Fundamental Groups of Schemes

Fundamental Groups of Schemes

Bachelorarbeit

von

Noah Held

betreut durch

Prof. Dr. Richard Pink

Februar 2018

Contents

1 Finite ´Etale Morphisms 1

1.1 Finite ´etale schemes over a field . . . 1

1.2 Finite locally free morphisms . . . 2

1.3 Finite ´etale morphisms . . . 3

1.4 Permanence properties . . . 5

2 Galois Covers 9 2.1 Galois covers . . . 9

2.2 Quotients of schemes . . . 11

2.3 Galois correspondence . . . 13

3 Profinite groups 16 3.1 Continuous group actions . . . 16

3.2 Profinite groups . . . 16

3.3 Automorphism groups of functors . . . 18

4 Fundamental Group 22 4.1 Definition . . . 22

4.2 Profinite structure of the fundamental group . . . 22

4.3 Classification theorem . . . 26

5 Analytic Topology 29 6 Comparison Theorem 33 6.1 Topological theory . . . 33

6.2 Analytification of finite ´etale covers . . . 34

6.3 Comparison theorem . . . 36

Introduction

The text at hand is a first look at the theory of fundamental groups of schemes. As the name suggests, this theory has many similarities with the theory of fundamental groups in topology. On the other hand, it also encompasses classical Galois theory, thereby generalizing it to arbitrary arithmetic schemes.

We briefly recall the topological theory. LetX be a connected topological space, and letx PX be a point. Let Fx be the functor from the category of covers of X to the category of sets which sends a cover Y Ñ X to its fiber overx. Each fiber is aπ1pX, xq-set via the monodromy action. If X has a universal cover Xr Ñ X, then Fx is represented by Xr and factors through an equivalence of categories between the category of covers of X and the category ofπ₁pX, xq-sets. In this way,π₁pX, xq completely classifies the covers ofX. By the Yoneda Lemma, the automorphism group of Fx is isomorphic to Aut_XpXqr ^op, which in turn is isomorphic to π₁pX, xq. Hence we recover the fundamental group of X as the automorphism group of F_x.

The fundamental group of a connected scheme S is defined using an analogous framework. The first step is to identify the class of morphisms replacing topological covers; these are the finite ´etale covers of S. The base scheme S is equipped with a geometric base points, i.e. a morphism from the spectrum of an algebraically closed field. Working with geometric fibers, we then construct a functor F_s from the category of finite ´etale covers of S to the category of sets. Reversing the topological situation, the fundamental group π₁pS, sq of S with base point s is defined to be the automorphism group ofF_s.

After setting up the general theory, we discuss the classification theorem:

F_s induces an equivalence of categories with the category of finite continuous π₁pS, sq-sets. There are two aspects of the theory which facilitate this discussion. Firstly, a special role is played by connected finite ´etale Galois covers, which are those connected finite ´etale covers whose automorphism group acts transitively on geometric fibers. Every finite ´etale cover is an intermediate cover of a Galois cover, and we can describe automorphisms of Fs as compatible families of automorphisms of the Galois covers. Secondly, because of the finiteness condition placed on finite ´etale covers, F_s takes values in the category of finite sets; hence π1pS, sq has a natural profinite

structure. Chapter 3 develops the necessary notions for this point of view.

The analogy drawn above can be made precise using the axiomatic framework of Galois categories, as developed by Grothendieck [1]. Although we do not introduce this notion, it will be clear that the proofs only use formal properties of finite ´etale covers and the functorFs.

The second part of the text is devoted to the study of finite ´etale covers of schemes which are locally of finite type over the complex numbers C. We associate with such a scheme S a topological space S^an, called the analytification of S, whose topology is obtained by gluing the topologies locally inherited from the analytic topology on C^m. This construction is functorial, and transforms finite ´etale covers ofS into topological covers of S^an. The natural question is now whether any topological cover ofS^an arises from a finite ´etale cover ofS.

Perhaps surprisingly, this is the case. For smooth projective curves and their associated compact Riemann surfaces, this question was already studied by Riemann. Grothendieck [1] gave a proof in the general setting introduced above. More precisely, the functor which maps a finite ´etale cover ofS to the associated topological cover of S^an with finite fibers is an equivalence of categories. It follows formally, using the material developed in Chapter 3 and the classification theorem in the topological setting, that there is an isomorphism of topological groups between the fundamental group ofS and the profinite completion of the fundamental group ofS^an.

The author would like to thank Prof. Dr. Richard Pink for his patient guidance, and Alexandre Puttick for helpful comments on an earlier version of the text.

Chapter 1

Finite ´ Etale Morphisms

We define finite ´etale morphisms and explore some of their properties, start- ing from the case where the base scheme is the spectrum of a field. The development of the theory in this and the next chapter follows Lenstra [4, Chapters 4 and 5] and Szamuely [7, Chapters 5.2 and 5.3].

1.1 Finite ´ etale schemes over a field

Definition 1.1. Letk be a field. A k-algebra A is called ´etale overk if it is isomorphic to a finite product of finite separable field extensions ofk.

Proposition 1.2. Let k be a field, let Ωbe an algebraically closed field con- taining k, and let A be a k-algebra. The following conditions are equivalent:

(a) A is ´etale over k,

(b) AbkΩis isomorphic to a finite product of copies of Ω, (c) AbkΩis reduced and finite-dimensional over Ω.

Proof. Assume first that (a) holds. Since the functor´ bkΩ preserves finite products, we may assume thatA is a finite separable field extension ofk.

By the primitive element theorem,A is isomorphic over k to krTs{pfq for a monic irreducible separable polynomialf PkrTs. Letf “ś_m

i“1pT ´aiq be its factorization in ΩrTs, where the factorsT ´a_i are distinct becausef is separable. Then

krTs{pfq bkΩ–

m

ź

i“1

ΩrTs{pT ´aiq –Ω^m by the Chinese Remainder Theorem.

It is clear that (b) implies (c). Assume now that (c) holds. Let m be the dimension ofAbkΩ over Ω. It coincides with the dimension ofA over k,

soA is finite-dimensional. Hence it is Artinian, which in turn implies that it is isomorphic to a finite product of Artinian local k-algebras. Because AbkΩ is reduced, so isA. ThusA is in fact isomorphic to a finite product k₁ ˆ ¨ ¨ ¨ ˆk_r of finite field extensions of k. Each k_i is separable, because kibkΩ is reduced. Hence Ais ´etale overk.

Proposition 1.3. Ifkis a field andAis an ´etale k-algebra, then the module of relative differentials Ω_A{k of A over k is zero.

Proof. By Proposition 1.2 and the compatibility of relative differentials with base change, it suffices to consider the case whereA“k^m for a nonnegative integer m. Let M be an A-module, and let d: A Ñ M be a k-derivation.

Denote by e₁, . . . , e_m the canonical basis of A over k. Note that each e_i is idempotent; we claim that this implies dpeiq “ 0. Indeed, applying d to both sides of the equation e²_i “e_i yields 2e_idpe_iq “dpe_iq. Multiplying bye_i on both sides turns this into 2e_idpe_iq “e_idpe_iq, so e_idpe_iq “0. Thus dpeiq “2eidpeiq “0. Because te1, . . . , emuspans Aas a vector space overk, it follows thatd“0.

