Fundamental Exact Sequence for the Pro-\'Etale Fundamental Group

The pro-\'etale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups -- the usual \'etale fundamental group $\pi_1^{\mathrm{et}}$ defined in SGA1 and the more general group defined in SGA3. It controls local systems in the pro-\'etale topology and leads to an interesting class of"geometric covers"of schemes, generalizing finite \'etale covers. We prove the homotopy exact sequence over a field for the pro-\'etale fundamental group of a geometrically connected scheme $X$ of finite type over a field $k$, i.e. that the sequence $$1 \rightarrow \pi_1^{\mathrm{proet}}(X_{\bar{k}}) \rightarrow \pi_1^{\mathrm{proet}}(X) \rightarrow \mathrm{Gal}_k \rightarrow 1$$ is exact as abstract groups and the map $\pi_1^{\mathrm{proet}}(X_{\bar{k}}) \rightarrow \pi_1^{\mathrm{proet}}(X)$ is a topological embedding. On the way, we prove a general van Kampen theorem and the K\"unneth formula for the pro-\'etale fundamental group.


INTRODUCTION
In [BS15], the authors introduced the pro-étale topology for schemes. The main motivation was that the definitions of ℓ-adic sheaves and cohomologies in the usual étale topology are rather indirect. In contrast, the naive definition of e.g. a constant Q ℓ -sheaf in the pro-étale topology as X proét ∋ U ↦ Maps cts (U, Q ℓ ) is a sheaf and if X is a variety over an algebraically closed field, then H i (Xé t , Q ℓ ) = H i (X proét , Q ℓ ), where the right hand side is defined "naively" by applying the derived functor RΓ(X proét , −) to the described constant sheaf.
Along with the new topology, the authors of [BS15] introduced a new fundamental group -the proétale fundamental group. It is defined for a connected locally topologically noetherian scheme X with a geometric pointx and denoted π proét 1 (X,x). The name "pro-étale" is justified by the fact that there is an equivalence π proét 1 (X,x) − Sets ≃ Loc X proét between the categories of (possibly infinite) discrete sets with continuous action by π proét 1 (X,x) and locally constant sheaves of (discrete) sets in X proét . This is analogous to the classical fact that πé t 1 (X,x) − FSets is equivalent to the category of lcc sheaves on Xé t , where G − FSets denotes finite sets with a continuous G action. This is the first striking difference between these fundamental groups: π proét 1 allows working with sheaves of infinite sets. In fact, the authors of [BS15] study abstract "infinite Galois categories", which are pairs (C, F ) satisfying certain axioms that (together with an additional tameness condition) turn out to be equivalent to a pair (G − Sets, F G ∶ G − Sets → Sets) for a Hausdorff topological group G and the forgetful functor F G . In fact, one takes G = Aut(F ) with a suitable topology. This generalizes the usual Galois categories, introduced by Grothendieck to define πé t 1 (X,x). In Grothendieck's approach, one takes the category FÉt X of finite étale coverings together with the fibre functor Fx and obtains that πé t 1 (X,x) − FSets ≃ FÉt X . Discrete sets with a continuous π proét 1 (X,x)-action correspond to a larger class of coverings, namely "geometric coverings", which are defined to be schemes Y over X such that Y → X: (1) is étale (not necessarily quasi-compact!) (2) satisfies the valuative criterion of properness. We denote the category of geometric coverings by Cov X (seen as a full subcategory of Sch X ). It is clear that FÉt ⊂ Cov X . As Y is not assumed to be of finite type over X, the valuative criterion does not imply that Y → X is proper (otherwise we would get finite étale morphisms again) and so in general we get more. A basic example of a non-finite covering in Cov X can be obtained by viewing an infinite chain of (suitably glued) P 1 k 's as a covering of the nodal curve X = P 1 {0, 1} obtained by gluing 0 and 1 on P 1 k (to formalize the gluing one can use [Sch05]). Then, if k =k, π proét 1 (X,x) = Z and πé t 1 (X,x) =Ẑ. In this example, the prodiscrete group π SGA3 1 defined in Chapter X.6 of [SGA70] would give the same answer. This is essentially because our infinite covering is a torsor under a discrete group in Xé t . However, for more general schemes (e.g. an elliptic curve with two points glued), the category Cov X contains more. So far, all the new examples were coming from non-normal schemes. This is not a coincidence, as for a normal scheme X, any Y ∈ Cov X is a (possibly infinite) disjoint union of finite étale coverings. In this case, π proét 1 (X,x) = π SGA3 1 (X,x) = πé t 1 (X,x). In general πé t 1 can be recovered as the profinite completion of π proét 1 and π SGA3 1 is the prodiscrete completion of π proét 1 . The groups π proét 1 belong in general to a class of Noohi groups. These can be characterized as Hausdorff topological groups G that are Raȋkov complete and such that the open subgroups form a basis of neighbourhoods at 1 G . However, open normal subgroups do not necessarily form a basis of open neighborhoods of 1 G in a Noohi group. In the case of π proét 1 , this means that there might exist a connected Y ∈ Cov X that do not have a Galois closure. Examples of Noohi groups include: profinite groups, (pro)discrete groups, but also Q ℓ and GL n (Q ℓ ). A slightly different example would be Aut(S), where S is a discrete set and Aut has the compact-open topology.
The fact that groups like GL n (Q ℓ ) are Noohi (but not profinite or prodiscrete) makes π proét 1 better suited to work with Q ℓ (or Q ℓ ) local systems. Indeed, denoting by Loc X proét (Q ℓ ) the category of Q ℓlocal systems on X proét , i.e. locally constant sheaves of finite-dimensional Q ℓ -vector spaces (again, the "naive" definition works in X proét ), one has an equivalence Rep cts,Q ℓ (π proét 1 (X,x)) ≃ Loc X proét (Q ℓ ).
This fails for πé t 1 , as any Q ℓ -representation of a profinite group must stabilize a Z ℓ -lattice, while Q ℓ -local systems (in the above sense) stabilize lattices only étale locally. The group π SGA3 1 is not enough either; as shown by [BS15,Example 7.4.9] (due to Deligne), if X is the scheme obtained by gluing two points on a smooth projective curve of genus g ≥ 1, there are Q ℓ -local systems on X that do not come from a representation of π SGA3 1 (X). We will often dropx from the notation for brevity. This usually does not matter much, as a different choice of the base point leads to an isomorphic group.
Classical results. In [SGA71], Grothendieck proved some foundational results regarding the étale fundamental group. Among them: (1) The fundamental exact sequence, i.e. the comparison between the "arithmetic" and "geometric" fundamental groups: Exp. IX, Théorème 6.1]) Let k be a field with algebraic closurek. Let X be a quasi-compact and quasi-separated scheme over k. If the base change Xk is connected, then there is a short exact sequence 1 → πé t 1 (Xk) → πé t 1 (X) → Gal k → 1 of profinite topological groups.
(2) The homotopy exact sequence: Exp. X, Corollaire 1.4]) Let f ∶ X → S be a flat proper morphism of finite presentation whose geometric fibres are connected and reduced. Assume S is connected and lets be a geometric point of S. Then there is an exact sequence πé t 1 (Xs) → πé t 1 (X) → πé t 1 (S) → 1 of fundamental groups.
(3) "Künneth formula": Exp. X, Cor. 1.7]) Let X, Y be two connected schemes locally of finite type over an algebraically closed field k and assume that Y is proper. Letx,ȳ be geometric points of X and Y respectively with values in the same algebraically closed field extension K of k. Then the map induced by the projections is an isomorphism πé t 1 (X × k Y, (x,ȳ)) ∼ → πé t 1 (X,x) × πé t 1 (Y,ȳ) (4) Invariance of πé t 1 under extensions of algebraically closed fields for proper schemes ([SGA71, Exp. X, Corollaire 1.8]); (5) General van Kampen theorem (proved in a special case in [SGA71, IX §5] and generalized in [Sti06]); The aim of this and the subsequent article [Lar21] is to generalize statements (1) and (2), correspondingly, to the case of π proét 1 . In the present article, we also establish the generalizations of all the other points besides (2). The main difficulties in trying to directly generalize the proofs of Grothendieck are as follows: • geometric coverings of schemes (i.e. elements of Cov X defined above) are often not quasicompact, unlike elements of FÉt X . For example, for X a variety over a field k and connected Y ∈ Cov Xk , there may be no finite extension l k such that Y would be defined over l. Similarly, some useful constructions (like Stein factorization) no longer work (at least without significant modifications). • for a connected geometric covering Y ∈ Cov X , there is in general no Galois geometric covering dominating it. Equivalently, there might exist an open subgroup U < π proét 1 (X) that does not contain an open normal subgroup. This prevents some proofs that would work for π SGA3 1 to carry over to π proét 1 .
• The topology of π proét 1 is more complicated than the one of πé t 1 , e.g. it is not necessarily compact, which complicates the discussion of exactness of sequences.
Our results. Our main theorem is the generalization of the fundamental exact sequence. More precisely, we prove the following.
Moreover, the map π proét 1 (Xk) → π proét 1 (X) is a topological embedding and the map π proét 1 (X) → Gal k is a quotient map of topological groups.
The most difficult part is showing that π proét 1 (Xk) → π proét 1 (X) is injective or, more precisely, a topological embedding. This is Theorem 4.13.
As in the case of usual Galois categories, statements about exactness of sequences of Noohi groups translate to statements on the corresponding categories of G − Sets. If the groups involved are the proétale fundamental groups, this translates to statements about geometric coverings. We give a detailed dictionary in Prop. 2.37. As Noohi groups are not necessarily compact, the statements on coverings are equivalent to some weaker notions of exactness (e.g. preserving connectedness of coverings is equivalent to the map of groups having dense image). In fact, we first prove a "near-exact" version of Theorem 4.14 and obtain the above one as a corollary using an extra argument.
For π proét 1 (Xk) → π proét 1 (X) to be a topological embedding boils down to the following statement: every geometric covering Y of Xk can be dominated by a covering Y ′ that embeds into a base-change tok of a geometric covering Y ′′ of X (i.e. defined over k).
For finite coverings, the analogous statement is easy to prove; by finiteness, the given covering is defined over a finite field extension l k and one concludes quickly. This is also the case for infinite coverings detected by π SGA3 1 , see Prop. 4.8. But for general geometric coverings, the situation is much less obvious; as we show by counterexamples (Ex. 4.5 and Ex. 4.6), it is not true in general that a connected geometric covering of Xk is isomorphic to a base-change of a covering of X l for some finite extension l k. This property is crucially used in the proof of [SGA71, Exp. IX, Theorem 6.1], and thus trying to carry the classical proof of SGA over to π proét 1 fails. This last statement is, however, stronger than what we need to prove, and so does not contradict our theorem.
A useful technical tool across the article is the van Kampen theorem for π proét 1 . Its abstract form is proven by adapting the proof in [Sti06] to the case of Noohi groups and infinite Galois categories. For a morphism of schemes X ′ ↠ X of effective descent for Cov (satisfying some extra conditions), it allows one to write the pro-étale fundamental group of X in terms of the pro-étale fundamental groups of the connected components of X ′ and certain relations. By the results of [Ryd10], one can take X ′ = X ν → X to be the normalization morphism of a Nagata scheme X. As π proét 1 and πé t 1 coincide for normal schemes, this allows us to present π proét 1 (X) in terms of πé t 1 (X ν w ), where X ν = ⊔ w X ν w , and the (discrete) topological fundamental group of a suitable graph. In this case, the van Kampen theorem takes on concrete form and generalizes [Lav18,Thm. 1.17].
Theorem (van Kampen theorem, Cor. 3.19 + Rmk. 3.21 + Prop. 3.12, cf. [Sti06]). Let X be a Nagata scheme and X ν = ⊔ w X ν w its normalization written as a union of connected components. Then, after a choice of geometric points, étale paths between them and a maximal tree T within a suitable "intersection" graph Γ, there is an isomorphism π proét 1 (X,x) ≃ * top w πé t 1 (X ν w ,x w ) * top π top 1 (Γ, T ) ⟨R 1 , R 2 ⟩ Noohi where R 1 , R 2 are two sets of relations described in Cor. 3.19 and (−) Noohi is the Noohi completion defined in Section 2.
In the proof of the main theorem, the van Kampen theorem allows us to construct π proét 1 (Xk)and π proét 1 (X)-sets in more concrete terms of graphs of groups involving πé t 1 's. We "explicitly" construct a Galois invariant open subgroup of a given open subgroup U < π proét 1 (Xk,x) in terms of "regular loops" (with respect to U ), see Defn. 4.20.
In fact, the existence of elements that are too far from being a product of regular loops is tacitly behind the counterexamples Ex. 4.5 and 4.6, while the fact that, despite this, there is still an abundance of (products of) regular loops (i.e. their closure is open) is behind our main proof. We also sketch a quicker but less constructive approach in Rmk. 4.27.
Another interesting result proven with the help of the van Kampen theorem is the Künneth formula.
Proposition (Künneth formula for π proét 1 , Prop. 3.29). Let X, Y be two connected schemes locally of finite type over an algebraically closed field k and assume that Y is proper. Letx,ȳ be geometric points of X and Y respectively with values in the same algebraically closed field extension K of k. Then the map induced by the projections is an isomorphism Along the way, we prove the invariance of π proét 1 under extensions of algebraically closed fields for proper schemes (see Prop. 3.31) and give a short direct proof of the fact that π SGA3 1 (Xk,x) ↪ π SGA3 1 (X,x), see Cor. 4.10.
