$ of a curve is genuinely larger than in characteristic $0$ (Artin–Schreier covers, obtained by adjoining solutions of $z^p-z=a$, already account for the abelian $p$-part), so the expected exact presentation is known only for $\pi_1'$. Higher étale homotopy groups are defined via the étale homotopy type of Artin and Mazur, refined by Friedlander: a pro-object in homotopy types built from étale hypercoverings (certain simplicial resolutions), which over $\mathbb C$ agrees with the classical homotopy type; a scheme is called a $K(\pi,1)$ for the étale topology if its higher étale homotopy groups vanish. Finally, two enlargements of $\pi_1$ towards the 'true' fundamental group exist: the fundamental group scheme (Nori's construction), which classifies torsors under finite group schemes and captures the infinitesimal covers, and the pro-étale fundamental group of Bhatt and Scholze, which classifies 'geometric covers', inverse limits of finite étale covers along open transition maps. By 2024, items (iv) and (v) are settled and item (iii)'s precise conjecture is refuted; the present problem is the surviving core — items (i) and (ii), and the broad form of (iii).

Problem Statement

The problem is to complete the specialization theory of the fundamental group as left open by SGA 1, Exposé X, by settling the three components that remain open. Their common objective is a purely algebraic theory of the behavior of fundamental groups in smooth proper families, valid over algebraically closed fields of every characteristic; the three components are jointly what the source demands of a 'satisfactory theory of specialization'.

(1) Full homotopy exact sequence. Construct a theory of higher homotopy groups for schemes in positive characteristic — either the Artin–Mazur–Friedlander étale homotopy groups or a precisely defined replacement, pinned by a comparison with the classical theory in characteristic zero — and prove that for every smooth proper morphism $f\colon X\to Y$ of connected, locally noetherian schemes with geometrically connected fibers (fibers that remain connected over an algebraic closure of the residue field), pointed compatibly, there is a long exact homotopy sequence

$$\cdots\to\pi_2(Y)\to\pi_1(\overline X_y)\to\pi_1(X)\to\pi_1(Y)\to 1,$$

extending Corollary X.1.4 of the source and exact in the appropriate profinite-topological sense, with the kernel of $\pi_1(\overline X_y)\to\pi_1(X)$ identified as the image of $\pi_2(Y)$. The characteristic-zero case obtained by comparison with topology does not by itself solve this; the target is characteristic $p>0$.

(2) Purely algebraic presentation. Give a purely algebraic proof, valid over every algebraically closed field $k$ and in particular in characteristic $p>0$, that for every connected smooth proper curve $C$ of genus $g\ge 2$ over $k$, the prime-to-$p$ fundamental group $\pi_1'(C,\bar x)$ is the maximal prime-to-$p$ quotient of the profinite completion of the free group on the $2g$ generators $\gamma_1,\sigma_1,\dots,\gamma_g,\sigma_g$ modulo the relation $\prod_{i=1}^{g}[\gamma_i,\sigma_i]=1$. 'Purely algebraic' means: no passage through the complex-analytic comparison between the étale and the topological fundamental group, and no passage through specialization from characteristic zero by means of the specialization homomorphism; every ingredient must be a statement valid over arbitrary algebraically closed fields. This is Problem 1.1 of Lieblich–Olsson, who made the source's Remark X.2.7 precise in this prime-to-$p$ form; the corresponding statement for punctured (affine) curves, with the analogous free-group presentation, may be treated together with it.