1.2 Finite locally free morphisms

Definition 1.4. A morphism of schemes ϕ:X ÑS is called finite locally free if it is affine andϕ_˚OX is a finite locally free OS-module.

Proposition 1.5. The image of a finite locally free morphism of schemes ϕ:X ÑS is open and closed.

Proof. Since ϕis finite,ϕpXq is closed. HenceϕpXq “supppϕ_˚OXq, which is open becauseϕ˚OX is finite locally free.

Corollary 1.6. If ϕ:XÑS is a finite locally free morphism of schemes to a connected scheme S, then ϕis surjective if and only if X is nonempty.

Proof. The image of ϕ is open and closed in S by Proposition 1.5. Since S is connected, this means that ϕ is surjective if and only if its image is nonempty.

Definition 1.7. Let ϕ:XÑS be a finite locally free morphism of schemes, and let sPS be a point. Since ϕ is finite locally free, the stalk pϕ_˚OXqs is free of finite rank over OS,s. Its rank is called the degree of ϕat s, and is denoted by deg_spϕq.

Proposition 1.8. The degree of a finite locally free morphism of schemes ϕ:X ÑS is a locally constant function of sPS. If S is connected, then the degree of ϕ is constant.

Proof. Every point sPS has an open neighborhoodU such thatpϕ_˚OXq|U

is free of rank deg_spϕq overOS|U. Then the stalk ofϕ˚OX at every point of U is free of rank deg_spϕq over the stalk ofOS at that point, so the degree of ϕis constant on U. The second assertion is a direct consequence of the first assertion and the definition of connectedness.

Lemma 1.9. A finite locally free morphism of schemes ϕ: X Ñ S is an isomorphism if and only if its degree at every point ofS is 1.

Proof. Being an isomorphism is local on the target, and a ring homomorphism A Ñ B is an isomorphism if and only if it makes B a free A-module of rank 1.

1.3 Finite ´ etale morphisms

Definition 1.10. A morphism of schemes ϕ:X ÑS is called finite ´etale if it is finite locally free and for every pointsPS the fiberX_s of ϕ over sis the spectrum of an ´etale κpsq-algebra, where κpsq denotes the residue field of s. A surjective finite ´etale morphism XÑS is also called a finite ´etale

cover ofS.

Let S be a scheme. We denote by Sch{S the category ofS-schemes, and byFin ´Et{Sits full subcategory whose objects are the finite ´etale morphisms XÑS. A geometric point ofS is a morphism of schemess: SpecpΩq ÑS, where Ω is an algebraically closed field. The image of sconsists of a single points; we say that slies over s. The geometric fiber overs of a morphism XÑS is X_s :“XˆSSpecpΩq.

Definition 1.11. LetS be a scheme, and lets: SpecpΩq ÑS be a geometric point. We view SpecpΩq as an S-scheme via s. The fiber functor associated withs is the functor

Fs:Fin ´Et{S ÝÑSet,

pXÑSq ÞÝÑMorSpSpecpΩq, Xq, ψÞÝÑ pxÞÑψ˝xq.

Proposition 1.12. Let ϕ:X Ñ S be a finite ´etale morphism of schemes, and lets: SpecpΩq ÑS be a geometric point ofS. Then FspXq is in natural bijection with the underlying set of X_s.

Proof. By Proposition 1.2, all points ofXs are Ω-rational; hence the underly- ing set of X_s is in natural bijection with Mor_ΩpSpecpΩq, X_sq. The claim fol- lows from the natural bijection MorΩpSpecpΩq, Xsq –MorSpSpecpΩq, Xq.

Definition 1.13. Letϕ:XÑS be a finite ´etale morphism of schemes, and let s be a geometric point of S. The degree of ϕats is the degree of ϕ at the point s over which slies, and is denoted by deg_spϕq.

Remark 1.14. Let ϕ:XÑS be a finite ´etale morphism of schemes, and let s be a geometric point of S. It follows from Proposition 1.2 that the degree ofϕatsis equal to the number of points ofXs. By Proposition 1.12, it is therefore also equal to the cardinality ofF_spXq.

Example 1.15. A locally closed embedding is finite ´etale if and only if it is an open and closed embedding.

Example 1.16. LetK ĂL be a finite field extension. The corresponding morphism SpecpLq ÑSpecpKqis finite locally free of degree dim_KpLq; it is finite ´etale if and only ifL is separable overK.

Example 1.17. LetA be a ring, and let f PArTs be a monic polynomial of degreem such thatpf,Bf{BTq “ p1q inArTs, whereBf{BT is the formal derivative of f with respect toT. Because f is monic, ArTs{pfq is free of rankm overA. If p is a prime ideal ofA and Ω is an algebraic closure of its residue field κppq, then ArTs{pfq bAΩ–Ω^m as Ω-algebras by the Chinese Remainder Theorem and the fact thatf splits into distinct linear factors over Ω. Hence the canonical morphismAÑArTs{pfq induces a finite ´etale morphism of degreem on spectra.

Example 1.18. Letkbe a field, letA“krT, T^´1s, and letGm,k “SpecpAq be the multiplicative group overk. For every nonzero integern, the morphism ofk-algebras ψ_n:AÑAwith ψ_npTq “Tⁿ corresponds to a surjective finite locally free morphism of schemes ϕn: Gm,k Ñ Gm,k. If n ą0, then ψn is isomorphic to the canonical morphism A Ñ ArUs{pUⁿ´Tq. The formal derivative ofUⁿ´T with respect to U isnU^n´1. If the characteristic ofk does not dividen, thenpUⁿ´T, nU^n´1q “ pTq “ p1qin ArUs; hence ϕn is finite ´etale of degree nby Example 1.17. If nă0, thenϕn is the composite ofϕ_´n and the automorphism ϕ_´1. Provided thatnis not divisible by the characteristic ofk, the morphismϕ_nis also finite ´etale of degree ´n in that case.

Example 1.19. Let k be a field, and let n ą 1 be an integer. Consider the morphism of k-algebras ϑn:krTs Ñ krTs with ϑnpTq “ Tⁿ, and the corresponding morphismA¹_kÑA¹_k. The latter is finite locally free of degree n, but not finite ´etale. Indeed, its fiber over the origin consist of a single nonreduced point.

Example 1.20. Let p be a prime number, and let A be an Fp-algebra.

Given an elementaPA, consider the polynomial f “T^p´T´aPArTsand the schemeX “SpecpArTs{pfqq. The formal derivative off with respect to T is ´1, so the canonical morphismAÑArTs{pfq corresponds to a finite

´etale morphismX ÑSpecpAq of degree p by Example 1.17.

1.4 Permanence properties

We now discuss permanence properties of finite locally free and finite ´etale morphisms, such as being stable under composition and base change. As a technical tool, we need the following algebraic result.

Proposition 1.21. LetA be a ring. For every A-module M the following conditions are equivalent:

(a) M is finitely generated and projective,

(b) M is finitely presented and Mp is free for every pPSpecpAq, (c) M is finite locally free.