In a separate article [Lar21], we discuss the homotopy exact sequence for π proét 1 . It is proven by constructing an infinite (i.e. non-quasi-compact) analogue of the Stein factorization. Although the construction does not use the main results of this article, the auxiliary results on Noohi groups and π proét 1 have proven to be very handy.
We hope that our techniques, with some extra tweaks and work, will allow to draw similar conclusions about other Noohi fundamental groups arising from the infinite Galois formalism. One such example could be the de Jong fundamental group π dJ 1 , defined in the rigid-analytic setting in [dJ95]. In a later joint work [ALY21], we have proven the existence of a specialization morphism between π proét 1 and π dJ 1 , relating π proét 1 to this more established fundamental group.
Acknowledgements. The main ideas and results contained in this article are a part of my PhD thesis. I express my gratitude to my advisor Hélène Esnault for introducing me to the topic and her constant encouragement. I would like to thank my co-advisor Vasudevan Srinivas for his support and suggestions. I am thankful to Peter Scholze for explaining some parts of his work to me via e-mail. I thank João Pedro dos Santos for for his comments and feedback. I owe special thanks to Fabio Tonini, Lei Zhang and Marco D'Addezio from our group in Berlin for many inspiring mathematical discussions. I thank Piotr Achinger and Jakob Stix for their support. I would also like to thank the referee for careful reading, valuable remarks and urging me to write a more streamlined version of the main proof.
My PhD was funded by the Einstein Foundation. This work is a part of the project "Kapibara" supported by the funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 802787).
The major revision was prepared at JU Kraków and GU Frankfurt. I was supported by the Priority Research Area SciMat under the program Excellence Initiative -Research University at the Jagiellonian University in Kraków. This research was also funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) TRR 326 Geometry and Arithmetic of Uniformized Structures, project number 444845124.
• H < ○ G will mean that H is an open subgroup of G.
• For subgroups H < G, H nc will denote the normal closure of H in G, i.e. the smallest normal subgroup of G containing H. We will use ⟨⟨−⟩⟩ to denote the normal closure of the subgroup generated by some subset of G, i.e. ⟨⟨−⟩⟩ = ⟨−⟩ nc . • For a field k, we will usek to denote its (fixed) algebraic closure and k sep or k s to denote its separable closure (ink). • The topological groups are assumed to be Hausdorff unless specified otherwise or appearing in a context where it is not automatically satisfied (e.g. as a quotient by a subgroup that is not necessarily closed). We will usually comment whenever a non-Hausdorff group appears. • We assume (almost) every base scheme to be locally topologically noetherian. This does not cause problems when considering geometric coverings, as a geometric covering of a locally topologically noetherian scheme is locally topologically noetherian again -this is [BS15, Lm. 6.6.10]. • A "G-set" for a topological group G will mean a discrete set with a continuous action of G unless specified otherwise. We will denote the category of G-sets by G − Sets. We will denote the category of sets by Sets. • We will often omit the base points from the statements and the discussion; by Cor. 3.18, this usually does not change much. In some proofs (e.g. involving the van Kampen theorem), we keep track of the base points.
2. INFINITE GALOIS CATEGORIES, NOOHI GROUPS AND π proét 1 2.1. Overview of the results in [BS15]. Throughout the entire article we use the language and results of [BS15], especially of Chapter 7, as this is where the pro-étale fundamental group was defined. Some familiarity with the results of [BS15, §7] is a prerequisite to read this article. We are going to give a quick overview of some of these results below, but we recommend keeping a copy of [BS15] at hand.
Definition 2.1. ([BS15, Defn. 7.1.1]) Fix a topological group G. Let G− Sets be the category of discrete sets with a continuous G-action, and let F G ∶ G − Sets → Sets be the forgetful functor. We say that G is a Noohi group if the natural map induces an isomorphism G → Aut(F G ) of topological groups.
Here, S ∈ Sets are considered with the discrete topology, Aut(S) with the compact-open topology and Aut(F G ) is topologized using Aut(F G (S)) for S ∈ G−Sets. More precisely, the stabilizers Stab The following groups are Noohi: Q ℓ , Q ℓ for the colimit topology induced by expressing Q ℓ as a union of finite extensions (in contrast with the situation for the ℓ-adic topology), GL n (Q ℓ ) for the colimit topology (see [BS15, Example 7.1.7]).
The notion of a Noohi group is tightly connected to a notion of an infinite Galois category, which we are about to introduce. Here, an object X ∈ C is called connected if it is not empty (i.e., initial), and for every subobject (1) C is a category admitting colimits and finite limits.
(2) Each X ∈ C is a disjoint union of connected (in the sense explained above) objects.
(3) C is generated under colimits by a set of connected objects. (4) F is faithful, conservative, and commutes with colimits and finite limits. The fundamental group of (C, F ) is the topological group π 1 (C, F ) ∶= Aut(F ), topologized by the compact-open topology on Aut(S) for any S ∈ Sets.
An infinite Galois category (C, F ) is tame if for any connected X ∈ C, π 1 (C, F ) acts transitively on F (X).
Example 2.4. If G is a topological group, then (G − Sets, F G ) is a tame infinite Galois category. (1) π 1 (C, F ) is a Noohi group.
(2) There is a natural identification of Hom cont (G, π 1 (C, F )) with the groupoid of functors C → G − Sets that commute with the fibre functors. (3) If (C, F ) is tame, then F induces an equivalence C ≃ π 1 (C, F ) − Sets.
The "tameness" assumption cannot be dropped as there exist infinite Galois categories that are not of the form (G − Sets, F G ), see [BS15,Ex. 7.2.3]. This was overlooked in [Noo08], where a similar formalism was considered.
Remark 2.6. The above formalism was also studied in [Lep10,Chapter 4] under the names of "quasiprodiscrete" groups and "pointed classifying categories".
In Section 2.2 below we will study "Noohi completion" and the dictionary between Noohi groups and G − Sets (see Section 2.3). For now, let us return to gathering the results from [BS15].
Definition 2.7. Let X be a locally topologically noetherian scheme. Let Y → X be a morphism of schemes such that: (1) it is étale (not necessarily quasi-compact!) (2) it satisfies the valuative criterion of properness. We will call Y a geometric covering of X. We will denote the category of geometric coverings by Cov X .
As Y is not assumed to be of finite type over X, the valuative criterion does not imply that Y → X is proper (otherwise we would simply get a finite étale morphism).
Example 2.8. For an algebraically closed fieldk, the category Cov Spec(k) consists of (possibly infinite) disjoint unions of Spec(k) and we have Cov Spec(k) ≃ Sets.
More generally, one has: Lemma 2.9. ([BS15, Lm. 7.3.8]) If X is a henselian local scheme, then any Y ∈ Cov X is a disjoint union of finite étale X-schemes.
Let us choose a geometric pointx ∶ Spec(k) → X on X. By Example 2.8, this gives a fibre functor Fx ∶ Cov X → Sets. By [BS15, Lemma 7.4.1]), the pair (Cov X , Fx) is a tame infinite Galois category. Then one defines Definition 2.10. The pro-étale fundamental group is defined as π proét 1 (X,x) = π 1 (Cov X , Fx).
In other words, π proét 1 (X,x) = Aut(Fx) and this group is topologized using the compact-open topology on Aut(S) for any S ∈ Sets.
One can compare the groups π proét 1 (X,x), πé t 1 (X,x) and π SGA3 1 (X,x), where the last group is the group introduced in Chapter X.6 of [SGA70].
Lemma 2.11. For a scheme X, the following relations between the fundamental groups hold (1) The group πé t 1 (X,x) is the profinite completion of π proét 1 (X).
As shown in [BS15, Example 7.4.9], π proét 1 (X,x) is indeed more general than π SGA3 1 (X,x). This can be also seen by combining Example 4.5 with Prop. 4.8 below.
The following lemma is extremely important to keep in mind and will be used many times throughout the paper. Recall that, for example, a normal scheme is geometrically unibranch.
(1) A map f ∶ Y → X of schemes is called weakly étale if f is flat and the diagonal (2) The pro-étale site X proét is the site of weakly étale X-schemes, with covers given by fpqc covers.
This definition of the pro-étale site is justified by a foundational theorem -part c) of the following fact.
We write Loc X for the corresponding full subcategory of Shv(X proét ).
We are ready to state the following important result.
Topological invariance of the pro-étale fundamental group. We note that universal homeomorphisms of schemes induce equivalences on the corresponding categories of geometric coverings.
Proof. As Cov X ≃ Loc X , the theorem follows by the same proof as in [BS15,Lm. 5 Alternatively, one can argue more directly (i.e. avoiding the equivalence with Loc X ) as follows. By [Sta20, Theorem 04DZ], V ↦ V ′ = V × X X ′ induces an equivalence of categories of schemes étale over X and schemes étale over X ′ . By [Ryd10,Proposition 5.4.], this induces an equivalence between schemes étale and separated over respectively X and X ′ . The only thing left to be shown is that if for an étale separated scheme Y → X, the map Y × X X ′ → X ′ satisfies the existence part of the valuative criterion of properness, then so does Y → X. But this property can be characterized in purely topological terms (see [Sta20,Lemma 01KE]) and so the result follows from the fact that h is a universal homeomorphism.

Noohi completion. Let
HausdGps denote the category of Hausdorff topological groups (recall that we assume all topological groups to be Hausdorff, unless stated otherwise) and NoohiGps to be the full subcategory of Noohi groups. Let G be a topological group. Denote C G = G − Sets and let F G ∶ C G → Sets be the forgetful functor. Observe that (C G , F G ) is a tame infinite Galois category. Thus, the group Aut(F G ) is a Noohi group. It is easy to see that a morphism G → H defines an induced morphism of groups Aut(F G ) → Aut(F H ) and check that it is continuous. Let ψ N ∶ HausdGps → NoohiGps be the functor defined by G ↦ Aut(F G ). Denote also the inclusion i N ∶ NoohiGps → HausdGps.
Definition 2.18. We call ψ N (G) the Noohi completion of G and will denote it G Noohi .
Example 2.19. In [BS15, Example 7.2.6], it was explained that the category of Noohi groups admits coproducts. Let G 1 , G 2 be two Noohi groups and let G 1 * N G 2 denote their coproduct as Noohi groups. Let G 1 * top G 2 be their topological coproduct. It exists and it is a Hausdorff group ( [Gra48]). Then Proposition 2.20. For a topological group G, the functor F G induces an equivalence of categories Moreover, α * G ○F G ≃ id, and thus α * is an equivalence of categories, too. Proof. The first part follows directly from [BS15, Theorem 7.2.5]. The natural isomorphism α * G ○F G ≃ id is clear from the definitions. It follows that α * G is an equivalence.
Lemma 2.21. For any topological group G, the image of α G ∶ G → G Noohi is dense.
Proof. Let U ⊂ G Noohi be open. As G Noohi is Noohi, there exists q ∈ G Noohi and an open subgroup V < ○ G Noohi such that qV ⊂ U . The quotient G Noohi V gives a G Noohi -set. It is connected in the category G Noohi − Sets and, by Prop. 2.20, α * G (G Noohi V ) is connected. Thus, the action of G on G Noohi V is transitive and so there exists g ∈ G such that α G (g) ⋅ [V ] = [qV ], i.e. α G (g) ∈ qV . Thus, the image of α G is dense.
Observation. Let f ∶ H → G be a map of topological groups. Directly from the definitions, one sees that the following diagram commutes: Corollary 2.23. The functor ψ N is a left adjoint of i N .
Remark 2.24. There are few places, where we write G Noohi for a non-Hausdorff group G. This is mostly to avoid a large overline sign over a subgroup described by generators. In these cases, we mean where G Hausd is the maximal Hausdorff quotient. As (−) Hausd is a left adjoint as well, this usually does not cause problems. This also provides a left adjoint to the forgetful functor NoohiGps → TopGps to all topological groups.
We now move towards a more explicit description of the Noohi completion.
Lemma 2.25. Let (G, τ ) be a topological group. Denote by B the collection of sets of the form Then B is a basis of a group topology τ ′ on G that is weaker than τ and open subgroups of (G, τ ) form a basis of open neighbourhoods of 1 G in (G, τ ′ ).

Moreover, the natural map
Proof.
where τ ′ denotes the topology described in the previous lemma and . . . denotes the Raȋkov completion.
Proof. We combine Fact 2.26 with the last lemma and get (G, τ ) Noohi ≃ (G, τ ′ ) Noohi ≃ (G, τ ′ ). Observation 2.28. Let G be a topological group and H a normal subgroup. Then the full subcategory of G − Sets of objects on which H acts trivially is equal to the full subcategory of G − Sets on which its closureH acts trivially and it is equivalent to the category of G H − Sets. So, it is an infinite Galois category with the fundamental group equal to (G H ) Noohi .
Lemma 2.29. Let X be a connected, locally path-connected, semilocally simply-connected topological space and x ∈ X a point. Let F x be the functor taking a covering space Y → X to the fibre Y x over the point x ∈ X. Then (TopCov(X), F x ) is a tame infinite Galois category and π 1 (TopCov(X), F x ) = π top 1 (X, x), where we consider π top 1 (X, x) with the discrete topology. Here, TopCov(X) denotes denotes the category of covering spaces of X.
Proof. We first claim that there is an isomorphism: (TopCov(X), F x ) ≃ (π top 1 (X, x)−Sets, F π top 1 (X,x) ). This is in fact a classical result in algebraic topology, which can be recovered from [Ful95,Ch. 13]  In a topological group open subgroups are also closed, so a thickly closed subgroup is also an intersection of closed subgroups, so it is closed in G. Observe also that an arbitrary intersection of thickly closed subgroups is thickly closed. This justifies, for example, the existence of the smallest normal thickly closed subgroup containing a given group. In fact, we can formulate a more precise observation.