(3) The continuous component of the full fundamental group. Construct, for connected varieties over algebraically closed fields, an enlarged ('true') fundamental group classifying covers beyond the finite étale ones — encompassing in particular the principal covers under infinitesimal group schemes that the source singles out, the infinitesimal covers measured by Nori's fundamental group scheme — together with a decomposition into a profinite part and a 'continuous component', and prove a specialization theory for it: for a smooth proper family as in (1), a compatibility between the specialization homomorphisms of the enlarged groups of the geometric fibers and of the total space, including a homotopy exact sequence. Here the enlarged fundamental group is to be a group scheme over the base field that classifies the finite torsors under the additional structural groups — in particular the torsors under finite, possibly infinitesimal, group schemes, as measured by Nori's fundamental group scheme — and that admits the profinite étale fundamental group $\pi_1$ as a quotient; and 'a decomposition into a profinite part and a continuous component' means that the enlarged group sits in a short exact sequence $1\to\pi^{\mathrm{cont}}\to\pi^{\mathrm{true}}\to\pi_1\to 1$ whose quotient is the profinite part and whose kernel $\pi^{\mathrm{cont}}$ is the continuous component. The precise local-system conjecture recorded at X.2.5 of the source is false (M. Artin's example, recorded there); the completion criterion for this component is therefore not that statement but the following: the construction of such an enlarged group, with the short exact sequence above, for every connected variety over an algebraically closed field; and, for a smooth proper family as in (1), a specialization homomorphism between the enlarged groups of the geometric generic and special fibers, compatible with the classical specialization homomorphism on the profinite parts, together with the associated homotopy exact sequence. Further covers beyond the finite étale ones, such as the inverse-limit ('geometric') covers classified by the pro-étale fundamental group of Bhatt–Scholze, are relevant modern points of reference for a constructed theory, but the source's explicit demand concerns covers under infinitesimal group schemes, and encompassing the pro-étale enlargement is not itself imposed as a completion requirement.

Settled by the literature and therefore not part of this problem: the topological finite presentability of $\pi_1$ of every connected proper scheme over an algebraically closed field (proved positively, 2018–2024), and the exposition of purity of the branch locus (supplied by SGA 2).