Proof. We indicate the main steps in the proof, and refer to Lenstra [4, Section 4.6] for a more complete explanation. That (a) implies (b) follows from the fact that a finitely generated projective module over a local ring is free. Suppose thatM satisfies (b), and let pPSpecpAq. A straightforward calculation shows that any basis of M_p over A_p lifts to a basis of M_f over A_f for some f PArp. Hence M satisfies (c). Finally, in order to show that (c) implies (a), first prove thatM is finitely presented. Consequently, the functors Hom_ApM,´qf and Hom_AfpM_f,p´qfq are isomorphic for every f PA. SinceM is locally projective, it follows thatM is projective.

Proposition 1.22. (a) The composite of two finite locally free morphisms of schemes ϕ:XÑY and ψ:Y ÑZ is finite locally free.

(b) Let ϕ:X ÑS and ψ:Y ÑS be morphisms of schemes. Ifϕ is finite locally free, the so is the base change pr₂:XˆSY ÑY of ϕ by ψ.

Proof. Because affine morphisms are stable under composition and base change, we may reduce to the affine case. Both assertions then follow from the equivalence of conditions (a) and (c) in Proposition 1.21 and the characterization of projective modules as direct summands of free modules.

Proposition 1.23. (a) The composite of two finite ´etale morphisms of schemesϕ:XÑY and ψ:Y ÑZ is finite ´etale.

(b) Let ϕ:X ÑS and ψ:Y ÑS be morphisms of schemes. Ifϕ is finite

´etale, the so is the base change pr₂:XˆSY ÑY of ϕ by ψ.

Proof. (a) The morphismψ˝ϕis finite locally free by Proposition 1.22. In order to show that it is finite ´etale, we use Proposition 1.2. LetzPZ be a point, let Ω be an algebraic closure ofκpzq, and letz: SpecpΩq ÑZ be the resulting geometric point lying overz. We need to show that the geometric fiber X_z is isomorphic to the spectrum of a finite product of copies of Ω.

Note thatX_z is naturally isomorphic to XˆY Y_z over Y_z, so in particular over Ω. Since ψ is finite ´etale, Yz is isomorphic over Ω to the spectrum of a finite product of copies of Ω; sinceϕis finite ´etale and fiber products of schemes commute with coproducts, so isX_z.

(b) The morphism pr₂ is finite locally free by Proposition 1.22. As in the proof of part (a), choose a pointyPY and a geometric pointx: SpecpΩq ÑY lying overy. We need to show, by Proposition 1.2, that the geometric fiber pXˆSYqx is isomorphic to the spectrum of a finite product of copies of Ω.

Note that pXˆSYqx is naturally isomorphic to X_ψ˝x “XˆSSpecpΩq over Ω. Becauseϕis finite ´etale,X_ψ˝x is isomorphic to the spectrum of a finite product of copies of Ω by Proposition 1.2; the claim follows.

Corollary 1.24. Let ϕ: X Ñ Y and ψ: Y Ñ Z be finite locally ´etale morphisms of schemes. IfY and Z are connected, then the degree ofψ˝ϕis equal to the product of the degrees of ϕand ψ.

Proof. See the proof of part (a) of the preceding proposition, which gives a formula for the geometric fibers ofψ˝ϕ.

Remark 1.25. Let S be a scheme. For every finite family of schemes X₁, . . . , X_r which are finite ´etale over S, their coproduct š_r

i“1X_i in the category Sch{S is finite ´etale over S. Hence it is also their coproduct in the full subcategory Fin ´Et{S. The same thing is true for their product X₁ˆS¨ ¨ ¨ ˆSX_r by Proposition 1.23. Note that the fiber functor preserves both finite coproducts and finite products.

Proposition 1.26. Let ϕ:X ÑS be a finite ´etale morphism to a connected scheme S.

(a) The number of connected components of X is less than or equal to the degree of ϕ.

(b) Every connected component ofX is open.

Proof. Assertion (b) is a purely topological consequence of (a), since the degree of ϕ is finite. We now prove (a). Let s be a geometric point of S.

We induct on deg_spϕq, which is independent of the choice ofs because S is connected. If deg_spϕq “ 0, then X is empty; if deg_spϕq “ 1, then ϕ is an isomorphism. Assume now that deg_spϕq ą 1. IfX is connected, then the claim holds. OtherwiseX is the union of two disjoint nonempty open and closed subsetsU₁ andU₂. Being composites of finite ´etale morphisms, the restrictionsϕ|_U1 andϕ|_U2 are finite ´etale; hence they are surjective by Corollary 1.6. SinceFs preserves coproducts, FspXq is the disjoint union of the nonempty setsF_spU₁q andF_spU₁q. In particular the degrees of ϕ|_U1 and ϕ|_U2 are strictly smaller than that ofϕ. By induction the claim holds for U₁ and U2; but then it holds forX.

Proposition 1.27. If ϕ:X ÑS is a finite ´etale morphism, then the sheaf of relative differentials Ω_X{S of X over S is zero.

Proof. We may assume that X “ SpecpBq andS “SpecpAq are affine, in which case Ω_X{S is the quasi-coherent OX-module associated with Ω_B{A. Sinceϕis of finite type, Ω_B{Ais finitely generated. Let qbe a prime ideal of B, and letp“ϕpqq. By Nakayama’s Lemma,pΩ_B{Aqq “0 if and only if Ω_B{AbBκpqq “0. The latter is isomorphic to Ω_{BbAκppq{κppq}bBb_Aκppqκpqq, which is zero by Proposition 1.3. Thus Ω_B{A“0.

Proposition 1.28. Ifϕ:XÑS is a finite ´etale morphism, then the diago- nal morphism ∆_ϕ:XÑXˆSX is an open and closed embedding.

Proof. Since ϕ is affine, it is separated, so ∆_ϕ induces an isomorphism of X with a closed subscheme Y of XˆSX. We now wish to show that Y is an open subscheme of XˆSX. We may assume that X“SpecpBq and S“SpecpAq are affine. Let I be the kernel of the codiagonalBbAB ÑB.

It is finitely generated because B is finitely generated over A, and the associated quasi-coherent ideal I of OXˆSX is the ideal of definition of Y. The quotient I{I² is isomorphic to Ω_B{A, which is zero by Proposition 1.27.

Letp be a prime ideal ofBbAB containing I. ThenIp is contained in the unique maximal ideal ofpBbABq_p. SinceI_p²“I_p, we must have I_p“0 by Nakayama’s Lemma. In other words, the stalk of Iat every point yPY is trivial; because it is finitely generated, Ivanishes on an open neighborhood ofy. ThusY is an open subscheme ofXˆSX.

Proposition 1.29. Letϕ:X ÑSand ψ:Y ÑX be morphisms of schemes.

If ϕ˝ψ and ϕ are finite ´etale, then so isψ.

Proof. The graph morphism Γ_ψ: Y Ñ Y ˆSX is the base change of the diagonal morphism ∆_ϕ by ψˆSid_X, and ψ “ pr₂˝Γ_ψ. As the diagonal morphism is an open and closed embedding by Proposition 1.28, it is finite

´etale. But then so is Γ_ψ, since finite ´etale morphisms are stable under base change by Proposition 1.23. Similarly, pr₂ is finite ´etale as the base change of ϕ˝ψbyϕ. Thusψis finite ´etale as a composite of finite ´etale morphisms.

Proposition 1.30. Let Y be a connected S-scheme, and let ϕ₁, ϕ₂:Y ÑX be S-morphisms to a finite ´etale S-scheme X. If there exists a nonempty S-scheme T and an S-morphism ψ:T ÑY such that ϕ1˝ψ“ϕ2˝ψ, then ϕ1 “ϕ2.