Observation 2.31. Let H < G be a subgroup of a topological group G. Then the smallest normal thickly closed subgroup of G containing H is equal to (H nc ), where H nc is the normal closure of H in G.
Observation 2.32. Let G be a topological group such that the open subgroups form a local base at 1 G . Let W ⊂ G be a subset. Then the topological closure of W can be written as W = ∩ V < ○ G W V .
The following lemma can be found on p.79 of [Lep10]. Let us make an easy observation, that will be useful to keep in mind while reading the proof of the technical proposition below.  ]. This justifies using words "injective" or "surjective" when speaking about maps in (C, F ).
Recall the following fact.
Observation. Let f ∶ G ′ → G be a surjective map of topological groups. Then the induced morphism ← C G the corresponding maps of the infinite Galois categories. Then the following hold: (1) The map h ′ ∶ G ′′ → G ′ is a topological embedding if and only if for every connected object X in C G ′′ , there exist connected objects X ′ ∈ C G ′′ and Y ∈ C G ′ and maps X ′ ↠ X and X ′ ↪ H ′ (Y ).
(2) The following are equivalent The functor H maps connected objects to connected objects.
if and only if the composition H ′ ○ H maps any object to a completely decomposed object. (5) Assume that h ′ (G ′′ ) ⊂ ker(h) and that h ∶ G ′ → G has dense image. Then the following conditions are equivalent: (a) the induced map (G ′ ker(h)) Noohi → G is an isomorphism and the smallest normal thickly closed subgroup containing Im(h ′ ) is equal to ker(h), Proof.
(1) The proof is virtually the same as for usual Galois categories, but there every injective map is automatically a topological embedding (as profinite groups are compact). Assume that For the other implication: we want to prove that G ′′ → G ′ is a topological embedding under the assumption from the statement. It is enough to check that the set of preimages h ′ −1 (B) of some basis B of opens of e G ′ forms a basis of opens of e G ′′ . Indeed, assume that this is the case. Firstly, observe that it implies that h ′ is injective, as both G ′′ and G ′ are Hausdorff (and in particular T 0 ). If U is an open subset of G ′′ , then we can write The surjectivity of the first map means that we can assume (up to replacingŨ by a conjugate)Ũ ⊂ U . The injectivity of the second means that we can assume (up to replacing V by a conjugate) that h ′−1 (V ) ⊂Ũ . Indeed, the injectivity implies that if h ′ (g ′′ )V = V , then g ′′Ũ =Ũ which translates immediately to h ′−1 (V ) ⊂Ũ . So we have also h ′−1 (V ) ⊂ U , which is what we wanted to prove.
(2) The equivalence between (a) and (b) follows from the observation that a map between Noohi groups G ′ → G has a dense image if and only if for any open subgroup U of G, the induced map on sets G ′ → G U is surjective. Here, we only use that open subgroups form a basis of open neighbourhoods of 1 G ∈ G. Now, the functor H is automatically faithful and conservative (because F G ′ ○H = F G is faithful and conservative). Assume that (b) holds. Let S, T ∈ G−Sets and let g ∈ Hom G ′ −Sets (H(S), H(T )). We have to show that g comes from g 0 ∈ Hom G−Sets (S, T ). We can and do assume S, T connected for that. Let Γ g ⊂ H(S) × H(T ) be the graph of g. It is a connected subobject. As . This shows that G ′ U pulls back to a completely decomposed object.
The other way round: assume that for every connected object Y of C G ′ such that H ′ (Y ) contains a final object, H ′ (Y ) is completely decomposed. Let U be an open subgroup of G ′ containing h ′ (G ′′ ). Then G ′′ fixes [U ] ∈ G U and so, by assumption, fixes every [g ′ U ] ∈ G U . This implies that for any As this is true for any U containing h ′ (G ′′ ) we get that h ′ (G ′′ ) = (h ′ (G ′′ ) nc ) and the last group is the smallest normal thickly closed subgroup of G ′ containing h ′ (G ′′ ) (Observation 2.31).
(4) The same as for usual Galois categories, we use that Sets G ′′ acts trivially on S}, the assumption of (b) implies that the functor G − Sets → G ′ ker(h) − Sets is essentially surjective. By the global assumption that G ′ → G has dense image, it is fully faithful (see (2)).
As ker(h) acts trivially on H(Z), we conclude that it also acts trivially on Y . Thus, by abuse of notation, . We give two proofs of this fact.
First proof: We have proven above that (b) ⇒ (G ′ ker(h)) Noohi ≃ G. Let N be the smallest normal thickly closed subgroup of G ′ containing h ′ (G ′′ ). Observe that N ⊂ ker h (as ker(h) is thickly closed). Let U be an open subgroup containing N . We want to show that U contains ker h. This will finish the proof as both N and ker h are thickly closed. Write Y = G ′ U . Observe that G ′ U pulls back to a completely decomposed G ′′ -set if and only if . So N ⊂ U implies that Y pulls back to a completely decomposed G ′′ -set and, by assumption, Y is isomorphic to a pull-back of some G-set and so ker(h) acts trivially on Y . This implies that ker h ⊂ U , which finishes the proof. Alternative proof: We already know that (b) ⇒ (G ′ ker(h)) Noohi ≃ G. Let N ⊂ ker(h) be as in the first proof above. Consider the map G N ↠ G ker(h). The assumption (b) and full faithfulness of H (by the global assumption and using (2)) imply that completely decomposed object. As we have seen while proving "(b) ⇒ (a)", this implies Observation 2.31, there is N = (H nc ). By assumption, we have N = ker h and so we conclude that ker h ⊂ U . But then, by assumption To distinguish between exactness in the usual sense (i.e. on the level of abstract groups) and notions of exactness appearing in Prop. 2.37, we introduce a new notion. It will be mainly used in the context of Noohi groups.
. Then we will say that the sequence is (1) nearly exact on the right if h has dense image, equal to the kernel of h, (3) nearly exact if it is both nearly exact on the right and nearly exact in the middle.
We end this subsection with a lemma on topological groups and their Noohi completions that will be used later in the proof of the main theorem.
Lemma 2.39. Let G be a topological group andG be a subgroup of G Noohi such that the canonical map be the discrete set that comes naturally with an abstract action byG. If the induced abstract G-action on S is continuous, Proof. By the universal property, the G-action on S extends to G Noohi and this action is transitive. is closed in V . But asG contains the image of G, it is dense in G Noohi , and from the definition of V it follows that V 0 has to be dense in V . Putting this together, we

2.4.
A remark on valuative criteria. We will sometimes shorten "the valuative criterion of properness" to "VCoP". It is useful to keep in mind the precise statements of different parts of the valuative criterion, see [Sta20, Lemma 01KE], [Sta20, Section 01KY] and [Sta20, Lemma 01KC]. Let us prove a lemma (which is implicit in [BS15]), that VCoP chan be checked fpqc-locally.
Lemma 2.40. Let g ∶ X → S be a map of schemes. The properties: (a) g is étale (b) g is separated (c) g satisfies the existence part of VCoP can be checked fpqc-locally on S.
Moreover, the property (c) can be also checked after a surjective proper base-change.
Proof. The cases of étale and separated morphisms are proven in [Sta20, Section 02YJ]. For the last part: satisfying the existence part of VCoP is equivalent to specializations lifting along any base-change of g ([Sta20, Lemma 01KE]). It is easy to see that this property can be checked Zariski locally. Thus, if S ′ → S is an fpqc cover such that the base-change g ′ ∶ X ′ → S ′ satisfies specialization lifting for any base-change, we can assume that S, S ′ are affine with S ′ → S faithfully flat. Let T → S be any morphism. Consider the diagram: It is enough to show, that W is stable under specialization or, equivalently, that T ∖ W is stable under generalization. But, from flatness ([Sta20, Lemma 03HV]), generalizations lift along S ′ × S T → T . Thus, it is enough to show that the preimage of T ∖ W in S ′ × S T is stable under generalizations or, equivalently (using the surjectivity of S ′ × S T → T ), that the preimage of W in S ′ × S T is closed under specializations. But an easy diagram chasing (using the fact that the right square of the diagram above is cartesian) shows that the preimage of W in S ′ × S T is the image of a closed subset of S ′ × S X × T . We conclude, because specializations lift along S ′ × S X × S T → S ′ × S T by assumption.
The last part of the statement is proven in an analogous way.
Lemma 2.41. Let f ∶ Y → X be a geometric covering of a locally topologically noetherian scheme. Then f is separated.
3. SEIFERT-VAN KAMPEN THEOREM FOR π proét 1 AND ITS APPLICATIONS 3.1. Abstract Seifert-van Kampen theorem for infinite Galois categories. We aim at recovering a general version of van Kampen theorem, proven in [Sti06], in the case of the pro-étale fundamental group. Most of the definitions and proofs are virtually the same as in [Sti06], after replacing "Galois category" with "(tame) infinite Galois category" and "profinite" with "Noohi", but still some additional technical difficulties appear here and there. We make the necessary changes in the definitions and deal with those difficulties below. Denote by ∆ ≤2 a category whose objects are By a 2-complex E we mean a 2-complex in the category of sets. We often think of E as a category: its objects are the elements of E n for n = 0, 1, 2 and its morphisms are obtained by defining ∂ ∶ s → t where s ∈ E n and t = E(∂)(s).
and its corresponding linear map d ∶ ∆ m → ∆ n sending e i to e ∂(i) , and s ∈ E n and x ∈ ∆ m . We call E connected if E is a connected topological space.
Definition 3.1. Noohi group data (G , α) on a 2-complex E consists of the following: (1) A mapping (not necessarily a functor!) G from the category E to the category of Noohi groups: to a complex s ∈ E n is attributed a Noohi group G (s) and to a map an element α vef ∈ G (v) (its existence is a part of the definition) such that the following diagram commutes: Observe that there is an obvious notion of a morphism of (G , α)-systems: a collection of G (s)-equivariant maps that commute with the m's. Let us denote by lcs(E, (G , α)) the category of locally constant (G , α)-systems.
Let M ∈ lcs(G , α) for Noohi group data (G , α) on some 2-complex E. We define oriented graphs E ≤1 and M ≤1 (which will be an oriented graph over E ≤1 ) as in [Sti06], but our graphs M ≤1 are possibly infinite. For E ≤1 the vertices are E 0 and edges E 1 such that ∂ 0 (resp. ∂ 1 ) map an edge to its target (resp. origin). For M ≤1 the vertices are ⊔ v∈E 0 M v and edges are ⊔ e∈E 1 M e serves as the set of edges. The target/origin maps are induced by the m ∂ and the map M ≤1 → E ≤1 is the obvious one.
There is an obvious topological realization functor for graphs ⋅ . By applying this functor to the above construction we get a topological covering (because M is locally constant) M ≤1 → E ≤1 . This gives a functor Choosing a maximal subtree T of E ≤1 gives a fibre functor is an infinite Galois category and the resulting fundamental group π 1 (Cov( E ≤1 ), F T )) is isomorphic to π top 1 ( E ≤1 ) (see Lemma 2.29) which is in turn isomorphic to Fr(E 1 ) ⟨⟨{⃗ e e ∈ T } Fr(E 1 ) ⟩⟩ = Fr(⃗ e e ∈ E 1 ∖T ), where Fr(. . .) denotes a free group on the given set of generators and ⟨⟨{⃗ e e ∈ T } F r(E 1 ) ⟩⟩ denotes the normal closure in Fr(E 1 ) of the subgroup generated by {⃗ e ∈ T }. Here, ⃗ e acts on F T (M ) via π 0 (p −1 ( T )) ≅ π 0 (p −1 (∂ 0 (e))) ≅ π 0 (p −1 ( e )) ≅ π 0 (p −1 (∂ 1 (e)) ≅ π 0 (p −1 ( T )) As in [Sti06], for every s ∈ E 0 and M ∈ lcs(E, (G , α)) we have that F T (M ) can be seen canonically as a G (s)-module by M s = π 0 (p −1 (s)) ≅ π 0 (p −1 (T )). Denote π 1 (E ≤1 , T ) = Fr(E 1 ) ⟨⟨{⃗ e e ∈ T } Fr(E 1 ) ⟩⟩. Putting the above together we get a functor In the setting of usual ("finite") Galois categories, it is usually enough to say that a particular morphism between two Galois categories is exact, because of the following fact ([Sta20, Tag 0BMV]): Let G be a topological group. Let F ∶ Finite−G−Sets → Sets be an exact functor with F (X) finite for all X. Then F is isomorphic to the forgetful functor.
As we do not know if an analogous fact is true for infinite Galois categories, given two infinite Galois categories (C, F ), (C ′ , F ′ ) and a morphism φ ∶ C → C ′ , we are usually more interested in checking whether F ≃ F ′ ○φ. If φ satisfies this condition, it also commutes with finite limits and arbitrary colimits. Indeed, we have a map colimφ(X i ) → φ(colimX i ) that becomes an isomorphism after applying F ′ (as F ′ and F = F ′ ○ φ commute with colimits) and we conclude by conservativity of F ′ . Similarly for finite limits.
Proposition 3.5. Let (E, (G , α)) be a connected 2-complex with Noohi group data. Define a functor F ∶ lcs(E, (G , α)) → Sets in the following way: pick any simplex s and define F by M ↦ M s . Then (lcs(E, (G , α)), F ) is a tame infinite Galois category.