References

1. A. Grothendieck, 'Théorie de la spécialisation du groupe fondamental', Exposé X in: A. Grothendieck and Michèle Raynaud, Revêtements étales et groupe fondamental (SGA 1), Lecture Notes in Mathematics 224, Springer-Verlag (1971), pp. 261–284, chapter DOI 10.1007/BFb0058665, https://doi.org/10.1007/BFb0058665 (volume DOI 10.1007/BFb0058656, https://doi.org/10.1007/BFb0058656); new updated and annotated edition with updating remarks by M. Raynaud, Documents Mathématiques 3, Société Mathématique de France (2003), arXiv:math/0206203, https://arxiv.org/abs/math/0206203
2. A. Grothendieck and M. Raynaud, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), North-Holland, Amsterdam (1968); new updated and enlarged edition, Documents Mathématiques 4, Société Mathématique de France (2005), arXiv:math/0511279, https://arxiv.org/abs/math/0511279
3. Michèle Raynaud, 'Propriétés de finitude du groupe fondamental', Exposé II in: Groupes de Monodromie en Géométrie Algébrique (SGA 7 I), Lecture Notes in Mathematics 288, Springer-Verlag (1972), pp. 25–31, DOI 10.1007/bfb0068690, https://doi.org/10.1007/bfb0068690
4. M. Artin and B. Mazur, Étale Homotopy, Lecture Notes in Mathematics 100, Springer-Verlag (1969), DOI 10.1007/bfb0080957, https://doi.org/10.1007/bfb0080957
5. Eric M. Friedlander, Étale Homotopy of Simplicial Schemes (AM-104), Annals of Mathematics Studies 104, Princeton University Press (1982), DOI 10.1515/9781400881499, https://doi.org/10.1515/9781400881499
6. Fabrice Orgogozo, 'Altérations et groupe fondamental premier à p', Bulletin de la Société Mathématique de France 131 (2003), 123–147, DOI 10.24033/bsmf.2438, https://doi.org/10.24033/bsmf.2438
7. Max Lieblich and Martin Olsson, 'Generators and relations for the etale fundamental group', Pure and Applied Mathematics Quarterly 6 (2010), no. 1, 209–243, DOI 10.4310/pamq.2010.v6.n1.a6, https://doi.org/10.4310/pamq.2010.v6.n1.a6; arXiv:math/0703139, https://arxiv.org/abs/math/0703139
8. Niels Borne and Michel Emsalem, 'Note sur la détermination algébrique du groupe fondamental pro-résoluble d'une courbe affine', Journal of Algebra 320 (2008), no. 6, 2615–2623, DOI 10.1016/j.jalgebra.2008.04.030, https://doi.org/10.1016/j.jalgebra.2008.04.030; arXiv:0708.0093, https://arxiv.org/abs/0708.0093
9. Hélène Esnault, Phùng Hô Hai, and Eckart Viehweg, 'On the homotopy exact sequence for Nori's fundamental group', arXiv:0908.0498, https://arxiv.org/abs/0908.0498 (2009)
10. Bhargav Bhatt and Peter Scholze, 'The pro-étale topology for schemes', Astérisque 369, Société Mathématique de France (2015), DOI 10.24033/ast.960, https://doi.org/10.24033/ast.960; arXiv:1309.1198, https://arxiv.org/abs/1309.1198
11. Marcin Lara, 'Fundamental Exact Sequence for the Pro-Étale Fundamental Group', Algebra & Number Theory 18 (2024), 631–683, DOI 10.2140/ant.2024.18.631, https://doi.org/10.2140/ant.2024.18.631; arXiv:1910.14015, https://arxiv.org/abs/1910.14015
12. Marcin Lara, 'Homotopy exact sequence for the pro-étale fundamental group', International Mathematics Research Notices 2022 (2022), no. 19, 15355–15389, DOI 10.1093/imrn/rnab101, https://doi.org/10.1093/imrn/rnab101; arXiv:1911.01884, https://arxiv.org/abs/1911.01884 (preprint version titled 'Homotopy Exact Sequence for the Pro-Étale Fundamental Group II')
13. Piotr Achinger, 'Wild ramification and K(π, 1) spaces', Inventiones Mathematicae 210 (2017), no. 2, 453–499, DOI 10.1007/s00222-017-0733-5, https://doi.org/10.1007/s00222-017-0733-5; arXiv:1701.03197, https://arxiv.org/abs/1701.03197
14. Alexander Schmidt and Jakob Stix, 'Anabelian geometry with étale homotopy types', Annals of Mathematics 184 (2016), no. 3, 817–868, DOI 10.4007/annals.2016.184.3.5, https://doi.org/10.4007/annals.2016.184.3.5; arXiv:1504.01068, https://arxiv.org/abs/1504.01068
15. Katharina Hübner and Alexander Schmidt, 'The tame site of a scheme', Inventiones Mathematicae 223 (2021), no. 2, 379–443, DOI 10.1007/s00222-020-00993-4, https://doi.org/10.1007/s00222-020-00993-4; arXiv:2004.05468, https://arxiv.org/abs/2004.05468
16. Mark Shusterman, 'Balanced presentations for fundamental groups of curves over finite fields', Mathematical Research Letters 29 (2022), no. 4, 1251–1259, DOI 10.4310/mrl.2022.v29.n4.a13, https://doi.org/10.4310/mrl.2022.v29.n4.a13; arXiv:1811.04192, https://arxiv.org/abs/1811.04192
17. Hélène Esnault, Mark Shusterman, and Vasudevan Srinivas, 'Finite presentation of the tame fundamental group', Selecta Mathematica (New Series) 28 (2022), no. 2, article 37, DOI 10.1007/s00029-021-00732-4, https://doi.org/10.1007/s00029-021-00732-4; arXiv:2102.13424, https://arxiv.org/abs/2102.13424
18. Marco Antei and Michel Emsalem, 'Models of torsors and the fundamental group scheme', Nagoya Mathematical Journal 230 (2018), 18–34, DOI 10.1017/nmj.2016.67, https://doi.org/10.1017/nmj.2016.67; arXiv:1603.01198, https://arxiv.org/abs/1603.01198
19. Marcin Lara, Vasudevan Srinivas, and Jakob Stix, 'Fundamental groups of proper varieties are finitely presented', Comptes Rendus. Mathématique 362 (2024), 51–54, DOI 10.5802/crmath.518, https://doi.org/10.5802/crmath.518; arXiv:2303.11161, https://arxiv.org/abs/2303.11161
20. Piotr Achinger, Marcin Lara, and Alex Youcis, 'Specialization for the pro-étale fundamental group', Compositio Mathematica 158 (2022), no. 8, 1713–1745, DOI 10.1112/s0010437x2200759x, https://doi.org/10.1112/s0010437x2200759x; arXiv:2107.06761, https://arxiv.org/abs/2107.06761
21. Hélène Esnault, Vasudevan Srinivas, and Jakob Stix, 'An obstruction to lifting to characteristic 0', Algebraic Geometry 10 (2023), no. 3, 327–347, DOI 10.14231/ag-2023-011, https://doi.org/10.14231/ag-2023-011; arXiv:2106.08381, https://arxiv.org/abs/2106.08381