Proof. Denote by eqpϕ₁, ϕ₂q the equalizer of ϕ₁ andϕ₂ in the category of S-schemes. The following diagram is easily checked to be cartesian:

eqpϕ1, ϕ2q Y

X XˆSX,

j

pϕ1,ϕ2q

∆

where ∆ is the diagonal morphism ofX overS,jis the canonical embedding, and eqpϕ1, ϕ2q ÑX is the composite ϕ1˝j “ϕ2˝j. Since ∆ is an open and closed embedding by Proposition 1.28, so is j. But Y is connected and eqpϕ₁, ϕ₂qis nonempty by assumption, which means that j must be an isomorphism. Thusϕ1 “ϕ2.

Chapter 2

Galois Covers

2.1 Galois covers

Having introduced finite ´etale morphisms, we now study their automorphism groups.

Construction 2.1. Given a finite ´etale morphism XÑS and a geometric pointsof S, there is a canonical left action of AutSpXq onFspXq. Namely, f PAut_SpXq acts on xPF_spXq by f¨x:“F_spfqpxq.

Proposition 2.2. Let X Ñ S be a connected finite ´etale cover, and let s be a geometric point of S. Then the left action of Aut_SpXq on F_spXq as defined in Construction 2.1 is free, and the cardinality of Aut_SpXq is less than or equal to the degree of ϕ.

Proof. Suppose thatf PAutSpXqandxPFspXqare such thatFspfqpxq “x.

Then f˝x“idX ˝x, so f “idX by Proposition 1.30. Hence the action is free. Since X Ñ S is surjective, there is a point x PF_spXq. We have the injective map

AutSpXqãÑFspXq, gÞÑFspgqpxq;

hence the cardinality of AutSpXq is at most that of FspXq. The second assertion follows from this and the natural bijection betweenF_spXq andX_s, see Proposition 1.12.

Proposition 2.3. Let ϕ: X Ñ S be a connected finite ´etale cover. Then the following conditions are equivalent:

(a) The order ofAut_SpXq is equal to the degree ofϕ,

(b) Aut_SpXq acts transitively onF_spXq for every geometric point s of S, (c) Aut_SpXq acts transitively onFspXq for one geometric point sof S.

Proof. Assume first that (a) holds. Let s be a geometric point of S, and letxPFspXq be a lift ofs. By Proposition 2.2, the action of AutSpXq on FspXq is free, so the map

u: AutSpXq ÑFspXq, gÞÑFspgqpxq

is injective. Because the degree of ϕ is equal to the cardinality of F_spXq, the mapu is a bijection. Therefore (a) implies (b). Since S is nonempty, (b) implies (c). Suppose that (c) holds, so Aut_SpXq acts transitively on

F_spXq for a geometric point sofS. Choose a lift xPF_spXqand defineu as above. Because the action of AutSpXqon FspXqis free and transitive, u is a bijection. Hence (a) follows.

Definition 2.4. A morphism of schemesX ÑS is called a connected finite

´etale Galois cover if it is a connected finite ´etale cover and satisfies the equivalent conditions of Proposition 2.3.

Remark 2.5. Given a finite ´etale morphism of schemes X Ñ S and a geometric point s of S, one can also consider the left action of AutSpXq onXs arising by base change of its left action on X. Under the bijection between X_s withF_spXqfrom Proposition 1.12, it corresponds to left action of AutSpXq on Fs.

Example 2.6. If K is a field, then a connected finite ´etale Galois cover of SpecpKq is a morphism SpecpLq ÑSpecpKqcorresponding to a finite Galois extensionK ĂL.

Example 2.7. Let us reexamine Example 1.18. Let nbe a positive integer not divisible by the characteristic ofk, let A“B “krT, T^´1s, and viewB as an A-algebra viaψ_n. There is an isomorphism of groups

u: Aut_ApBq „Ýѵ_npkq, f ÞÑfpTq{T,

where µnpkqdenotes the group ofnth roots of unity ink. Hence the corre- sponding connected finite ´etale coverϕ_n:Gm,k ÑGm,k is Galois if and only ifkcontains a primitive nth root of unity.

Proof. Anyf PAut_ApBqsatisfiesψ_n˝f “ψ_n, sofpTqⁿ“Tⁿ. HencefpTq{T is annth root of unity ink. We construct an inversev ofu. Givenζ Pµnpkq, definevpζq to be the morphism ofk-algebras krT, T^´1s ÑkrT, T^´1ssending T toζT. Then ψ_n˝vpζq “ψ_n, since

ψ_npvpζqpTqq “ pζTqⁿ“Tⁿ.

The map v is clearly a group homomorphism and an inverse ofu. The last assertion follows from the above isomorphism and Proposition 2.3.

Proposition 2.8. Ifϕ:X ÑS is a connected finite ´etale Galois cover, then every S-endomorphism of X is an automorphism.

Proof. Let s be a geometric point of S, let x PF_spXq be a lift of s, and let f be an S-endomorphism of X. Because ϕ is Galois, there exists an S-automorphismg ofX such thatF_spgqpxq “F_spfqpxq. But then g“f by Proposition 1.30.

Definition 2.9. Let ϕ:X Ñ S be a finite ´etale cover. An intermediate cover ofϕis a factorization XÑZ ÑS of ϕ. A morphism of intermediate covers pX Ñ Z Ñ Sq Ñ pX ÑZ¹ Ñ Sq is a morphism Z ÑZ¹ such that the diagram

X

Z Z¹

S commutes.

As in Galois theory, there is a correspondence between intermediate covers of a connected finite ´etale Galois cover and subgroups of its automorphism group. In order to state the correspondence, we need to introduce quotients of schemes by groups of automorphisms.

2.2 Quotients of schemes

Definition 2.10. Let C be a category, let X be an object of C, and let G be a subgroup of AutpXq. A quotient of X by G is an object GzX of C together with a universal G-invariant morphism π:XÑGzX, i.e. for every G-invariant morphismψ:X ÑY there is a unique morphismψ¹:GzX ÑY satisfying ψ¹˝π“ψ.

Remark 2.11. If X Ñ S is a morphism in C and G is a subgroup of AutSpXq such that the quotient GzX exists, then X ÑS factors through the canonical morphismX ÑGzX.

Remark 2.12. Let X “ SpecpAq be an affine scheme, and let G be a subgroup of AutpAq. The functor Spec induces a bijection of G with a subgroupG¹of AutpXq. Denote byA^Gthe ring of invariants of the canonical left action of G on A. The affine scheme SpecpA^Gq together with the morphism X Ñ SpecpA^Gq corresponding to the inclusion A^G Ă A is a quotient ofX by G¹ in the category of affine schemes.

Construction 2.13. LetXbe a scheme, and letGbe a subgroup of AutpXq.

Consider the quotient spaceGzX, whose points are the orbits of the points of Xunder the canonical left action ofG, and letπ:XÑGzXbe the canonical projection. Sinceπ_˚OX “π_˚g_˚OX for every gPG, there is a canonical right action of Gonπ˚OX. Defining OGzX to be the sheaf of G-invariant sections ofπ_˚OX, we obtain the ringed spacepGzX,OGzXq.