Moreover, the obtained functor Colimits and finite limits: they exist simplexwise and taking limits and colimits is functorial so we get a system as candidate for a colimit/finite limit. This will be a locally constant system, as the colimit/finite limit of bijections between some G-sets is a bijection.
Each M is a disjoint union of connected objects: let us call N ∈ lcs(G , α) a subsystem of M if there exists a morphism N → M such that for any simplex s the map N s → M s is injective (we then identify, for any simplex s, N s with a subset of M s ). We can intersect such subsystems in an obvious way and observe that it gives another subsystem. So for any element a ∈ M v there exists the smallest subsystem N of M such that a ∈ N v . We see readily that for any vertices v, v ′ and a ∈ M v , a ′ ∈ M v ′ the smallest subsystems N and N ′ containing one of them are either equal or disjoint (in the sense that, for each simplex s, N s and N ′ s are disjoint as subsets of M s ). It is easy to see that in this way we have obtained a decomposition of M into a disjoint union of connected objects.
F is faithful, conservative and commutes with colimits and finite limits: observe that φ s ∶ lcs(E, (G , α)) ∋ M ↦ M s ∈ G (s) − Sets is faithful, conservative and commutes with colimits and finite limits and It is obvious that F ≃ F forget ○ Q. We are now going to show that Q preserves connected objects. Take a connected object M ∈ lcs(E, (G , α)) and suppose that N is a non-empty subset of F T (M ) stable under the action of π 1 (E ≤1 , T ) and G (v) for v ∈ E 0 . Stability under the action of π 1 (E ≤1 , T ) shows that N can be extended to a subgraph N ≤1 ⊂ M ≤1 : for an edge e of M ≤1 we declare it to be an edge of N ≤1 if one of its ends touches a connected component of p −1 ( T ) corresponding to an element of N . This is well defined, as in this case both ends touch such a component -this is because the action of m ∂ 1 m −1 ∂ 0 equals the action of → e∈ π 1 (E ≤1 , T ). Now we want to show that it extends to 2-simplexes. This is a local question and we can restrict to simplices in the boundary of a given face f ∈ E 2 . Define N f as a preimage of N s via any ∂ such that ∂(f ) = s. We see that if the choice is independent of s, then we have extended N to a locally constant system. To see the independence it is enough to prove that if (vef ) is a barycentric subdivision (i.e. we have ∂ and ∂ ′ such that ∂ ′ (f ) = e and ∂(e) = v), then m −1 and thus N can be seen as an element of lcs(E, (G , α)) which is a subobject of M , which contradicts connectedness of M .
To see that lcs(E, (G , α)) is generated under colimits by a set of connected objects, observe that in the above proof of the fact that Q preserves connected objects, we have in fact shown the following statement. We want to show that there exists a set of connected objects in lcs(G , α) such that any connected object of lcs(G , α) is isomorphic to an element in that set. As an analogous fact is true in Looking at the graph of this isomorphism, we find a connected subobject Z ⊂ QX × QY that maps isomorphically on QX and QY via the respective projections. By the above fact, we know that there exists W ⊂ X × Y such that QW = Z. Because F ≃ F forget ○ Q and F is conservative, we see that the projections W → X and W → Y must be isomorphisms. This shows X ≃ Y as desired.
The only claim left is that lcs(E(G , α)) is tame, but this follows from tameness of ( * N v∈E 0 G (v) * N π 1 (E ≤1 , T )) − Sets, the equality F ≃ F forget ○ Q and the fact that Q maps connected objects to connected objects.
Let us denote by π 1 (E, G , s) the fundamental group of the infinite Galois category (lcs(E, G ), F s ). The proposition above tells us that there is a continuous map of Noohi groups with dense image * N v∈E 0 G (v) * N π 1 (E ≤1 , T )) → π 1 (E, G , s). We now proceed to describe the kernel.
Theorem 3.7. (abstract Seifert-Van Kampen theorem for infinite Galois categories) Let E be a connected 2-complex with group data (G , α). With notations as above, the functor Q induces an isomorphism of Noohi groups where ⟨⟨−⟩⟩ denotes the normal closure of the subgroup generated by the indicated elements and α's come from the definition of a (G , α)-system for each given f .
Proof. The same proof as the proof of [Sti06, Thm. 3.2 (2)] shows that Q induces an equivalence of categories between the infinite Galois categories (lcs(E, G ), F s ) and the full subcategory of objects of * N v∈E 0 G (v) * N π 1 (E ≤1 , T ) − Sets on which H acts trivially. We conclude by Observation 2.28. Remark 3.8. It is important to note that we can replace free Noohi products by free topological products in the statement above, as we take the Noohi completion of the quotient anyway. More precisely, the canonical map of a group having the same generators as H. This is because the categories of G − Sets are the same for those two Noohi groups.
Fact 3.9. The topological free product * top i G i of topological groups has as an underlying space the free product of abstract groups * i G i . This follows from the original construction of Graev [Gra48].

Application to the pro-étale fundamental group.
Descent data. Let T • be a 2-complex in a category C and let F → C be a category fibred over C , with F (S) as a category of sections above the object S.
Definition 3.10. The category DD(T • , F ) of descent data for F C relative T • has as objects pairs such that the cocycle condition holds, i.e., the following commutes in F (T 2 ): Definition 3.11. In the above context h ∶ S ′ → S is called an effective descent morphism for F if h * is an equivalence of categories. Proof. This was proven by Lavanda and relies on the results of [Ryd10]. More precisely, this follows from Prop. 5.4 and Thm. 5.19 of [Ryd10], then checking that the obtained algebraic space is a scheme (using étaleness and separatedness, see [Sta20, Tag 0417]) and that it still satisfies the valuative criterion (see Lemma 2.40).
Discretisation of descent data. We would like to apply the procedure described in [Sti06, §4.3] but to the pro-étale fundamental group. However, in the classical setting of Galois categories, given a category C and functors F, F ′ ∶ C → Sets such that (C, F ) and (C, F ′ ) are Galois categories (i.e. F, F ′ are fibre functors), there exists an isomorphism (not unique) between F and F ′ . Choosing such an isomorphism is called "choosing a path" between F and F ′ . However, it is not clear whether an analogous statement is true for tame infinite Galois categories as the proof does not carry over to this case (see the proof of [Sta20,Lemma 0BN5]  Question 3.13. Let C be a category and F, F ′ ∶ C → Sets be two functors such that (C, F ) and (C, F ′ ) are tame infinite Galois categories. Is it true that F and F ′ are isomorphic?
As we do not know the answer to this question, we have to make an additional assumption when trying to discretise the descent data. Fortunately, it will always be satisfied in the geometric setting, which is our main case of interest.
Definition 3.14. Let (C, F ), (C ′ , F ′ ) be two infinite Galois categories and let φ ∶ C → C ′ be a functor. We say that φ is compatible if there exists an isomorphism of functors F ≃ F ′ ○ φ.
Let F → C be fibred in tame infinite Galois categories. More precisely, we have a notion of connected objects in C and any T ∈ C is a coproduct of connected components. Over connected objects F takes values in tame infinite Galois categories (i.e. over a connected Y ∈ C there exists a functor F Y ∶ F (Y ) → Sets such that (F (Y ), F Y ) is a tame infinite Galois category but we do not fix the functor).
Definition 3.15. Let T • be a 2-complex in C . Let E = π 0 (T • ) be its 2-complex of connected components: the 2-complex in Sets built by degree-wise application of the connected component functor. We will say that T • is a compatible 2-complex if one can fix fibre functors F s of F (s) for each simplex s ∈ E such that (F (s), F s ) is tame and for any boundary map ∂ ∶ s → s ′ there exists an isomorphism of fibre functors F s ○ T (∂) * ∼ → F s ′ . The 2-complexes that will appear in the (geometric) applications below will always be compatible. From now on, we will assume all 2-complexes to be compatible, even if not stated explicitly. Let T • be a compatible 2-complex in C . Fix fibre functors F s and isomorphisms between them as in the definition of a compatible 2-complex. For any ∂, denote the fixed isomorphism by ⃗ ∂. For a 2-simplex (vef ) of the barycentric subdivision with ∂ ′ ∶ f → e and ∂ ∶ e → v we define or, more precisely, We define Noohi group data (G , α) on E in the following way: G (s) = π 1 (F (s), F s ) for any simplex . We define elements α as described above and we easily check that this gives Noohi group data.
Proposition 3.16. The choice of functors F s and the choice of ⃗ ∂ as above fix a functor which is an equivalence of categories.
Proof. Given a descent datum (X ′ , φ) relative T • we have to attach a locally constant (G , α)-system on E in a functorial way. For v ∈ E 0 , e ∈ E 1 and f ∈ E 2 , the definition of suitable G (v) (or G (e) or G (f )) sets and maps m ∂ between them can be given by the same formulas as in [Sti06,Prop. 4.4] and also the same computations as in [Sti06,Prop. 4.4] show that we obtain an element of lcs(E, (G , α)). Again, the reasoning of [Sti06,Prop. 4.4] gives a functor in the opposite direction: given M ∈ lcs(E, (G , α)) we define Maps from edges to vertices define a map φ ∶ T (∂ 0 ) * X ′ → T (∂ 1 ) * X ′ and to check the cocycle condition one reverses the argument of the proof that discr gives a locally constant system.
To apply the last proposition we need to know that the compatibility condition holds in the setting we are interested in. ). Let f ∶ X ′ → X be a morphism of two connected locally topologically noetherian schemes and letx ′ ,x be geometric points on X ′ , X, correspondingly. Then the functor f * ∶ Cov X → Cov X ′ is a compatible functor between infinite Galois categories (Cov X , Fx) and (Cov X ′ , Fx′ ), i.e. the functors Fx and Fx′ ○ f * are isomorphic.
Proof. Looking at the image ofx ′ (as a geometric point) on X, we reduce to the case when bothx ′ and x lie on the same scheme X. In that case we proceed as in the second part of the proof of [BS15, Lm. 7.4.1].
The above results combine to recover the analogue of [Sti06,Cor. 5.3] in the pro-étale setting.
Corollary 3.19. Let h ∶ S ′ → S be an effective descent morphism for geometric coverings. Assume that S is connected and S, S ′ , S ′ × S S ′ , S ′ × S S ′ × S S ′ are locally topologically noetherian. Let S ′ = ⊔ v S ′ v be the decomposition into connected components. Lets be a geometric point of S, lets(t) be a geometric point of the simplex t ∈ π 0 (S • (h)), and let T be a maximal tree in the graph Γ = π 0 (S • (h)) ≤1 . For every boundary map ∂ ∶ t → t ′ let γ t ′ ,t ∶s(t ′ ) → S • (h)(∂)s(t) be a fixed path (i.e. an isomorphism of fibre functors as in Lm. 3.17). Then canonically with respect to all these choices where H is the normal subgroup generated by the cocycle and edge relations for all parameter values e ∈ S 1 (h), g ∈ π proét 1 (e,s(e)), and f ∈ S 2 (h). The map π proét 1 (∂ i ) uses the fixed path γ ∂ i (e),e and α Remark 3.20. Similarly as in Rmk. 3.8, we could replace * N by * top in the above, as we take the Noohi completion of the whole quotient anyway.
Remark 3.21. We will often use Cor. 3.19 for h -the normalization map (or similar situations), where the connected components S ′ v are normal. In this case π proét . This implies that π proét 1 (∂ 1 ) factorizes through the profinite completion of π proét 1 (e,s(e)), which can be identified with πé t 1 (e,s(e)). Moreover, the map π proét 1 (e,s(e)) → πé t 1 (e,s(e)) has dense image and, in the end, we take the closure H of H. The upshot of this discussion is that in the definition of generators of H we might consider g ∈ πé t 1 (e,s(e)) instead of g ∈ π proét 1 (e,s(e)) and πé t 1 where H is the normal subgroup generated by (R 1 ) πé t 1 (∂ 1 )(g)⃗ eπé t 1 (∂ 0 )(g) −1 ⃗ e −1 for all e ∈ S 1 (h), g ∈ πé t 1 (e,s(e)) and Let us move on to some applications.
Ordered descent data. Let F be a category fibred over C with a fixed splitting cleavage (i.e. the associated pseudo-functor is a functor). Assume that C is some subcategory of the category of locally topologically noetherian schemes with the property that finite fibre products in C are the same as the finite fibre products as schemes. Let h = ⊔ i∈I h i ∶ S ′ = ⊔ i S ′ i∈I → S be a morphism of schemes and let < be a total order on the set of indices I.
be the open and closed sub-2-complex of schemes in C of ordered partial products S be a morphism of schemes such that, for every i, j ∈ I, the maps induced by the diagonal morphisms ∆

fully faithful. Then the natural open and closed immersion
. We first claim that there is exactly one isomorphism ∂ 0 * , so there is at most one map φ as above (we use here and below that we work with a splitting cleavage and so, by definition, the pullback functors preserve compositions of maps). Moreover, our assumptions imply that ∆ * 2,i is fully faithful as well, which shows that φ ∶ ∂ * 0 Y S i → ∂ * 1 Y S i corresponding to id Y S i will satisfy the condition. A similar reasoning shows that if we have φ ij specified for i < j, then φ ji is uniquely determined and the if φ ij 's satisfy the cocycle condition on S ijk for i < j < k, then φ ij 's together with φ ji 's obtained will satisfy the cocycle condition on any S αβγ , α, β, γ ∈ {i, j, k} is an isomorphism, then (still assuming splitting of the cleavage) the assumptions of the proposition are satisfied.
Two examples.