Proposition 2.14. Letϕ:XÑS be an affine morphism of schemes, and letGbe a finite subgroup ofAut_SpXq. Then the ringed spaceGzX as defined in Construction 2.13 is an S-scheme, and is a quotient of X by G in the category Aff{S of schemes which are affine over S.

Proof. See Szamuely [7, Proposition 5.3.6] for the first assertion, and Lenstra [4, Paragraph 5.18] for the second.

Proposition 2.15. Let ϕ:X Ñ S be a finite ´etale morphism of schemes, and let G be a subgroup of AutSpXq. Let XÑGzX be a quotient of X by Gin Aff{S. Then both morphisms in the factorizationX ÑGzX ÑS of ϕ are finite ´etale.

Proof. See Lenstra [4, Proposition 5.20].

Corollary 2.16. Let ϕ:X ÑS be a finite ´etale morphism of schemes, and letG be a subgroup of Aut_SpXq. Then a quotient ofX by G in the category Fin ´Et{S exists.

Proof. This is immediate from Proposition 2.15, because Fin ´Et{S is a full subcategory ofAff{S.

Proposition 2.17. Let ϕ:X Ñ S be a finite ´etale morphism of schemes, let G be a subgroup of AutSpXq, and let Z ÑS be a morphism of schemes.

Then the canonical morphismGzpXˆSZq Ñ pGzXq ˆ_SZ is an isomorphism.

Proof. See Lenstra [4, Proposition 5.21].

Proposition 2.18. Let ϕ:X Ñ S be a connected finite ´etale cover, and let G be a subgroup of Aut_SpXq. Then X ÝÑ^π GzX Ñ S is a connected intermediate cover of ϕ, and Aut_GzXpXq “G.

Proof. The first part follows from Proposition 2.15. SinceπisG-invariant, we haveGĂAut_GzXpXq. The degree of finite ´etale morphisms is multiplicative by Corollary 1.24, soπ has degree equal to the orderG. Hence we must have G“Aut_GzXpXqby Proposition 2.2.

Proposition 2.19. A connected finite ´etale cover ϕ:X Ñ S is Galois if and only if the canonical morphism Aut_SpXqzXÑS is an isomorphism.

Proof. Letsbe a geometric point of S. The morphism Aut_SpXqzXÑS is an isomorphism if and only if it is of degree 1. By Proposition 2.17, we have Aut_SpXqzXs – pAut_SpXqzXqs. Hence the degree of Aut_SpXqzX Ñ S is 1 if and only if Aut_SpXq acts transitively onX_s. By Remark 2.5, this is the case if and only if AutSpXq acts transitively onFspXq.

2.3 Galois correspondence

Theorem 2.20 (Galois correspondence). Let ϕ: X Ñ S be a connected finite ´etale Galois cover. The assignments

pXÑZ ÑSq ÞÝÑAut_ZpXq pXÑHzXÑSq ÐÝ[H.

extend to an equivalence of categories between the category of intermediate covers of ϕ and the category of subgroups of Aut_SpXq.

Proof. Given a morphism of intermediate covers pX ÑZ ÑSq Ñ pX ÑZ¹ ÑSq,

the inclusion Aut_ZpXq ĂAut_Z¹pXqholds. Conversely, an inclusion HĂH¹ among subgroups of Aut_SpXq yields a factorizationX ÑHzXÑH¹zX of the canonical morphismXÑH¹zX, which is readily seen to be a morphism of intermediate covers

pXÑHzXÑSq Ñ pX ÑH¹zXÑSq.

That these constructions are inverse to each other is immediate from Propo- sitions 2.18 and 2.19.

Proposition 2.21. Letϕ:XÑS be a connected finite ´etale Galois cover, and let

X ÝÑ^π Z ÝÑ^ψ S

be an intermediate cover of ϕsuch that ψ is a connected finite ´etale Galois cover. Then there is surjective group homomorphism

u: AutSpXq ÑAutSpZq with kernel AutZpXq.

Proof. Let s be a geometric point of S, and let x P F_spXq be a lift of s.

Givenf PAut_SpXq, we want to find upfq PAut_SpZq making the diagram

X X

Z Z

f

π π

upfq

commute. By Proposition 1.30, this is equivalent to F_spupfqqpF_spπqpxqq “F_spπ˝fqpxq.

Since ψis Galois, there exists a unique such upfq. By uniqueness ofupfq, the resulting mapu: Aut_SpXq ÑAut_SpZq is a group homomorphism. Now let us show thatu is surjective. Note that F_spπq is surjective. Because ϕ is Galois, this implies that for everygPAutSpZq there exists f PAutSpXq such thatF_spπ˝fqpxq “F_spg˝πqpxq. Then upfq “ g by construction, so u is surjective. The kernel of u consists of those f P Aut_SpXq satisfying π“π˝f, i.e.f PAutZpXq.

Example 2.22. By Examples 1.16 and 2.6, the Galois correspondence for a Galois extensionK ĂL is a special case of Theorem 2.20.

Example 2.23. We return to Example 2.7. Assume that there exists a primitiventh root of unityζ ink. Then the automorphism group ofψn is cyclic with generatorvpζq, and its subgroups are generated by powers ofvpζq.

LetH be such a subgroup, say generated byvpζq^dfor a divisordofn. Denote byH¹ the corresponding group of automorphisms ofGm,k. By Remark 2.12, H¹zGm,k is given by the spectrum ofA^H. The latter consist of all Laurent polynomials f “ř_r

i“´raiTⁱ such that fpζ^dTq “f, i.e. ai is nonzero only ifdiis divisible by n. HenceA^H “krT^n{d, T^´n{ds. The intermediate cover Gm,k ÑH¹zGm,k ÑGm,k of ϕ_n corresponds to the canonical factorization A Ñ A^H Ă A of ψn. Its isomorphism class is the same as that of the intermediate cover

Gm,k ^ϕn{d Gm,k ^ϕd Gm,k.

By Theorem 2.20, every connected intermediate cover ofϕn lies in such an isomorphism class.

We now show that every connected finite ´etale cover ofSis an intermediate cover of a connected finite ´etale Galois cover of S. The proof is taken from Szamuely [7, Proposition 5.3.9].

Proposition 2.24 (Galois closure). Let ϕ:X Ñ S be a connected finite

´etale cover. Then there exists a connected finite ´etale Galois cover P ÑS which factors through ϕ.

Proof. Letsbe a geometric point ofS, and letmbe the degree ofϕ. Choose an enumerationF_spXq “ tx₁, . . . , x_mu. We denote byX^mthem-fold product XˆS¨ ¨ ¨ ˆSX. By the universal property of the fiber product, there is a natural bijection

u:FspX^mq „ÝÑFspXq^m, xÞÑ ppr₁˝x, . . . ,pr_m˝xq.

Letx be the element ofF_spX^mqcorresponding to px₁, . . . , x_mq under u, and letP be the connected component of X^m over which x lies. Let π be the composite of the embedding P ãÑX^m with the projection pr₁: X^m ÑX.

By Proposition 1.23, bothπ andϕ˝π are finite ´etale.

We now show that the image of FspPq in FspXq^m consists of tuples with pairwise distinct entries. Suppose that there exists x¹ P FspPq such that upx¹q “ px¹₁, . . . , x¹_mq has entries x¹_i “ x¹_j for distinct indices i and j. By Proposition 1.30, this implies that the projections pr_i,pr_j:P ÑX are equal.

Since the entries of upxq are pairwise distinct, this is impossible.

We prove that every x¹ PF_spPq lies in the Aut_SpPq-orbit of x. By the above,upx¹q “ px_σp1q, . . . , x_σpmqq for a permutationσPSpt1, . . . , muq. This permutation induces an automorphismf of X^m by permuting the factors.

Then fpPq is a connected component ofX; the point pPP over which x¹ lies is contained in bothP andfpPq, so we must havefpPq “P. Hencef restricts to an automorphismf¹ of P such thatFspf¹qpxq “x¹.

Corollary 2.25. LetS be a connected scheme, let s be a geometric point of S, letϕ:X ÑS be a finite ´etale cover, and letx PFspXq. There exists a connected finite ´etale Galois cover P ÑS, anS-morphism π:P ÑX, and

pPFspPq such that Fspπqppq “x.

Proof. Let Z be the connected component of X over whichx lies. It is open and closed by Proposition 1.26, so the canonical embedding j:Z ãÑ X is finite ´etale. Applying Proposition 2.24 to ϕ˝jyields a connected finite ´etale Galois cover P ÑS which factors through a finite ´etale coverπ¹:P Ñ Z.

Defineπ “j˝π¹. Since Fspπ¹q is surjective,x lies in the image ofFspπq.

Chapter 3

Profinite groups

3.1 Continuous group actions

A left action of a topological groupGon a setE without a topology is called continuous if the corresponding map GˆE Ñ E is continuous, where E is equipped with the discrete topology. This is the case if and only if the stabilizer of everyxPE is open inG.

Remark 3.1. IfE is finite and the symmetric groupSpEqis equipped with the discrete topology, then an actionGˆE ÑE is continuous if and only if the corresponding group homomorphismGÑSpEq is continuous.

3.2 Profinite groups

By acofiltered diagram in a categoryCwe mean a functor P:IÑC, where I is a small cofiltered category. In order to simplify our notation, we will write iPIfor iPObpIq. A topological group Gis called profinite if it is a limit of a cofiltered diagram of finite discrete topological groups. In particular Gis quasi-compact, Hausdorff, and totally disconnected. Profinite groups form a full subcategory ofTopGrp, the category of topological groups. The inclusion functor from profinite to topological groups has a left adjoint, which we construct in several steps.

Construction 3.2. Let G be a topological group. Define I to be the category whose objects are the open normal subgroups ofGof finite index, with a unique morphismM ÑN ifM ĂN, and no morphism from M toN otherwise. With the obvious composition of morphisms,Iis a small cofiltered category. There is a functor P: I Ñ TopGrp that maps an object M of I to the finite discrete quotient group G{M. Given a morphism M Ñ N in I, define PpM ÑNq to be the unique morphism G{M Ñ G{N that is compatible with the projections fromG. The profinite completion Gp of Gis

the limit ofP; it is a profinite group, and comes with a natural morphism ηG:GÑG.p

Lemma 3.3. The image of η_G is dense in G.p

Proof. Let h“ ph_MMqMPI be an element ofG, and letp U “hkerppr_Nq be a fundamental open neighborhood ofh; see Lemma 3.4. Then η_Gph_Nq PU, so ηGpGq is dense inG.p

Lemma 3.4. Let G be the limit of a cofiltered diagramP:IÑTopGrp of finite discrete groups. For everyiPI, denote bypr_i:GÑPpiqthe canonical projection. The open normal subgroups kerppr_iq of G form a fundamental system of open neighborhoods of the identity element ePG.

Proof. LetU be an open neighborhood ofeinG. By definition of the topology of the limit, there is a nonnegative integer m and objectsip1q, . . . , iprq PI such thatŞr

k“1kerppr_ipkqqis an open neighborhood of einU. BecauseI is cofiltered, there existsjPIsuch that there is a morphism jÑipkq inIfor everykP t1, . . . , ru. Then kerppr_jq is an open neighborhood ofe inU. Proposition 3.5. The assignment G ÞÑ Gp extends to a functor from the category of topological groups to the category of profinite groups.

Proof. Letϕ:GÑH be a morphism of topological groups. For every open normal subgroupN ofHof finite index,ϕ^´1pNqis an open normal subgroup ofG of finite index. The family of morphisms

Gp ÑG{ϕ^´1pNqãÑH{N,

where the first morphism is the canonical projection, induces a morphism ϕ:p GpÑHp by the universal property of the limit. The diagram

G Gp

H Hp

ηG

ϕ ϕp

ηH

commutes; since the image of G Ñ Gp is dense by Lemma 3.3 and Gp is Hausdorff, this uniquely characterizesϕ. It follows thatp

GÞÝÑG,p ϕÞÝÑϕp

is a functor.

Proposition 3.6. A topological group G is profinite if and only ifηG is an isomorphism.

Proof. The condition is clearly sufficient, so we only need to prove necessity.

Suppose thatGis a profinite group. The kernel ofηGis trivial by Lemma 3.4 and the fact that G is Hausdorff. Because G is quasi-compact and Gp is Hausdorff,η_G is a closed map. Its image is also dense by Lemma 3.3, so it is a homeomorphism.

Proposition 3.7. The profinite completion of a topological group G has the following universal property: given a profinite group H and a morphism ϕ:GÑH, there exists a unique morphismϕ¹:Gp ÑHsuch thatϕ“ϕ¹˝η_G. Proof. By Proposition 3.5 there exists a unique morphism ϕ:p GpÑHp such thatϕp˝η_G “η_H ˝ϕ. Since η_H is an isomorphism, the claim follows.

It follows formally that the functorGÞÑGp is left adjoint to the inclusion functor from profinite groups to topological groups.

Corollary 3.8. Let G be a profinite group, and let H be a closed subgroup of G, equipped with the induced topology. ThenH is a profinite group.

Proof. We apply Proposition 3.6. Since H is quasi-compact andHp is Haus- dorff, the morphismη_H is closed and surjective. Consider the commutative diagram

H Hp

G G,p

ηH

j pj

ηG

wherej is the inclusion. Since η_G is an isomorphism andj is injective,η_H is injective. Thus it is an isomorphism.

Proposition 3.9. Let G be a topological group. There is a natural isomor- phism of categories between the category of finite continuousG-sets and the category of finite continuous G-sets.p

Proof. LetE be a finite set. By Remark 3.1, continuous left actions ofGonE are in natural bijection with morphismsGÑSpEq. By Proposition 3.7, these are in natural bijection with morphismsGpÑSpEq. Applying Remark 3.1 again yields the desired isomorphism on objects. The correspondence on morphisms is immediate.

3.3 Automorphism groups of functors

Construction 3.10. Let Cbe a small category, and letF:CÑFinSetbe a functor to the category of finite sets. For every EPC, equip AutpFpEqq with the discrete topology. Let I be the category whose objects are the

finite subsets of ObpCq, with a unique morphismAÑA¹ ifA¹ ĂA, and no morphism fromA toA¹ otherwise. Consider the cofiltered diagram

P:IÝÑTopGrp, AÞÝÑ

ź

EPA

AutpFpEqq,

which sends a morphism AÑA¹ to the canonical projection ź

EPA

AutpFpEqq Ñ ź

E¹PA¹

AutpFpE¹qq.