Example 3.24. Let k be a field and C be P 1 k with two k-rational closed points p 0 and p 1 glued (see [Sch05] for results on gluing schemes). Denote by p the node (i.e. the image of p i 's in C). We want to compute π proét 1 (C). By the definition of C, we have a map h ∶C = P 1 → C (which is also the normalization). It is finite, so it is an effective descent map for geometric coverings. Thus, we can use the van Kampen theorem. This goes as follows: • Check thatC × CC ≃C ⊔p 01 ⊔p 10 as schemes over C, where p αβ are equal to Spec(k) and map to the node of C via the structural map. This can be done by checking that Hom C (Y,C⊔p 01 ⊔p 10 ) ≃ Hom C (Y,C) × Hom C (Y,C); • Similarly, check thatC × CC × CC ≃C ⊔ p 001 ⊔ p 010 ⊔ p 011 ⊔ p 100 ⊔ p 101 ⊔ p 110 , where the projectionC × CC × CC →C × CC omitting the first factor maps p abc to p bc and so on; • We fix a geometric pointb = Spec(k) over the base scheme Spec(k) and fix geometric pointsp 0 andp 1 over p 0 and p 1 that map tob. Then we fix geometric points onC, p 01 , p 10 ⊂C ⊔p 01 ⊔p 10 ≃ C × CC in a compatible way and similarly for connected components ofC × CC × CC (i.e. let us say thatp αβγ ↦p α via v 0 andp αβ ↦p α ). We fix a path γ fromp 0 top 1 that becomes trivial on Spec(k) via the structural map (this can be done by viewingp 0 andp 1 as geometric points oñ Ck, choosing the path onCk first and defining γ to be its image). Letp be the fixed geometric point on C given by the image ofp 0 (or, equivalently,p 1 ).
With this setup, the α (f ) ijk 's (defined as in Cor. 3.19) are trivial for any f and so the relation (2) in this corollary reads gives that the image of π 1 (Γ, T ) ≃ Z * 3 in π proét 1 (C,p) is generated by a single edge (in our case only one maximal tree can be chosen -containing a single vertex). The choice of paths made guarantees π proét 1 (∂ 0 )(g) = π proét 1 (∂ 1 )(g) in π proét 1 (C,p 0 ) for any g ∈ π proét 1 (p ab ,p ab ) = Gal(k). So relation (1) in Cor. 3.19 implies that the image of π proét 1 (C,p 0 ) ≃ Gal(k) in π proét 1 (C,p 0 ) commutes with the elements of the image of π 1 (Γ, T ). Putting this together we get Example 3.25. Let X 1 , . . . , X m be geometrically connected normal curves over a field k and let Y m+1 , . . . , Y n be nodal curves over k as in Ex. 3.24. Let x i ∶ Spec(k) → X i be rational points and let y j denote the node of Y j . Let X ∶= ∪ • X i ∪ • Y j be a scheme over k obtained via gluing of X i 's and Y j 's along the rational points x i and y j (in the sense of [Sch05]). The notation ∪ • denotes gluing along the obvious points. The point of gluing gives a rational point x ∶ Spec(k) → X. We choose a geometric pointb = Spec(k) over the base Spec(k) and choose a geometric pointx over x such that it maps tob. The maps X i → X and Y j → X are closed immersions (this is basically [Sch05, Lm. 3.8]). We also get geometric pointsx i and y j over x i and y j that map tob as well. Denote It is a copy of Gal k in the sense that the induced map πé t 1 (x i ,x i ) → πé t 1 (Spec(k),b) is an isomorphism. Let us denote by ι i ∶ Gal k → Gal k,i the inverse of this isomorphism. The group πé t 1 (x i ,x i ) acts on πé t 1 (X i ,x i ) and allows to write πé t 1 (X i ,x i ) ≃ πé t 1 (X i ,x i ) ⋊ Gal k,i . After some computations (as in the previous example), using Cor. 3.19 and Ex. 3.24, one gets Let us describe the category of group-sets.
Lemma 3.26. Let K and Q be topological groups and assume we have a continuous action K × Q → K respecting multiplication in K. Then K ⋊ Q with the product topology (on K × Q) is a topological group and there is an isomorphism Proof. That K ⋊Q becomes a topological group is easy from the continuity assumption of the action. The isomorphism is obtained as follows: from the universal property we have a continuous homomorphism K * top Q → K ⋊ Q and the kernel of this map is the smallest normal subgroup containing the elements qkq −1 ( q k) −1 (this follows from the fact that the underlying abstract group of K * top Q is the abstract free product of the underlying abstract groups, similarly for K ⋊ Q and that we know the kernel in this case). So we have a continuous map that is an isomorphism of abstract groups. We have to check that the inverse map K ⋊ Q ∋ kq ↦ kq ∈ K * top Q ⟨⟨qkq −1 = q k⟩⟩ is continuous. It is enough to check that the map K × Q ∋ (k, q) ↦ kq ∈ K * top Q (of topological spaces) is continuous, but this follows from the fact that the maps K → K * top Q and Q → K * top Q are continuous and that the multiplication map (K * top Q) × (K * top Q) → K * top Q is continuous.
Let us also state a technical lemma concerning the "functoriality" of the van Kampen theorem. It is important that the diagram formed by the schemes X 1 , X 2 , X, X 1 in the statement is cartesian.
Lemma 3.27. Let f ∶ X 1 → X 2 be a morphism of connected schemes and h ∶ X → X 2 be a morphism of schemes. Denote by h 1 ∶ X 1 → X 1 the base-change of h via f . Assume that h and h 1 are effective descent morphisms for geometric coverings and that local topological noetherianity assumptions are satisfied for the schemes involved as in the statement of Cor. 3.19. Assume that for any connected component W ∈ π 0 (S • (h)), the base-change W 1 of W via f is connected. Choose the geometric points on W 1 ∈ π 0 (S • (h 1 )) and paths between the obtained fibre functors as in Cor. 3.19 and choose the geometric points and paths on W ∈ π 0 (S • (h)) as the images of those chosen for X 1 . Identify the graphs Γ = π 0 (S • (h)) ⩽1 and Γ 1 = π 0 (S • (h 1 )) ⩽1 (it is possible thanks to the assumption made) and choose a maximal tree T in Γ. Using the above choices, use Cor. 3.19. to write the fundamental groups π proét 1 (X 1 ) ≃ ( * top W ∈π 0 ( X) π proét 1 (W 1 )) * top π 1 (Γ 1 , T ) ⟨R ′ ⟩ Noohi and π proét 1 (X 2 ) ≃ ( * top W ∈π 0 ( X) π proét 1 (W )) * top π 1 (Γ, T ) ⟨R⟩ Noohi .
Proof. It is clear that on (the image of) π proét 1 (W 1 ) (in π proét 1 (X 1 )) the map is the one induced from The part about π 1 (Γ 1 , T ) follows from the fact that π 1 (Γ 1 , T ) < π proét 1 (X 1 ) acts in the same way as π 1 (Γ, T ) < π proét 1 (X 2 ) on any geometric covering of X 2 . This follows from the choice of points and paths on W ∈ π 0 (S • (h)) as the images of the points and paths on the corresponding connected components W 1 ∈ π 0 (S • (h 1 )). The maps as in the statement give a morphism π proét 1 (W )) * top π 1 (Γ, T ) and it is easy to check that φ(R ′ ) ⊂ R, which finishes the proof.
3.3. Künneth formula. In this subsection we use the van Kampen formula to prove the Künneth formula for π proét 1 . Let X, Y be two connected schemes locally of finite type over an algebraically closed field k and assume that Y is proper. Letx,ȳ be geometric points of X and Y respectively with values in the same algebraically closed field extension K of k. With these assumptions, the classical statement says that the "Künneth formula" for πé t 1 holds, i.e. Fact 3.28. ([SGA71, Exp. X, Cor. 1.7]) With the above assumptions, the map induced by the projections is an isomorphism We want to establish analogous statement for π proét 1 . Proposition 3.29. Let X, Y be two connected schemes locally of finite type over an algebraically closed field k and assume that Y is proper. Letx,ȳ be geometric points of X and Y respectively with values in the same algebraically closed field extension K of k. Then the map induced by the projections is an isomorphism π proét Choosing a path between (x,ȳ) and some fixed k-point of X × k Y (seen as a geometric point) and looking at the images of this path via projections onto X and Y reduces us (by Cor. 3.18 and compatibility of the chosen paths), to the situation where we can assume thatx andȳ are k-points. We are going to assume this in the proof. Before we start, let us state and prove the surjectivity of the above map as a lemma. Properness is not needed for this.
Lemma 3.30. Let X, Y be two connected schemes over an algebraically closed field k with k-points on them:x on X andȳ on Y . Then the map induced by the projections It is easy to check that the map induced on fundamental groups π proét Proof. (of Prop. 3.29) As X, Y are locally of finite type over a field, the normalization maps are finite and we can apply Prop. 3.12. Let X → X be the normalization of X and let X = ⊔ v X v be its decomposition into connected components and let us fix a closed point x v ∈ X v for each v. Similarly, let ⊔ uỸu =Ỹ → Y be the decomposition into connected components of the normalization of Y with closed points y u ∈Ỹ u .
We first deal with a particular case. Claim: the statement of Prop. 3.29 holds under the additional assumption that • either, for any v, the projections induce isomorphisms • or, for any u, the projections induce isomorphisms Proof of the claim. Apply Cor. 3.19 to h ∶ X → X. We choosex and x v 's as geometric pointss(t) of the corresponding simplexes t ∈ π 0 (S • (h)) 0 and chooses(t) to be arbitrary closed points (of suitable double and triple fibre products) for t ∈ π 0 (S • (h)) 2 . We fix a maximal tree T in Γ = π 0 (S • (h)) ≤1 and fix paths γ t ′ ,t ∶s(t ′ ) → S • (h)(∂)s(t). Thus, we get π proét where H is defined as in Cor. 3.19.
Observe now that X v × k Y are connected (as k is algebraically closed) and that h × id Y ∶ X × Y → X × Y is an effective descent morphism for geometric coverings. So we might use Cor. 3.19 in this setting.
and similarly for triple products, we can identify in a natural way ). In particular we can identify the graph Γ Y = π 0 (S • (h × id Y )) ≤1 with Γ and we choose the maximal tree T Y of Γ Y as the image of T via this identification. For t ∈ π 0 (S • (h)) choose (s(t),ȳ) as the closed base points for i(t) ∈ π 0 (S • (h × id Y )). Denote by α ijk elements of various π proét 1 ( X v ) defined as in Cor. 3.19 and by ⃗ e elements of π 1 (Γ, T ). By the choices and identifications above we can identify π 1 (Γ Y , T Y ) with π 1 (Γ, T ). Using van Kampen and the assumption, we write Here π proét 1 (Y,ȳ) v denotes a "copy" of π proét 1 (Y,ȳ) for each v. By Lm. 3.30, for T ∈ π 0 (S • (h)) the natural map π proét 1 (T × Y, (s(T ),ȳ)) → π proét 1 (T,s(T )) × π proét 1 (Y,ȳ) is surjective. It follows that the relations defining H Y (as in Cor. 3.19) can be written as where α's in the second relation are elements of suitable π proét 1 ( X v )'s and are the same as in the corresponding generators of H. The h y,i denotes a copy of element h y ∈ π proét 1 (Y,ȳ) in a suitable π proét 1 (Y,ȳ) v . Varying e and h y while choosing g = 1 ∈ π proét 1 (e,s(e)) for every e, gives that h y,1 ⃗ e = ⃗ eh y,0 . For e ∈ T we have ⃗ e = 1 and so the first relation reads h y,1 = h y,0 , i.e. it identifies π proét relations of H. Using notations from the above discussion, we can sum it up by writing Putting this together, we get equivalences of categories where equality ♠ follows from the fact that for topological groups G 1 , G 2 there is equivalence This finishes the proof of the Claim in the "either" case. After noting that eachỸ u is still proper, the "or" case follows in a completely symmetrical manner. We have proven a particular case of the proposition.
Let us now go ahead and prove the full statement. General case. The general case follows from the claim proven above in the following way: let ⊔ v X v = X → X and ⊔ uỸu =Ỹ → Y be decompositions into connected components of the normalizations of X and Y . Fix v and note that π proét (Y ) by applying the claim to Y and X v . This is possible, asỸ u 's, X v and the productsỸ u × k X v (for all u) are normal varieties and so their pro-étale fundamental groups are equal to the usual étale fundamental groups (by Lm. 2.12) for which the equality πé t is known (see Fact 3.28). Thus, for any v, we have that π proét We can now apply the claim to X and Y and finish the proof in the general case.