Then ś

EPCAutpFpEqq is a limit of P, so in particular a profinite group.

Note that AutpFq is a closed subgroup of this product, because the groups AutpFpEqq are Hausdorff. We equip AutpFq with the induced topology, which makes it a profinite group by Corollary 3.8.

Proposition 3.11. In the situation of Construction 3.10, the canonical left action of AutpFq onE is continuous for everyE PC.

Proof. The left action AutpFq ˆEÑE factors as

AutpFq ˆE ^prEˆidE AutpFpEqq ˆE E,

where the second map is the canonical left action of AutpFpEqq onE. This action is continuous because the topology of AutpFpEqqis discrete; the map pr_E ˆid_E is continuous by definition of the topology of AutpFq.

Proposition 3.12. For every topological groupG, the categoryFinCont-G-Set of finite continuous G-sets is essentially small, i.e. equivalent to a small category.

Proof. Let E be a finite continuousG-set, and let E₁, . . . , E_r be itsG-orbits.

For everyiP t1, . . . , ru, choose xi PEi and denote byUi its stabilizer. Since U_i is open, the canonical left action ofG onG{U_i is continuous. Since E_i is a transitive G-set, the map gÞÑ g¨x_i induces an isomorphism between G{Ui and Ei. Hence E is isomorphic to š_r

i“1G{Ui. Such G-sets form a set, soFinCont-G-Set has a full subcategory whose inclusion functor is an equivalence of categories.

Whenever necessary, e.g. when applying Construction 3.10, we replace FinCont-G-Set by the full subcategory described in the preceding proof.

Proposition 3.13. Let G be a profinite group, and let U:FinCont-G-SetÑFinSet

be the forgetful functor. Endow AutpUq with the topology from Construc- tion 3.10. There is a natural isomorphism of topological groups

u:GÝÑ„ AutpUq

mapping gPGto the automorphism upgq of U whose component at EPCis upgq_E:UpEq ÑUpEq, xÞÑg¨x.

Proof. Because the morphisms in FinCont-G-Set are G-equivariant maps, u is well-defined. It follows from the definition of a left action that u is a group homomorphism.

We now construct an inverse ofu. For every open normal subgroupM ofG we have the continuous map

AutpUq ÑG{M, f ÞÑf_G{MpMq.

The family of these maps is compatible with the projection G{N¹ ÑG{N for every inclusionN¹ ĂN among open normal subgroups ofG, so it induces a continuous map

v: AutpUq ÑG.p

Since AutpUqis quasi-compact andGpis Hausdorff,vis closed. It is immediate from the respective constructions thatv˝u“ηG, which is an isomorphism by Proposition 3.6. Hencepη_G^´1˝vq ˝u“id_G.

It remains to show thatu˝pη_G^´1˝vq “id_AutpUq. Letfbe an automorphism ofU, and let M be an open normal subgroup ofG. For everygPG there is aG-equivariant map

ωg:G{M ÑG{M, g¹M ÞÑg¹gM.

This is well-defined becauseM is normal. Since f is a morphism of functors, f_G{MpgMq “f_G{MpωgpMqq “ωgpf_G{MpMqq “f_G{MpMqgM.

Hence f_G{M is just left-multiplication by f_G{MpMq. By construction of u and v, this implies

upη_G^´1pvpfqqq_G{M “f_G{M.

In order to finish the proof, it suffices to show thatf is already determined by the componentsf_G{N, whereN ranges over all open normal subgroups of G. LetE be a finite continuousG-set. By Proposition 3.12, we may suppose thatE is of the formš_r

i“1G{H_i for open subgroupsH_i of G. Denote byj the inclusion ofG{H_i intoE. Since f is a morphism of functors,

f_E|G{Hi “f_E˝Upjq “Upjq ˝f_G{Hi.

Hence we further reduce to the case thatE “G{H for an open subgroup H of E. By Lemma 3.4, H contains a normal open subgroup N of G. Then the diagram

UpG{Nq UpG{Hq

UpG{Nq UpG{Hq

f_G{N f_G{H

commutes, sof_G{H can be recovered fromf_G{N. Thusu˝ pη_G^´1˝vq “id_AutpUq, as desired.

Chapter 4

Fundamental Group

4.1 Definition

Fix a schemeS and a geometric point sofS. In order to avoid set-theoretic difficulties, we prove thatFin ´Et{S is essentially small. Whenever necessary, we replace Fin ´Et{S by the full subcategory described in the proof.

Proposition 4.1. The category Fin ´Et{S is essentially small.

Proof. Up to isomorphism, a finite locally free morphismX Ñ S is deter- mined by

(a) an affine open coveringpViqiPI ofS consisting of pairwise distinct sets, (b) a nonnegative integerm_i for every iPI,

(c) the structure of an OSpV_iq-algebra on every module OSpV_iq^mi,

(d) and gluing data for the schemesUi :“SpecpOSpViq^miq, i.e. open sub- schemes U_ij ĂU_i for all i, j PI and S-isomorphismsϕ_ij:U_ij ÝÑ„ U_ji satisfying the cocycle conditions.

The schemes obtained by this process form a set, which implies that the category of finite locally free morphismsX ÑS is essentially small. Hence so is the full subcategory of finite ´etale morphismsXÑS.

Definition 4.2. The fundamental group of S with base pointsis defined to be the automorphism group of F_s, equipped with the topology from Con- struction 3.10, and is denoted by π₁pS, sq.

4.2 Profinite structure of the fundamental group

From now on, we assume that the base schemeS is connected.

Definition 4.3. LetC be a category. A functor F:CÑSet is called pro- representable if there exists a cofiltered diagramP:IÑC together with an isomorphism of functors

colim

iPI Mor_CpPpiq,´q „ÝÑF.

If this is the case, then we say that P pro-representsF.

Construction 4.4. We construct a cofiltered diagram pro-representingFs. The index category I is a subcategory of Fin ´Et{S whose objects are the connected finite ´etale Galois covers of S. For everyiPI, choose p_i PF_spiq.

Giveni, jPI, there is at most one S-morphismϕij:iÑj satisfying F_spϕ_ijqpp_iq “p_j.

We define Mor_Ipi, jq “ tϕiju if ϕij exists, and Mor_Ipi, jq “ H otherwise.

Denote the inclusion functorIÑFin ´Et{S by P.

Giveni, jPI, the canonical morphismPpiqˆSPpjq ÑSis finite ´etale. By Corollary 2.25, there existskPIand anS-morphismπ:Ppkq ÑPpiqˆSPpjq such thatF_spπqpp_kq “ pp_i, p_jq. Composingπ with the respective projections, we obtain morphismskÑiand kÑj inI. Hence Iis cofiltered.

We retain the diagram P and the pointsp_i for the rest of this chapter.

The following proof is taken from Szamuely [7, Proposition 5.4.6].

Proposition 4.5. The cofiltered diagram P defined in Construction 4.4 pro-representsF_s.

Proof. For every iPIthere is a morphism of functors ηi: Mor_SpPpiq,´q ÑFs

whose component at an objectX ofFin ´Et{S is

pη_iqX: Mor_SpPpiq, Xq ÑF_spXq, gÞÑF_spgqpp_iq.