3.4. Invariance of π proét 1 of a proper scheme under a base-change K ⊃ k of algebraically closed fields.
Proposition 3.31. Let X be a proper scheme over an algebraically closed field k. Let K ⊃ k be another algebraically closed field. Then the pullback induces an equivalence of categories Proof. Let X ν → X be the normalization. It is finite, and thus a morphism of effective descent for geometric coverings. Let us show that the functor F is essentially surjective. Let Y ′ ∈ Cov X K . As k is algebraically closed and X ν is normal, we conclude that X ν is geometrically normal, and thus the base change (X ν ) K is normal as well (see [Sta20, Tag 038O]). Pulling Y ′ back to (X ν ) K we get a disjoint union of schemes finite étale over (X ν ) K with a descent datum. It is a classical result ([SGA71, Exp. X, Cor. 1.8]) that the pullback induces an equivalence Fét X ν → Fét X ν K of finite étale coverings and similarly for the double and triple products X ν 2 = X ν × X X ν , X ν 3 = X ν × X X ν × X X ν . These equivalences obviously extend to categories whose objects are (possibly infinite) disjoint unions of finite étale schemes (over X ν , X ν 2 , X ν 3 respectively) with étale morphisms as arrows. These categories can be seen as subcategories of Cov X ν and so on. These subcategories are moreover stable under pullbacks between Cov X ν i . Putting this together we see, that Y ′′ = Y ′ × X K (X ν ) K with its descent datum is isomorphic to a pullback of a descent datum from X ν . Thus, we conclude that there exists Y ∈ Cov X such that Y ′ ≃ Y K . Full faithfulness of F is shown in the same way. If X is connected, it can be also proven more directly, as F being fully faithful is equivalent to preserving connectedness of geometric coverings, but any connected Y ∈ Cov X is geometrically connected, and thus Y K remains connected by Lm. 2.37 (2). Note that in the above argument we do not claim that the double and triple intersections X ν 2 , X ν 3 are normal, as this is in general false. Instead, we are only using that all the considered geometric coverings of those schemes came as pullbacks from X ν , and thus were already split-to-finite.

Statement of the results and examples. The main result of this chapter is the following theorem.
Theorem. (see Theorem 4.14 below) Let k be a field and fix an algebraic closurek. Let X be a geometrically connected scheme of finite type over k. Then the sequence of abstract groups Moreover, the map π proét 1 (Xk) → π proét 1 (X) is a topological embedding and the map π proét 1 (X) → Gal k is a quotient map of topological groups.
One shows the near exactness first and obtains the above version as a corollary with an extra argument. The most difficult part of the sequence is exactness on the left. We will prove it as a separate theorem and its proof occupies an entire subsection.
Theorem. (see Theorem 4.13 below) Let k be a field and fix an algebraic closurek of k. Let X be a scheme of finite type over k such that the base change Xk is connected. Then the induced map is a topological embedding.
By Prop. 2.37, it translates to the following statement in terms of coverings: every geometric covering of Xk can be dominated by a covering that embeds into a base-change tok of a geometric covering of X (i.e. defined over k). In practice, we prove that every connected geometric covering of Xk can be dominated by a (base-change of a) covering of X l for l k finite.
For finite coverings, the analogous statement is very easy to prove simply by finiteness condition. But for general geometric coverings this is non-trivial and maybe even slightly surprising as we show by counterexamples (Ex. 4.5 and Ex. 4.6) that it is not always true that a connected geometric covering of Xk is isomorphic to a base-change of a covering of X l for some finite extension l k. This last statement is, however, stronger than what we need to prove, and thus does not contradict our theorem. Observe, that the stronger statement is true for finite coverings and, even more generally, whenever π proét 1 (Xk) is prodiscrete, as proven in Prop. 4.8.
Let us proceed to proving the easier part of the sequence first.
Observation 4.1. By Prop. 2.17, the category of geometric coverings is invariant under universal homeomorphisms. In particular, for a connected X over a field and k ′ k purely inseparable, there is π proét 1 (X k ′ ) = π proét 1 (X). Similarly, we can replace X by X red and so assume X to be reduced when convenient. In this case, base change to separable closure X k s is reduced as well. We will often use this observation without an explicit reference.
We start with the following lemmas.
Lemma 4.2. Let k be a field. Let k ⊂ k ′ be a (possibly infinite) Galois extension. Let X be a connected scheme over k. Let T 0 ⊂ π 0 (X k ′ ) be a non-empty closed subset preserved by the Gal(k ′ k)-action.
Proof. Let T be the preimage of T 0 in X k ′ (with the reduced induced structure). By [Sta20, Lemma 038B], T is the preimage of a closed subset T ⊂ X via the projection morphism p ∶ X k ′ → X. On the other hand, by [Sta20, Lemma 04PZ], the image p(T ) equals the entire X. Thus, T = X and T = X k ′ , and so T 0 = π 0 (X k ′ ).
Lemma 4.3. Let X be a connected scheme over a field k with an l ′ -rational point with l ′ k a finite field extension. Then π 0 (X k sep ) is finite, the Gal k action on π 0 (X k sep ) is continuous and there exists a finite separable extension l k such that the induced map π 0 (X k sep ) → π 0 (X l ) is a bijection. Moreover, there exists the smallest field (contained in k sep ) with this property and it is Galois over k.
Proof. Let us first show the continuity of the Gal k -action. The morphism Spec(l ′ ) → X gives a Gal kequivariant morphism Spec(l ′ ⊗ k k sep ) → X k sep and a Gal k -equivariant map π 0 (Spec(l ′ ⊗ k k sep )) → π 0 (X k sep ). Denote by M ⊂ π 0 (X k sep ) the image of the last map. It is finite and Gal k -invariant, and by Lm. 4.2, M = π 0 (X k ′ ). We have tacitly used that M is closed, as π 0 (X k ′ ) is Hausdorff (as the connected components are closed). As Gal k acts continuously on π 0 (Spec(l ′ ⊗ k k sep )) (for example by [Sta20, Lemma 038E]), we conclude that it acts continuously on π 0 (X k sep ) as well. From Lm. 4.2 again and from [Sta20, Tag 038D], we easily see that the fields l ⊂ k sep such that π 0 (X k sep ) → π 0 (X l ) is a bijection are precisely those that Gal l acts trivially on π 0 (X k sep ). To get the minimal field with this property we choose l such that Gal l = ker(Gal k → Aut(π 0 (X k sep ))).
Theorem 4.4. Let k be a field and fix an algebraic closurek. Let X be a geometrically connected scheme of finite type over k. Letx ∶ Spec(k) → Xk be a geometric point on Xk. Then the induced sequence of topological groups is nearly exact in the middle (i.e. the thick closure of im(ι) equals ker(p)) and π proét 1 (X) → Gal k is a topological quotient map. Proof.
( On the other hand, this image is dense as we have the following diagram where . . . prof means the profinite completion. In the diagram, the left vertical map has dense image and the lower horizontal is surjective. This shows that π proét 1 (X) → Gal k is surjective.
2.37 and the fact that the map Xk → Spec(k) factorizes through Spec(k).
(3) The thick closure of im(ι) is normal: as remarked above, π proét 1 (Xk) = π proét 1 (X k s ), where k s denotes the separable closure. Thus, we are allowed to replacek with k s in the proof of this point. Moreover, by the same remark, we can and do assume X to be reduced. Let Y → X be a connected geometric covering such that there exists a section s ∶ X k s → Y × X X k s = Y k s over X k s . Observe that any such section is a clopen immersion: this follows immediately from the equivalence of categories of π proét 1 (X k s ) − Sets and geometric coverings. DefineT ∶= ⋃ σ∈Gal(k) σ s(X k s ) ⊂ Y k s . Observe that two images of sections in the sum either coincide or are disjoint as X k s is connected and they are clopen. Now,T is obviously open, but we claim that it is also a closed subset. This follows from Lm. 4.3 (which implies that π 0 (Y k s ) is finite), but one can also argue directly by using that Y k s is locally noetherian and σ s(X k s ) are clopen. Now by [Sta20, Tag 038B],T descends to a closed subset T ⊂ Y . It is also open as T is the image ofT via projection Y k s → Y which is surjective and open map. Indeed, surjectivity is clear and openness is easy as well and is a particular case of a general fact, that any map from a scheme to a field is universally open ([Sta20, Tag 0383]). By connectedness of Y we see that T = Y . So Y k s =T . But this last one is a disjoint union of copies of X k s , which is what we wanted to show by Prop. 2.37. (4) The smallest normal thickly closed subgroup of π proét 1 (X) containing im(ι) is equal to ker(p): as we already know that this image is contained in the kernel and that the map π proét 1 (X) → Gal k is a quotient map of topological groups, we can apply Prop. 2.37. Let Y be a connected geometric covering of X such that Yk = Y × X Xk splits completely. Denote Yk = ⊔ α Xk ,α , where by Xk ,α we label different copies of Xk. By Lm. 4.3, π 0 (Yk) is finite, and thus the indexing set {α} and the covering Y → X are finite. But in this case, the statement follows from the classical exact sequence of étale fundamental groups due to Grothendieck.
As promised above, we give examples of geometric coverings of Xk that cannot be defined over any finite field extension l k.
Example 4.5. Let X i = G m,Q , i = 1, 2. Define X to be the gluing X = ∪ • X i of these schemes at the rational points 1 i ∶ Spec(Q) → X i corresponding to 1. Fix an algebraic closure Q of Q and so a geometric pointb over the base Spec(Q). This gives geometric pointsx i on X i = X i,Q and X i lying over 1 i , which we choose as base points for the fundamental groups involved. Similarly, we get a geometric pointx over the point of gluing x that maps tob. Then Example 3.25 gives us a description of the fundamental group Noohi and of its category of sets: Recall that the groups πé t 1 (X i ,x i ) are isomorphic toẐ(1) = lim ← µ n as Gal Q -modules. Fix these isomorphisms. Let S = N >0 . Let us define a π proét 1 (X,x)-action on S, which means giving actions by πé t 1 (X 1 ,x 1 ) and πé t 1 (X 2 ,x 2 ) (no compatibilities of the actions required). Let ℓ be a fixed odd prime number (e.g. ℓ = 3). We will give two different actions of Z ℓ (1) on S which will define actions ofẐ(1) by projections on Z ℓ (1). We start by dividing S into consecutive intervals labelled a 1 , a 3 , a 5 , . . . of cardinality ℓ 1 , ℓ 3 , ℓ 5 , . . . respectively. These will be the orbits under the action of πé t 1 (X 1 ,x 1 ). Similarly, we divide S into consecutive intervals b 2 , b 4 , b 6 , . . . of cardinality ℓ 2 , ℓ 4 , . . ..
We still have to define the action on each a m and b m . We choose arbitrary identifications b m ≃ µ ℓ m as Z ℓ (1)-modules. Now, fix a compatible system of ℓ n -th primitive roots of unity ζ = (ζ ℓ n ) ∈ Z(1). For a m 's, we choose the identifications with µ ℓ m arbitrarily with one caveat: we demand that for any even number m, the intersection b m ∩a m+1 contains the elements 1, ζ ℓ m+1 ∈ µ ℓ m+1 via the chosen identification a m+1 ≃ µ ℓ m+1 . As b m ∩ a m+1 > 0 and b m ∩ a m+1 ≡ 0 mod ℓ, the intersection b m ∩ a m+1 contains at least two elements and we see that choosing such a labelling is always possible.
Assume that S corresponds to a covering that can be defined over a finite Galois extension K Q. Fix s 0 ∈ a 1 ∩ b 2 . By increasing K, we might and do assume that Gal K fixes s 0 . Let p be a prime number ≠ ℓ that splits completely in K and p be a prime of O K lying above p. Let φ p ∈ Gal K be a Frobenius element (which depends on the choice of the decomposition group and the coset of the inertia subgroup). It acts on Z ℓ (1) via t ↦ t p and this action is independent of the choice of φ p . Choose N > 0 such that p N ≡ 1 mod ℓ 2 and let m be the biggest number such that p N ≡ 1 mod ℓ m . If m is odd, we look at p ℓN instead. In this case m + 1 is the biggest number such that p ℓN ≡ 1 mod ℓ m+1 and so, by changing N if necessary, we can assume that m is even, > 1. The whole point of the construction is the following: if s ∈ a i ∩ b j with i, j < m is fixed by φ N p , then so are g ⋅ s and h ⋅ s (for h ∈ πé t 1 (X 1 ,x 1 ) and g ∈ πé t 1 (X 2 ,x 2 )). Then moving such s with g's and h's to b m ∩ a m+1 leads to a contradiction. Indeed, let s 1 ∈ b m ∩ a m+1 ⊂ S correspond to 1 ∈ µ ℓ m+1 ≃ a m+1 (it is possible by the choices made in the construction of S). Write s 1 = g m h m−1 . . . h 3 g 2 h 1 ⋅ s 0 with h i ∈ πé t 1 (X 1 ,x 1 ) and g j ∈ πé t 1 (X 2 ,x 2 ) (this form is not unique, of course). This is possible thanks to the fact that the sets a i , b j form consecutive intervals separately such that b j intersects non-trivially a j−1 and a j+1 . By the construction of S again, there is an element s 2 ∈ b m ∩ a m+1 corresponding to ζ ℓ m+1 ∈ µ ℓ m+1 via a m+1 ≃ µ ℓ m+1 . We can now write s 2 in two ways: s 2 = ζ ⋅ s 1 = g ⋅ s 1 , where g ∈ πé t 1 (X 2 ,x 2 ) and ζ is the chosen element in πé t 1 (X 1 ,x 1 ) ≃Ẑ(1). By the choices made, the action of φ N p fixes the elements s 1 and g ⋅ s 1 , while it moves ζ ⋅ s 1 .
Example 4.6. Let X i = G m,Q , i = 1, 2, 3 and let X 4 , X 5 be the nodal curves obtained from gluing 1 and −1 on P 1 Q (see Ex. 3.24). Define X to be the gluing X = ∪ • X i of all these schemes at the rational points corresponding to 1 (or the image of 1 in the case of the nodal curves). We fix an algebraic closure Q of Q and so fix a geometric pointb over the base Spec(Q). We get geometric pointsx i on X i = X i × Q Q lying over 1. We have and fix the following isomorphisms of Gal Q -modules. For 1 ≤ i ≤ 3, πé t 1 (X i ,x i ) ≃Ẑ(1) and for 4 ≤ j ≤ 5, we have π proét 1 (X j ,x j ) ≃ ⟨t Z ⟩ (i.e. Z written multiplicatively).
Let t i ∈ π proét 1 (X i ,x i ) be the elements corresponding via these isomorphisms to a fixed inverse system of primitive roots ζ ∈Ẑ(1) (for i = 1, 2, 3) and to t ∈ ⟨t Z ⟩ (for i = 4, 5). Example 3.25 gives a description of the fundamental group and of its category of sets: be the subgroup of upper triangular matrices. Fix u 1 ∈ Z × ℓ such that u p 1 ≠ u 1 . Let H = * top i πé t 1 (X i ,x i ) and define a continuous homomorphism ψ ∶ H → G by: It is easy to see that ψ is surjective.
Let U ⊂ G be the subgroup of matrices with elements in Z ℓ , i.e. U = * * * ⊂ GL 2 (Z ℓ ). It is an open subgroup of G. Thus, using ψ and the fact that H Noohi = π proét 1 (X,x), we get that S ∶= G U defines a π proét 1 (X,x)-set. It is connected (i.e. transitive) and so corresponds to a connected geometric covering of X. Assume that it can be defined over a finite extension L of Q. We can assume L Q is Galois. By the description above, it means that there is a compatible action of groups Z(1) i , Z * 2 and Gal L on S. By increasing L, we can assume moreover that Gal L fixes [U ].
Choose p ≠ ℓ that splits completely in Gal L , fix a prime p of L dividing p and let φ p ∈ Gal L denote a fixed Frobenius element. Let t u 1 3 denote the unique element of ψ −1 As n ≫ 0 and u p 1 ≠ u 1 , it follows that φ p ⋅ [U ] ≠ [U ] -a contradiction.
It is important to note, that the above (counter-)examples are possible only when considering the geometric coverings that are not trivialized by an étale cover (but one really needs to use the pro-étale cover to trivialize them). In [BS15], the category of geometric coverings trivialized by an étale cover on X is denoted by Loc Xé t and the authors prove the following We are now going to prove: Proposition 4.8. Let X be a geometrically connected separated scheme of finite type over a field k. Let Y ∈ Cov Xk be such that Y ∈ Loc (Xk)é t . Then there exists a finite extension l k such and Y 0 ∈ Cov X l such that Y ≃ Y 0 × X l Xk.
Proof. By the topological invariance (Prop. 2.17), we can replacek by k sep if desired. By the assumption Y ∈ Loc (Xk)é t , there exists an étale cover of finite type that trivializes Y . Being of finite type, it is a base-change X ′ k = X ′ × Spec(l) Spec(k) → Xk of an étale cover X ′ → X l for some finite extension l k. Thus, Y X ′ k is constant (i.e. ≃ ⊔ s∈S X ′ = S) and the isomorphism between the pull-backs of we use the fact that X ′ k is étale over Xk, and thus π 0 (X ′ k × Xk X ′ k ) is discrete, in this case even finite). By enlarging l, we can assume that the connected components of the schemes involved: X ′ , X ′ × X l X ′ etc. are geometrically connected over l. Define Y ′ 0 = ⊔ s∈S X ′ . The discussion above shows that the descent datum on Y X ′ k with respect to X ′ k → Xk is in fact the pull-back of a descent datum on Y ′ 0 with respect to X ′ → X l . As étale covers are morphisms of effective descent for geometric coverings (this follows from the fpqc descent for fpqc sheaves and the equivalence Cov X l ≃ Loc X l of [BS15, Lemma 7.3.9]), the proof is finished.
Remark 4.9. Over a scheme with a non-discrete set of connected components, Aut(S) might not be equal to Aut(S).
Proposition 4.8 shows that our main theorem is significantly easier for π SGA3 1 . Corollary 4.10. Let X be a geometrically connected separated scheme of finite type over a field k. Fix an algebraic closurek of k. Then π SGA3 is a topological embedding.

4.2.
Preparation for the proof of Theorem 4.13. We are going to use the following proposition.
Proposition 4.11. Let X be a scheme of finite type over a field k with a k-rational point x 0 and assume that Xk is connected. Let Y 1 , . . . , Y N be a set of connected finite étale coverings of Xk. Then there exists a finite Galois étale covering Y of X such that for all 1 ≤ i ≤ N , there exists a surjective map Yk ↠ Y i of coverings of Xk.
Proof. There is a finite connected Galois covering of Xk dominating Y 1 , . . . , Y N . Thus, we can assume N = 1 and Y 1 is Galois. Fix a geometric pointx 0 over x 0 . The k-rational point x 0 gives a splitting s ∶ Gal k → πé t 1 (X,x 0 ), allowing to write πé t 1 (X,x 0 ) ≃ πé t 1 (Xk,x 0 ) ⋊ Gal k and so an action of Gal k on πé t 1 (Xk,x 0 ). Fix a geometric pointȳ on Y 1 overx 0 . The group U = πé t 1 (Y 1 ,ȳ) is a normal open subgroup of πé t 1 (Xk,x 0 ). As Y 1 is defined over a finite Galois field extension l k (contained ink), it is easy to check that Gal l ⊂ Gal k fixes U , i.e. σ U = U for σ ∈ Gal l . It follows that the set of conjugates σ U is finite, of cardinality bounded by [l ∶ k]. Define V = ∩ σ∈Gal k σ U . It follows that this is an open subgroup of πé t 1 (Xk,x 0 ) fixed by the action of Gal k . Moreover, it is normal, as g(∩ σ∈Gal k σ U )g −1 = ∩ σ∈Gal k g σ U g −1 = ∩ σ∈Gal k σ (( σ −1 g)U ( σ −1 g −1 )) = ∩ σ∈Gal k g σ U g −1 , due to normality of U . The open normal subgroup V ⋅ Gal k = V ⋊ Gal k < πé t 1 (Xk,x 0 ) ⋊ Gal k corresponds to a covering with the desired properties.
Before starting the proof, we need to collect some facts about the Galois action on the geometric πé t 1 . They are discussed, for example, in [Sti13,Ch. 2]. The existence, functoriality and compatibility with compositions of the action can be readily seen to generalize to π proét 1 as well, but note (see the last point below) that one has to be careful when discussing continuity . For a connected topologically noetherian scheme W and geometric pointsw 1 ,w 2 , let π proét 1 (W,w 1 ,w 2 ) = Isom Cov Wk (Fw 1 , Fw 2 ) denote the set of isomorphisms of the two fibre functors, topologized in a way completely analogous to the case whenw 1 = w 2 . By Cor. 3.18, it is a bi-torsor under π proét 1 (W,w 1 ) and π proét 1 (W,w 2 ). The bi-torsors under profinite groups πé t 1 (W,w 1 ,w 2 ) are defined similarly and are rather standard. For a geometrically unibranch W , the two notions match.
g) The action Gal k × πé t 1 (Wk,w 1 ,w 2 ) → πé t 1 (Wk,w 1 ,w 2 ) is continuous. Note, however, that at this stage of the proof we do not know whether this is true for π proét 1 . In fact, this is closely related to the main result we need to prove.
Theorem 4.13. Let k be a field and fix an algebraic closurek of k. Let X be a scheme of finite type over k such that the base-change Xk is connected. Letx be a Spec(k)-point on Xk. Then the induced map is a topological embedding.
Then, we will derive the final form of the fundamental exact sequence.
Theorem 4.14. With the assumptions as in Thm. 4.13, the sequence of abstract groups Moreover, the map π proét 1 (Xk,x) → π proét 1 (X,x) is a topological embedding and the map π proét 1 (X,x) → Gal k is a quotient map of topological groups.
In the proof, after some preparatory steps (e.g. extending the field k), we define the set of regular loops in π proét 1 (Xk) with respect to a fixed open subgroup U < ○ π proét 1 (Xk,x) and use it to construct an Galois invariant open subgroup V inside of U (see Steps II and III below). There is also an alternative approach to proving the existence of V that avoids the direct construction involving regular loops. We sketch it in Rmk. 4.27. While this latter approach is quicker, it is less instructive: as explained in Rmk. 4.26 below, the notion of a regular loop provides an insight of what goes wrong in the counterexample Ex. 4.5. Still, it might be worth having a look at, as our main approach is rather lengthy.
Step have tacitly liftedx to X l . Thus, we can start by replacing k by a finite extension. Considering the normalization X ν → X, base-changing the whole problem to a finite extension l of k, considering the factorization l l ′ k into separable and purely inseparable extension of fields, and using first that the base-change along a separable field extension of a normal scheme is normal and then the topological invariance of π proét 1 , we can assume that we have a surjective finite morphism h ∶ X → X such that the connected components of X, X × X X, X × X X × X X are geometrically connected, have rational points and for each W ∈ π 0 ( X), there is π proét 1 (W ) = πé t 1 (W ) and π proét 1 (Wk) = πé t 1 (Wk). Let X = ⊔ v∈Vert X v be the decomposition into connected components. Note that the indexing set Vert is finite. For each t ∈ π 0 ( X)∪π 0 ( X × X X)∪π 0 ( X × X X × X X)), we fix a k-rational point x(t) on t and ak-pointx(t) ont = tk lying over x(t). We will often writex t to meanx(t). Let us fix vx ∈ Vert for the rest of the text and say that the image ofx( X vx,k ) in Xk will be the fixed geometric pointx of Xk and its image in X the fixed geometric point of X. For any Wk, W ′ k ∈ π 0 (S • (h)) and every boundary map ) between the chosen geometric points, as in Cor. 3.19. This is possible thanks to Lm. 3.17. We define γ W ′ ,W to be the image of this path.
Leth ∶ Xk → Xk be the base-change of h. We choose a maximal tree T (resp. T ′ ) in the graph Γ = π 0 (S • (h)) ⩽1 (resp. Γ ′ = π 0 (S • (h)) ⩽1 ). After making these choices, we can apply Cor. 3.19 with Rmk. 3.21 to write the fundamental groups of (X,x) and (Xk,x). This way we get a diagram * top v πé t where (. . .) denotes the topological closure, ⟨R⟩ nc denotes the normal subgroup generated by the set R, and R 1 , R ′ 1 , R 2 , R ′ 2 are as in Rmk. 3.21. Note that, while the (connected components of the) fibre products X × X X, X × X X × X X are not necessarily normal nor satisfy π proét 1 (W ) = πé t 1 (W ), we can effectively work as if this was the case, see Rmk. 3.21.
Observation 4.15. The maps and groups above enjoy the following properties. a) By Lm. 3.27, the left vertical map is the Noohi completion of the obvious map of the underlying quotients of free topological products. b) By geometrical connectedness of the schemes in sight, we can (and do) identify π 0 (S • (h)) = π 0 (S • (h)), Γ ′ = Γ and T ′ = T c) As γ W ′ ,W 's are chosen to be the images of γ W ′ k ,Wk 's, we see that α (f ) abc 's appearing in R 2 , and so a priori elements of πé t 1 ( X v ,x v )'s, are in fact in πé t 1 ( X v,k ,x v ). It follows that R ′ 2 = R 2 d) The k-rational points x(W ) give identification πé t 1 (W,x W ) ≃ πé t 1 (Wk,x W ) ⋊ Gal k When W = X v for v ∈ Vert, we will write Gal k,v in the identification above to distinguish between different copies of Gal k in the van Kampen presentation of π proét 1 (X,x). e) As γ W ′ ,W is the image of the path γ W ′ k ,Wk on W ′ k , it maps to the trivial element of πé t 1 (Spec(k),x(W ),x(W ′ )) = Gal k . It implies, that the following diagram commutes πé t 1 (Wk,x(W )) πé t 1 (W,x(W )) Let P be a walk in Γ, i.e. a sequence of consecutive edges (with possible repetitions) e 1 , . . . , e m in Γ with an orientation such that the terminal vertex of e i is the initial vertex of e i+1 . Using the orientation of Γ, it can be written as ǫ 1 e 1 . . . ǫ m e m with ǫ i ∈ {±} indicating whether the orientation agrees or not. This will come handy as follows: define In the following, we will use ○ ? to denote the "composition of étale paths" and • ? to denote the multiplication in some group(oid) ?. When ? = π proét 1 (Xk,x) or π proét 1 (X,x), we will skip the subscript.
While we could just use ○ ? everywhere, it is sometimes convenient to keep track of when some paths "have been closed" by using • ? .
Step II: Defining regular loops in π proét 1 (Xk,x) Definition 4.16. An element γ ∈ Isom Cov Xk (Fx w , Fx v ) is called an étale path of special form supported on P if it lies in the image of the composition map above for some walk P starting in w and ending in v.
Any element (γ 2m , . . . , γ 1 ) in the preimage of such γ will be called a presentation of γ with respect to P .
For a walk P , denote by l(P ) the length of P , i.e. the number of consecutive edges (not necessarily different) it is composed of.
Observation 4.17. A useful example of a path of special form is the following. In the van Kampen presentation, the maps πé t (Xk,x) are given by is defined as follows: if P vx,v ⊂ T denotes the unique shortest path in the tree T ⊂ Γ (forgetting the orientation) from vx to v, then the choices of paths γ W ′ k ,Wk made when applying the van Kampen theorem give a unique étale path of special form γ v supported on P vx,v .
Before introducing the main objects of the proof, we note a simple result.
Proof. This follows from the continuity of the composition maps of paths and the fact that the statement is true for πé t 1 . To prove Thm. 4.13, it is enough to prove the following statement: any connected geometric covering Y of Xk can be dominated by a covering defined over a finite separable extension l k.
Indeed, let Y ′ ∈ Cov X l be a connected covering that dominates Y after base-change tok. By looking at the separable closure of k in l and using the topological invariance of π proét 1 , we can assume l k is separable. The composition Y ′′ = Y ′ → X l → X is an element of Cov X and there is a diagonal embedding Y ′ × Spec(l) Spec(k) → Y ′′ × Spec(k) Spec(k). By Prop. 2.37(5), the proof will be finished.
Let us fix a connected Y ∈ Cov till the end of the proof and denote by S = Yx the corresponding π proét 1 (Xk,x)-set. Fix some point s 0 ∈ S and let U = Stab π proét 1 (Xk,x) (s 0 ).
supp. on P s = γ ⋅ s 0 and call it the set of "elements at v reachable in at most N steps".
The following is a crucial observation regarding O N v . Lemma. For any v and N , the set O N v is finite. Proof. We proceed by induction on N . For N = 1, the walks of length not greater than N starting in v 0 (are either trivial or) consist of a single edge whose initial vertex is necessarily vx. As Γ is finite, there are only finitely many such edges. Let us fix one, named e, with vertices v 0 , w. We need to show that the set 0 (x e ),x w ) is finite. However, as in general the sets πé t 1 (W,x 1 ,x 2 ) are (bi-)torsors under profinite groups (namely πé t 1 (W,x 1 ) and πé t 1 (W,x 2 )) and the maps and actions in sight are continuous, we see that the finiteness of this last set follows directly from finiteness of orbits of points in discrete sets under an action by a profinite group. Now, to see the inductive step, assume that the claim is true for N . To prove it for N + 1, note that any element in O N +1 v can be connected by a single edge to an element of O N w (for some vertex w). As O N w is finite and as we have just explained that, starting from a fixed point, one can only reach finitely many points by applying étale paths of special forms supported on a single edge, the result follows.
x v )-sets. We can find sets satisfying the first two conditions by applying Prop. 4.11, and the last condition can be guaranteed by choosing the C N v 's inductively (for a given v). We now proceed to define a subgroup of π proét 1 (X,x) that will lead to the desired π proét 1 (X,x)-set.
For that we need to find a suitably large subgroup of elements of U that are well behaved under the Galois action.
Here, the larger bullets correspond tox v i 's and the smaller ones to ∂ ǫ 0 or 1 (x(e i )). Remark 4.21. We find the definition involving C N v 's quite convenient. One could, however, avoid introducing C N v 's and make a slightly different definition. Define O N,+ v to be the set of (isomorphism classes of) Gal k -conjugates of the πé t 1 ( Let V 0 < π proét 1 (Xk,x) denote the subgroup generated by the set of regular loops and let V be its topological closure.
denote the topological group appearing in the van Kampen presentation above. We have that G Noohi = π proét 1 (Xk,x). LetG ⊂ π proét 1 (Xk,x) denote the subgroup of all étale paths (or "loops", rather) of special form, supported on walks from vx to vx.
Observation 4.22. By Obs. 4.17, the map G → π proét 1 (Xk,x) = G Noohi factorizes throughG. Directly from the definitions, there is V 0 <G. We are thus in the situation of Lm. 2.39. We will use it below.
For brevity, let us denoteḠ v = πé t 1 ( X v,k ,x v ) and G v = πé t 1 ( X v ,x v ) ≃Ḡ v ⋊ Gal k in the proofs below. Proposition 4.23. The following statements about the subgroup V hold: (1) There is a containment V < U . Proof.
(1) As any open subgroup is automatically closed, it is enough to show that any regular loop lies in U . Let g be a regular loop and write g = γ ′ ○ β ○ γ with γ, γ ′ étale paths of special form supported on some walk (and its inverse) from vx to v of length m, with presentations (γ 1 , . . . , γ 2m ) and (γ ′ 2m , . . . , γ ′ 1 ) of γ and γ ′ , and β ∈ ker(πé t 1 ( X v,k ,x v ) → C m v ), as in the definition of a regular loop. Let us introduce the following notation (and analogously for γ ′ ) For i = m, it follows from the condition on β that (β ○ γ) ⋅ s 0 = γ ⋅ s 0 . Similarly, the condition on The process continues in a similar fashion to show that g stabilizes s 0 , and thus belongs to U . (2) By Lm. 2.39, it is enough to check that the map G → Aut(G V 0 ) is continuous whenG V 0 is considered with the discrete topology.
Using the universal property of free topological products, continuity can be checked separately forḠ v and D. For D, this is automatic, as D is discrete. To see the result forḠ v 's, we need to show that the stabilizers of the action ofḠ v onG V 0 induced byḠ v → G are open. Fix [gV 0 ] ∈G V 0 and g ∈G representing it. The element g is represented by some étale path (or a "loop", in fact) of special form ρ supported on a walk P ρ of length l(P ρ ). By Obs. 4.17, the morphismḠ v →G ⊂ π proét 1 (Xk,x) is also defined using an étale path of special form γ v supported on a walk P vx,v in the tree T ⊂ Γ.
Then H v is open inḠ v and its image inG can be written as It follows from the setup that for β ∈ H v , (3) For each σ, the map ψ σ is continuous. As V = V 0 G Noohi , it is thus enough to prove that V 0 is Gal k -invariant. By Lm. 4.12, it follows that under the action of Gal k , an étale path of special form supported on a walk P is mapped again to an étale path of special form supported on P . Consequently, checking that the action of σ ∈ Gal k maps a regular loop g to another regular loop boils down to checking the following fact. If g has a presentation g = γ ′ ○β ○γ as in the definition of a regular loop, then • ψ σ (β) still acts trivially on C , depending on parity, still acts trivially on C ⌈i⌉ v ⌊i⌋+1 for every i. However, as the automorphism ψ σ on πé t 1 ( X v j ,k ,x v j ) matches conjugation by σ in πé t 1 ( X v j ,x v j ) restricted to its normal subgroup πé t 1 ( X v j ,k ,x v j ) and the sets C j v j were Galois as πé t 1 ( X v j ,x v j )sets, the result follows.
Lemma 4.24. For each v ∈ Vert, define an (abstract) Gal k,v -action on π proét 1 (Xk,x) to be Then there exists a finite extension l k, such that for all v ∈ Vert, there is a) Gal l,v fixes V ; b) The obtained induced Gal l,v -action on S ′ can be written as c) The induced Gal l,v action on S ′ is continuous and compatible with theḠ v -action.
Proof. As there are finitely many vertices v, it is enough to prove the statements for a single fixed v. Let g ∈ V . By definition of ρ v , there is By Prop. 4.23, we have ψ σ (g) ∈ V and we only need to show that γ −1 v ○ ψ σ (γ v ) ∈ V . By Lm. 4.18 and Obs. 4.17, the map Gal k ∋ σ ↦ ψ σ (γ v ) ∈ π proét 1 (Xk,x,x v ) is continuous, and we conclude that for an open subgroup of σ ∈ Gal k we have the desired containment.
It follows from the previous point that we get an induced action of Gal l,v on S ′ . Using that γ −1 v ○ ψ σ (γ v ) ∈ V , the alternative formula in the statement follows from the computation Let us move to the last point. Compatibility with theḠ v -action follows from Lm. 4.12(d) and the fact that the mapḠ v → π proét 1 (Xk,x) is defined by postcomposing with ρ v . To check continuity, fix [gV ].
By Lm. 2.39, this class is represented by a path (loop) of special form, and so we can assume this about g. Checking that the stabilizer of [gV ] is open boils down to checking that for an open subgroup of σ's in Gal l,v , one has g −1 • (γ −1 v ○ ψ σ (γ v )) • ψ σ (g) ∈ V . However, this follows from the openness of V and Lm. 4.18.
Proof. By the van Kampen theorem for π proét 1 (X l ,x), it is enough to show that there are continuous actions ofḠ v ⋊ Gal l,v 's and D compatible with theḠ v and D actions that S ′ is already equipped with, and such that the van Kampen relations are satisfied. We already have a continuous action by D on S ′ , and by Lm. 4.24, we get an action ofḠ v ⋊ Gal l,v .
We have finished our main proof, and thus the most difficult part of the exact sequence is now proven. We now obtain the final form of the fundamental exact sequence.
Proof. (End of the proof of Thm. 4.14) We already know the statements of the "moreover" part and the near exactness in the middle of the sequence. All we have to prove is that π proét can be performed after replacing π proét 1 (X,x) by any open subgroup U such that π proét 1 (Xk,x) < U < ○ π proét 1 (X,x). Choosing a suitably large finite field extension l k and looking at U = π proét 1 (X l ,x), we are reduced to the situation as in the proof of Thm. 4.13, i.e. we have enough rational points on the connected components we are interested in when applying van Kampen. LetG < π proét 1 (Xk,x) be the dense subgroup defined above Prop. 4.23. Note that by the van Kampen theorem applied to π proét 1 (X,x) together with the observations in Obs. 4.15, it follows that the subgroup generated byG and Gal k,v 's is dense in π proét 1 (X,x). Putting this together, it follows that it is enough to check that, for each v, conjugation by elements of Gal k,v fixesG in π proét 1 (X,x). This, however, follows from Lm. 4.12 c) d) and the fact that Gal k,v → π proét 1 (X,x) is defined as the composition Gal k,v → π proét 1 (X,x v ) ρv → π proét 1 (X,x), Remark 4.26. Let us revisit the counterexample of Ex. 4.5 from the point of view of the proof above. We will freely use the notation set there. In this example, we have started from the fixed point s 0 , and used the group elements to reach point s 1 = g m h m−1 . . . h 3 g 2 h 1 ⋅ s 0 . We have then concluded that s 2 = ζ ℓ m+1 ⋅ s 1 = g ⋅ s 1 and justified that the setup forces that this equality contradicts the possibility of extending the Galois action to the set S. The problem here is caused by the fact that, denoting γ = g m h m−1 . . . h 3 g 2 h 1 , the element γ −1 ○ g −1 ○ ζ ℓ m+1 ○ γ stabilizes s 0 , but it is not a "regular loop" in the language introduced above.
Of course, this only means that this particular "obvious" presentation is not as in the definition of a regular loop. But, by now, we know that it provably cannot be a regular loop with any presentation.
Remark 4.27. We sketch a slightly different approach to the central part of the main proof. It is a bit quicker, but less constructive, i.e. does not "explicitly" construct the desired Galois invariant open subgroup in terms of regular loops. We will freely use the fact that a surjective map from a compact space onto a Hausdorff space is a quotient map.
Assume that we have already done the preparatory steps of the main proof, i.e. we have increased the base field to have many rational points and applied the van Kampen theorem. We want to prove that the action Gal k × π proét 1 (Xk,x) → π proét 1 (Xk,x) given by ψ σ is continuous. Let G,G be as introduced above Obs. 4.22.
Firstly, one checks that any element ofG, so a path of special form, can be in fact rewritten with a presentation that makes it visibly an image of an element of G, at the expense of the presentation possibly getting longer. Another words, the map G →G is surjective. By default,G is considered with the subspace topology from π proét 1 (Xk,x). Let us denote (G, quot) the same group but considered with the quotient topology from G. We thus have a continuous bijection (G, quot) →G.
The group G is a topological quotient of the free topological product of finitely many compact groups G v and a finitely generated free group D ≃ Z * r . One checks from the universal properties that this free product can be written as a quotient of the free topological group F (Z) (see [AT08,Ch. 7.]) on a compact space of generators Z = ⊔ v G v ⊔ {1,...,r} * , i.e. the disjoint union of G v 's and r singletons.
By [AT08, Thm. 7.4.1], F (Z) is, as a topological space, a colimit of an increasing union . . . ⊂ B n ⊂ B n+1 ⊂ . . . of compact subspaces. These spaces are explicitly described as words of bounded length in F (Z) (this makes sense, as the underlying group of F (Z) is the abstract free group on Z). From this, it follows that (as a topological space) (G, quot) = colimK n , with K n = im(B n ).
Working directly with K n 's is inconvenient for our purposes, as these sets are not necessarily preserved by the Galois action. The reason is that the van Kampen presentation as a quotient of a free product uses fixed paths, while applying Galois action will usually move the paths. One then has to conjugate by a suitable element to "return" to the paths fixed in van Kampen, possibly increasing the length of the word.
Instead, we can consider subsets K ′ n ⊂G of elements that are paths of special form of length ⩽ n, i.e. possessing a presentation as a path of special form of length ⩽ n (see Defn. 4.16). By a reasonably simple combinatorics, one can cook up "brute force" bounds f (n, d), g(n, d) ∈ N in terms of n and the diameter d = diam(Γ) of Γ such that there is and K ′ n ⊂ K g(n,d) In conclusion, (G, quot) = colimK ′ n in Top. By Lm. 4.12, the Gal k -action preserves the sets K ′ n and Gal k × K ′ n → K ′ n is continuous. As Gal k is compact, Gal k × (−) has a right adjoint Maps cts (Gal k , −) in Top and so Gal k × (colim n∈N K ′ n ) = colim n∈N (Gal k × K ′ n ). From this, we immediately get that Gal k × (G, quot) → (G, quot) is continuous. As Gal k -action respects the group action ofG, it quickly follows that the action is still continuous when (G, quot) is equipped with the weakened topology τ making open subgroups a base at 1, as in Lm. 2.25.
By (the easier part of) Lm. 2.39, this weakened topology on (G, quot) matches that ofG. It follows that Gal k ×G →G is continuous.
By Lm. 2.25 again, one has to check that the continuity is not lost when passing to the Raǐkov completion of the maximal Hausdorff quotient of (G, τ ). This in turn can be justified by similar arguments as in the proof of Lm. 2.39. This finishes the sketch. See also [BS15,Prop. 4