For every morphismϕij:iÑj in Iwe have ηi˝Mor_Spϕij,´q “ηj, since Fspg˝ϕijqpp_iq “Fspgqpp_jq

for everygPMor_SpPpiq, Xq. The universal property of G:“colim

iPI Mor_SpPpiq,´q

yields a unique morphism η:GÑFs induced by the morphisms ηi.

We construct an inverse of η. Given a finite ´etale cover X Ñ S and xPFspXq, there existsiPIand an S-morphismπ:Ppiq ÑX such that

F_spπqpp_iq “x

by Corollary 2.25. Letϑ_X:F_spXq ÑGpXqbe the map that sends x to the equivalence class of π; it is well-defined because I is cofiltered. We claim thatϑ_X is an inverse of η_X. On the one hand, we have

pηX ˝ϑXqpxq “Fspπqpp_iq “x;

hence ηX ˝ϑX is the identity map. On the other hand, if π¹:Ppi¹q Ñ X represents an element of GpXq, then

pϑ_X˝η_Xqprπ¹sq “ϑ_XpFspπ¹qpp_i1qq “ rπ¹s.

Note thatϑX is natural in X, so we have a morphism of functors ϑ:FsÑG with componentϑ_X atX. It follows from the above calculations thatϑ is the desired inverse ofη.

Given a group G, we writeG^op for the group with the same underlying set and the opposite group law; it is naturally isomorphic to GviaxÞÑx^´1. Construction 4.6. Let ϕij:i Ñ j be a morphism in the category I, i.e.

Ppjq ÑS is an intermediate connected finite ´etale Galois cover ofPpiq ÑS.

By Proposition 2.21, there is a surjective group homomorphism u_ij: Aut_SpPpiqq^op ÑAut_SpPpjqq^op;

it mapsg_iPAut_SpPpiqq^op to the uniqueg_j PAut_SpPpjqq^op satisfying gj˝ϕij “ϕij˝gi.

Note that uii is the identity and u_jk˝uij “u_ik for all i, j, kPI, so we have a functor

IÝÑTopGrp, iÞÝÑAutSpPpiqq^op, ϕij ÞÝÑuij,

where the finite groups AutSpPpiqq^opare equipped with the discrete topology.

Proposition 4.7. There is an isomorphism of topological groups u:π1pS, sq „ÝÑlim

iPI AutSpPpiqq^op induced by the morphisms

ui:π1pS, sq ÑAutSpPpiqq^op such that

F_spu_ipfqqpp_iq “f_Ppiqpp_iq for everyf Pπ₁pS, sq.

Proof. We construct an inversew of u. As in the proof of Proposition 4.5, denote by Gthe functor colim_iPIMorSpPpiq,´qand by η the isomorphism GÝÑ„ Fs. The diagram

iÞÝÑMorSpPpiq,´q

is the composite ofP with the contravariant Yoneda embedding Ppiq ÞÝÑMor_SpPpiq,´q

and AutpPq^op“lim_iPIAut_SpPpiqq^op, so there is a canonical group homomor- phism

v: lim

iPI AutSpPpiqq^opÑAutpGq.

Letw be the composite ofv with

AutpGq „ÝÑπ1pS, sq, hÞÑη˝h˝η^´1.

Let us check that u and ware inverse to each other. Given f Pπ₁pS, sq, let X Ñ S be a finite ´etale morphism of schemes, and x P FspXq. By Corollary 2.25, there existsiPI and anS-morphismπ:Ppiq ÑX such that F_spπqpp_iq “x. Then

wpupfqq_Xpxq “wpupfqqXpF_spπqpp_iqq

“F_spπqpwpupfqqPpiqpp_iqq

“Fspπqpf_Ppiqpp_iqq

“f_XpF_spπqpp_iqq

“f_Xpxq.

Hence pw˝uqpfq “ f. On the other hand, starting out with an element g“ pg_iqiPI of lim_iPIAut_SpPpiqq^op, we have

Fspuipwpgqqqpp_iq “wpgq_Ppiqpp_iq

“ pη˝vpgq ˝η^´1q_Ppiqpp_iq

“η_Ppiqpvpgq_Ppiqprid_Ppiqsqq

“η_Ppiqpgiq

“F_spg_iqpp_iq;

by Proposition 1.30, this impliesu_ipwpgqq “g_i.

Thusuis a bijective group homomorphism. It is also continuous, because eachuiis continuous. Sinceπ1pS, sqis quasi-compact and lim_iPIAutSpPpiqq^op is Hausdorff,u is also a closed map. Hence it is a homeomorphism.

The following technical result shows that the morphismsui are surjective.

Lemma 4.8. Let G be the limit of a cofiltered diagramP:IÑTopGrp of quasi-compact Hausdorff topological groups with surjective transition mor-

phism. Suppose moreover that between any two objects of I there are only finitely many parallel morphisms. Then the projection pr_j:G Ñ Ppjq is surjective for everyjPI.

Proof. Leth_j PPpjq. For everyiPI, defineE_i to be the subset ofś

i¹PIPpi¹q consisting of all elementspgi¹qi¹PI such that gj “hj and Ppϕqpgiq “ gi¹ for everyi¹ PI and every morphism ϕ: iÑi¹. Then pr^´1_j phjq “ Ş

iPIEi, and eachE_i is closed because the groupsPpi¹qare Hausdorff. Since ś

i¹PIPpi¹q is quasi-compact, it suffices to show that the family pEiq_iPI has the finite intersection property. To prove that each Ei is nonempty, use that I is cofiltered, that there are only finitely many parallel morphisms between any two objects ofI, and that the transition morphisms are surjective. Given finitely many objectsip1q, . . . , iprq of I, there existslPIsuch that there is a morphism lÑipkq in Ifor every kP t1, . . . , ru. ThenE_l is contained in Ş_r

k“1E_ipkq, so the latter is nonempty.

4.3 Classification theorem

Let f be an automorphism of Fs, and let ψ:X Ñ Y be a morphism in Fin ´Et{S. Then f_Y ˝F_spψq “F_spψq ˝f_X, i.e. F_spψq isπ₁pS, sq-equivariant.

Thus F_s factors through the functor

Fib_s:Fin ´Et{S ÝÑFinCont-π₁pS, sq-Set, pX ÑSq ÞÝÑF_spXq,

ψÞÝÑFspψq.

Proposition 4.9. The group π1pS, sq acts transitively on FspXq for every connected finite ´etale coverX ÑS.

Proof. Let π: QÑ X be a Galois closure ofX Ñ S, and q P FspQq. We show that every point of F_spXq lies in the orbit of x :“ F_spπqpqq. Given x¹ P FspXq, let q¹ P FspQq be such that Fspπqpq¹q “ x¹. Since π is Galois, it has an automorphism h takingq to q¹. Combining Proposition 4.7 with Lemma 4.8, we see thathcan be lifted toπ₁pS, sq, i.e. there existsf Pπ₁pS, sq such thatfQpqq “q¹. Since f is a morphism of functors, we have

f_Xpxq “f_XpF_spπqpqqq

“F_spπqpf_Qpqqq

“x¹. Henceπ1pS, sq acts transitively onFspXq.

We are now in a position to prove that the fundamental group ofS clas- sifies its finite ´etale covers. More precisely, we have the following statement: