Arithmetic Fundamental Group

Andrew Kobin Spring 2017 Contents Contents

Contents

0 Introduction 1 0.1 Topology Review ...... 1 0.2 Finite Etale´ Algebras ...... 4 0.3 Locally Constant Sheaves ...... 5 0.4 Etale´ Morphisms ...... 8

1 Fundamental Groups of Algebraic Curves 11 1.1 Curves Over Algebraically Closed Fields ...... 11 1.2 Curves Over Arbitrary Fields ...... 12 1.3 Proper Normal Curves ...... 15 1.4 Finite Branched Covers ...... 16 1.5 The Fundamental Group of Curves ...... 22 1.6 The Outer Galois Action ...... 25 1.7 The Inverse Galois Problem ...... 30

2 Riemann’s Existence Theorem 36 2.1 Riemann Surfaces ...... 36 2.2 The Existence Theorem over 1 ...... 42 PC 2.3 The General Case ...... 48 2.4 The Analytic Existence Theorem ...... 50

3 Scheme Theory 54 3.1 Aﬃne Schemes ...... 54 3.2 Schemes ...... 56 3.3 Properties of Schemes ...... 58 3.4 Sheaves of Modules ...... 64 3.5 Group Schemes ...... 70

4 Fundamental Groups of Schemes 73 4.1 Galois Theory for Schemes ...... 74 4.2 The Etale´ Fundamental Group ...... 78 4.3 Properties of the Etale´ Fundamental Group ...... 81 4.4 Structure Theorems ...... 84 4.5 Specialization and Characteristic p Results ...... 87

i 0 Introduction

0 Introduction

The following notes are taken from a reading course on ´etalefundamental groups led by Dr. Lloyd West at the University of Virginia in Spring 2017. The contents were presented by stu- dents throughout the course and mostly follow Szamuely’s Galois Groups and Fundamental Groups. Main topics include: A review of covering space theory in topology (universal covers, monodromy, locally constant sheaves)

Covers and ramiﬁed covers of normal curves

The algebraic fundamental group for curves

An introduction to schemes

Finite ´etalecovers of schemes and the ´etalefundamental group

Grothendieck’s main theorems for the fundamental group of a scheme

Applications.

0.1 Topology Review

There are two key concepts in algebraic topology that, for various reasons, one might want to consider in an algebraic setting. These are covering spaces and fundamental groups, and they are intimately connected. The more familiar concept might be that of the fundamental group, which at the beginning is usually defined in terms of homotopy classes of based loops in a given topological space. To define an algebraic analogue, we will need an alternative perspective on the fundamental group. Let X be a connected, locally simply connected topological space. Definition. A cover of X is a space Y and a map p : Y → X that is a local homeomorphism. That is, for every x ∈ X there is a neighborhood U ⊆ X of x such that p−1(U) is a disjoint union of open sets in Y and p restricts to a homeomorphism on each open set. One consequence of this definition is that for all x, y ∈ X, p−1(x) is a discrete space and p p−1(x) ∼= p−1(y). A primary goal in topology is to study and classify all such covers Y −→ X.

p p0 Deﬁnition. A morphism of covers between Y −→ X and Y 0 −→ X is a map f : Y → Y 0 making the following diagram commute:

f Y1 Y2

p1 p2

1 0.1 Topology Review 0 Introduction

This deﬁnes a category CovX of covers over X. In this category, we will abbreviate

HomCovX (Y,Z) by HomX (Y,Z).

Example 0.1.1. The unit interval [0, 1] ⊆ R has no nontrivial covers. However, S1 ⊆ C does: for each n ∈ Z, the map

1 1 n pn : S −→ S , z 7−→ z is a cover. Further, there is a special cover

1 2πit π : R −→ S , t 7−→ e such that for every n ∈ Z, the following diagram commutes:

R S1

π pn

All covers of the circle arise in this way.

The special property of R → S1 leads to the notion of a universal cover. Deﬁnition. A covering space π : Xe → X is a universal cover for X if for every other cover p : Y → X, there is a unique map f : Xe → Y making the diagram commute: f Xe Y

π p

It is equivalent to say that a universal cover is any simply connected cover of X, and one shows easily that universal covers are unique up to equivalence of covers. An important result is that a universal cover exists, under certain mild conditions on X. In topology, the topological fundamental group is deﬁned using homotopy:

homotopy classes of loops πtop(X, x) := . 1 in X based at x

The universal cover has important connections to this fundamental group. In particular, consider an automorphism α ∈ AutX (Xe) = HomX (X,e Xe). Fix x ∈ X and a liftx ˜ ∈ Xe of x. Then πα(˜x) = x. Moreover, since Xe is simply connected, any pathx ˜ → αx˜ is unique up to top homotopy. This determines a map AutX (Xe) → π1 (X, x).

top Theorem 0.1.2. For any x ∈ X, AutX (Xe) → π1 (X, x) is an isomorphism.

2 0.1 Topology Review 0 Introduction

Unfortunately, such a space Xe does not exist in the algebraic world. Thus we describe a slightly diﬀerent interpretation of the fundamental group.

Deﬁnition. The ﬁbre functor over x ∈ X is the assignment

Fibx : CovX −→ Sets p (Y −→ X) 7−→ p−1(x).

By the universal property of a universal cover Xe ∈ CovX , to give a morphism of covers −1 f : Xe → Y is the same as to choose a point y = f(˜x) ∈ p (x). In other words, Fibx is a representable functor, i.e. for x ∈ X, there is a natural isomorphism ∼ Fibx(−) = HomX (Xex˜, −),

where the Hom set consists of morphisms based atx ˜. This ﬁbre functor is constructible in algebraic categories, though it fails to be representable. Going further, there is a natural left action of AutX (Xe) on Xe; however, it will be more op convenient to view this as a right action of AutX (Xe) on Xe. This induces a left action of top AutX (Xe) = π1 (X, x) on HomX (X,Ye ):

α · f = f ◦ α for any α ∈ AutX (Xe), f : Xe → Y.

top This action is called the monodromy action. Often, one views this as an action of π1 (X, x) on the ﬁbre Fibx(Y ) given by lifting paths. In any case, we get a map

top π1 (X, x) −→ Aut(Fibx), where Aut(Fibx) is the automorphism group of the ﬁbre functor in the following sense. For any functor F : C → D, an automorphism of F is a natural transformation of F that has a two-sided inverse. The set Aut(F ) of all automorphisms of F is then a group under composition. Moreover, Aut(F ) has a natural action on F (C) for any object C ∈ C.

top Theorem 0.1.3. For all x ∈ X, π1 (X, x) → Aut(Fibx) is an isomorphism. Theorem 0.1.4. Let X be a connected, locally simply connected space and fix x ∈ X. Then the fibre functor Fibx defines an equivalence of categories

∼ CovX −→{left π1(X, x)-sets} with connected covers corresponding to transitive π1(X, x)-sets and Galois covers to coset spaces of Xex by normal subgroups.

Proof. For a transitive π1(X, x)-set S, deﬁne YS = X/He where H = Stabπ1(X,x)(s) for any point s ∈ S. This deﬁnes a Galois cover YS → X and one can extend this to arbitrary π1(X, x)-sets orbitwise for the full correspondence.

3 0.2 Finite Etale´ Algebras 0 Introduction

The picture gets more interesting if we restrict ourselves to ﬁnite covers. Given such a cover p : Y → X, there is an exact sequence of groups

top −1 1 → N → π1 (X, x) → AutX (p (x)) → 1 where N is some finite index kernel. This shows that the monodromy action factors through a finite quotient. As a result, this action can be defined on the level of a profinite group, top namely the profinite completion of π1 (X, x):

top\ top π1 (X, x) := lim π1 (X, x)/N, ←−

top where the inverse limit is over all ﬁnite index subgroups N ≤ π1 (X, x).

Corollary 0.1.5. The ﬁbre functor Fibx deﬁnes an equivalence of categories

∼ top\ {ﬁnite covers of X} −→{ﬁnite, continuous π1 (X, x)-sets}. Moreover, the correspondence restricts to

∼ top\ {connected covers} −→{ﬁnite π1 (X, x)-sets with transitive action} ∼ top\ and {Galois covers} −→{π1 (X, x)/N | N an open normal subgroup}.

0.2 Finite Etale´ Algebras

Fix a ﬁeld k and an algebraic closure k¯, which comes equipped with a separable closure ¯ ks ⊆ k. Set Gk = Gal(ks/s) and let L/k be any ﬁnite, separable extension.

Lemma 0.2.1. Homk(L, ks) is a ﬁnite, continuous, transitive Gk-set.

Proof. By Galois theory, # Homk(L, ks) = [L : k], so this is finite when the extension is assumed to be finite. The Gk-action is defined by σ·f = σ◦f for σ ∈ Gk and f ∈ Homk(L, ks); it is routine to verify that this is indeed a group action. Now to show the action is continuous, since Homk(L, ks) is discrete, this is equivalent to showing the stabilizer StabGk (f) is open for each f ∈ Homk(L, ks). Notice that

StabGk (f) = {σ ∈ Gk | σ ◦ f = f} = {σ ∈ Gk | σ ﬁxes f(L)} and this is open by Galois theory / the topology of the proﬁnite group Gk. Finally, since L/k is separable, we may pick a minimal polynomial h(t) for a primitive element of L/k. Then Gk permutes the roots of h(t) transitively, so it follows that Gk acts transitively on Homk(L, ks). ∼ Corollary 0.2.2. There exists an open subgroup H ≤ Gk such that Homk(L, ks) = Gk/H as Gk-sets. Further, if L/k is Galois, H may be chosen to be an open normal subgroup.

Proof. We may pick H = StabGk (h), the stabilizer of the minimal polynomial of a primitive element of L/k. The Galois case follows from the fundamental theorem of Galois theory.

4 0.3 Locally Constant Sheaves 0 Introduction

Theorem 0.2.3. The assignment L/k 7→ Homk(L, ks) is a contravariant functor

{ﬁnite separable extensions of k} −→ {ﬁnite, continuous, transitive Gk-sets}

which is an anti-equivalence of categories. Moreover, Galois extensions L/k correspond to ﬁnite quotients of Gk.

Proof. For any finite, continuous, transitive Gk-set S, define a finite separable extension

LS/k by taking the subﬁeld of ks/k ﬁxed by the stabilizer StabGk (s) of any point s ∈ S. One now checks that this is an inverse functor to the Homk(−, ks) functor.

To study all finite continuous Gk-sets, we replace separable field extensions with a slightly more general object. ∼ Definition. A k-algebra A is a finite étalealgebra if A = L1 ×· · ·×Lr for finite separable extensions Li/k.

Corollary 0.2.4 (Grothendieck). The assignment A/k 7→ Homk(A, ks) is an anti-equivalence of categories ∼ {finite étale k-algebras} −→{finite continuous Gk-sets} which reduces to the above case when A = L is a finite separable extension of k.

Topologically, we may view k as a point space covering the ‘smaller’ point space ks, and any intermediate extension L/k as an intermediate cover. In this setting, Gk plays the role of the deck transformations of the ‘universal cover’ k → ks and Homk(L, ks) plays the role of the fibre of a cover. Also, under this analogy, a finite separable extension L represents a connected cover while a finite étalealgebra A may be viewed as a disconnected cover. Finally, the choice of a algebraic (and separable) closure of k is analagous to the choice of a basepoint of a topological space, which also determines a universal cover. We will see that this subtlety conceals a wealth of information about the algebraic fundamental group.

0.3 Locally Constant Sheaves

In this setting we unite the topological and ﬁeld-theoretic approaches to the fundamental group introduced in Sections 0.1 and 0.2. Let X be a space.

Deﬁnition. A presheaf on X with values in a category of sets C (e.g. Sets, Groups, AbGps, Rings, etc.) is a functor

F : TopX −→ C U 7−→ F(U)

(V,→ U) ρ ∈ Hom (F(U), F(V )), UV C

such that for each object U, ρUU = idU , and for any inclusions W,→ V,→ U, ρUW = ρVW ◦ ρUV .

5 0.3 Locally Constant Sheaves 0 Introduction

For an element s ∈ F(U) and an inclusion V,→ U, we will write s|V = ρUV (s). Sugges- tively, these elements are called sections over U and the ρUV restriction maps. Deﬁnition. Two presheaves F and G on X are said to be isomorphic if there exists a natural isomorphism F → G.

Example 0.3.1. The classic example of a presheaf on a space X is the assignment to each open U ⊆ X of the space of continuous (or smooth or holomorphic, when appropriate) functions on U, with ρUV given by restriction of functions. This example in fact enjoys several important properties, e.g. a function is 0 on an open set if and only if it is 0 on all open covers of the set. These properties are generalized below.

Deﬁnition. A presheaf F on X is a sheaf if:

(1) For every covering {Ui} of an open set U ⊆ X and sections s, t ∈ F(U) such that

s|Ui = tUi for all i, we must have s = t.

(2) For every covering {Ui} of U and sections {si ∈ F(Ui)}, if si|Ui∩Uj = sj|Ui∩Uj whenever

Ui ∩ Uj 6= ∅, then there exists a section s ∈ F(U) such that s|Ui = si for each i. Remark. Let F be a sheaf on X. For any inclusion of an open set U,→ X, there is an obvious restriction of F to a sheaf F|U deﬁned on U. Deﬁnition. Let S be a set. Then the constant sheaf on X with values in S is the sheaf U 7→ FS(U), where FS(U) = C(U, S) is the set of continuous functions U → S.

Notice that if S is a discrete set and U is a connected, open set, then FS(U) = S, hence the name “constant sheaf”.

Deﬁnition. A sheaf F on X is locally constant if for each x ∈ X, there is a neighborhood U of x such that F|U is isomorphic to a constant sheaf. We now rephrase the theory of covers from Section 0.1 in terms of locally constant sheaves. For a space X, let LCShX be the category of locally constant sheaves on X (as a subcategory of ShX ). Deﬁnition. Let p : Y → X be a cover and U ⊆ X an open set. Then a section of p over U is a continuous map s : U → Y satisfying p ◦ s = idU . The set of all sections over U is denoted Γ(U, p).

Proposition 0.3.2. For a cover p : Y → X, the assignment U 7→ Fp(U) := Γ(U, p) is a locally constant sheaf on X.

Proof. The sheaf axioms are easily veriﬁed using properties of continuous functions. To prove the locally constant condition, ﬁx x ∈ X and choose a connected, locally trivial neighborhood U ⊆ X of x such that p−1(U) ∼= U × p−1(x). For a section s : U → Y , p maps s(U) homeomorphically onto U, so since U is connected, s(U) must be a connected −1 ∼ −1 component of p (U). In this manner Γ(U, p) = p (x) and therefore Fp|U is isomorphic to the constant sheaf Fp−1(x). Hence Fp is locally constant.

6 0.3 Locally Constant Sheaves 0 Introduction

We will show that the section functor establishes an equivalence between covers of X and locally constant sheaves on X. To do so requires the construction of a covering space from the data of a locally constant sheaf F. Such a cover is called the étalespace of F. To define this, we need: Definition. Let F be a presheaf on X and fix x ∈ X. The stalk of F at x is the direct limit Fx := lim F(U) −→ over all open neighborhoods U of x, directed by inclusion.

Lemma 0.3.3. The assignment F 7→ Fx is a functor for all x ∈ X. p Theorem 0.3.4. The functor (Y −→ X) 7→ Fp induces an equivalence of categories ∼ CovX −→ LCShX . ` Proof. Given a presheaf F on X, we construct its ´etalespace XF by XF = x∈X Fx, the disjoint union of the stalks over each x in the base. The maps Fx → {x} induce a map p : XF → X. Further, we endow XF with the topology induced by the maps p and {is}s∈F(U) for all open U ⊆ X, where is : U → XF is the map sending x to the unique element of s(U) ∩ Fx. Now suppose F is a locally constant sheaf; we claim p : XF → X is a topological cover. If U is a locally constant neighborhood, then Fx = S for all x ∈ U and some ﬁxed discrete −1 ∼ set S. Thus p (U) = U × S so this shows XF is in fact a cover of X. Now consider the assignments

CovX −→ LCShX and LCShX −→ CovX p p (Y −→ X) 7−→ Fp F 7−→ (XF −→ X). We claim that these functors are naturally inverse isomorphisms. It is enough to prove that the natural transformations

A : F −→ FXF and B : Y −→ XFY −1 s ∈ F(U) 7−→ is y ∈ p (x) 7−→ y ∈ (Fp)x are isomorphisms. Choose a covering {Ui} such that F|Ui is isomorphic to a constant sheaf on some F . Then (U ) ∼= U × F and Γ(U ,U × F → U ) = F for each i. Thus A i i F|Ui i i i i i i Fi and B are isomorphisms. Checking naturality is straightforward, so we have the desired equivalence of categories. Corollary 0.3.5. Let X be a connected, locally simply connected space and ﬁx x ∈ X. Then the functor F 7→ Fx induces an equivalence of categories ∼ LCShX −→{left π1(X, x)-sets}. With a little more work, we can also prove: Corollary 0.3.6. For a space X and x ∈ X as above and any commutative ring R, there is an equivalence of categories ∼ {locally constant sheaves of R-modules on X} −→{left modules over R[π1(X, x)]}.

7 0.4 Etale´ Morphisms 0 Introduction

0.4 Etale´ Morphisms

Let us now shift focus to the algebraic setting, meaning a category of algebro-geometric objects such as nonsingular varieties over a field or schemes. The Zariski topology makes it tricky to define a cover in terms of a local homeomorphism – in fact, by Zariski’s theorem, such a map is only ever a trivial inclusion of an open set, so nothing terribly interesting. However, in the geometric setting, we have the notion of the differential on tangent spaces.

Definition. For a morphism p : Y → X of nonsingular varieties X and Y over an algebraically closed field k, we say p is étaleat y ∈ Y if the differential dpy : TyY → Tp(y)X is an isomorphism. The map is said to be étale if it is étaleat every y ∈ Y . ¯ ¯ If k is not algebraically closed, consider the base change pk¯ : Y (k) → X(k). Then we say p is étaleat y ∈ Y if for all geometric points y¯ → y, pk¯ is étaleat y¯. As above, p is étaleif it is étaleat every y ∈ Y .

A consequence of this definition is that the fibres of an étalemorphism p : Y → X are finite and their cardinality is constant (at least, on connected components). The definition of an étalemorphism for schemes requires a little more care.

Deﬁnition. Suppose f : A → B is a morphism of local rings. We say f is ﬂat if the functor M → M ⊗A B is exact.

Definition. A morphism f : A → B of local rings is unramified if B/f(mA)B is a finite, separable field extension of A/mA.

We will typically encounter the case when f(mA)B = mB, so that unramified is the same as B/mB being a finite, separable extension of A/mA. Definition. A morphism of schemes p : X → Y is étaleat y ∈ Y if the induced morphism # # on local rings p : OX,p(y) → OY,y is an étalemorphism of local rings, that is, p is flat and unramified. We say p is étale if it is étaleat every y ∈ Y .

Let FétX be the category consisting of finite étale covers of X, together with morphisms of covers defined as in the topological case (i.e. commuting with covering maps over X). For a geometric pointx ¯ : Spec Ω → X in a scheme X, define the fibre functor

Fibx¯ : F´etX −→ Sets p (Y −→ X) 7−→ Spec Ω ×X Y.

Definition. The étalefundamental group of a scheme X at a geometric point x¯ : Spec Ω → X is the automorphism group of the fibre functor over x¯,

´et π1 (X, x¯) = Aut(Fibx¯). As in the topological case, we will prove:

Theorem 0.4.1 (Grothendieck). For X a connected scheme and x¯ : Spec Ω → X a geometric point,

8 0.4 Etale´ Morphisms 0 Introduction

ét (1) π1 (X, x¯) is a profinite group which acts continuously on each fibre Fibx¯(Y ) for Y → X a cover.

ét (2) Fibx¯ : FétX → {continuous, finite π1 (X, x¯)-sets} is an equivalence of categories.

ét Further, we will see that π1 (X, x¯) = lim AutX (Fibx¯(Y )), where the inverse limit may be ←− taken over all Galois covers p : Y → X. Example 0.4.2. Let k be a field and consider the scheme X = Spec k which is a point. A geometric pointx ¯ : Spec Ω → X is a choice of algebraic closure Ω ⊇ k; such a choice also defines a separable closure ksep ⊇ k. In this case, the correspondence of Theorem 0.4.1 is:

FétX ←→ {Spec A | A is a finite étale k-algebra}. ∼ Here, the functor Fibx¯ is representable by Homk(A, Ω) = Homk(A, ksep). Moreover, we have a Galois action of Gk := Gal(ksep/k) on Homk(A, ksep) given by σ · f = σ ◦ f for any f : A → ksep and σ ∈ Gk. In particular, π1(X, x¯) = Gk and this action is the monodromy action as described in Section 0.1.

Theorem 0.4.3. For a ﬁeld k, Homk(−, ksep) induces an equivalence of categories

∼ {finite étale k-algebras} −→{finite continuous Gk-sets}.

Example 0.4.4. Let (A, m, k) be a complete DVR and set X = Spec A. There is an equivalence between étalecovers FétA ←→ Fétk. (One direction is obvious; the other is an application of Hensel’s Lemma.) Therefore ét ∼ ét π1 (Spec A, x¯) = π1 (Spec k, x¯) for any geometric pointx ¯ of A. Theorem 0.4.5. Suppose X is connected and normal, K = k(X) is the function field of X and L/K is a finite, separable field extension. Let Y be the normalization of X in L. Then p : Y → X is a ramified cover and for some set S ⊂ Y , p : Y r S → X is étale. Further, every étalecover of X arises in this way. Corollary 0.4.6. If X is a connected, normal scheme with function field K and Kur is the compositum of all extensions of K in which normalization of X is unramified, then

´et ∼ ur π1 (X) = Gal(K /K).

Example 0.4.7. A famous result in number theory says that Q has no nontrivial unramified ét extensions. Therefore π1 (Spec Z) = 1. However, ‘removing some points’ from Spec Z, e.g. ét 1 localizing, yields nontrivial fundamental groups, such as π1 Spec Z n . ét Example 0.4.8. We will show that π1 (AC) = 1 (compare to the topological case!), but for ét char k > 0, we may have π1 (Ak) 6= 1. There is a classic and important connection between curves and Riemann surfaces which we will explored further in Chapter 2. The arithmetic version of this correspondence is borne out by the following theorem.

9 0.4 Etale´ Morphisms 0 Introduction

Theorem 0.4.9. Let X be a normal curve of genus g over k. If char k = p > 0, then

* g + ´et (p0) ∼ Y π1 (X) = x1, y1, . . . , xg, yg : [xi, yi] = 1 [p] i=1

0 where (·)(p ) denotes the prime-to-p part of a proﬁnite group. Further, if k = C, then ´et ∼ \top π1 (X) = π1 (X).

10 1 Fundamental Groups of Algebraic Curves

1 Fundamental Groups of Algebraic Curves

1.1 Curves Over Algebraically Closed Fields

In the first two sections of Chapter 1, we will review the basics of algebraic curves. Fix an n algebraically closed field k. Denote by Ak the affine n-space over k, which as a set is equal n to k . Let R = k[x1, . . . , xn] be the polynomial ring in n variables over k. An ideal I ⊆ R defines an algebraic set X = V (I), whose coordinate ring is O(X) = R/I. Lemma 1.1.1. O(X) is a finitely generated, reduced k-algebra.

Deﬁnition. If I is a prime ideal of R = k[x1, . . . , xn], then we call X = V (I) an integral aﬃne variety.

n Definition. Let Y = V (J) ⊆ Ak be an integral affine variety. A morphism (or regular m m map) ϕ : Y → Ak is a choice of functions (f1, . . . , fm) ∈ O(Y ) . In general, for another m integral affine variety X ⊆ Ak , we say ϕ : Y → X is a morphism if ϕ = (f1, . . . , fm) such that (f1(P ), . . . , fm(P )) ∈ X for all P ∈ Y . Every morphism ϕ : Y → X induces a k-algebra homomorphism ϕ∗ : O(X) → O(Y ) given by f 7→ f ◦ ϕ. When we equip X and Y with the Zariski topology, a morphism ϕ : Y → X is continuous with respect to these topologies. Proposition 1.1.2. The assignments X 7→ O(X) and ϕ 7→ ϕ∗ define an anti-equivalence of categories

{aﬃne varieties over k} −→ {ﬁnitely generated, reduced k-algebras} X 7−→ O(X)

V (I) 7−→ A = k[x1, . . . , xn]/I.

Let X be an integral affine variety. Then O(X) is a domain, so we can define the following: The fraction field of X, k(X) := Frac O(X). For any point P ∈ X, the local ring at P , OX,P := O(X)mP , where mP = (x1 − α1, . . . , xn − αn) is the maximal ideal of O(X) corresponding to P = (α1, . . . , αn). T For any open subset U ⊆ X, the ring OX (U) := P ∈U OX,P . T Lemma 1.1.3. For any integral affine variety X, O(X) = P ∈X OX,P . Proof. By commutative algebra, any domain can be written as the intersection of its local- izations at maximal ideals. Thus for O(X), we have \ \ O(X) = O(X)m = OX,P . max. ideals P ∈X m⊂O(X)

11 1.2 Curves Over Arbitrary Fields 1 Fundamental Groups of Algebraic Curves

Definition. The dimension of an integral affine variety X is dim X := tr degk k(X). We say X is an integral affine curve if dim X = 1.

By dimension theory, dim X = dim O(X), the Krull dimension of the coordinate ring of X.

Deﬁnition. An integral aﬃne variety X is normal at a point P ∈ X if the local ring OX,P is integrally closed in k(X). We say X is a normal variety if it is normal at every point P ∈ X.

Proposition 1.1.4. X is normal if and only if O(X) is an integrally closed domain.

Proof. ( ⇒ = ) follows from the fact that if a domain is integrally closed, then so is every localization. T ( =⇒ ) By Lemma 1.1.3, O(X) = P ∈X OX,P so if each local ring is integrally closed, then so is their intersection.

Remark. When dim X = 1, X is normal at P if and only if OX,P is a DVR.

Deﬁnition. An integral aﬃne variety X is nonsingular (or smooth) at P ∈ X if OX,P is a regular local ring. Equivalently (though this requires proof), X is nonsingular at P if the dimension of the tangent space to X at P is equal to dim X.

Proposition 1.1.5. If X is an integral aﬃne curve, then X is nonsingular at P ∈ X if and only if X is normal at P .

2 Example 1.1.6. Let X ⊂ Ak be an integral affine plane curve. Then X is smooth at P ∈ X as long as dxf(P ) 6= 0 or dyf(P ) 6= 0. Proposition 1.1.7. Let X be an integral affine curve and P ∈ X a normal point. Then there exists an affine open neighborhood U of P in X which is isomorphic to an open neighborhood of some smooth point in a plane curve.

1.2 Curves Over Arbitrary Fields

When k is not algebraically closed, there is no guarantee of a correspondence between ideals of k[x , . . . , x ] and points in aﬃne space. For example, there are no -points (x, y) ∈ 2 1 n R AR satisfying x2 + y2 + 1 = 0, and therefore there are ideals of R[x, y] which do not correspond to points in aﬃne space. To deal with this complication, we generalize our notion of curves considerably.

Deﬁnition. A ringed space is a pair (X, F) consisting of a space X and a sheaf of rings F on X.

Let A be a ﬁnitely generated k-algebra which is an integral domain with ﬁeld of frac-

tions K, and suppose tr degk K = 1. Consider the space X = Spec A = {p ⊂ A | p is a prime ideal}. Taking closed sets to be of the form V (I) = {p ∈ Spec A | I ⊇ p} for ideals I ⊂ A deﬁnes a Zariski topology on X. The zero ideal 0 deﬁnes a generic point

12 1.2 Curves Over Arbitrary Fields 1 Fundamental Groups of Algebraic Curves

of X: for any nonempty open subset U ⊆ X, 0 ∈ U. It follows that A0 = K. On the other hand, any nonzero prime p is maximal (since dim A = tr degk K = 1), so V (p) = {p}. Thus prime ideals correspond to closed points in X. In particular, all proper closed subsets of X are ﬁnite.

Deﬁnition. For P ∈ X, let OX,P := AP be the local ring at P . For any open U ⊆ X, set \ OX (U) := OX,P . P ∈U

Proposition 1.2.1. U 7→ OX (U) deﬁnes a sheaf of functions on X.

Definition. We call the ringed space (X, OX ) an integral affine curve. Definition. A morphism of sheaves is a natural transformation F → G of sheaves on a common space X, that is, a ring homomorphism F(U) → G(U) for each open U ⊆ X such that for every inclusion of open sets V ⊆ U, the following diagram commutes:

F(U) G(U)

F(V ) G(V )

Definition. For a sheaf G on Y and a continuous map f : Y → X, the pushforward sheaf −1 is a sheaf on X defined by U 7→ f∗G(U) := G(f (U)) for all open U ⊆ X. Lemma 1.2.2. f ∗G is a sheaf on X. Definition. A morphism of ringed spaces (Y, G) → (X, F) is a pair (ϕ, ϕ#) consisting # of a continuous map ϕ : Y → X and a morphism of sheaves ϕ : F → ϕ∗G, where ϕ∗G is the pushforward.

Remark. Let (X, OX ) and (Y, OY ) be integral affine curves. We will say ϕ : X → Y is a # morphism of curves if it is a morphism of ringed spaces such that ϕ : OX → ϕ∗OY is a morphism of sheaves of local rings, i.e. one that sends maximal ideals in the local rings OY,Q to maximal ideals in the OX,ϕ(Q). Proposition 1.2.3. There is an anti-isomorphism of categories finitely generated integral domains {integral affine curves over k} −→ A/k with tr degk K = 1

(X, OX ) 7−→ O(X) (ϕ : Y → X) (ϕ# : O(X) → O(Y )) Spec A 7−→ A (f ∗ : Spec B → Spec A) (f : A → B).

13 1.2 Curves Over Arbitrary Fields 1 Fundamental Groups of Algebraic Curves

Let X = Spec A be an integral affine curve. Suppose L/k is a field extension such that A ⊗k L is still a domain. Then XL := Spec(A ⊗k L) is an integral affine curve over L, called the base change of X to L. The natural map A → A ⊗k L, a 7→ a ⊗ 1 induces a morphism of ringed spaces XL → X. ¯ Definition. If A⊗k k is an integral domain, we say X = Spec A is geometrically integral. ¯ In this case, XL is integral for all intermediate extensions k ⊇ L ⊇ k. For any of these L, there is a functor geometrically integral integral affine F : −→ . L affine curves over k curves over k

Example 1.2.4. Let X = Spec(R[x, y]/(x2 + y2 + 1)). Then closed points of X are in correspondence with maximal ideals of the ring R[x, y]/(x2+y2+1). These in turn correspond 2 2 with Galois orbits of XC. Explicitly, each closed point (x , y + 1) of X is covered by exactly two points (x, y − i) and (x, y + i) via XC → X.

Definition. A curve X is normal at P ∈ X if O(X)P is integrally closed. As before, X is normal if it is normal at every P ∈ X. Definition. A morphism ϕ : Y → X of integral affine curves is finite if O(Y ) is a finitely generated O(X)-module via ϕ# : O(X) → O(Y ). Lemma 1.2.5. Let ϕ : Y → X be a finite morphism. Then (1) O(Y ) is integral over O(X). (2) ϕ is surjective. (3) ϕ# is injective. (4) k(Y )/k(X) is a finite extension. (5) ϕ has finite fibres. Theorem 1.2.6. Let X be an integral normal affine curve. Then the assignments Y 7→ k(Y ) and ϕ 7→ ϕ∗ induce an anti-equivalence of categories normal affine curves over k with embeddings of fields −→ . finite maps Y → X k(X) ,→ L Proof. Given an embedding i : k(X) ,→ L, let B be the integral closure of O(X) in L. Then by commutative algebra, B is a finitely generated k-algebra which is integrally closed, and

since L/k(X) is a finite extension, tr degk L = 1. Hence by Proposition 1.2.3, Y = Spec B is a normal affine curve with a morphism ϕ = i∗ : Y → X. One now checks that these assignments are natural inverses. Definition. For an integral affine curve X and a field extension L/k(X), the curve Y constructed above is called the normalization of X in L.

Remark. If X is geometrically integral, then XL is equal to the normalization of X in L for any L ⊇ k(X).

14 1.3 Proper Normal Curves 1 Fundamental Groups of Algebraic Curves

1.3 Proper Normal Curves

To fill out the analogy with the topological case (Section 0.1), we want to consider some version of compact curves. Let X be an affine curve with function field K = k(X).

Lemma 1.3.1. The set {OX,P | P 6= 0} is equal to the set of DVRs of K containing O(X). Proof. On one hand, (⊆) is obvious. For the other containment, let R be such a DVR and call its maximal ideal m. Then p := m ∩ O(X) is a nonzero maximal ideal in O(X). This means O(X)p ⊆ R but each of these is a DVR with the same function ﬁeld, so we must have OX,P = R.

1 1 Example 1.3.2. Let X = Ak so that O(Ak) = k[t], the polynomial ring in a single 1 −1 indeterminate. If Y is also Ak, we may take O(Y ) = k[t ]. Let R be a DVR with Frac(R) = k(X) ∼= k(t) ∼= k(t−1). Then either R ⊃ O(X) or R ⊃ O(Y ). In fact, the only −1 0 DVR not containing O(X) is R = k[t ](t−1), while for O(Y ) the exception is R = k[t](t). We want some way of “gluing” these two DVRs together so that all DVRs of the function field over k are considered. Now assume X is a normal affine curve. By Noether normalization, there exists a function f ∈ O(X) such that O(X) is a finitely generated k[f]-module. Define X− to be the normal h 1 i curve associated to the integral closure of k f in K. Then every DVR R with fraction field + − 1 K is a local ring for X = X or for X , corresponding to whether f ∈ R or f ∈ R. More abstractly, for a finitely generated field extension K/k of transcendence degree 1, let XK be the set of DVRs of K/k. Define a topology on XK by declaring the complements T of finite subsets to be open. Define a sheaf of rings on XK by U 7→ OXK (U) := R∈U R.

Theorem 1.3.3. (XK , OXK ) is a ringed space which is “locally aﬃne” in the sense that + − + − there is a decomposition XK = XK ∪ XK where XK and XK are aﬃne curves.

Deﬁnition. The ringed space (XK , OXK ) is called an integral proper normal curve.

Definition. A morphism of (integral) proper normal curves ϕ : YL → XK is a # morphism of the underlying ringed spaces in which the morphism of sheaves ϕ : OXK → OYL is a morphism of local rings, i.e. sends maximal ideals to maximal ideals. ϕ Proposition 1.3.4. The functor that sends a morphism of proper normal curves YL −→ XK to the induced injection of fields ϕ∗ : K,→ L is an anti-equivalence of categories proper normal curves with finite ∼ finite field extensions −→ . surjections YL → XK K,→ L over k

We will now drop the subscript on XK denoting the field unless it becomes convenient to have it. Take a morphism of proper normal curves ϕ : Y → X with affine covers X = X+ ∪ X− and Y = Y + ∪ Y −. Then there are induced morphisms of affine curves ϕ+ : Y + → X+ and ϕ− : Y − → X−.

Definition. We say an open set U ⊆ X in a proper normal curve is affine if OX (U) is a finitely generated k-algebra.

15 1.4 Finite Branched Covers 1 Fundamental Groups of Algebraic Curves

Lemma 1.3.5. Let X be a proper normal curve. Then a subset U ⊆ X is open if and only if U ∼= Spec A for some finitely generated k-algebra A. Proposition 1.3.6. There is an equivalence of categories ∼ affine open subsets of {normal affine curves over k} −→ . proper normal curves over k

Proof. (Sketch) Given a normal affine curve X, set X+ = X and embed X into X0 = X+ ∪X− as shown above. This yields a proper normal curve, and to show the bijection is an equivalence of categories, one need only check that a morphism of affine curves ϕ : Y → X 0 0 extends uniquely to ϕe : Y → X . Definition. We say a morphism of proper normal curves ϕ : Y → X is finite if for any affine U ⊆ X, (1) ϕ−1(U) is affine in Y .

(2) ϕ∗O(U) is a finitely generated O(U)-module. Notice that for any affine set U ⊆ X and finite morphism ϕ : Y → X, the restriction −1 ϕ|ϕ−1(U) : ϕ (U) → U is a finite morphism of affine curves in the sense of Section 1.2. Thus Lemma 1.2.5 yields: Corollary 1.3.7. Any finite morphism is surjective. Conversely, we have:

Proposition 1.3.8. Let ϕ : YL → XK be a surjective morphism of proper normal curves. Then ϕ is finite. Proof. Let U ⊆ X be affine. Then ϕ−1(U) = {R | R is a DVR of L containing O(U)}. For any R ∈ ϕ−1(U), the integral closure B of O(U) is contained in R since R is itself integrally closed. But by Theorem 1.2.6, this B is precisely the integral closure of the normalization V of U in L. Hence V = ϕ−1(U) so (1) is satisfied. It also follows from the identification of −1 V as the normalization of U in L that ϕ∗O(U) = O(ϕ (U)) = O(V ) is finitely generated over O(U). Corollary 1.3.9. For any proper normal curve X, the assignment Y 7→ k(Y ) induces an anti-equivalence of categories proper normal curves with finite ∼ finitely generated field extensions k(X) ,→ L −→ . morphisms Y → X with tr degk L = 1

1.4 Finite Branched Covers

In this section we describe the analogue of branched covers in the algebraic setting. We begin with the case of aﬃne curves and later generalize to proper normal curves. Fix a morphism ϕ : Y → X of integral aﬃne curves.

16 1.4 Finite Branched Covers 1 Fundamental Groups of Algebraic Curves

Definition. We say ϕ : Y → X is separable if ϕ∗ : k(X) ,→ k(Y ) is a separable field extension. If ϕ is finite and separable, then we say ϕ is étaleat a closed point P ∈ X if O(Y )/P O(Y ) is a finite étalealgebra over the residue field κ(P ) = O(X)/P . For an open set U ⊆ X, ϕ is étaleover U if it is étaleat every point P ∈ U. Further suppose X and Y are normal affine curves. Then O(X) and O(Y ) are Dedekind domains, so P O(Y ) can be written

r Y ei P O(Y ) = Qi i=1

for distinct closed points Qi ∈ Y and integers ei. By the Chinese remainder theorem,

r ∼ Y ei O(Y )/P O(Y ) = (O(Y )/Qi ). i=1

Definition. The integer ei is called the ramification index of ϕ at Qi (over P ). If ei > 1 for any Qi, we say ϕ is ramified at P (or P is a branch point of ϕ). Proposition 1.4.1. For a morphism of normal affine curves ϕ : Y → X and a point P ∈ X Qr ei such that P O(Y ) = i=1 Qi , the following are equivalent: (1) ϕ is étaleat P .

(2) ei = 1 for each 1 ≤ i ≤ r and each residue ﬁeld κ(Qi) = O(Y )/Qi is separable over κ(P ).

(3) For each Qi over P , κ(Qi)/κ(P ) is separable and mP OY,Qi = mQi , where mP ⊂ OX,P

and mQi ⊂ OY,Qi are the maximal ideals in the given local rings. ∼ Note that O(Y )/P O(Y ) = (ϕ∗OY )P ⊗OX,P κ(P ). We can view the set {Qi} as the geometric fibre ϕ−1(P ), and under this identification O(Y )/P O(Y ) acts as the space of regular functions on ϕ−1(P ). Example 1.4.2. Let k = and X = 1 . Then the space of regular functions at the points C AC 0 and 2 is given by ∼ C[t]/t(t − 2) = C × C. Example 1.4.3. Let k = C and define the map

n n n ϕn : AC → AC, x 7→ x .

This induces the ring extension C[tn] ,→ C[t]. Here, the maximal ideals are (tn − a) for a ∈ C, and we have the following ramiﬁcation behavior: ( √ Qn−1 i n ∼ n n ∼ i=0 (C[t]/(t − ζn a)) = C , a 6= 0 C[t]/(t − a) = C[t]/(tn), a = 0.

Thus ϕn is ´etaleat each a 6= 0 but not at a = 0.

17 1.4 Finite Branched Covers 1 Fundamental Groups of Algebraic Curves

ψ ϕ Lemma 1.4.4. If Z −→ Y −→ X are finite separable morphisms of affine curves and P ∈ X is a closed point, then (1) If ϕ is étaleat P and ψ is étaleover ϕ−1(P ), then ψ ◦ ϕ is étaleat P .

(2) If X,Y and Z are normal affine curves, then the converse holds. Proof. Let X = Spec A, Y = Spec B and Z = Spec Z for k-algebras A, B and C. Then we ϕ∗ ψ∗ have ring extensions A −→ B −→ C as κ(P )-algebras. By the above comments, ∼ C/P C = C ⊗A κ(P ) ∼ = C ⊗B (B ⊗A κ(P )) ! ∼ Y = C ⊗B κ(Q) Q→P Y ∼= κ(R). R→P Thus the composition is étaleat P if each morphism is. Now suppose the curves are all Pr normal. Then the ramification indices satisfy i=1 eifi = [Frac(B) : Frac(A)], where fi = [κ(Qi): κ(P )] and likewise for the ring extensions C/B and C/A. It follows easily that if one of ϕ, ψ is ramified, then so is the composition. Proposition 1.4.5. Let ϕ : Y → X be a finite separable morphism of affine curves. Then there is a nonempty open set U ⊆ X such that ϕ is étaleover U. In particular, ϕ has finitely many branch points. Proof. Let X = Spec A and Y = Spec B. By hypothesis, B is a finitely generated A-module, so write B = A[f1, . . . , fr] for integral elements f1, . . . , fr. Then there is a tower of ring extensions A ⊆ A[f1] ⊆ A[f1, f2] ⊆ · · · ⊆ A[f1, . . . , fr−1] ⊆ B. This determines a sequence of morphisms

Y → Xr−1 → · · · → X2 → X1 → X

whose composition is ϕ, where Xi = Spec A[f1, . . . , fi] for 1 ≤ i ≤ r − 1. By induction and using Lemma 1.4.4, we may reduce to the case Y = X1 → X, i.e. B = A[f] for some integral f = f1 ∈ B. Write B = A[t]/(F (t)) where F ∈ A[t] is the minimal polynomial over A of f. Since ϕ is separable, k(Y )/k(X) is a separable extension of ﬁelds and thus 0 0 (F ,F ) = 1 in k(X)[t]. Thus there exist G1,G2 ∈ k(X)[t] such that G1F + G2F = 1. Cancelling denominators, there exists some g ∈ A such that H1 = G1g and H2 = G2g in A[t], and so we have 0 H1F + H2F = g in A[t]. Let U = D(g) = {P ∈ X | g(P ) 6= 0} which is a nonempty open subset of X. We claim ϕ is ´etaleover U. For any P ∈ U, take the reduction of the above equation in κ(P )[t] to see that

0 H1F + H2F =g ¯ 6= 0 in κ(P )[t].

18 1.4 Finite Branched Covers 1 Fundamental Groups of Algebraic Curves

0 Thus for all α ∈ A, F (α) = 0 implies F (α) 6= 0, or in other words, F has no multiple roots. Therefore we can write s ∼ ∼ Y B/P B = κ(P )[t]/(F ) = κ(P )[t]/(F i) i=1 where F 1,..., F s are the irreducible factors of F in κ(P )[t]. The work above shows that each κ(P )[t]/(F i) is separable, so B/P B is a finite étale κ(P )-algebra. The result follows. In light of Proposition 1.4.5, the following definitions make sense: Definition. We call a morphism ϕ : Y → X of integral affine curves a finite branched cover if it is finite and separable. Further, we call ϕ a (finite) Galois branched cover if the field extension k(X) ,→ k(Y ) is Galois. If, in addition to ϕ being a finite Galois cover, X and Y are normal curves, then Gal(k(Y )/k(X)) acts transitively on the fibres ϕ−1(P ) of ϕ, just as in the case of a Ga- lois topological cover. Proposition 1.4.6. If ϕ : Y → X is a Galois branched cover over a perfect field k and X

and Y are normal affine curves, then ϕ is étaleat P ∈ X if and only if IQi = {1} for all −1 Qi ∈ ϕ (P ), where IQi is the inertia subgroup of Gal(k(Y )/k(X)). −1 Proof. Recall the definitions of the decomposition and inertia groups for a point Qi ∈ ϕ (P ):

DQi = {σ ∈ Gal(k(Y )/k(X)) | σ(Qi) = Qi}

IQi = {σ ∈ Gal(k(Y )/k(X)) | σ acts trivially on κ(Qi)}.

IQ Since k is perfect, we know from algebraic number theory that ei = [κ(Qi): κ(Qi) i ], where IQ κ(Qi) i is the subﬁeld of κ(Qi) ﬁxed by the inertia group, but this index is precisely the order

of IQi . Hence ei = 1 if and only IQi = {1}, so the result follows by Proposition 1.4.1. For a normal affine curve X and a tower of fields (finite over k(X))

k(X) ⊂ K1 ⊂ K2 ⊂ · · · S∞ set L = j=1 Kj and let Aj be the integral closure of O(X) in Kj for each j ≥ 1. From this we get a sequence of morphisms of curves

· · · → X2 → X1 → X.

Fix P ∈ X. Choose maximal ideals {Pj}j≥1 with Pj ∈ Aj such that Pj ∩ O(X) = P and Pj+1 ∩ Aj = Pj for all j ≥ 1. If Ij is the inertia subgroup of Pj in Gj := Gal(Ki/k(X)), then the Ij form an inverse system with the natural maps coming from each quotient of Galois groups. Deﬁne the inertia subgroup of P to be the inverse limit

IP := lim Ij. ←−

Since each Ij is a normal subgroup of Gj, it follows that IP is a closed subgroup of the proﬁnite group G := Gal(L/k(X)). Note that IP depends on the choices of P1,P2,... but a diﬀerent choice of points lying over P yields a conjugate subgroup.

19 1.4 Finite Branched Covers 1 Fundamental Groups of Algebraic Curves

Corollary 1.4.7. Let ϕ : Y → X be a Galois branched cover of normal affine curves over a perfect field and assume k(Y ) ⊆ L where L is as above. Then ϕ is étaleat P ∈ X if and only if the image of IP in Gal(k(Y )/k(X)) = G/ Gal(L/k(Y )) is trivial.

Proof. Pick ` ≥ 1 so that k(Y ) ⊆ K`. Let Q = P` ∩ O(Y ). Then the image of IP in Gal(k(Y )/k(X)) is precisely the inertia subgroup IQ ≤ Gal(k(Y )/k(X)). Now apply Propo- sition 1.4.6. We now extend the notion of finite branched covers to proper normal curves. Suppose ϕ : Y → X is a finite separable morphism of proper normal curves (as defined in Section 1.3).

Definition. We say ϕ : Y → X is étaleat P ∈ X if there exists an affine open neighborhood −1 U ⊆ X of P such that ϕ|ϕ−1(U) : ϕ (U) → U is étaleat P as a morphism of affine curves. As in the affine case, call ϕ étaleover a subset S ⊆ X if it is étaleat every point P ∈ S, and simply étale if it is étaleat every P ∈ X.

Note that by Lemma 1.4.1, this definition does not depend on the neighborhood U of P chosen. The last few results for affine curves generalize to proper normal curves as an immediate consequence of the definition:

Theorem 1.4.8. Let ϕ : Y → X be a morphism of proper normal curves. Then

(1) There exists a nonempty open set U ⊆ X such that ϕ is ´etaleover U.

(2) If ϕ is a Galois branched cover and k is perfect, then ϕ is ´etaleat P ∈ X if and only if the inertia group is trivial for each point Qi over P . S∞ (3) If ϕ is a Galois branched cover, k is perfect L = j=1 Kj and k(Y ) ⊆ L, then ϕ is ´etaleat P ∈ X if and only if IP maps trivially into Gal(k(Y )/k(X)).

Example 1.4.9. For a proper normal curve X over C, X(C) has the structure of a compact Riemann surface such that the cover X(C) → P1(C) corresponding to X → P1 is proper and holomorphic.

Proposition 1.4.10. For a proper normal curve X over C, the following categories are anti-equivalent:

 compact connected Riemann  proper normal curves Y ﬁnite extensions   ←→ ←→ surfaces Y with proper . with ﬁnite morphism Y → X k(X) ,→ L holomorphic maps Y → X(C) Example 1.4.11. Consider the squaring map

1 1 2 ρ2 : PR −→ PR, t 7→ t corresponding to the ring extension R[t2] ,→ R[t]. The maximal ideals of R[t] are of the form

(1)( t − a) for a ∈ R (2)( t2 + bt + c) for b2 − 4c < 0.

20 1.4 Finite Branched Covers 1 Fundamental Groups of Algebraic Curves

For (1), we have the following ramification behavior:  × , a > 0 R R 2 ∼ 2 R[t]/(t − a) = R[t]/(t ), a = 0  C, a < 0 while for (2), we get 4 2 ∼ R[t]/(t + bt + c) = C × C. 2 Note that κ(t − a) = R, while κ(t + bt + c) = C so we see that ρ2 is étaleoff of {0, ∞}. (To see that ρ2 is not étaleat ∞, repeat the above argument replacing t with 1/t.) Example 1.4.12. Let E be an elliptic curve over an algebraically closed field k of characteristic char k = p. Take m ∈ Z such that p - m and consider the multiplication-by-m isogeny: [m]: E −→ E,P 7→ mP. We know that [m] is separable when p - m, and moreover that E[m] = (Z/mZ)2 and thus #E[m] = m2. For each P ∈ E, we have

m2 m2 Y Y O(E)/[m]P ∼= κ(P ) ∼= k i=1 i=1

and hence [m] is étale.In general, any degree-m isogeny ϕ : E1 → E2 of elliptic curves over an algebraically closed field of characteristic not dividing m is separable and therefore étale by the same argument. Example 1.4.13. Let k be any field of characteristic p > 0 and consider the field L = k(x)[y]/(yp − (f(x)g(x))p−1y − f(x)). Note that yp − (f(x)g(x))p−1y − f(x) is irreducible, e.g. by Eisenstein’s criterion, so L is indeed a field. Let B be the integral closure of k[x] in L and let X be the proper normal curve 1 corresponding (via Proposition 1.3.4) to L. Then there is a morphism of curves ϕ : X → Pk coming from k(x) ,→ L. For a ∈ k, consider the maximal ideal (x − a) ∈ B. We consider three cases: (1) If f(a) = 0, then (x − a) = (yp) in B, so we have B/(x − a)B ∼= B/(yp)B. Thus ϕ is not étaleat (x − a). (2) If f(a) 6= 0 but g(a) = 0, (x − a) = (yp − f(a)) so ( B/(y − f(a)1/p)p, f(a)1/p ∈ k B/(x − a)B ∼= B/(yp − f(a)), otherwise.

In the first case, B/(y − f(a)1/p)p is not a field extension of κ(x − a) = k, while in the second, B/(yp − f(a)) is an inseparable field extension of k. Thus ϕ is not étale at any of these (x − a).

21 1.5 The Fundamental Group of Curves 1 Fundamental Groups of Algebraic Curves

(3) Finally, if f(a) 6= 0 and g(a) 6= 0, then (x − a) = (yp − (f(a)g(a))p−1y − f(a)). If there is a c ∈ k such that f(a) = cp − c, then

p−1 p−1 Y Y (x − a) = (x − (c + if(a)g(a))) =⇒ B/(x − a)B ∼= k. i=0 i=0 Thus ϕ is étaleat (x − a) in this case. On the other hand, if there does not exist such a c, then (x − a) remains a prime ideal in B and thus B/(x − a) is a finite separable extension of k. Hence ϕ is étaleat the point (x − a) if and only if f(a) 6= 0 and g(a) 6= 0.

1.5 The Fundamental Group of Curves

Let k be a perfect field, X an integral proper normal curve over k, K = k(X) its function field and fix a separable closure Ks of K. For a nonempty open subset U ⊆ X, define KU to be the compositum of all finite subextensions Ks ⊇ L ⊇ K corresponding to finite morphisms of proper normal curves Y → X which are étaleover U.

Proposition 1.5.1. For any nonempty open subset U ⊆ X, KU /K is a Galois extension and each ﬁnite subextension corresponds to a cover of X which is ´etaleover U.

Proof. To show KU /K is a Galois extension, it’s enough to see that KU is stable under the Gal(Ks/K)-action, but this follows from the definition of the field KU . To prove the second statement, it’s enough to show for a finite subextension L/K of KU /K that: (i) If L comes from a cover of X which is étaleover U, then so does any subfield L ⊇ L0 ⊇ K.

(ii) If M/K is any other finite subextension of KU /K coming from a cover which is étale over U, then LM/K also comes from a cover étaleover U.

Then the result will follow since KU is the compositum of a tower of finite extensions L1 ⊆ L1L2 ⊆ L1L2L3 ⊆ · · · ϕ For (i), let L/K correspond to a cover Y −→ X which is étaleover U. Then for any ϕ0 L ⊇ L0 ⊇ K, we get a composition of covers Y → Y 0 −→ X. Applying Lemma 1.4.4, 0 we get that ϕ is étaleover U. For (ii), fix P ∈ U and consider the inertia group IP ≤ Gal(Ks/K) as defined in Section 1.4. Then by (3) of Theorem 1.4.8, IP is trivial in both Gal(L/K) and Gal(M/K). Since Gal(L/K) = Gal(Ks/K)/ Gal(Ks/L), this means IP ⊆ Gal(Ks/L); likewise IP ⊆ Gal(Ks/M). Hence IP ⊆ Gal(Ks/LM), which means IP is trivial in Gal(LM/K), so by Theorem 1.4.8 again, the curve corresponding to LM/K is étaleat P . Definition. For a proper normal curve X over a perfect field k and a nonempty open subset U ⊆ X, we define the algebraic fundamental group over U to be

π1(U) := Gal(KU /K).

22 1.5 The Fundamental Group of Curves 1 Fundamental Groups of Algebraic Curves

Note that π1(U) is a proﬁnite group which depends on the choice of separable closure Ks. Here, this choice of Ks acts as the “basepoint” in analogy with the topological case (see Section 0.1).

Definition. Extending the notion of curves from the last section, we define a proper normal curve to be a finite disjoint union of integral proper normal curves. ` The ring of rational functions on a proper normal curve X = Xi is defined to be the L direct sum of the function fields of each component, O(X) = k(Xi); this is naturally a finite dimensional algebra over any of the components k(Xi). In general, we say a morphism Y → X, with X integral, is separable if k(Y ) is étaleover k(X).

Theorem 1.5.2. Suppose X is an integral proper normal curve over a perfect ﬁeld k and U ⊆ X is a nonempty open subset. Then there is an equivalence of categories

ϕ covers of proper normal curves Y −→ X ∼ −→{finite continuous left π (U)-sets}. where ϕ is finite, separable, étaleover U 1

Proof. Propositions 1.3.4 and 1.5.1 extend to direct sums of fields, so for a cover Y → X ` L which is a proper normal curve Y = Yi, let A = Li be the finite étale K-algebra corresponding to these Yi → X. Finally, by Corollary 0.2.4, Hom(A, Ks) is a finite continuous Gal(Ks/K)-set. Now, for an integral normal affine curve U over k, we know U is an affine open subset of − an integral proper normal curve X = U ∪ U . Thus π1(U) is defined. Corollary 1.5.3. For any integral normal affine curve U over a perfect field k, there is an equivalence of categories normal affine curves with ∼ −→{finite continuous left π (U)-sets}. finite étalecovers V → U 1

Proof. Any finite étalecover of normal affine curves V → U extends by Proposition 1.3.6 to a cover of proper normal curves Y L → XK which is in turn finite by Proposition 1.3.8. Finally, Theorem 1.5.2 gives the desired correspondence. We now discuss how to capture the notion of a “universal cover” as we have in the topological case (Section 0.1). Let U ⊆ X be a nonempty open subset of an integral proper normal curve and let A = O(U) be its ring of rational functions. Take Ae be the integral closure of A in KU . Then for a finite étalecover V → U with fraction field K(V ), we have O(V ) = Ae ∩ K(V ).

23 1.5 The Fundamental Group of Curves 1 Fundamental Groups of Algebraic Curves

KU Ae

K(V ) O(V )

K A

Note that for any maximal ideal m ⊆ Ae, m ∩ O(V ) is a closed point in V . Although Ae is not itself a finitely generated ring over A, it is a compositum of such rings, i.e. the O(V ) are finite over A. Define Ue = MaxSpec Ae and equip this with the inverse limit topology with respect to the inverse system V → V 0 → U. We define a sheaf of rings O on U stalkwise Ue e by setting O = A for any Q ∈ U. Then for any open set V ⊆ U, we put U,e Qe eQe e e e e \ O (V ) = O . Ue e U,e Qe Qe∈Ue Definition. Ue is called the pro-étalecover of U. Lemma 1.5.4. (U, O ) is a locally ringed space. e Ue Remark. Note that Ue is not an object in the category of proper normal curves (or even in the category of schemes, as we shall see!) However, constructing a “universal cover” is useful to illustrate the analogy with the topological case. For example, we have the following theorem which will be proven in Section 2.3. Theorem 1.5.5. Let X be an integral proper normal curve over C and U ⊆ X a nonempty top\ open subset. Then π1(U) is isomorphic to π1 (U(C)), the profinite completion of the topological fundamental group of the Riemann surface U(C). In particular, π1(U) is the profinite completion of

ha1, b1, . . . , ag, bg, γ1, . . . , γn | [a1, b1] ··· [ag, bg]γ1 ··· γn = 1i, where n is the number of points in X r U and g is the genus of the Riemann surface U(C).

Let X/C and U be as above. If G is any finite quotient of π1(U), then G corresponds to a finite Galois branched cover Y → X. Here, the image of each generator γi ∈ π1(U) in G is a cyclic generator of the stabilizer of a point Qi → Pi, where Pi ∈ X r U is the point corresponding to γi. Let X be an integral proper normal curve over an arbitrary (perfect) field k. If L/k is a finite extension, then k(X) ⊗k L is a finite dimensional algebra over L(t) which is not, in general, a field. Assume k(X) ⊗k L is a finite direct product of fields L1,...,Ln, each of which is a finitely generated extension of L(t) of transcendence degree 1 over L. Then each Li corresponds to an integral proper normal curve Xi/L by Proposition 1.3.4. Define the ` base change XL = Xi. This comes equipped with a natural morphism of proper normal curves XL → X. We will prove:

24 1.6 The Outer Galois Action 1 Fundamental Groups of Algebraic Curves

Theorem 1.5.6. If k is an algebraically closed field of characteristic 0, L/k is a finite extension, X is an integral proper normal curve over k and U ⊆ X is open, then the base change functor X 7→ XL induces an equivalence of categories finite branched Galois covers Y → X ∼ finite branched Galois covers Y → X −→ L L . étaleover U étaleover UL ∼ Corollary 1.5.7. For U ⊆ X as above, there is an isomorphism π1(UL) = π1(U) for all finite extensions L/k. Theorem 1.5.5 generalizes to curves over an algebraically closed field k of characteristic 0 in the following sense. Corollary 1.5.8. If k is algebraically closed of characteristic 0 and U ⊆ X is a nonempty open subset of an integral proper normal curve X over k, then π1(U) is isomorphic to the profinite completion of a group with presentation

ha1, b1, . . . , ag, bg, γ1, . . . , γn | [a1, b1] ··· [ag, bg]γ1 ··· γn = 1i,

where g = g(X) is the genus of X and n is the number of points in X r U. 1 Example 1.5.9. If k is an algebraically closed field of characteristic 0 and X → Pk is a finite étalecover, the Riemann-Hurwitz inequality says that 2g(X) − 2 = n(0 − 2) + 0

1 but this is only possible if g(X) = 0 and n = 1. Thus X → Pk is a birational isomorphism, but since X is complete, it must be a regular isomorphism. Hence there are no nontrivial 1 1 extensions L of k(Pk) = k which proves π1(Pk) = 1. 1 Example 1.5.10. A similar argument shows π1(Ak) = 1 when k is algebraically closed and char k = 0. Both this and the previous result also follow easily from Theorem 1.5.5 and Corollary 1.5.8. 1 ∼ Example 1.5.11. Theorem 1.5.5 and Corollary 1.5.8 also imply that π1(Pk r {0, ∞}) = Zb, the profinite completion of the integers. Using Theorem 1.5.2, we get a finite étalecover 1 Xn → Pk r {0, ∞} for each n ≥ 1 having Galois group Z/nZ. 1 Example 1.5.12. By Theorem 1.5.5 and Corollary 1.5.8, π1(Pk r {0, 1, ∞}) is the free profinite group on two generators.

1.6 The Outer Galois Action

Let X be an integral proper normal curve over a perfect field k which is not necessarily algebraically closed. Let K = k(X). Fixing an algebraic closure k¯, we assume X is geometrically ¯ ¯ integral, that is, K⊗k k is a field. Then Ks, the separable closure of K containing k, is defined. ¯ For any nonempty open U ⊆ X, the base change Uk¯ is integral and we have Kk ⊆ KU , where the compositum is taken in KU . By construction, for any finite extension L/k, the cover ∼ ¯ ∼ ¯ XL → X is finite étalewith Gal(k(XL)/K) = Gal(L/k). Thus Gal(Kk/K) = Gal(k/k) so the latter may be thought of as a quotient of π1(U) for any open U ⊆ X.

25 1.6 The Outer Galois Action 1 Fundamental Groups of Algebraic Curves

π1(U) Kk¯

Gal(k/k¯ )

Proposition 1.6.1. For X a geometrically integral, proper normal curve over a perfect field k and for any nonempty open subset U ⊆ X, there is a short exact sequence of profinite groups ¯ 1 → π1(Uk¯) → π1(U) → Gal(k/k) → 1. ¯ Proof. The surjection π1(U) → Gal(k/k) was described above, so it remains to identify π1(Uk¯) with the kernel of this map. Suppose G is a finite quotient of π1(Uk¯). Then there is ¯ a corresponding finite extension K0/Kk:

KUk¯

π1(Uk¯) K0

Kk¯

This also determies a finite branched cover Y0 → Xk¯ which is étaleover Uk¯ ⊆ Xk¯. Let ¯ ¯ f ∈ Kk[t] be the minimal polynomial of K0/Kk. Then we can find a finite extension L/k such that KL/k contains all the coefficients of f. Let L0/KL be the finite extension with Gal(L0/KL) = G.

K0 G

π1(U) L0 Kk¯ G KL k¯

K L

26 1.6 The Outer Galois Action 1 Fundamental Groups of Algebraic Curves

¯ By construction, L0k = K0 and the tower L0 ⊇ KL ⊇ L determines a composition of covers Y → XL → X which is étaleover U, and such that Yk¯ = Y0. This implies L0 ⊆ KU and then ¯ K0 = L0k ⊆ KU . This shows that we can identify any such G = Gal(L0/KL) with a finite ¯ quotient of ker(π1(U) → Gal(Kk/K)). Varying the finite quotients G of π1(Uk¯), we conclude ¯ ¯ ¯ that KUk¯ ⊆ KU and hence there is a surjection Gal(KU /Kk) → Gal(KUk¯ /Kk) = π1(Uk) which is bijective on finite quotients. Taking inverse limits – which is a left exact functor – ¯ gives π1(Uk¯) = ker(π1(U) → Gal(Kk/K)) as required. ¯ Corollary 1.6.2. There is a continuous homomorphism pU : Gal(k/k) → Out(π1(Uk¯) for any nonempty open set U ⊆ X. Proof. Any short exact sequence of groups 1 → N → G → P → 1 defines a homomorphism G → Aut(N) whose restriction to the subgroup N ≤ G has image lying in Inn(N). By definition, Out(N) = Aut(N)/ Inn(N) so we get a map P → Out(N). Applying this to the short exact sequence from Proposition 1.6.1 gives the result. Definition. For an open subset U of a geometrically integral proper normal curve X over a

perfect ﬁeld k, the subgroup π1(Uk¯) ≤ π1(U) is called the geometric fundamental group ¯ of U. The map pU : Gal(k/k) → Out(π1(Uk¯) is called the outer Galois action on the geometric fundamental group.

We next study the action of π1(U) on Xe, the pro-étalecover of X defined in Section 1.5. Let Q be a pro-point in X lying over a point P ∈ X. We define D to be the stabilizer e e Qe of Q under the π (U)-action. Then κ(Q) = O /QO = κ(P ), the algebraic closure of e 1 e X,e Qe e X,e Qe the residue field κ(P ) = O /P O . We have a natural surjection D → Gal(κ(Q)/κ(P )) X,P X,P Qe e with kernel denoted I , called the inertia group of Q. We may alternatively view X as Qe e e a profinite Galois cover of Xk¯ with Galois group π1(Uk¯), corresponding to the short exact sequence from Proposition 1.6.1:

π1(Uk¯)

π1(U) Xk¯

Gal(k/k¯ )

Deﬁnition. Let P ∈ X be a closed point such that κ(P ) ∼= k. Then we say P is k-rational. Lemma 1.6.3. Let P ∈ X be a k-rational (closed) point. Then the stabilizer of any pro-point Q ∈ X lying over P in π (U¯) is equal to the inertia group I . e e 1 k Qe Proof. We have a surjection D → Gal(κ(Q)/κ(P )) = Gal(k/k¯ ) by assumption, but this Qe e ¯ is just the restriction of the map π1(U) → Gal(k/k) to the action on Qe. By deﬁnition the kernel of the latter map is I . Qe

27 1.6 The Outer Galois Action 1 Fundamental Groups of Algebraic Curves

Corollary 1.6.4. If U contains a k-rational point of X then the short exact sequence ¯ 1 → π1(Uk¯) → π1(U) → Gal(k/k) → 1

is split.

Proof. If Qe is a pro-point over some k-rational point P ∈ U, then the stabilizer of Qe in π1(Uk¯) is trivial. Therefore Lemma 1.6.3 says that I = 1, but by deﬁnition of the inertia group this Qe implies D ∼ Gal(k/k¯ ). Thus there is a map Gal(k/k¯ ) → π (U) mapping isomorphically Qe = 1 onto D ⊆ π (U) which deﬁnes the required splitting. Qe 1

1 1 Example 1.6.5. Let X = Pk and consider the open set U = Pkr{0, ∞}. By Example 1.5.11, ∼ ¯ π1(U) = Zb, a free abelian profinite group. Hence Gal(k/k) acts on π1(Uk¯) directly, not just by ∼ outer automorphisms. Let n ≥ 1. Then the quotient π1(Uk¯)/nπ1(Uk¯) = Z/nZ corresponds ¯ ¯ n to Gal(Ln/k(t)) where Ln/k(t) is the field extension defined by x − t. Consider the short exact sequence of profinite groups ¯ ¯ 1 → Gal(Ln/k(t)) → Gal(Ln/k(t)) → Gal(k/k) → 1. ¯ Identifying π1(Uk¯)/nπ1(Uk¯) = Z/nZ, we see that this sequence√ defines√ the Gal(k/k)-action ¯ n n ¯ on π1(Uk¯)/nπ1(Uk¯). Explicitly, Gal(Ln/k(t)) is generated by t 7→ ζn t, where ζn ∈ k is a primitive nth root of unity. Then for σ ∈ Gal(k/k¯ ), the Galois action is given by √ √ √ √ n n n n σ · ( t 7→ ζn t) = t 7→ σ(ζn) t.

Fixing a compatible system of primitive roots of unity (ζn)n∈N, we get isomorphisms ∼ ¯ ¯ cyc ¯ ∼ 1 b = Gal lim Ln/k(t) = Gal(k(t) /k(t)) = π1( k {0, ∞}). Z ←− P r

This deﬁnes a representation Gal(k/k¯ ) → Aut(Zb) = Zb× called the cyclotomic character of Gal(k/k¯ ). Notice that this descends to a character Gal(k/k¯ ) → (Z/nZ)× for each n ≥ 1.

1 One of the most important objects in arithmetic geometry is π1( {0, 1, ∞}). If PQ r U = 1 {0, 1, ∞}, there is an outer Galois action ρ : Gal( / ) → Out(π (U )). We will PQ r Q Q 1 Q prove that this action is faithful, but ﬁrst we need:

Theorem 1.6.6 (Belyi). Suppose X is an integral proper normal curve defined over an 1 algebraic closed field k of characteristic 0. Then there exists a morphism X → Pk which is 1 étaleover Pk r {0, 1, ∞} if and only if X is defined over Q. Proof. ( =⇒ ) follows from Theorem 1.5.6. ( ⇒ = ) If X is defined over , there exists a map π : X → 1 which is étaleover 1 S Q PQ PQ r for some finite set S ⊆ 1 . We first show S consists of -rational points. Since S is finite, PQ Q we can find a point P ∈ S with [κ(P ): Q] = n maximal among the points in S. Let 1 1 f ∈ [t] be the minimal polynomial of κ(P )/ . Define ϕf : → by x 7→ F (x), where Q Q PQ PQ 1 F is the homogenization of f. Then ϕf is defined over and is étale over Sf , where Q PQ r

28 1.6 The Outer Galois Action 1 Fundamental Groups of Algebraic Curves

1 0 0 Sf = {R ∈ | f (R) = 0}. Notice that deg f = n − 1 so for all R ∈ Sf ,[κ(R): ] < n. AQ Q 1 0 The composition ϕf ◦ π : X → is now ´etaleaway from S := ϕf (S) ∪ {∞} ∪ ϕf (Sf ). But PQ now ∞ has degree 1, points in ϕf (S) have degree at most n and points in ϕf (Sf ) have degree 0 at most n − 1, so because ϕf (P ) = 0, there are strictly less points of degree n in S than in S. Repeat this procedure until n = 1, at which time all points in S0 will be Q-rational. Next, suppose S has more than three points. We may assume 0, 1, ∞ ∈ S and take α ∈ S r {0, 1, ∞}. Deﬁne the Belyi function ϕ : 1 −→ 1 PQ PQ x 7−→ xA(x − 1)B

A 0 where A, B ∈ Z r {0} are such that α = A+B . As above, ϕ is ´etaleaway from S = 1 0 0 A−1 B ϕ(S) ∪ {∞} ∪ ϕ(Sϕ) where Sϕ = {R ∈ | ϕ (R) = 0}. Note that ϕ (x) = Ax (x − 1) + AQ BxA(x − 1)B−1 so ϕ0(R) = 0 precisely when R ∈ {0, 1, ∞, α}. This shows that ϕ(S) = S0. Composing with π, we get a map ϕ ◦ π : X → 1 which is ´etaleoutside ϕ(S) and maps PQ {0, 1, ∞} to {0, ∞}. Hence |ϕ(S)| < |S|, so repeating this process reduces to the case when S0 has (at most) three elements.

1 Theorem 1.6.7. The outer Galois representation ρ : Gal( / ) → Out(π1( {0, 1, ∞})) Q Q PQ r is faithful.

ker ρ Proof. Let L = Q be the subfield fixed by the kernel of the outer Galois representation. Then for U = 1 {0, 1, ∞}, Proposition 1.6.1 gives us a short exact sequence of profinite PQ r groups 1 → π (U ) → π (U ) → Gal( /L) → 1. 1 Q 1 L Q By construction, ρ : Gal( /L) → Out(π (U )) has trivial kernel, which means any auto- UL Q 1 Q morphism of π (U ) in π (U ) is given by conjugation by some y ∈ π (U ). That is, for any 1 Q 1 L 1 Q x ∈ π (U ), there exists a y ∈ π (U ) such that for all a ∈ π (U ), we have xax−1 = yay−1. 1 L 1 Q 1 Q This is equivalent to y−1xax−1y = a, i.e. y−1x belongs to C, the centralizer of π (U ) in 1 Q π (U ). Hence π (U ) is generated by π (U ) and C. However, Corollary 1.5.8 says that 1 L 1 L 1 Q π (U ) is a free profinite group on 2 generators, and such a group has trivial center, so we 1 Q must have π (U ) ∩ C = {1}. Thus π (U ) ∼ π (U ) × C as a direct product, or in other 1 Q 1 L = 1 Q words, π (U ) → π (U ) is a continuous retraction of profinite groups. This says that any 1 L 1 Q continuous left π (U )-set comes from a continuous left π (U )-set, but by Theorem 1.5.6, 1 Q 1 L any finite cover of U is obtained by base change from some finite cover of U . Finally, Q L Belyi’s theorem says that any such cover defined over Q can be defined over L, but we will demonstrate that this is impossible unless L = Q, in which case ker ρ is trivial. If L 6= Q, take x ∈ Q r L. Let E0 be an elliptic curve over Q with j-invariant j(E0) = x. If E0 can be obtained by base change from some genus 1 curve X defined over L, as Belyi’s theorem indicates, then X has a Jacobian E/L which is an elliptic curve. There exists an 2 2 embedding φ : E → PL sending the distinguished L-point of E to [0, 1, 0] ∈ PL. Moreover, it is known that j(E) ∈ L and this j-invariant is preserved by any Q-isomorphism. Since E0 is the base change of X, we have E0 ∼ X ,→ E, but this implies j(E) = j(E0) = x 6∈ L, a = Q contradiction. Hence L = Q, so the outer Galois representation of U is faithful.

29 1.7 The Inverse Galois Problem 1 Fundamental Groups of Algebraic Curves

Example 1.6.8. One can similarly prove that for any elliptic curve E over Q (or any number ﬁeld K), the outer Galois representation ρ : Gal(Q/Q) → π1(E r {0}) is faithful.

1.7 The Inverse Galois Problem

¯ For a ﬁeld k, let Gk = Gal(k/k) be the absolute Galois group. We showed in the previous section (Theorem 1.6.7) that there is a faithful representation G ,→ Out(π ( 1 {0, 1, ∞})), Q 1 PQ r but this group of automorphisms is inordinately complicated so we don’t have much under-

stand of GQ. A related question we might answer is: what are the finite quotients of GQ. That is, Question (Inverse Galois Problem). Which groups arise as Galois groups Gal(L/Q) for some finite Galois extension L/Q? Definition. If L/Q has Galois group G, we say L is a G-extension of Q.

Example 1.7.1. Let Cn = Z/nZ be a cyclic group of order n. We know that the cyclotomic ∼ extension Q(ζp) has Galois group Gal(Q(ζp)/Q) = Z/(p−1)Z, so by group theory, if n | p−1, H this Galois group has a subgroup H of order (p−1)/n. Hence the subextension Q(ζp) /Q has H group Gal(Q(ζp) /Z) = Z/nZ. Thus the inverse Galois problem for cyclic groups reduces to finding a prime p such that p ≡ 1 (mod n), but by Dirichlet’s theorem, there are infinitely many such primes p. Thus all cyclic groups arise as Galois groups over Q. Using the theory of cyclotomic extensions, this can be generalized to give a positive solution to the inverse Galois problem for all finite abelian groups. Combined with the Kronecker-Weber theorem (every finite abelian extension of Q lies in some cyclotomic extension Q(ζm)/Q), the conclusion of Example 1.7.1 gives a complete description of abelian extensions of Q, namely that the maximal abelian extension of the ab cyc S ab rationals Q is equal to the compositum Q := m≥1 Q(ζm). The study of Gal(Q /Q) and its subgroups is the content of global class field theory. If a group G arises as a Galois group over Q, we may then ask for all G-extensions of Q. This may be a much harder question to answer. Example 1.7.2. One can show that the splitting field of f(x) = x3 + x2 − 2x − 1 defines a C3-extension of Q. How do we determine all C3-extensions? Well notice that C3 embeds as a subgroup of P GL ( ) = Aut( 1 ) via 2 Q PQ 1 C3 = hσi −→ Aut(P ) 1 σ 7−→ x 7→ . 1 − x 1 1 2 Then P /C3 is birationally equivalent to P and the rational expression T = x + σx + σ x = x3 − 3x − 1 is an invariant under the C -action. Then the fixed subfield (x)C3 = (T ) x2 − x 3 Q Q corresponds to a cover 1 1 ∼ 1 P −→ P /C3 = P x3 − 3x − 1 x 7−→ T = . x2 − x

30 1.7 The Inverse Galois Problem 1 Fundamental Groups of Algebraic Curves

∼ By construction, Gal(Q(x)/Q(T )) = C3 and this extension has minimal polynomial f(x, T ) = x3 − T x2 + (T − 3)x + 1.

Notice that the discriminant of f is (T 2 −3T −9)2, so for any t ∈ Q such that t2 −3t−9 6= 0, ∼ f(x, t) becomes an irreducible polynomial over Q with Galois group Gal(f(x, t)) = C3. In fact, one can show that f(x, T ) parametrizes all C3-extensions of Q.

Definition. We say a polynomial f(x, T ), with T = (T1,...,Tn) an n-tuple of indetermi- nates, is a generic polynomial for a group G over a field K if for all extensions F/K, ∼ (1) The splitting field Ff /F (T ) has group Gal(Ff /F (T )) = G. (2) Every G-extension L/F is obtained as the splitting field of f(x, t) for some t = n (t1, . . . , tn) ∈ F . A polynomial f(x, T ) that satisfies these conditions for some fixed F is called a versal family for G-extensions of F . Thus a generic polynomial for K is one that is a versal family for all extensions of K.

Example 1.7.3. We saw in Example 1.7.2 that f(x, T ) = x3 − T x2 + (T − 3)x + 1 is a versal family for C3-extensions of Q. One can show (see Theorem 2.2.1 of Jensen, Ledet and Yui’s Generic Polynomials: Constructive Aspects of the Inverse Galois Problem) that in fact this f(x, T ) is a generic polynomial for C3-extensions of Q.

Example 1.7.4. For C8-extensions of Q, there is a versal family for Q, but there is no generic polynomial for all extensions of Q.

Remark. Why could we ﬁnd a one-parameter generic polynomial for C3-extensions? It is a fact that if there exists a one-parameter generic polynomial, i.e. f(x, T1) = f(x, T ), for G-extensions of K, then G embeds as a subgroup of P GL2(K).

Example 1.7.5. By the Remark, the fact that C4 does not embed as a subgroup of P GL2(Q) implies that there does not exist a 1-parameter generic polynomial for C4-extensions of Q. However, there does exist an embedding C4 ,→ P GL2(Q(i)), so there may be a 1-parameter generic polynomial over Q(i).

Let G be a ﬁnite group. By Cayley’s theorem, G embeds as a subgroup of some Sn, inducing a G-action on the set {x , . . . , x }. Deﬁne E = (x , . . . , x )G and let π : n → 1 n Q 1 n AQ An/G be the corresponding covering map.

G Question (Noether’s Problem). For which groups G is the ﬁeld E = Q(x1, . . . , xn) tran- scendental over Q? The inverse Galois problem is known to be a consequence of the so-called regular inverse Galois problem:

Question (Regular Inverse Galois Problem). Which groups G arise as Galois groups of ﬁnite Galois extensions K/Q(t) which are regular over Q?

31 1.7 The Inverse Galois Problem 1 Fundamental Groups of Algebraic Curves

(A regular extension K/Q(t) is one which does not contain a subextension of the form L(t) for L/Q a nontrivial extension.) To see this implication, we need Hilbert’s irreducibility theorem:

Theorem 1.7.6 (Hilbert Irreducibility). Let G be a ﬁnite group acting on {x1, . . . , xn}. If n /G is rational over , then there exist inﬁnitely many points P ∈ n /G with Gal( (Q)/ ) ∼ AQ Q AQ Q Q = G for each Q ∈ π−1(P ).

Example 1.7.7. Let G = Sn be the symmetric group on n symbols. Let F = Q(x1, . . . , xn). Sn Then E = F = Q(σ1, . . . , σn) where σi is the ith elementary symmetric polynomial and the extension F/E has minimal polynomial

n n−1 n f(x, σ) = x − σ1x + ... + (−1) σn for σ = (σ , . . . , σ ). This deﬁnes an S -cover n → n /S . By Hilbert’s irreducibility 1 n n AQ AQ n n n−1 n theorem, f(x, s) = x − s1x + ... + (−1) sn is irreducible for inﬁnitely many tuples n s = (s1, . . . , sn) ∈ Q . It is a fact that

n n−1/2 #{s = (s1, . . . , sn) ∈ Q | 1 ≤ si ≤ N, f(x, s) is irreducible} = O(N log N).

Using this, one can prove that this polynomial f(x, σ) is a generic polynomial for Sn- extensions of Q. Similar proofs using Hilbert’s irreducibility theorem can be used to verify generic polynomials for other classes of G-extensions. A question related to Noether’s Problem and the regular IGP is:

n Question. For which groups G and ﬁelds K is AK /G rational? The answer to this was demonstrated to be no the general case: Lenstra provided coun- terexamples for K = Q and G = C8, and Saltman even showed a counterexample for K = C. The question is even unknown for G = An, n > 5. Here is a strategy to solving the regular Inverse Galois Problem for a given group G. (1) If G has a generating set consisting of n − 1 elements, then G arises as a quotient of the topological fundamental group π ( 1 {P ,...,P }), since this is the free group on 1 PC r 1 n n − 1 elements. Set πtop(n) = π ( 1 {P ,...,P }). In particular, suppose we can ﬁnd 1 PC r 1 n top g1, . . . , gn ∈ G such that G = hg1, . . . , gni, g1 ··· gn = 1 and create a map ϕ : π (n) → G with ϕ(γi) = gi for 1 ≤ i ≤ n. We call (g1, . . . , gn) ∈ G a generating n-tuple for G.

1 top (2) Set π(n) = π1( {P1,...,Pn}) so that by Corollary 1.5.8, π(n) = π\(n). We may PQ r even pick P ,...,P ∈ 1 ( ). 1 n PQ Q (3) Also set Π(n) = π ( 1 {P ,...,P }) so that Proposition 1.6.1 gives us a short exact 1 PQ r 1 n sequence of proﬁnite groups

1 → π(n) → Π(n) → G → 1. Q Given ϕ : π(n) → G as in Step 1, we want to ﬁnd a lift of the following diagram:

32 1.7 The Inverse Galois Problem 1 Fundamental Groups of Algebraic Curves

γi π(n) Π(n)

ϕ ϕb

gi G

In particular, can we extend a continuous, surjective homomorphism ϕ : π(n) → G to a continuous, surjective ϕb : Π(n) → G? In general, consider a lifting problem of the form

N Γ

where N is a normal subgroup of a proﬁnite group Γ and G is any ﬁnite group. The set Homcts(N,G) comes equipped with two actions:

G × Homcts(N,G) −→ Homcts(N,G) (g, ϕ) 7−→ gϕ(−)g−1

and Homcts(N,G) × Γ −→ Homcts(N,G) −1 (ϕ, σ) 7−→ (ϕσ : n 7→ ϕ(σnσ )) which are compatible in the sense that gϕσ = (gϕ)σ.

Lemma 1.7.8. Let S ⊆ Homcts(N,G) be a set of maps which are stable under the G- and Γ-actions and such that G acts freely and transitively on S. Then any ϕ ∈ S extends to a continuous homomorphism ϕb :Γ → G. −1 −1 Proof. For ϕ ∈ S and σ ∈ Γ, ϕσ ∈ S by stability, but ϕσ(n) = ϕ(σnσ ) = gσϕ(n)gσ for some gσ ∈ G, since G acts transitively. Moreover, freeness of the G-action implies this gσ is unique. Deﬁne ϕb(σ) = gσ. We claim that this is the desired extension. First, for σ, τ ∈ Γ, −1 −1 −1 ϕστ (n) = ϕ(στnτ σ ) = ϕσ(τnτ ) = ϕσϕτ (n)

so ϕb is a group homomorphism. Next, for σ ∈ N ⊆ Γ, −1 −1 ϕσ(n) = ϕ(σnσ ) = ϕ(σ)ϕ(n)ϕ(σ)

since ϕ is a homomorphism on N, so by uniqueness we have gσ = ϕ(σ). Thus ϕb extends ϕ as claimed.

33 1.7 The Inverse Galois Problem 1 Fundamental Groups of Algebraic Curves

Now to access Lemma 1.7.8, we want to construct such a set S ⊆ Homcts(π(n),G). First, we need S to be stable under the action of π(n). Note that for ϕ ∈ S, having ϕσ ∈ S −1 for all σ ∈ π(n) is equivalent to having ϕσ(n) = ϕ(σ)ϕ(n)ϕ = (ϕ(σ) · ϕ)(n), under the left action of G on ϕ. Let (g1, . . . , gn) be a generating n-tuple for G. Then we must have gi = ϕ(γi) stable under the conjugacy action of G. In particular, we will specify conjugacy classes C1,..., Cn in G and consider sets

S = {ϕ ∈ Homcts(π(n),G) | ϕ(γi) ∈ Ci, (ϕ(γ1), . . . , ϕ(γn)) is a gen. n-tuple}.

By construction, this S is stable under the π(n)- and G-actions. Now we impose further conditions on the Ci to force (i) transitivity;

(ii) free action;

(iii)Π( n)-stability.

Deﬁnition. Let G be a ﬁnite group. We say a set of conjugacy classes C1,..., Cn in G is a n rigid system if there exists a generating n-tuple (g1, . . . , gn) ∈ G for G with gi ∈ Ci and G acts transitively on the set of all such tuples.

To ensure (i), we pick a rigid system of conjugacy classes in G represented by the ϕ(γi). To guarantee (ii), it is enough to assume that Z(G) = 1 (G has trivial center). Next, set

n Σ = {(g1, . . . , gn) ∈ G | g1 ··· gn = 1 and gi ∈ Ci}

Σ = {(g1, . . . , gn) ∈ Σ | hg1, . . . , gni = G}.

Deﬁnition. If Σ = Σ, we say the Ci are strictly rigid. G acts on Σ and Σ, so supposing Z(G) = 1, it is clear that rigidity is equivalent to |G| = |Σ|. One can even write down formulas relating |G| and |Σ| (see Serre).

Example 1.7.9. Consider G = Sn for n ≥ 3. For 1 ≤ k ≤ n, let kA be the conjugacy (k) class of k-cycles in Sn. Also, let C be the conjugacy class of (1 2 ··· k)(k + 1 ··· n). In particular, C(1) = (n − 1)A. We claim that the system of conjugacy classes {nA, 2A, C(1)} is strictly rigid. By deﬁnition,

3 (1) Σ = {(x, y, z) ∈ Sn | xyz = 1, x ∈ nA, y ∈ 2A, z ∈ C }. Thus determining the size of Σ comes down to determining when xy is an (n − 1)-cycle. (n−k) In general, one can see through elementary calculations that xy ∈ C if y = (y1 y2) with y1 and y2 exactly k numbers apart (e.g. if y = (2 4) then k = 2). One then shows that the conjugacy action of Sn is transitive on these 3-tuples, so we get |Σ| = |G|. But now it is obvious that this system of conjugacy classes is strictly rigid, since by group theory hnA, 2Ai = G. Hence {nA, 2A, C(1)} is a good choice of conjugacy classes. Even {nA, 2A, C(k)} will work in most situations. To ensure that the chosen S is stable under the action of Π(n), we make a ﬁnal deﬁnition.

34 1.7 The Inverse Galois Problem 1 Fundamental Groups of Algebraic Curves

Deﬁnition. A conjugacy class C in G is said to be rational if for every g ∈ C, gm ∈ C for every m ∈ Z coprime to |G|.

Lemma 1.7.10. If C1,..., Cn are rational conjugacy classes of G and ϕ : π(n) → G is a continuous homomorphism with ϕ(γi) ∈ Ci for each 1 ≤ i ≤ n, then for all σ ∈ Π(n), we have ϕσ(γi) ∈ Ci.

Proof. Topologically, each γ generates an inertia group I in π(n) for some pro-point Q → i Qei ei P . Also, for each σ ∈ Π(n), σ(Q ) is another pro-point over P and σγ σ−1 generates I . i ei i i σ(Qei) 1 1 But Pi ∈ ( ) so Qei and σ(Qei) are both pro-points above the same point P i ∈ which PQ Q PQ 1 lies over Pi. Thus I and I are both stabilizers in π1( ) of points above some P i. Qei σ(Qei) PQ This implies these groups are conjugate in π(n). Write I = αI α−1 for α ∈ π(n). Then σ(Qei) Qei ϕ(I ) = ϕ(α)ϕ(I )ϕ(α)−1 so ϕ(I ) and ϕ(I ) are conjugate in G and cyclically σ(Qei) Qei Qei σ(Qei) −1 generated by ϕ(γi) and ϕ(σγiσ ), respectively. So for some g ∈ G and m prime to |G|, we −1 m −1 m have ϕ(σγiσ ) = gϕ(γi) g . Finally, since Ci is rational, this means ϕ(γi) ∈ Ci and so −1 ϕ(σγiσ ) ∈ Ci. Hence ϕ is Π(n)-stable. Theorem 1.7.11. For a ﬁnite group G with Z(G) = 1, suppose there exists a rigid system of rational conjugacy classes C ,..., C in G. Then for any P ,...,P ∈ 1 ( ), there is a 1 n 1 n PQ Q continuous, surjective homomorphism

1 ϕ : π1(PQ r {P1,...,Pn}) −→ G

with ϕ(γi) ∈ Ci for 1 ≤ i ≤ n. In particular, G = Gal(F/Q(T )) for some regular Galois extension F/Q(T ).

(1) Example 1.7.12. For Sn, we saw that {nA, 2A, C } is a strictly rigid system. Also, any m conjugacy class in Sn is rational since g 7→ g preserves cycle type. In general, one can prove this holds for {nA, 2A, C(k)}, 1 ≤ k ≤ n. This choice of rigid system corresponds to the cover

1 1 P −→ P x 7−→ xk(x − 1)n−k

with t = 0 corresponding to C(k), t = ∞ corresponding to nA and t = kn(k − n)n−kn−n k n−k corresponding to 2A. Taking a splitting ﬁeld of x (x−1) −T determines an Sn-extension of Q(T ), so the regular Inverse Galois Problem is solvable for G = Sn.

35 2 Riemann’s Existence Theorem

2 Riemann’s Existence Theorem

2.1 Riemann Surfaces

Riemann surfaces are a mix of the topology of covering spaces and the complex analysis of analytic continuation. The main problem one encounters in the latter setting is that a holomorphic function does not always admit a uniquely deﬁned analytic continatuion. The normal strategy then is to employ ‘branch cuts’, but this tactic seems ad hoc and not suited to generalization. Riemann’s idea was to replace the branches of a function with a covering space on which the analytic continuation is an actual function. Deﬁnition. Let X be a surface, i.e. a two-dimensional manifold. A complex atlas on X is a choice of open covering {Ui} of X together with homeomorphisms ϕi : Ui → ϕi(Ui) ⊆ C such that for each pair of overlapping charts Ui,Uj, the transition map

−1 ϕij := ϕj ◦ ϕi : ϕi(Ui ∩ Uj) −→ ϕj(Ui ∩ Uj) and its inverse are holomorphic. A complex structure on X is the choice of a complex atlas, up to holomorphic equivalence of charts, deﬁned by a similar condition to the above. A connected surface which admits a complex structure is called a Riemann surface.

Example 2.1.1. The complex plane C is a trivial Riemann surface. Any connected open subset U in C is also a Riemann surface via the given embedding U,→ C. Example 2.1.2. The complex projective line 1 = 1 = ∪{∞} admits a complex structure P PC C 1 1 × deﬁned by the open sets U0 = P r {∞} = C and U1 = P r {0} = C ∪ {∞}, together with charts 1 ϕ : U → , z 7→ z and ϕ : U → , z 7→ , 0 0 C 1 1 C z 1 −1 1 × where ∞ = 0 by convention. Note that ϕ1 ◦ ϕ0 is the function z 7→ z on C which is holomorphic.

Example 2.1.3. Let Λ ⊆ C be a lattice with basis [ω1, ω2].

ω1

ω2

36 2.1 Riemann Surfaces 2 Riemann’s Existence Theorem

Then the quotient C/Λ admits a complex structure as follows. Let π : C → C/Λ be the quotient map and suppose Π ⊆ C is a fundamental domain for Λ, meaning no two points in Π are equivalent mod Λ. Set U = π(Π) ⊆ C/Λ. Then π|Π :Π → U is a homeomorphism, so let ϕ : U → Π be its inverse. Letting {Ui} be the collection of all images under π of fundamental domains for Λ, we get a complex atlas on C/Λ (one can easily check that the transition functions between the Ui are locally constant, hence holomorphic). Topologically, C/Λ is homeomorphic to a torus.

Deﬁnition. A function f : U → C on an open subset U of a Riemann surface X is holomorphic if for every complex chart ϕ : V → ϕ(V ) ⊆ C, the function f ◦ ϕ−1 : ϕ(U ∩ V ) → U ∩ V → C is holomorphic.

Let O(U) denote the set of all holomorphic functions U → C.

Lemma 2.1.4. For any open set U of a Riemann surface X, O(U) is a commutative C- algebra.

Proposition 2.1.5 (Holomorphic Continuation). For any open set U ⊆ X of a Riemann surface and any x ∈ U, if f ∈ O(U r {x}) is bounded in a neighborhood of x, then f extends uniquely to some f˜ ∈ O(U).

More generally, we can deﬁne holomorphic maps between two Riemann surfaces.

Deﬁnition. A continuous map f : X → Y between Riemann surfaces is called holomorphic if for every pair of charts ϕ : U → ϕ(U) ⊆ C on X and ψ : V → ψ(V ) ⊆ C on Y such that f(U) ⊆ V , the map ψ ◦ f ◦ ϕ−1 : ϕ(U) → U → V → ψ(V ) is holomorphic. We say f is biholomorphic if it is a bijection and its inverse f −1 is also holomorphic. In this case X and Y are said to be isomorphic as Riemann surfaces.

f g Lemma 2.1.6. If X −→ Y −→ Z are holomorphic maps between Riemann surfaces, then g ◦ f : X → Z is also holomorphic.

Proposition 2.1.7. Let f : X → Y be a holomorphic map. Then for all open U ⊆ X, there is an induced C-algebra homomorphism f ∗ : O(U) −→ O(f −1(U)) ψ 7−→ f ∗ψ := ψ ◦ f.

Proof. The fact that f ∗ψ is an element of O(f −1(U)) follows from the above deﬁnitions of O and a holomorphic map between Riemann surfaces. The ring axioms are also easy to verify.

Theorem 2.1.8. Suppose f, g : X → Y are holomorphic maps between Riemann surfaces such that there exist a set A ⊆ X containing a limit point a ∈ A and f|A = g|A. Then f = g.

37 2.1 Riemann Surfaces 2 Riemann’s Existence Theorem

Proof. Let U ⊆ X be the set of all x ∈ X with an open neighborhood W on which f|W = g|W . Then U is open and a ∈ U; we will show it is also closed. If x ∈ ∂U, we have f(x) = g(x) since f and g are continuous. Choose a neighborhood W ⊆ X of x and charts ϕ : W → ϕ(W ) ⊆ C and ψ : W 0 → ψ(W 0) ⊆ C in Y with f(W ) ⊆ W 0 and g(W ) ⊆ W 0. Consider F = ψ ◦ f ◦ ϕ−1 : ϕ(W ) → ψ(W 0) and G = ψ ◦ g ◦ ϕ−1 : ϕ(W ) → ψ(W 0).

Then F and G are holomorphic and W ∩ U 6= ∅, so we must have F = G. Therefore f|W = g|W , so x ∈ U after all. This implies U = X. Deﬁnition. A meromorphic function on an open set U ⊆ X consists of an open subset V ⊆ U and a holomorphic function f : V → C such that U rV contains only isolated points, called the poles of f, and limx→p |f(x)| = ∞ for every pole p ∈ U r V . Denote the set of meromorphic functions on U by M(U). Then M(U) is a C-algebra, where f + g and fg are deﬁned by meromorphic continuation.

n Example 2.1.9. Any polynomial f(z) = c0 + c1z + ... + cnz is a holomorphic function C → C. Viewing C ⊆ P1, f is a meromorphic function on P1 with only a pole at ∞ of order n (assuming cn 6= 0). Example 2.1.10. Any meromorphic function f ∈ M(X) may be represented by a Laurent series expansion about any of its poles p by choosing a complex chart U → C containing p, lifting z to a parameter t on U and writing

∞ X n f(t) = cnt for some cn ∈ C. n=−N Theorem 2.1.11. Suppose X is a Riemann surface. Then the set of meromorphic functions M(X) is in bijection with the set of holomorphic maps X → P1. Proof. If f ∈ M(X) is a meromorphic function, then setting f(p) = ∞ for every pole p of f deﬁnes a holomorphic map f : X → P1. Indeed, it is clear that f is continuous. Let P be the set of its poles. If ϕ : U → ϕ(U) ⊆ C is a chart on X and ψ : V → ψ(V ) ⊆ C is a chart on P1 with f(U) ⊆ V , then since f is holomorphic on X r P , ψ ◦ f ◦ ϕ−1 : ϕ(U) → ψ(V ) is holomorphic on ϕ(U) r ϕ(P ). By Proposition 2.1.5, ψ ◦ f ◦ ϕ−1 is actually holomorphic on ϕ(U), so f is a holomorphic map of Riemann surfaces. Conversely, if g : X → P1 is holomorphic, then by Theorem 2.1.8, either g(X) = {∞} or g−1(∞) is a set of isolated points in X. It is then easy to see that g : X r g−1(∞) → C is meromorphic. Corollary 2.1.12 (Meromorphic Continuation). For any open set U ⊆ X and any x ∈ U, if f ∈ M(U r {x}) is bounded in a neighborhood of x, then f extends uniquely to some f˜ ∈ M(U). Proof. Apply Proposition 2.1.5 and Theorem 2.1.11. Corollary 2.1.13. Any nonzero function in M(X) has only isolated zeroes. In particular, M(X) is a ﬁeld.

38 2.1 Riemann Surfaces 2 Riemann’s Existence Theorem

Theorem 2.1.14. Let f : X → Y be a nonconstant holomorphic map between Riemann surfaces. Then for every x ∈ X with y = f(x) ∈ Y , there exists k ∈ N and complex charts ϕ : U → ϕ(U) ⊆ C of X and ψ : V → ψ(V ) ⊆ C of Y with f(U) ⊆ V such that (1) x ∈ U with ϕ(x) = 0 and y ∈ V with ψ(y) = 0. (2) F = ψ ◦ f ◦ ϕ−1 : ϕ(U) → ψ(V ) is given by F (z) = zk for all z ∈ ϕ(U). Proof. It is easy to arrange (1) by replacing (U, ϕ) with another chart obtained by composing ϕ with an automorphism of C taking ϕ(x) 7→ 0. So without loss of generality assume (1) is satisfied. By Theorem 2.1.8, F = ψ ◦ f ◦ ϕ−1 is nonconstant. Thus since f(0) = 0, we may write F (z) = zkg(z) for some k ≥ 1 and some g ∈ O(ϕ(U)) with g(0) 6= 0. Then g(z) = h(z)k for some holomorphic function h on ϕ(U), and H(z) = zh(z) defines a biholomorphic map α of some open neighborhood W ⊆ ϕ(U) of 0 onto another open neighborhood of 0. Finally, replace (U, ϕ) by (ϕ−1(W ), α ◦ ϕ). By construction, F = ψ ◦ fϕ−1 is now of the form F (z) = zk. Definition. The integer k for which F can be written F (z) = zk about x ∈ X is called the multiplicity of f at x. Corollary 2.1.15. If f : X → Y is a nonconstant holomorphic map between Riemann surfaces, then f takes open sets to open sets. Corollary 2.1.16. If f : X → Y is an injective holomorphic map, then f is biholomorphic X → f(X). Proof. If f is injective, then locally F (z) = zk with k = 1. Hence f −1 is holomorphic. Theorem 2.1.17. If f : X → Y is a nonconstant holomorphic map and X is compact, then Y is also compact and f is surjective. Proof. By Corollary 2.1.15, f(X) is open but since X is compact, f(X) is also compact and in particular closed. Therefore f(X) = Y . Corollary 2.1.18. Every holomorphic function on a compact Riemann surface is constant.

Proof. C is not compact, so Theorem 2.1.17 implies that every holomorphic function from a compact space into C must be constant. Corollary 2.1.19. Every meromorphic function on P1 is rational. Proof. First, note that the only way for such an f ∈ M(P1) to have inﬁnitely many poles is if it had a limit point, but then Theorem 2.1.8 would imply f ≡ ∞. Thus f has ﬁnitely 1 many poles, say a1, . . . , an ∈ P ; we may assume ∞ is not one of the poles, or else consider 1 the function f instead. For 1 ≤ i ≤ n, expand f as a Laurent series about ai:

mi X −j fi(z) = cij(z − ai) for cij ∈ C. j=1

1 Then g = f − (f1 + ... + fn) is holomorphic on P and thus constant by Corollary 2.1.18 since P1 is compact. This shows f is rational.

39 2.1 Riemann Surfaces 2 Riemann’s Existence Theorem

Corollary 2.1.20 (Liouville’s Theorem). Every bounded holomorphic function on C is constant.

Proof. By Proposition 2.1.5, f has a holomorphic continuation to f˜ : P1 → C, but by Corollary 2.1.18, f˜ must be constant. The idea in the rest of the section is to relate holomorphic maps between Riemann surfaces to covering space theory. Theorem 2.1.21. If p : Y → X is a nonconstant holomorphic map between Riemann surfaces then p is open and has discrete ﬁbres. Proof. By Corollary 2.1.15, p is open and Theorem 2.1.8 implies each ﬁbre is discrete. Let p : Y → X be a cover of Riemann surfaces. Traditionally, holomorphic functions f : Y → C are treated as multi-valued functions on X by setting f(x) = {f(y1), . . . , f(yn)} −1 where p (x) = {y1, . . . , yn}.

Example 2.1.22. Let exp : C → C× be the exponential map z 7→ ez and f = id : C → C the identity map. Then the resulting multi-valued function C× → C is the complex logarithm, which is only defined as a function after making a particular choice of branch of the function. We can describe this idea more cleanly with Riemann surfaces and branched covers. Definition. Suppose p : Y → X is a nonconstant holomorphic map. A ramification point of p is a point y ∈ Y such that for every neighborhood V ⊆ Y of y, p|V : V → p(V ) is not injective. The image x = p(y) is called a branch point of p. If p has no ramification points (and hence no branch points), then we call p an unramified map. Theorem 2.1.23. A nonconstant holomorphic map p : Y → X is unramified if and only if it is a local homeomorphism. Proof. Suppose p is unramified. Then for any y ∈ Y , there exists a neighborhood V ⊆ Y of y such that p|V : V → p(V ) is injective and open. Therefore p|V is a homeomorphism onto p(V ). The converse follows from basically the same argument.

n Example 2.1.24. For each n ≥ 2, the map pn : C → C defined by pn(z) = z is ramified × at 0 ∈ C and unramified everywhere else. Therefore pn : C → C is an unramified cover. Moreover, Theorem 2.1.14 says that every ramified cover of Riemann surfaces Y → X is locally of the form C → C, z 7→ zn. Example 2.1.25. The exponential map exp : C → C×, z 7→ ez is an unramified cover. In fact, as in the topological case, exp gives a universal cover of C via the inverse system of the covers pn.

Example 2.1.26. The quotient map π : C → C/Λ from Example 2.1.3 is an unramiﬁed cover of Riemann surfaces. Theorem 2.1.27. Suppose p : Y → X is a local homeomorphism of Hausdorﬀ topological spaces and X is a Riemann surface. Then Y admits a unique complex structure making p a holomorphic map.

40 2.1 Riemann Surfaces 2 Riemann’s Existence Theorem

Proof. Let ϕ : V → C be a chart of X. Then there exists an open subset U ⊆ V over which −1 −1 p|U : p (U) → U is a homeomorphism. Set Ue = p (U) and ϕe = ϕ ◦ p|U : Ue → C. Then ϕe is a complex chart on Y and the collection {U,e ϕe} obtained in this way forms a complex atlas on Y . Since p : Y → X is locally biholomorphic by construction, it is a holomorphic map between Riemann surfaces. Uniqueness is easy to check. Example 2.1.28. Now that we can view nonconstant holomorphic maps as local homeomorphisms, and in most cases covering spaces, we can rephrase the language of branch cuts as a lifting problem. For example, let exp : C → C× be the exponential map and suppose f : X → C× is a holomorphic map of Riemann surfaces, with X simply connected. Then by z0 covering space theory, for each ﬁxed x0 ∈ X and z0 ∈ C such that f(x0) = e , there exists a unique lift F : X → C making the diagram C F exp

X × f C

commute. Theorem 2.1.27 can be used to show that any such F is holomorphic. Moreover, any other lift G of f diﬀers from F by 2πin for some n ∈ Z. For the special case of a simply connected open set X ⊆ C×, any lift F is a branch of the complex logarithm on X. Example 2.1.29. Similarly, one can construct the complex root functions z 7→ z1/n, n ≥ 2, × as lifts along the cover pn : C → C. Let f : Y → X be a nonconstant holomorphic map that is proper, i.e. the preimage of any compact set in X is compact in Y . For each x ∈ X, deﬁne the multiplicity of f at x to be X ordx(f) = vy(f) y∈f −1(x)

where vy(f) is the multiplicity of f at y. −1 Example 2.1.30. If f : Y → X is unbranched at x ∈ X, then p (x) = {y1, . . . , yn} for some n and vyi (f) = 1 for each 1 ≤ i ≤ n. Thus ordx(f) = n. Theorem 2.1.31. If f : Y → X is a proper, nonconstant holomorphic map between Rie- mann surfaces, then there exists a number n ∈ N such that for every x ∈ X, ordx(f) = n. Proof. By Theorem 2.1.21, the set B of ramiﬁcation points of f is a closed, discrete subset of Y . Let A = f(B) ⊆ X. Then since f is proper, A is also closed and discrete. The

restriction f|Y rB : Y r B → X r A is unramified, so it is a finite-sheeted covering space; say −1 n is the number of sheets of f|Y rB, i.e. the size of any fibre f (x) for an unbranched point x ∈ X. By the above example, f has multiplicity n at every y ∈ Y r B. Suppose a ∈ A −1 and write f (a) = {b1, . . . , bk} ⊆ B and mi = vbi (f). For each 1 ≤ i ≤ k, we may choose −1 neighborhoods Vi ⊂ Y of bi and Ui ⊂ X of a such that for all x ∈ Ui r {a}, f (x) ∩ Vi consists of exactly mi points. Then there is a neighborhood U ⊆ U1 ∩ · · · ∩ Uk of a such that −1 −1 f (U) ⊆ V1 ∪· · ·∪Uk and for every x ∈ U ∩(X rA), f (x) consists of exactly m1 +...+mk −1 points. However we showed that |f (x)| = n, so n = m1 + ... + mk as required.

41 2.2 The Existence Theorem over 1 2 Riemann’s Existence Theorem PC

Corollary 2.1.32. Let X be a compact Riemann surface and f : X → C a nonconstant meromorphic function. Then the number of zeroes of f equals the number of poles of f, counted with multiplicity.

Proof. View f as a holomorphic function X → P1. Since X and P1 are compact, f is a proper map so ord0(f) = ord∞(f). But ord0(f) is precisely the number of zeroes of f, while ord∞(f) is the number of poles.

Corollary 2.1.33. Any complex polynomial f(z) ∈ C[z] of degree n has exactly n zeroes, counted with multiplicity.

1 1 Proof. We may view f as a holomorphic map P → P . Then it is easy to see ord∞(f) = n, so once again ord0(f) = n.

2.2 The Existence Theorem over 1 PC Consider the complex projective line 1 = ∪ {∞}, also known as the Riemann sphere, PC C and take a finite set of points P ⊂ 1 . Then 1 P is both a topological space and an PC PC r algebraic curve. In the topological case, the first invariant that describes the projective top 1 line is Γ := π1 (P r P ), while in the algebraic case we now have an algebraic invariant 1 ét 1 π1(P r P ) = π1 (P r P ). We will prove the following case of Theorem 1.5.5:

Theorem 2.2.1 (Riemann Existence). For any finite set P ⊂ P1, there is an isomorphism ét 1 ∼ π1 (P r P ) = Γb, where Γb is the profinite completion of the topological fundamental group top 1 Γ = π1 (P r P ).

Set K = C(P1) = C(t) be the function field of the projective line and let L(P )/K be the maximal Galois extension of K that is unramified outside P . Then by the definition of the algebraic fundamental group in Section 1.5, an equivalent statement to Theorem 2.2.1 is that Gal(L(P )/K) ∼= Γ.b This statement is quite useful for applications to the Inverse Galois Problem over K = C(t), as we saw in Section 1.7. Recall that each point Pi ∈ P corresponds to a generator in the fundamental group 1 Γ = hγ1, . . . , γr | γ1 ··· γr = 1i, for a fixed basepoint x ∈ P :

γ1 γ2 γr

P1 P2 ··· Pr

42 2.2 The Existence Theorem over 1 2 Riemann’s Existence Theorem PC

Specify a generating r-tuple g1, . . . , gr ∈ G – recall from Section 1.7 that this means hg1, . . . , gri = G and g1 ··· gr = 1. Then G is obtained as a quotient of Γ by mapping γi 7→ gi for each 1 ≤ i ≤ r. We have seen that the inertia groups at the points Pi ∈ P are

cyclicly generated, IPi = hgii. The following lemma allows us to dispense with worries about “canonical” generators γ1, . . . , γr for Γ.

Lemma 2.2.2. If Γ1 and Γ2 are two ﬁnitely generated groups with the same homomorphic images, then their proﬁnite completions Γb1 and Γb2 are isomorphic.

On the other hand, with K = C(t), we have that Gal(K/K) = Fbω is free profinite of (uncountably) infinite rank. By definition Gal(K/K) = lim Gal(L(P )/K) where the inverse ←− limit is taken over all finite sets P ⊂ P1, ordered by P ⊆ P 0 which induces L(P ) ⊆ L(P 0) and thus Gal(L(P 0)/K) → Gal(L(P )/K). In particular, for each containment of finite sets P ⊆ P 0 we have a diagram

Gal(L(P 0)/K) Gal(L(P )/K)

∼= ∼=

Fb|P 0|−1 Fb|P |−1

The topological side of the proof of Theorem 2.2.1 is essentially contained in the following theorem.

Theorem 2.2.3. Let P ⊂ 1 be a finite set and π : R → 1 P a finite-sheeted Galois PC PC r covering of topological spaces. Then there exists a smooth projective curve W with a regular map f : W → 1 which is unramified outside P and such that the covering PC

−1 1 W r f (P ) → PC r P is equivalent to π. Furthermore, under this identiﬁcation, the deck transformations of π become regular automorphisms of W → 1 . PC The proof requires a sequence of reduction steps, given by the following lemmas.

π Lemma 2.2.4. For a topological cover R −→ P1 r P , R may be given the structure of a compact Riemann surface such that π becomes analytic.

π Proof. Given R −→ P1 r P , we ﬁrst give R the structure of a Riemann surface. This can be done using Theorem 2.1.27, but the following explicit description will be of use in later arguments.

43 2.2 The Existence Theorem over 1 2 Riemann’s Existence Theorem PC

Ui q R

P2 B(z, ε) P3 1 P1 z P

Take q ∈ R r π−1(P ) and set z = π(q). Since P is ﬁnite, we can choose a ball B(z, ε) which −1 does not contain any point of P . Then π (B(z, ε)) = U1 q · · · q Un for disjoint open sets Ui for which π|Ui : Ui → B(z, ε) is a homeomorphism. One of these Ui contains q; let this Ui be the coordinate chart deﬁning the surface structure on R. For two of these charts, U = Ui and 0 0 V = Vj, the transition maps are just given by lifting the transition maps B(z, ε) → B(z , ε ) along the local homeomorphism π, and these are analytic on the disks B(z, ε),B(z0, ε0) so they are analytic on the U, V . This gives R the structure of a Riemann surface, but it may not be compact. In the next step, we show that there exists a compact Riemann surface R with an analytic mapπ ¯ : R → P1 and an open embedding R,→ R such that the diagram R R

π π¯

1 P r P P1 commutes. For any ball B = B(z, ε) ⊆ P1, let B0 = B0(z, ε) = B r {z} be the disk top 0 ∼ 0 punctured at its center. Then π1 (B ) = Z and in particular B has (up to isomorphism) only one connected, degree d cover for each d ≥ 1. Call this cover µd. When B = B(0, 1) is the unit ball, this degree d cover is explicitly given by

0 0 µd : B −→ B z 7−→ zd.

Moreover, µd extends to a branched cover of the entire disk B:

44 2.2 The Existence Theorem over 1 2 Riemann’s Existence Theorem PC

µd B0 B0

µd B B

The deck transformations of this cover are also easy to write down: they are generated by τ : z 7→ ζz where ζ = ζd is a primitive dth root of unity. Let z0 ∈ P and take a disk D = B(z0, ε) around z0 which does not contain any other 0 0 points in P . Let D = D r {z0}. Then as above, the preimage of D along π can be written as a disjoint union of connected components,

−1 0 0 0 π (D ) = D1 q · · · q Dm

0 0 where m ≤ n, such that for each 1 ≤ i ≤ m, π| 0 : D → D is a covering map. We now Di i 1 construct a Riemann surface R1 together with an analytic map π1 : R1 → P r (P r {z0}) and an embedding R,→ R1 such that |R1 r R| = m and the diagram

R R1

π π1

1 1 P r P P r (P r {z0})

commutes. After scaling by some λ ∈ C, we may replace D0 with B0 = B(0, 1) r {0}. Then 0 there exists an integer di and a homeomorphism τi : Di → B making the following diagram commute:

τi 0 0 Di B

π µdi

λ D0 B0

It is an easy exercise to show that τi is in fact biholomorphic. Notice that while τi is not

unique, it is deﬁned up to a deck transformation of µdi . We now construct R1 by taking m 0 0 copies of B, say B1,...,Bm, and the corresponding punctured disks B1,...,Bm and setting T = B1 q · · · q Bm. Obtain R1 by gluing R to T along the homeomorphisms

0 0 0 0 D1 q · · · q Dm −→ B1 q · · · q Bm.

Repeating this process, we obtain R after a ﬁnite number of steps, i.e. whenever all points of π−1(P ) have been “ﬁlled in”.

45 2.2 The Existence Theorem over 1 2 Riemann’s Existence Theorem PC

Lemma 2.2.5. If π : R → P1 r P is a Galois covering, then π¯ : R → P1 is Galois as well.

Proof. Suppose σ is a deck transformation of π : R → P1 r P . We claim σ extends uniquely 1 0 0 0 to a biholomorphic automorphism ofπ ¯ : R → P . Let D ,D1,...,Dm be as above. On the −1 0 0 0 0 disjoint union π (D ) = D1 q · · · q Dm, σ just acts by permuting the Di. For one of the 0 0 0 Di, set σ(Di) = Dj. Then the diagram

0 σ 0 Di Dj

0 σ 0 Bi Bj

commutes but by Proposition 2.1.5, since σ has bounded image it extends to Bi → Bj and 0 0 hence toσ ¯ : Di → Dj. Now for any z0 ∈ P , let D = D r {z0}. Then each Di in the disjoint −1 0 0 0 union π (D ) = D1 q · · · q Dm corresponds to a point qi ∈ R such that π(qi) = z0, and 0 which we glue to R to obtain R. Moreover, for each qi, qj, we may choose points xi ∈ Di and 0 xj ∈ Dj such that xi 6= qi and xj 6= qj. Now π is Galois, so there exists a deck transformation 0 0 σ ∈ ∆(π) mapping σ(xi) = xj. Then we must have σ(Di) = Dj and hence by the above, the extensionσ ¯ mapsσ ¯(qi) = qj. This provesπ ¯ is a Galois covering.

Suppose π : R → P1 r P is a Galois cover and set G = ∆(π) = Aut(π), the group of deck transformations. For any analytic curve Y , let M(Y ) be the ﬁeld of meromorphic functions on Y . For Y = R as above, we have an action of G on M(R) by (σf)(x) = f(σ−1(x)).

Lemma 2.2.6. For any f ∈ M(R) which is invariant under the G-action described above, then f ∈ M(P1) ⊆ M(R). In other words, f = g ◦ π for some g ∈ M(P1).

Proof. By the construction in Lemma 2.2.4, there exists a function g : P1 → P1 such that f = g ◦ π. Thus we need only show that g is meromorphic. Take p ∈ R and consider q = π(p) ∈ P1. If q 6∈ P , the set of branch points of π, then locally f is given by f = g ◦ π : D(p) → D(q) on local trivializations D(p) ⊆ R and D(q) ⊆ P1 of p, q, respectively. But locally, π is biholomorphic on these neighborhoods so we can invert π and write g = f ◦ π−1 and it follows that g is meromorphic. If q ∈ P , look at the punctured neighborhoods D(p) = D(p) r {0}, D(q) = D(q) r {0} and the cover π : D(p) → D(q). This is not holomorphic, but we know that it’s a degree m map that in fact just corresponds to the multiplication-by-m map on the unit punctured disk D(0, 1): π D(p) D(q)

∼= ∼= [m] D D

46 2.2 The Existence Theorem over 1 2 Riemann’s Existence Theorem PC

P∞ j Write f as a Laurent series f(z) = j=−N ajz for aj ∈ C and N ≥ 0. Then we must have f(ζmz) = f(z) for any root of unity ζm. Thus aj is nonzero only when m | j and we can rewrite ∞ X km f(z) = akmz . k=−K Let w = zm. Then w is a local parameter on D and we have

∞ X k f(z) = akmw = g(w). k=−K

Thus g(w) is meromorphic as required. In the next step, we need the so-called analytic version of the existence theorem. A proof will be provided in Section 2.4.

Theorem 2.2.7 (Riemann Existence – Analytic Version). Let X be a compact Riemann surface. Then for any distinct points p1, . . . , pn ∈ X and any numbers c1, . . . , cn ∈ C, possibly not distinct, there exists a meromorphic function g ∈ M(X) such that g(pi) = ci for each 1 ≤ i ≤ n.

Lemma 2.2.6 shows that there exists a ﬁnite group G acting on M(R) such that M(R)G = M(P1). By Serre’s GAGA principle, M(P1) = C(t), so therefore we have proven that [M(R): C(t)] < ∞.

Theorem 2.2.8. M(R)/C(t) is a Galois extension with Galois group G = ∆(π).

Proof. The paragraph above implies, by Artin’s Lemma, that M(R)/C(t) is Galois. Further, Gal(M(R)/C(t)) = G/N where N is the kernel of the action of G on M(R). To prove 1 −1 this, pick a point p ∈ P and write π (p) = {p1, . . . , pn}. By the analytic version of Riemann’s existence theorem (2.2.7), there exists a meromorphic g ∈ M(R) such that the values g(p1), . . . , g(pn) are all distinct. Suppose σ ∈ N is nontrivial. Then σ(pi) = pj 6= pi for some i 6= j, but σ ∈ N implies σ(g) = g, or g ◦ σ = g. Thus g(pi) = g(σ(pi)) = g(pj), which is impossible. Therefore N = {1} so Gal(M(R)/C(t)) = G as claimed. The ﬁnal step in the proof of Theorem 2.2.3 is contained the following lemma.

Lemma 2.2.9. There exists a smooth projective curve W over C and a biholomorphic iso- ∼ morphism R = W (C).

Proof. We know M(R) is a transcendence degree 1 extension of C, so by Corollary 1.3.9 there exists a proper normal curve W with C(W ) = M(R). Abstractly, there is a map ϕ : R → W which sends a point r ∈ R to the associated valuation vr on the field M(R), which is now identified with a valuation v on C(W ) that is trivial on C, and hence determines a point of W . We may assume W is smooth and projective over C. Let W,→ PN for large enough N. N Write P = {[x0, x1, . . . , xN ]: xi ∈ C, some xi 6= 0}. Then for any x ∈ R, there is an affine patch on which the map ϕ : R → W (C) is defined, so on this patch, ϕ = (f1, . . . , fN ) where

47 2.3 The General Case 2 Riemann’s Existence Theorem

xi fi = . But by properness, this must in fact be holomorphic. Moreover, Theorem 2.2.7 x0 implies ϕ is also injective. Further, properness also implies ϕ(R) ⊆ W is closed. Therefore to prove ϕ is a biholomorphic isomorphism, it suffices to show W (C) is connected. This is a rather standard property, but we repeat the proof here. Given that W is irreducible, smooth and projective over C, let us assume W (C) = M1 ∪ M2 for some open sets M1,M2 ⊂ W . Pick x0 ∈ M1. Then by the Riemann-Roch theorem, there exists a rational function f ∈ C(W ) that has a pole at x0 and is holomorphic everywhere else. View 1 f : M2 → P ; this cannot be constant since M2 is infinite and in particular dense in W (C). 1 On the other hand, f(M2) ⊆ P is open and W (C) is compact, so M2 is compact and 1 therefore f(M2) is also closed. Thus f(M2) = P so some point m ∈ M2 has f(m) = ∞, contradicting the fact that f was holomorphic on M2. Hence W (C) is connected, so we are finished. We derive the following important consequences.

Corollary 2.2.10. Let K = C(t) and E = C(W ). Then the embedding f ∗ : K,→ E satisﬁes [E : K] = deg π = |∆(π)|

where ∆(π) is the group of deck transformations of π.

Corollary 2.2.11. E/K is Galois and | Gal(E/K)| = |∆(π)|.

Proof. By Theorem 2.2.3, ∆(π) is a subgroup of Aut(E/K) but in general we have | Aut(E/K)| ≤ [E : K]. Therefore Corollary 2.2.10 implies Aut(E/K) = ∆(π) and therefore the extension is Galois.

Corollary 2.2.12. There is an anti-equivalence of categories

{compact Riemann surfaces} −→ {ﬁnite ´etale C(t)-algebras} Y 7−→ M(Y ).

Corollary 2.2.13. Let K = C(t). Then there is an equivalence of categories {compact Riemann surfaces} −→{∼ ﬁnite, continuous left Gal(K/K)-sets}.

Proof. Combine Corollaries 2.2.12 and 0.2.4.

2.3 The General Case

Let X and Y be Riemann surfaces and let M(X) and M(Y ) be their corresponding fields of meromorphic functions, as in the previous section. Suppose ϕ : Y → X is a holomorphic map which is not constant on any connected component. This induces a field homomorphism ϕ∗ : M(X) → M(Y ) defined by f 7→ ϕ∗(f) = f ◦ ϕ.

Proposition 2.3.1. If ϕ : Y → X is a nonconstant holomorphic map of degree d between compact, connected Riemann surfaces, then M(Y )/ϕ∗M(X) is a ﬁeld extension of degree d.

48 2.3 The General Case 2 Riemann’s Existence Theorem

Proof. Let f ∈ M(Y ) be a meromorphic function. We first prove f satisfies a polynomial equation of degree d over M(X). Let S be the set of branch points of the cover. Then −1 `d for each x 6∈ ϕ(S), there is a neighborhood U ⊆ X of x such that ϕ (U) = i=1 Vi for V1,...,Vd ⊆ Y homeomorphic to U via ψi : U → Vi. Set fi = f ◦ ψi, which is meromorphic on U. Define d Y d d−1 F (t) = (t − fi(t)) = t + an−1t + ... + a0 i=1 for ai meromorphic on U. One shows further that ai extend meromorphically to all of X, so ∗ F (t) is defined over M(X). Lastly, (ϕ F )(f) = 0 since it restricts to F (fi) = 0 on each Vi. This proves the claim. Now we show that some f ∈ M(Y ) satisfies an irreducible polynomial of degree d over −1 M(X). If x 6∈ ϕ(S), then ϕ (x) = {y1, . . . , yd}. By the analytic existence theorem (2.2.7), choose f ∈ M(Y ) which is holomorphic at y1, . . . , yd and has f(yi) all distinct. Then by the first paragraph, there exist ai ∈ M(X) such that

∗ n ∗ (ϕ an)f + ... + ϕ a0 = 0.

(Here, n ≤ d.) Suppose one of the ai has a pole at x. Then we may choose a small enough neighborhood about x not containing any branch points such that f is holomorphic and assigns distinct values to all preimages of each point. Replacing x with any of the points of this neighborhood, we may assume each ai is holomorphic at x. Finally, take g ∈ M(Y ). Then by the primitive element theorem, M(X)(f, g) = M(X)(h) for some h ∈ M(Y ). But then M(X)(f) ⊆ M(X)(h) and by the ﬁrst paragraph, h has degree ≤ d over M(X), so M(X)(f) = M(X)(h). In particular, M(Y ) = M(X)(f) so the proof is complete. We now generalize the results from Section 2.2 to arbitrary covers of Riemann surfaces.

Lemma 2.3.2. Suppose X is a Riemann surface, S ⊂ X a discrete set of points and ϕ : Y → X r S a proper holomorphic cover of Riemann surfaces. Then there is a Riemann surface Y and proper holomorphic map ϕ¯ : Y → X such that Y r ϕ−1(S) = Y . Proof. This is only a slight generalization of the proof of Lemma 2.2.4. For details, see Forster.

Lemma 2.3.3. If ϕ : Y → X r S is a Galois covering, then ϕ¯ : Y → X is Galois as well. Proof. A generalization of Lemma 2.2.5.

Theorem 2.3.4. Let X be a connected, compact Riemann surface with field of meromorphic functions M(X). Then for every finite étale M(X)-algebra A, there is a compact Riemann surface Y with M(Y ) ∼= A and a holomorphic map Y → X. Corollary 2.3.5. For any connected, compact Riemann surface X, there is an anti-equivalence of categories proper holomorphic maps of ∼ −→{finite étale M(X)-algebras}. Riemann surfaces Y → X

49 2.4 The Analytic Existence Theorem 2 Riemann’s Existence Theorem

Corollary 2.3.6. Let X be a connected, compact Riemann surface. Then there is an equivalence of categories compact Riemann surfaces ∼ −→{ﬁnite, continuous left Gal(M(X)/M(X)-sets}. Y → X

We can now give a proof of Theorem 1.5.5.

Proof. Let K = k(X) be the function field of X and KU the compositum in Ks of all finite extensions L/K coming from proper normal curves Y → X étale over U. Then by Theo- rem 2.3.4, for each of these L/K there is a compact connected Riemann surface YL → X étaleover U such that M(Y ) ∼= L. Let Mf be the compositum of these M(Y ). Then ∼ π1(U) = Gal(KU /K) = Gal(M/f M(X)) so it suffices to show Gal(M/f M(X)) is the profi- top nite completion of G := π1 (U(C)). But by Corollary 0.1.5, every finite quotient of Gb uniquely determines a finite Galois cover Y 0 → U; thus a covering Y → X by Lemma 2.3.2 which is Galois by Lemma 2.3.3; thus a finite Galois field extension M(Y )/M(X) by Propo- sition 2.3.1. It is clear that this correspondence is functorial on both sides, so taking inverse limits, we get the desired identification Gb = Gal(M/f M(X)).

2.4 The Analytic Existence Theorem

In this section we give a proof of Theorem 2.2.7. First, recall the following results for sheaf cohomology.

Theorem 2.4.1 (Leray). Let X be a space with a sheaf F and an open cover U. If the cover is acyclic, i.e. Hˇ 1(U, F) = 0, then H1(X, F) = Hˇ 1(U, F).

For a cover U = {Ui}i∈I , we can view Ui as a ball B(ai, εi) – using the biholomorphic map z. Writing ∞ 1 B(ai, εi) = ∪n=kB ai, εi − n ,

compactness of X implies there is a ﬁnite subcover of each B(ai, εi) and therefore each Ui. N Taking a ﬁnite subcover consisting of such sets, we obtain a cover W = {Wj}j=1 of X such that each W j is compactly contained in some Ui. We say the cover W is compactly contained in U.

Proposition 2.4.2. If W is compactly contained in U then the restriction map Hˇ 1(U, O) → Hˇ 1(W, O) has ﬁnite-dimensional image.

Proof. Consider the sheaf E of smooth functions on X. It is a fact that H1(B, E) = 0 for 1 any disk B ⊆ C – in technical terms, E is a ﬂabby sheaf. Given a cocycle (fij) ∈ Z (B, O), 0 we can resolve it using smooth functions: fij = gj − gi for gi, gj ∈ C (B, E). Now these may not be analytic, but this won’t be a problem. Consider the Cauchy-Riemann operator

∂ 1 ∂f ∂f : f 7→ + i . ∂z¯ 2 ∂x ∂y

50 2.4 The Analytic Existence Theorem 2 Riemann’s Existence Theorem

∂ By the Cauchy-Riemann equations, ∂z¯ is 0 on analytic functions, so in particular ∂f ∂g ∂g 0 = ij = j − i . ∂z¯ ∂z¯ ∂z¯

∂gi ˇ1 Now the functions ∂z¯ agree on any Ui ∩ Uj, so we get a lift h ∈ Z (B, E). By Dolbeault’s ∂g Lemma from complex analysis, there is some g ∈ E(B) such that ∂z¯ = h. By the above, ∂(gj −g ∂z¯ = 0 for all j. Writing fij = (gj − g) − (gi − g), we now see that (fij) is a coboundary. Hence Hˇ 1(B, O) = 0. To extend this to covers, suppose W is compactly contained in U. Take U ∈ U and W ∈ W with W ⊆ U. Then for any f ∈ O(U), f may be unbounded but by compactness, 2 f|W is bounded. Let || · || be the L -norm, so that ZZ ||f|| = |f(x + iy)|2 dx dy. U This may be inﬁnite, but deﬁne

L2(U, O) = {f ∈ O(U): ||f|| < ∞}.

As spaces, L2(U, O) ⊆ L2(U), the ordinary Hilbert space of square-integrable complex functions on U. Thus the above says that O(U) → O(W ) has image lying in L2(W, O). We need: Lemma 2.4.3. If W 0 is compactly contained in U 0 ⊆ C, then for every ε > 0 there exists a 2 closed subspace A ⊆ L (U, OC) of ﬁnite codimension such that for all f ∈ A,

||f||L2(W 0) ≤ ε||f||L2(U 0). ˇ We next construct a norm on the Cech cochain groups of X. First, each Ui ∈ U is biholo-

2 2 morphic to some disk Di ⊆ C via a map zi. For g ∈ O(Ui), set ||g||L (Ui) := ||zi(g)||L (Di). ˇ0 For a 0-cocycle η = (fi) ∈ C (U, O), where fi ∈ O(Ui), deﬁne

n !1/2 X 2 ||η|| := ||f || 2 . i L (Ui) i=1

ˇ1 Likewise, for ξ = (fij) ∈ C (U, O), deﬁne

!1/2 X 2 ||ξ|| := ||f || 2 . ij L (Ui∩Uj ) i,j

Continue in this fashion, we get a norm for every element of Cˇq(U, O), q ≥ 0. Write ˇq ˇq CL2 (U, O) for the subspace of C (U, O) where this norm is ﬁnite. Now for the compactly contained covers W ⊆ U and for any ε > 0, Lemma 2.4.3 provides a closed, ﬁnite-codimension ˇ1 subspace A ⊆ ZL2 (U, O) such that for all ξ ∈ A, ||ξ||L2(W) ≤ ε||ξ||L2(U). On the other hand, suppose we have a containment of covers W ⊆ V ⊆ U, with compact containments Wi ⊆ Vi ⊆ Ui for each i.

51 2.4 The Analytic Existence Theorem 2 Riemann’s Existence Theorem

Lemma 2.4.4. Given compactly contained covers W ⊆ V ⊆ U, there exists c > 0 such that ˇ1 ˇ1 for every cocycle ξ ∈ ZL2 (V, O), there is some ζ ∈ ZL2 (U, O) such that ζ = ξ + d(0)η

ˇ0 for some η ∈ CL2 (W, O) with ||ζ||, ||η|| ≤ c||ξ||. Now we play Lemmas 2.4.3 and 2.4.4 against each other. Use Lemma 2.4.4 to “extend” 1 the norm to U subject to the bound c > 0. Let ε = 2c and apply Lemma 2.4.3 to produce a ˇ1 closed subspace A ⊆ ZL2 (U, O) of ﬁnite codimension for which ||ξ||L2(V) ≤ ε||ξ||L2(U) for all ξ ∈ A. Then S = A⊥ is ﬁnite dimensional, where ⊥ is taken with respect to the L2-norm on Zˇ1(U, O). Consider the restriction map Hˇ 1(U, O) → Hˇ 1(W, O). We claim the image is equal to ˇ1 ˇ1 the image of S. Take ξ ∈ Z (U, O) and restrict to a bounded cocycle ξ|V ∈ ZL2 (V, O). Then ˇ1 (0) extend to U with Lemma 2.4.4 to get ζ0 ∈ ZL2 (U, O) such that ζ0 = ξ + d η on W. Write ζ = ξ0 + σ0 for ξ0 ∈ A and σ0 ∈ S. By orthogonal decomposition, ||ξ0||, ||σ0|| ≤ ||ζ0||. Now (0) ξ0 + σ0 = ζ0 = ξ + d η, so we can restrict ξ0 to V and extend to 1 ζ = ξ + d(0)η , with ||ζ || ≤ cε||ξ || = ||ξ ||. 1 0 1 1 0 2 0

Further decompose ζ1 = ξ1 + σ1 for ξ1 ∈ A, σ1 ∈ S and repeat. After k steps, we have

(0) ξk + (σ0 + ... + σk) = ξ0 + d (η0 + ... + ηk) 1 with ||ξ || ≤ ||ξ || k 2k 0 1 and ||σ ||, ||η || ≤ ||ζ || for all i. i i 2i 0 2 As k → ∞, σ0 + ... + σk converges in L -norm to some σ ∈ S, while η0 + ... + ηk converges ˇ0 (0) to some η ∈ CL2 (W, O). Thus we have σ = ξ + d η on W and hence the cohomology class of ξ is in the image of S. Theorem 2.4.5 (Finiteness). If X is a compact Riemann surface with sheaf of analytic 1 functions O = OX , then dimC H (X, O) < ∞. Proof. For any open cover U of X, we have a commutative diagram id Hˇ 1(X, O) Hˇ 1(X, O)

Hˇ 1(U, O) Hˇ 1(U, O)

By Leray’s theorem (2.4.1), the vertical arrows are isomorphisms but by Proposition 2.4.2, the restriction map along the bottom has ﬁnite dimensional image. This implies the identity has ﬁnite dimensional image, i.e. dim H1(X, O) < ∞. As a preliminary step to proving the analytic existence theorem, we have:

52 2.4 The Analytic Existence Theorem 2 Riemann’s Existence Theorem

Theorem 2.4.6. Let X be a compact Riemann surface. Then for any a0 ∈ X, there exists a meromorphic function f ∈ M(X) that has a pole at a0 and is holomorphic elsewhere.

Proof. Fix a0 ∈ X and choose an open cover {Up}p∈X such that p ∈ Up and a0 6∈ Up for any p 6= a0. Since X is compact, there is a ﬁnite subcover X = U0 ∪ U1 ∪ · · · ∪ Ur with r z a0 ∈ U0 = Ua0 . Set U = {Ui}i=0. We know the map U0 −→ B(0, 1) is biholomorphic. (m) (m) ˇ1 Now for each m ∈ N, we deﬁne a cocycle ξ = (fij ) ∈ Z (U, O) by

(m) fij = 0 when i, j 6= 0 (m) f00 = 0 (m) (m) −m f0j = −fj0 = z .

−m Note that since a0 ∈ U0, z is holomorphic. Moreover, we have the restriction map Hˇ 1(U, O) → Hˇ 1(X, O) and H1(X, O) is ﬁnite dimensional by Theorem 2.4.5, so there is a relation

n X (m) (0) cmξ = d η (*) m=1

ˇ0 (0) where η = (gi) ∈ C (U, O) with gi ∈ O(Ui) and d is the 0th boundary operator on the ˇ Cech complex. Deﬁne f : X r {a0} → C by ( g (z), z ∈ U and i ≥ 1 f(z) = i i Pn −m g0(z) + m=1 cmz , z ∈ U0.

By construction, f(z) will be meromorphic so we need only check that the definition is consistent on overlapping elements of the cover U. Suppose i, j 6= 0 and take z ∈ Ui ∩ Uj. (m) (m) Then by definition the ijth part of the left side of (∗) is 0 since (ξ )ij = fij = 0, whereas on the right it’s gi − gj (or the reverse). Hence gi = gj on Ui ∩ Uj. When j = 0, we have Pn −m m=1 cmz = gi − g0 on Ui ∩ U0 so again the definition of f(z) is consistent. We now prove Theorem 2.2.7.

Proof. Suppose we can find meromorphic functions h1, . . . , hn ∈ M(X) such that hi(pj) = δij for each 1 ≤ i, j ≤ n. Then the function g(z) = c1h1(z) + ... + cnhn(z) will satisfy the desired property. To produce these hi, first use Theorem 2.4.6 to find meromorphic functions f1, . . . , fn ∈ M(X) such that for each i, fi has a pole at pi and is holomorphic everywhere else. Choose dij ∈ C such that fi(z) − fi(pi) + dij is nonzero whenever we evaluate at z = pj. Then define Y fi(z) − fi(pi) hi(z) = . fi(z) − fi(pi) + dij j6=i

By the choice of dij, these functions are well-deﬁned, meromorphic and clearly hi(pi) = 1 and hi(pj) = 0 for all j 6= i. Therefore the theorem is proved.

53 3 Scheme Theory

3 Scheme Theory

This chapter follows a short course on scheme theory I gave at the University of Virginia in summer 2017. The topics covered are: Aﬃne schemes

General schemes and their properties

Fibre products and the ﬁbre functor

Sheaves of OX -modules Quasi-coherent and coherent sheaves

Diﬀerentials

Group schemes. The main references for this short course are Szamuely’s Galois Groups and Fundamental Groups, Hartshorne’s Algebraic Geometry and Vakil’s course notes on algebraic geometry. I have omitted many proofs in favor of covering more material, but these three references contain all the details that were left out.

3.1 Aﬃne Schemes

Hilbert’s Nullstellensatz is an important theorem in commutative algebra which is essentially the jumping off point for classical algebraic geometry (by which we mean the study of algebraic varieties in affine and projective space). We recall the statement here. Theorem 3.1.1 (Hilbert’s Nullstellensatz). If k is an algebraically closed field, then there is a bijection

n Ak ←→ MaxSpec k[t1, . . . , tn] P = (α1, . . . , αn) 7−→ mP = (t1 − α1, . . . , tn − αn),

n n where Ak = k is aﬃne n-space over k and MaxSpec denotes the set of all maximal ideals of a ring. Further, if f : A → B is a morphism of ﬁnitely generated k-algebras then we get a map f ∗ : MaxSpec B → MaxSpec A given by f ∗m = f −1(m) for any maximal ideal m ⊂ B. Note that if k is not algebraically closed, f −1(m) need not be a maximal ideal of A. Lemma 3.1.2. Let f : A → B be a ring homomorphism and p ⊂ B a prime ideal. Then f −1(p) is a prime ideal of A. This suggests a natural replacement for MaxSpec A,

Spec A = {p ⊂ A | p is a prime ideal}.

54 3.1 Aﬃne Schemes 3 Scheme Theory

Definition. An affine scheme is a ringed space with underlying topological space X = Spec A for some ring A. In order to justify this definition, I will now tell you the topology on Spec A and the sheaf of rings making it into a ringed space. For any subset E ⊆ A, define

V (E) = {p ∈ Spec A | E ⊆ p}.

Lemma 3.1.3. Let A be a ring and E ⊆ A any subset. Set a = (E), the ideal generated by E. Then (a) V (E) = V (a) = V (r(a)) where r denotes the radical of an ideal.

(b) V ({0}) = Spec A and V (A) = ∅. S T (c) For a collection of subsets {Ei} of A, V ( Ei) = V (Ei). (d) For any ideals a, b ⊂ A, V (a ∩ b) = V (ab) = V (a) ∪ V (b). As a result, the sets V (E) for E ⊆ A form the closed sets for a topology on Spec A, called the Zariski topology. Next, for any prime ideal p ⊂ A, let Ap denote the localization at p. For any open set U ⊆ Spec A, we define ( ) a f O(U) = s : U → A s(p) ∈ A , ∃ p ∈ V ⊆ U such that s(q) = for all q ∈ V, f, g ∈ A . p p g p∈U Theorem 3.1.4. (Spec A, O) is a ringed space. Moreover, ∼ (1) For any p ∈ Spec A, Op = Ap as rings. (2) Γ(Spec A, O) ∼= A as rings. (3) For any f ∈ A, define the open set D(f) = {p ∈ Spec A | f 6∈ p}. Then the D(f) ∼ form a basis for the topology on Spec A and O(D(f)) = Af as rings. Example 3.1.5. For any field k, Spec k is a single point ∗ corresponding to the zero ideal, with sheaf O(∗) ∼= k.

Example 3.1.6. Let A = k[t1, . . . , tn] be the polynomial ring in n variables over k. Then n Spec A = Ak , the aﬃne n-space over k. For example, when A = k[t] is the polynomial ring 1 in a single variable, Spec k[t] = Ak, the aﬃne line. 1 When k = C, Hilbert’s Nullstellensatz tells us that all the closed points of Ak correspond to maximal ideals of the form (t − α) for α ∈ C. But there is also a non-closed, ‘generic point’ corresponding to the zero ideal which was not detected before. closed points −2 0 1 + i generic point Spec C[t] (t + 2) (t) (t − (1 + i)) (0)

55 3.2 Schemes 3 Scheme Theory

On the other hand, if k = Q or another non-algebraically closed ﬁeld, the same closed points corresponding to linear ideals (t − α) show up, as well as the generic point corresponding to (0), but there are also points corrresponding to ideals generated by higher degree irreducible polynomials like t2 + 1. Thus the structure of Spec Q[t] is much diﬀerent than the algebraically closed case.

closed points −2 0 ?? generic point Spec Q[t] (t + 2) (t) (t2 + 1) (0)

Example 3.1.7. Let X be an algebraic variety over a field k, x ∈ X a point and consider the affine scheme Y = Spec(k[ε]/(ε2)). We can think of Y as a “big point” with underlying space ∗ corresponding to the zeor ideal, along with a “tangent vector” extending infinitesimally in every direction around ∗. Then any map Y → X determines a unique tangent vector in TxX, the tangent space of X at x. This idea is useful in intersection theory. For example, 2 2 consider the tangency of the x-axis and the parabola y = x in Ak:

y − x2

y (0, 0)

As a variety, this point (0, 0) corresponds to the quotient of k-algebras k[x, y]/r(y, y − x2) = k[x]/r(x2) = k[x]/(x) = k. Thus the information of tangency is lost. However, as an aﬃne scheme, (0, 0) corresponds to Spec(k[x, y]/(y, y − x2)) = Spec(k[x]/(x2)) so the intersection information is preserved.

3.2 Schemes

In this section we deﬁne a scheme and prove some basic properties resulting from this deﬁ- nition. Recall that a ringed space is a pair (X, F) where X is a topological space and F is a sheaf of rings on X.

Deﬁnition. A locally ringed space is a ringed space (X, F) such that for all P ∈ X, ∼ there is a ring A such that FP = Ap for some prime ideal p ⊂ A. Example 3.2.1. Any aﬃne scheme Spec A is a locally ringed space by (1) of Theorem 3.1.4. We will sometimes denote the structure sheaf O by OA.

56 3.2 Schemes 3 Scheme Theory

Deﬁnition. The category of locally ringed spaces is the category whose objects are locally ringed spaces (X, F) and whose morphisms are morphisms of ringed spaces (X, F) → # (Y, G) such that for each P ∈ X, the induced map fP : OY,f(P ) → OX,P is a morphism of # −1 local rings, i.e. (fP ) (mP ) = mf(P ) where mP (resp. mf(P )) is the maximal ideal of the local ring OX,P (resp. OY,f(P )). We are now able to deﬁne a scheme.

Definition. A scheme is a locally ringed space (X, OX ) that admits an open covering {Ui} ∼ such that each Ui is affine, i.e. there are rings Ai such that (Ui, OX |Ui ) = (Spec Ai, OAi ) as locally ringed spaces. The category of schemes Sch is defined to be the full subcategory of schemes in the category of locally ringed spaces. Denote the subcategory of affine schemes by AffSch. Also let CommRings denote the category of commutative rings with unity. Proposition 3.2.2. There is an isomorphism of categories

AffSch −→∼ CommRingsop

(X, OX ) 7−→ OX (X) (Spec A, O) 7−→ A.

Proof. (Sketch) First suppose we have a homomorphism of rings f : A → B. By Lemma 3.1.2 this induces a morphism f ∗ : Spec B → Spec A, p 7→ f −1(p) which is continuous since f −1(V (a)) = V (f(a)) for any ideal a ⊂ A. Now for each p ∈ Spec B, define the localization fp : Af ∗p → Bp using the universal property of localization. Then for any open set V ⊆ Spec A, we get a map # ∗ −1 f : OA(V ) −→ OB((f ) (V )). One checks that each is a homomorphism of rings and commutes with the restriction maps. # Thus f : OA → OB is defined. Moreover, the induced map on stalks is just each fp, so the ∗ # pair (f , f ) gives a morphism (Spec B, OB) → (Spec A, OA) of locally ringed spaces, hence of schemes. # Conversely, take a morphism of schemes (ϕ, ϕ ) : (Spec B, OB) → (Spec A, OA). This induces a ring homomorphism Γ(Spec A, OA) → Γ(Spec B, OB) but by (2) of Theorem 3.1.4, ∼ ∼ Γ(Spec A, OA) = A and Γ(Spec B, OB) = B so we get a homomorphism A → B. It’s easy to see that the two functors described give the required isomorphism of categories. Example 3.2.3. We saw in Example 3.1.5 that for any field k, Spec k = ∗ is a point with structure sheaf O(∗) = k. If A = L1 × · · · × Lr is a finite étale k-algebra, then ` ` Spec A = Spec L1 ··· Spec Lr is (schematically) a disjoint union of points.

Example 3.2.4. Let A be a DVR with residue field k. Then Spec A = {0, mA}, a closed point for the maximal ideal m and a generic point for the zero ideal. There are two open subsets here, {0} and Spec A, and we have OA({0}) = k and OA(Spec A) = A. Example 3.2.5. If k is a field and A is a finitely generated k-algebra, then the closed points of X = Spec A are in bijection with the closed points of an affine variety over k with coordinate ring A.

57 3.3 Properties of Schemes 3 Scheme Theory

Example 3.2.6. Let A = Z (or any Dedekind domain). Then dim A = 1 and it turns out that dim Spec A = 1 for some appropriate notion of dimension (see Section 3.3). Explicitly, Spec Z has a closed point for every prime p ∈ Z and a generic point for (0): closed points generic point Spec Z 2 3 5 7 11 (0)

1 Example 3.2.7. Let k be a field, X1 = X2 = Ak two copies of the affine line and U1 = U2 = 1 1 Ak r {0}, where 0 is the closed point of Ak corresponding to (x) in k[x]. Then we can glue together X1 and X2 along the identity map U1 → U2 to get a scheme X which looks like the affine line with the origin “doubled”. Note that X is not affine!

1 Ak r {0}

3.3 Properties of Schemes

Many deﬁnitions in ring theory can be rephrased for schemes. For example:

Deﬁnition. A scheme X is reduced if for all open U ⊆ X, OX (U) has no nilpotent elements.

Deﬁnition. A scheme X is integral if for all open U ⊆ X, OX (U) has no zero divisors. Lemma 3.3.1. X is integral if and only if X is reduced and irreducible as a topological space.

Proof. ( =⇒ ) Clearly integral implies reduced, so we just need to prove X is irreducible. Suppose X = U ∪ V for open subsets U, V ⊆ X. Then OX (U ∪ V ) = OX (U) × OX (V ) which is not a domain unless one of OX (U), OX (V ) is 0. In that case, U or V is empty, so this shows X is irreducible. ( ⇒ = ) Suppose X is reduced and irreducible, but there exists an open set U ⊆ X and f, g ∈ OX (U) with fg = 0. Deﬁne closed sets

C = {P ∈ U | fP ∈ mP ⊂ OX,P }

D = {P ∈ U | gP ∈ mP ⊂ OX,P }.

Then by deﬁnition of OX , we must have C ∪D = U. By irreducibility, C = U without loss of 0 0 generality. Thus for any aﬃne open set U ⊆ U with U = Spec A, we have (OX |U 0 )(D(f)) = ∼ 0 but by (3) of Theorem 3.1.4, OU 0 (D(f)) = Af , the localization of A at powers of f. When Af = 0, f is nilpotent but by assumption this means f = 0. Hence X is integral.

58 3.3 Properties of Schemes 3 Scheme Theory

Deﬁnition. The dimension of a scheme X (or any topological space) is

dim X = sup{n ∈ N0 | there exists a chain of irreducible, closed sets X0 ( X1 ( ··· ( Xn ⊆ X}. Proposition 3.3.2. Let A be a noetherian ring. Then dim Spec A = dim A, the Krull dimension of A. Be warned that the converse to Proposition 3.3.2 is false in general. Definition. Let X be any scheme. For a point P ∈ X, we define the codimension of P to be the Krull dimension of the local ring at P , that is codim P = dim OX,P . Note that by commutative algebra, the codimension of P is equal to the height of the prime ideal p ⊂ A associated to P for any choice of affine open neighborhood P ∈ U = Spec A. Definition. Let X be a scheme. Then

X is locally noetherian if each stalk OX,P is a local noetherian ring. X is noetherian if X is integral and locally noetherian.

An integral scheme X is normal if each stalk OX,P is integrally closed in its ﬁeld of fractions. 2 X is regular if each OX,P is regular as a local ring, that is, dim OX,P = dim mP /mP as OX,P /mP -vector spaces.

Definition. Let U ⊆ X be an open subset. Then (U, OX |U ) is a scheme which we call an # open subscheme of X. The natural morphism j : U,→ X, j : OX → j∗OX |U is called an open immersion. Example 3.3.3. For X = Spec A, let f ∈ A and recall the open set D(f) defined in Theorem 3.1.4. Then D(f) is an open subscheme of X and the open immersion D(f) ,→ X corresponds to the natural inclusion of prime ideals Spec Af ,→ Spec A (this is a property of any localization). Definition. Let A → A/I be a quotient homomorphism of rings. Then the induced morphism Spec(A/I) → Spec A is called an affine closed immersion. For a general morphism of schemes f : X → Y , f is called a closed immersion if f is injective, f(X) ⊆ Y is closed

and there exists a covering of X by aﬃne open sets {Ui} such that each f|Ui : Ui → f(Ui) is an aﬃne closed immersion. The set f(X) is called a closed subscheme of Y .

Deﬁnition. Let X be a scheme. The category of schemes over X, denoted SchX , consists p of objects Y −→ X, where Y is a scheme and p is a morphism, and morphisms Y → Z making the following diagram commute: Y Z

59 3.3 Properties of Schemes 3 Scheme Theory

S ∼ Example 3.3.4. Every scheme Y is a scheme over Spec Z. Write Y = Ui where Ui = Spec Ai for rings Ai. Then for each of these there is a canonical homomorphism ϕi : Z → Ai ∗ ∼ which induces ϕi : Spec Ai → Spec Z. Composing these with the isomorphisms Ui = Spec Ai, S we get a map Y = Ui → Spec Z. The ﬁbre of a topological cover p : Y → X can be interpreted as a ﬁbre product:

−1 p (x) := {x} ×X Y Y

{x} X

We next construct fibre products in the category SchX and use these to construct the algebraic analogue of a fibre. Definition. Let X be a scheme and Y,Z schemes over X.A fibre product of Y and Z over X, denoted Y ×X Z, is a scheme over both Y and Z such that the diagram

Y ×X Z

Y Z

X commutes and Y ×X Z is universal with respect to such diagrams, i.e. for any scheme W over both Y and Z, the following diagram can be completed uniquely: W

∃!

Y ×X Z

Y Z

Given f : Y → X and any scheme Z over X, the induced map fZ : Y ×X Z → Z is called the base change of f over Z.

60 3.3 Properties of Schemes 3 Scheme Theory

Theorem 3.3.5. For any schemes Y,Z over X, there exists a ﬁbre product Y ×X Z which is unique up to unique isomorphism.

Proof. (Sketch) First suppose X,Y and Z are all affine; write X = Spec A, Y = Spec B and C = Spec Z. Then Spec(B ⊗A C) is a natural candidate for the fibre product in this case. Indeed, the tensor product satisfies the universal property conveyed by the following diagrams: R

∃!

B ⊗A C B ⊗A C

B C B C

A A

Applying the functor Spec yields the right diagrams with arrows reversed, by Proposi- tion 3.2.2, so the fibre product exists in the affine case. In general, note that once we construct any fibre product, it will be unique up to unique isomorphism by the universal property, just as in every proof of the solution to a universal mapping problem. Now suppose X and Z are affine and Y is arbitrary. Write Y as a union S of affine open subschemes Y = Yi. Then by the affine case, Yi ×X Z exists for each Yi. For each pair of overlapping open sets Y ∩ Y , set U = p−1(Y ∩ Y ) ⊆ Y × Z, where p i j ij Yi i j i X Yi is the morphism Yi ×X Z → Yi. Then it’s easy to verify that Uij = (Yi ∩ Yj) ×X Z (that is, Uij satisfies the definition of the fibre product for Yi ∩ Yj and Z over X), and by the universal property, there are unique isomorphisms ϕij : Uij → Uji for each overlapping pair, commuting with all projections. Therefore we may glue together the fibre products Yi ×X Z along the isomorphisms ϕij to get a scheme Y ×X Z which then satisfies the definition of the fibre product for Y and Z over X. Now, covering Z by affine open subschemes and repeating this process will construct Y ×X Z for any schemes Y,Z over an affine scheme X. S Finally, let X be an arbitrary scheme and write X = Xi for affine open subschemes Xi. −1 Let q : Y → X and r : Z → X be the given morphisms and for each Xi, set Yi = q (Xi) −1 and Zi = r (Xi). By the affine case, each Yi ×Xi Zi exists, but any morphisms f : W → Yi and g : W → Z making the diagram

61 3.3 Properties of Schemes 3 Scheme Theory

Yi Z

commute must satisfy g(W ) ⊆ Zi. It follows that Yi ×Xi Z = Yi ×X Zi and the gluing procedure from above allows us to construct Y ×X Z from these. Hence the ﬁbre product exists in every case.

Definition. Let p : Y → X be a morphism of schemes, x ∈ X a point and k(x) = OX,x/mx the residue field at x, with natural map Spec k(x) ,→ X. Then the fibre of p at x is the fibre product Yx := Y ×X Spec k(x). Lemma 3.3.6. Let p : Y → X be a morphism of schemes and x ∈ X any point. Then

(a) The ﬁbre Yx = Y ×X Spec k(x) is a scheme over the point Spec k(x).

−1 (b) The underlying topological space of Yx is homeomorphic to the set p (x) of preimages of x. p (c) The assignment (Y −→ X) 7→ Yx is functorial. Example 3.3.7. Let A be a DVR and consider the aﬃne scheme X = Spec A. We saw in Example 3.2.4 that X has a closed point m = mA and a generic point (0). For any morphism p : Y → X, there are two ﬁbres:

The generic ﬁbre Y(0), which is an open subscheme of Y

The special ﬁbre Ym, which is a closed subscheme of Y .

Let Y be a scheme over X and deﬁne the diagonal map ∆ : Y → Y ×X Y coming from the universal property applied to the diagram Y

∆ id id Y ×X Y

Y Y

62 3.3 Properties of Schemes 3 Scheme Theory

Definition. A morphism Y → X is called separated if the diagonal ∆ : Y → Y ×X Y is a closed immersion of schemes. We will say a scheme Y over X is separated if the corresponding morphism Y → X is separated, and a scheme is simply separated if it is separated as a scheme over Spec Z. Example 3.3.8. Let X = Spec A and Y = Spec B be affine schemes, with Y → X a morphism between them. This corresponds to a ring homomorphism A → B which makes B into an A-module. The diagonal ∆ : Y → Y ×X Y corresponds to the multiplication map 0 0 B ⊗A B → B, b ⊗ b 7→ bb , which is a homomorphism of B-modules. This map is clearly surjective, so ∆ is a closed immersion. Hence every affine scheme (and morphism of affine schemes) is separated.

Example 3.3.9. One can show that the aﬃne line with the origin doubled (Example 3.2.7) is not separated as a scheme over Spec k.

One perspective on separatedness is that it is a suitable replacement for the Hausdorﬀ condition in algebraic geometry. In the Zariski topology on any scheme, there are always proper open subsets that are dense, so the Hausdorﬀ property usually fails to hold.

Definition. A morphism f : Y → X is of finite type if there exists an affine covering S −1 −1 X = Ui, with Ui = Spec Ai, such that each f (Ui) has an open covering f (Ui) = Sni j=1 Spec Bij for ni < ∞ and Bij a finitely generated Ai-algebra. Further, we say f is −1 a finite morphism if each ni = 1, i.e. f (Ui) = Spec Bi for some finitely generated Ai-algebra Bi. Definition. A separated morphism f : Y → X is proper if it is of finite type and for every morphism Z → X, the base change morphism Y ×X Z → Z is closed. Lemma 3.3.10. Let X,Y,Z be noetherian schemes. Then for any morphism f : Y → X,

(a) If f is an open immersion, then f is separated.

(b) If f is a closed immersion, then f is separated and proper.

(d) If Z is a scheme over X, the base change Y ×X Z → Z is separated and proper. (e) If f is ﬁnite, then f is proper.

Example 3.3.11. (Projective line over a scheme) Let X = Spec A be an affine scheme. 1 1 Then the “affine line” AX = Spec A[t] is an affine scheme over X. Set X1 = AX = Spec A[t] −1 −1 and X2 = A[t ]. Then each contains an open subscheme isomorphic to U = Spec A[t, t ], coming from applying Spec to the diagram of A-algebras

63 3.4 Sheaves of Modules 3 Scheme Theory

A[t, t−1]

A[t] A[t−1]

1 Gluing along these isomorphic open subschemes gives us a scheme PX = X1 ∪U X2, called 1 1 the projective line over X. In the affine case, we will write PX = PA. S When X is an arbitrary scheme, X has a covering by open affine subschemes X = Ui 1 and a gluing construction defines the projective line PX . n Example 3.3.12. More generally, one defines projective n-space over X, written PX , by 1 gluing together n + 1 copies of affine n-space AX = Spec A[t1, . . . , tn] along the isomorphic open subsets Xi = {ti 6= 0} (when X is affine; in the general case, glue affine subschemes n together as in the previous example). The natural morphism PX → X generalizes in the following way. Definition. A morphism of schemes Y → X is projective if it factors through a closed n immersion Y → PX for some n ≥ 1. Theorem 3.3.13. If f : Y → X is a projective morphism of noetherian schemes, then f is proper.

n The idea behind the proof of Theorem 3.3.13 is to ﬁrst prove Pk → Spec k is proper n for any n, which is a straightforward adaptation of the proof when Pk is considered as a projective algebraic variety. One can then modify this proof for n → Spec and then use PZ Z the properties of proper morphisms in Lemma 3.3.10 to obtain the general result.

3.4 Sheaves of Modules

Through Proposition 3.2.2, we are able to transfer commutative ring theory to the language of aﬃne schemes. In this section, we deﬁne a suitable setting for transferring module theory to the language of sheaves and schemes.

Deﬁnition. Let (X, OX ) be a ringed space. A sheaf of OX -modules, or an OX -module for short, is a sheaf of abelian groups F on X such that each F(U) is an OX (U)-module and for each inclusion of open sets V ⊆ U, the following diagram commutes:

OX (U) × F(U) F(U)

OX (V ) × F(V ) F(V )

64 3.4 Sheaves of Modules 3 Scheme Theory

If F(U) ⊆ OX (U) is an ideal for each open set U, then we call F a sheaf of ideals on X. Example 3.4.1. Let f : Y → X be a morphism of ringed spaces. Then the pushforward # sheaf f∗OY is naturally an OX -module on X via f : OX → f∗OY . Additionally, the kernel # # sheaf of f , deﬁned on open sets by (ker f )(U) = ker(OX (U) → f∗OY (U)), is a sheaf of ideals on X.

Most module terminology extends to sheaves of OX -modules. For example,

A morphism of OX -modules is a morphism of sheaves F → G such that each F(U) →

G(U) is an OX (U)-module map. We write HomX (F, G) = HomOX (F, G) for the set of morphisms F → G as OX -modules. This deﬁnes the category of OX -modules, written OX -Mod.

Taking kernels, cokernels and images of morphisms of OX -modules again give OX - modules.

Taking quotients of OX -modules by OX -submodules again give OX -modules.

0 00 0 An exact sequence of OX -modules is a sequence F → F → F such that each F (U) → 00 F(U) → F (U) is an exact sequence of OX (U)-modules.

Basically any functor on modules over a ring generalizes to an operation on OX -

modules, including Hom, written HomOX (F, G); direct product F ⊗ G; tensor product Vn F ⊗OX G; and exterior powers F. The most important of these constructions for our purposes will be the direct sum operation. ∼ ⊕r Deﬁnition. An OX -module F is free (of rank r) if F = O as OX -modules. F is locally S X free if X has a covering X = Ui such that each F|Ui is free as an OX |Ui -module.

Remark. The rank of a locally free sheaf of OX -modules is constant on connected components. In particular, the rank of a locally free OX -module is well-deﬁned whenever X is connected.

Definition. A locally free OX -module of rank 1 is called an invertible sheaf. Let A be a ring, M an A-module and set X = Spec A. To extend module theory to the language of schemes, we want to define an OX -module Mf on X. To start, for each p ∈ Spec A, let Mp = M ⊗A Ap be the localization of the module M at p. Then Mp is an m Ap-module consisting of ‘formal fractions’ s where m ∈ M and s ∈ S = A r p. For each open set U ⊆ X, define ( ) a m Mf(U) = h : U → Mp s(p) ∈ Mp, ∃ p ∈ V ⊆ U, m ∈ M, s ∈ A with s(q) = for all q ∈ V . s p∈U

(Compare this to the construction of the structure sheaf OA on Spec A in Section 3.1. Also, note that necessarily the s ∈ A in the deﬁnition above must lie outside of all q ∈ V .)

65 3.4 Sheaves of Modules 3 Scheme Theory

Proposition 3.4.2. Let M be an A-module and X = Spec A. Then Mf is a sheaf of OX - modules on X, and moreover, ∼ (1) For any p ∈ Spec A, Mfp = Mp as rings.

(2) Γ(X, Mf) ∼= M as A-modules. ∼ (3) For any f ∈ A, Mf(D(f)) = Mf = M ⊗A Af as A-modules. The proof is similar to the proof of Theorem 3.1.4; both can be found in Hartshorne.

Proposition 3.4.3. Let X = Spec A. Then the association

A-Mod −→ OX -Mod M 7−→ Mf deﬁnes an exact, fully faithful functor.

Proof. Similar to the proof of Proposition 3.2.2.

These Mf will be our affine model for modules over a scheme X. We next define the general notion, along with an analogue of finitely generated modules over a ring.

Deﬁnition. Let (X, OX ) be a scheme. An OX -module F is quasi-coherent if there is an S ∼ aﬃne covering X = Xi, with Xi = Spec Ai, and Ai-modules Mi such that F|Xi = Mfi as

OX |Xi -modules. Further, we say F is coherent if each Mi is a ﬁnitely generated Ai-module.

Example 3.4.4. For any scheme X, the structure sheaf OX is obviously a coherent sheaf on X.

Let QCohX (resp. CohX ) be the category of quasi-coherent (resp. coherent) sheaves of OX -modules on X.

Theorem 3.4.5. QCohX and CohX are abelian categories. Example 3.4.6. Let X = Spec A, I ⊆ A an ideal and Y = Spec(A/I). Then the natural ∼ inclusion i : Y,→ X is a closed immersion by deﬁnition, and it turns out that i∗OY = A/Ig as OX -modules, so i∗OY is a quasi-coherent, even coherent, sheaf on X.

We next identify the image of the functor M 7→ Mf from Proposition 3.4.3. Theorem 3.4.7. Let X = Spec A. Then there is an equivalence of categories

∼ A-Mod −→ QCohX .

Moreover, if A is noetherian, this restricts to an equivalence

∼ A-mod −→ CohX where A-mod denotes the subcategory of ﬁnitely generated A-modules.

66 3.4 Sheaves of Modules 3 Scheme Theory

Proof. (Sketch) The association M 7→ Mf sends an A-module to a quasi-coherent sheaf on X = Spec A by deﬁnition of quasi-coherence. Further, one can prove that a sheaf F on X is ∼ a quasi-coherent OX -module if and only if F = Mf for an A-module M. The inverse functor QCohX → A-Mod is given by F 7→ Γ(X, F). When A is noetherian, the above extends to say that F is coherent if and only if F ∼= Mf for a ﬁnitely generated A-module M. The rest of the proof is identical. The following lemma generalizes Example 3.4.6.

Lemma 3.4.8. Let f : Y → X be a morphism of schemes and let G be a quasi-coherent sheaf on Y . Then f∗G is a quasi-coherent sheaf on X. Further, if G is coherent and f is a ﬁnite morphism, then f∗G is also coherent. Note that the second statement is false in general. Next, we construct an important example of a quasi-coherent sheaf on a scheme. As always, we begin with a construction on rings.

Definition. Let A → B be a ring homomorphism. The module of relative differentials for B/A is defined to be 1 ΩB/A := Zhdb | b ∈ Bi/N, the quotient of the free B-module generated by formal symbols db for all b ∈ B by the submodule N = hda, d(b + b0) − db − db0, d(bb0) − b(db0) − (db)b0i. This is the universal B- module for these three relations.

Example 3.4.9. If A = k is a ﬁeld and B is a ﬁnitely generated k-algebra, write B = n k[t1, . . . , tn]/(f1, . . . , fr). Then B is the coordinate ring of the variety in Ak cut out by the fi and * n + X ∂fj Ω1 = khdt i/ dt B/k i ∂t i i=1 i is the module of total derivatives on this variety.

Lemma 3.4.10. Let A → B be a ring homomorphism. Then

(a) For any A-algebra C, Ω1 ∼= Ω1 ⊗ C. B⊗AC/C B/A A 1 ∼ −1 1 1 −1 (b) For any multiplicative set S ⊆ B, ΩS−1B/A = S ΩB/A = ωB/A ⊗B S B.

1 That is, the functor B 7→ ΩB/A commutes with base change and localization. We now give the analogous construction for OX -modules, starting in the affine case. Definition. Let A → B be a ring homomorphism. The sheaf of relative differentials is 1 the OB-module ΩeB/A on Spec B defined by the module ΩB/A. Lemma 3.4.11. Let A → B be a ring homomorphism. Then

(a) ΩeB/A is a quasi-coherent sheaf on Spec B.

67 3.4 Sheaves of Modules 3 Scheme Theory

∼ (b) For any element f ∈ B, ΩeB/A(D(f)) = ΩBf /A where Bf is the localization of B at powers of f.

Now consider the map m : B ⊗A B → B, m(b1 ⊗ b2) = b1b2 from Example 3.3.8. Let I ∼ be the kernel of m. Since m is surjective, this means B ⊗A B/I = B. Since I acts trivially 2 2 on I/I , there is an induced module action of B ⊗A B/I on I/I , and thus a corresponding B-module structure on I/I2. The proof of the following fact can be found in Eisenbud, among other places.

∼ 2 Lemma 3.4.12. ΩB/A = I/I .

∼ 2 Example 3.4.13. In Example 3.4.9, the isomorphism ΩB/k = I/I is induced by the map

B −→ ΩB/k

ti 7−→ dti.

Let Y → X be a separated morphism of schemes and let ∆ : Y → Y ×X Y be the # corresponding diagonal. This induces a morphism of sheaves ∆ : OY ×X Y → ∆∗OX which has kernel sheaf I (a sheaf on Y ×X Y ). This I in fact deﬁnes the closed subscheme ∆(Y ) ⊆ Y ×X Y . Lemma 3.4.14. For Y → X, ∆ and I as above, ∼ (a) O∆(Y ) = OY ×X Y /I as sheaves on ∆(Y ).

2 (b) I/I is an O∆(Y )-module.

Identifying Y with its image ∆(Y ) in the fibre product Y ×X Y allows us to define a sheaf analogue of the module of differentials by pulling back I/I2.

Deﬁnition. For a separated morphism Y → X, the sheaf of relative diﬀerentials ΩY/X is the pullback:

2 ΩY/X I/I

Y ∆(Y ) ∆

Remark. ΩY/X is a sheaf of OY -modules on Y . Moreover, on an affine patch Spec B ⊆ Y , the ∼ 2 sheaf of relative differentials restricts to ΩeB/A = I/Ig for some rings A → B. In particular, ΩY/X is quasi-coherent. We finish the section by discussing some applications of relative differentials. Let k be a field and X a connected scheme over Spec k of finite type and dimension d.

Deﬁnition. A k-scheme X is smooth over k if the sheaf of relative diﬀerentials ΩX/ Spec k is locally free of rank d.

68 3.4 Sheaves of Modules 3 Scheme Theory

Theorem 3.4.15. Assume k is algebraically closed and dim X = d. Then the following are equivalent:

(1) X is smooth over k. ∼ (2) For every aﬃne open subset U = Spec(k[t1, . . . , tn]/(f1, . . . , fm)), the Jacobian matrix ∂fi JP = (P ) ∂tj has rank n − d at all closed points P ∈ X.

(3) For every closed point P ∈ X, the stalk OX,P is a regular local ring.

This amazing result unites geometry (ΩX/ Spec k being locally free), algebra (OX,P being regular) and analysis (the vanishing of partial derivatives in JP ) into one concept of smooth- ness. Unfortunately, the theorem fails when k is not algebraically closed, but it still hints at a deep intersection between all three areas of math.

Theorem 3.4.16. Let X be any variety over an algebraically closed ﬁeld. Then there is a dense open subset which is smooth.

Recall that a finitely generated k-algebra A is finite étaleif A = L1 × · · · × Lr for finite, separable extensions Li/k.

1 Proposition 3.4.17. A finitely generated k-algebra A is finite étaleif and only if ΩA/k = 0. Proof. See Eisenbud. Finally, we give an important construction relating the geometry and algebra of a smooth scheme. Let X be smooth over a field k and let n = dim X.

Deﬁnition. The canonical sheaf of X is the nth exterior power sheaf

^n ωX := ΩX/ Spec k.

Here are some interesting facts about the canonical sheaf:

ωX is an invertible sheaf on X.

One can deﬁne the geometric genus of X by g(X) := dimk Γ(X, ωX ). Then when X is a curve (a smooth scheme of dimension 1), g(X) is equal to the arithmetic genus of X, another important algebraic invariant. These genera are not equal in general.

If X is a curve over C, then the genus of the corresponding Riemann surface X(C) is precisely g(X), so the canonical bundle carries important topological information about X(C).

69 3.5 Group Schemes 3 Scheme Theory

3.5 Group Schemes

Recall that a group is a set G together with three maps,

µ : G × G −→ G (multiplication) e : {e} ,−→ G (identity) i : G −→ G (inverse) satisfying associativity, identity and and inversion axioms. This generalizes to the notion of a group object in an arbitrary category C. We state the deﬁnition for scheme categories here.

p Deﬁnition. A group scheme over a scheme X is a scheme G −→ X with morphisms

µ : G × G −→ G e : {e} ,−→ G i : G −→ G satisfying the following axioms:

(1) (Associativity) µ ◦ (id × µ) = µ ◦ (µ × id).

(2) (Identity) µ ◦ (id × e) = id = µ ◦ (e × id).

(3) (Inversion) µ ◦ (id × i) = e ◦ p = µ ◦ (i × id).

Definition. A group scheme G over X is finite if p : G → X is a finite morphism, and flat if p : G → X is a flat morphism, i.e. p∗OG is a sheaf of flat OX -modules. Remark. When G is a finite group scheme over X, flat is equivalent to locally free.

The following describes an equivalent, and equally important, perspective on group schemes using the language of functors.

Proposition 3.5.1. Let G be a scheme over X. Then a choice of group scheme structure on G is equivalent to a compatible choice of group structure on the sets HomX (Y,G) for all schemes Y over X. That is, a group scheme structure is a functor SchX → Groups such that the composition with the forgetful functor Groups → Sets is representable.

Proof. This is a basic application of Yoneda’s Lemma.

Example 3.5.2. Let G be a finite group of order n and let X be any scheme. The constant Qn group scheme on G over X is defined as GX := i=1 X, with projection map induced by the identity on each disjoint copy of X. Multiplication GX ×X GX → GX is given by sending Qn (P,Q), where P = x ∈ X in the gith component of i=1 X and Q = x in the gjth component (gi, gj ∈ G) to the corresponding point PQ = x in the gigjth component of the disjoint union. (Note that P and Q must correspond to the same point x ∈ X by definition of the fibre product – draw the diagram!) Similarly, the identity is the morphism taking X onto the Qn copy of X indexed by the identity element eG ∈ G, e : X → XeG ⊆ i=1 X. Finally, the

70 3.5 Group Schemes 3 Scheme Theory

inversion morphism i : GX → GX takes P in the gith component to the corresponding point −1 −1 P in the gi th component. This definition can be extended to an arbitrary group G. Note that when G is finite, GX is a finite (étale)group scheme over X. A special case of this is the trivial group scheme {1}X = X. Thus every scheme is a group scheme.

1 Example 3.5.3. Let X = Spec A be affine and recall the affine line AX = Spec A[t] con- 1 structed in Example 3.3.11. Then AX is an affine group scheme over X, denoted Ga, called the additive group scheme over X, with morphisms induced by the following ring homomorphisms:

µ∗ : A[t] −→ A[t] ⊗ A[t] t 7−→ t ⊗ 1 + 1 ⊗ t e∗ : A[t] −→ A t 7−→ 0 i∗ : A[t] −→ A[t] t 7−→ −t.

Notice that these are just the axioms of a Hopf algebra! This construction generalizes to the aﬃne line over a non-aﬃne scheme as well.

Example 3.5.4. For X = Spec A, the multiplicative group scheme over X is Gm := Spec A[t, t−1] with morphisms induced by

µ∗ : A[t, t−1] −→ A[t, t−1] ⊗ A[t, t−1] t 7−→ t ⊗ t t−1 7−→ t−1 ⊗ t−1 e∗ : A[t, t−1] −→ A t, t−1 7−→ 1 i∗ : A[t, t−1] −→ A[t, t−1] t 7−→ t−1 t−1 7−→ t.

Note that when A = k is a field, these are just the schematic versions of the algebraic groups Ga,k and Gm,k. This shows that group schemes are a direct generalization of algebraic groups. Example 3.5.5. For X = Spec A, the nth roots of unity form a group scheme defined by −1 n µn = Spec(A[t, t ]/(t − 1)). This is a finite group subscheme of Gm.

p Example 3.5.6. If char A = p > 0, then αp = Spec(A[t]/(t )) deﬁnes a group scheme over Spec A which is isomorphic as a scheme to Spec A, but not as a group scheme!

Example 3.5.7. The Jacobian of a curve is a group scheme. In particular, an elliptic curve (a dimension 1 scheme with a speciﬁed point O) is a group scheme with identity O.

71 3.5 Group Schemes 3 Scheme Theory

p q Definition. Let G −→ X be a finite, flat group scheme. A left G-torsor is a scheme Y −→ X with q finite, locally free and surjective, together with a group action ρ : G ×X Y → Y , which satisfies:

(1) ρ ◦ (e × idY ) is equal to the projection map X ×X Y → Y .

(2) ρ ◦ (idG × ρ) = ρ ◦ (µ × idY ): G ×X G ×X Y → G ×X Y → Y .

(3) ρ × idY : G ×X Y → Y ×X Y is an isomorphism of X-schemes. Right G-torsors are deﬁned similarly.

Remark. The idea is that a G-torsor is exactly the same as G, except we have forgotten the “identity point” e (which is a morphism, not a point).

Example 3.5.8. Left multiplication deﬁnes a G-torsor structure on G itself.

Example 3.5.9. Let k be a field and m an integer such that char k - m. Let µm = −1 m Spec(k[t, t ]/(t − 1)) be the group scheme of mth roots of unity over√ Spec k. Take a ∈ k×/(k×)m (that is, a is not an mth power in k) and set L = k( m a), which is a finite field extension of k. We claim Y = Spec L is a µm-torsor over Spec k. Let ζ be a primitive mth root of unity. Up to finite base change, we may assume ζ ∈ k. Define √ √ ∗ m −1 m m ρ : k( a) −→ k[t, t ]/(t − 1) ⊗k k( a) √ √ m a 7−→ ζ ⊗ m a.

This deﬁnes a morphism ρ : µm ×Spec k Y → Y and one can prove that it satisﬁes axioms (1) ∗ and (2) of a torsor by checking the corresponding properties for ρ . When char k - m, µm ∼ is a reduced scheme over Spec k and it’s easy to see that µm = (Z/mZ)Spec k, the constant ∼ group scheme on Z/mZ over Spec k. Moreover, L/k is a Galois extension with Gal(L/k) = ∼ Qm Z/mZ, L ⊗k L = i=1 L and this has a corresponding Galois action which induces the ∼ −1 m isomorphism L⊗k L −→ k[t, t ]/(t −1)⊗k L. Applying Spec again, we get the isomorphism ∼ µm ×Spec k Y −→ Y ×Spec k Y so Y is indeed a µm-torsor. Using Artin-Schreier√ theory, one can show that every µm-torsor arises in this way, i.e. as Spec L for L = k( m a).

72 4 Fundamental Groups of Schemes

4 Fundamental Groups of Schemes

In this chapter we define the étalefundamental group of a scheme. This will generalize the construction for curves in Section 1.5 and allow us to fully translate the topological theory of covering spaces to an algebraic setting. We first recall the so-called Galois theory for covering spaces. Let p : Y → X be a topological cover and let Aut(Y/X) be the group of automorphisms of the cover, i.e. the homeomorphisms Y → Y making the diagram Y Y

X commute. If p is a ﬁnite, degree n cover, then # Aut(Y/X) ≤ n. When # Aut(Y/X) = n, we call p a regular, or Galois cover. In this case, we write Gal(Y/X) = Aut(Y/X). Proposition 4.0.1. For a covering space p : Y → X of degree n, there is a bijective correspondence {subgroups of Gal(Y/X)} ←→ {intermediate covers Y → Y 0 → X} p¯ H 7−→ (Y/H −→ X) Aut(Y/Y 0) 7−→ (Y 0 → X).

Example 4.0.2. Let X = Y = C r {0}, the punctured complex plane, with complex parameters x and y, respectively. Then an example of a Galois cover of degree n is the nth power map f : Y → X, y 7→ yn. For example, when n = 2 the real coordinates of ths cover take the form of a parabola covering the line in all but one point: Y

n ∼ In general, f : y 7→ y is a Galois cover with Galois group Gal(Y/X) = Z/nZ generated by the automorphism ϕ : Y → Y, y 7→ ζny, where ζn is a primitive nth root of unity. Viewing X and Y as complex varieties, f and ϕ are in fact morphisms of varieties which determine corresponding ﬁeld embeddings:

C(x) ,−→ C(y) and C(y) ,−→ C(y) n x 7−→ y y 7−→ ζny. We want to generalize this to any variety or scheme.

73 4.1 Galois Theory for Schemes 4 Fundamental Groups of Schemes

4.1 Galois Theory for Schemes

Let X and Y be schemes.

Definition. A finite morphism ϕ : Y → X is locally free if the direct image sheaf ϕ∗OY is locally free of finite rank as an OX -module.

Definition. Suppose ϕ : Y → X is finite and locally free. If the fibre YP over a point P ∈ X is equal to the ring spectrum Spec A of some finite étale κ(P )-algebra A, we say ϕ is étale at P . If this holds for each point P ∈ X, we call ϕ a finite étalemorphism. Finally, if ϕ is also surjective, it is called a finite étalecover of schemes. Remark. Note that if A is a local ring, a finitely generated A-module M is free if and only if it is flat. It follows that ϕ : Y → X is locally free if and only if ϕ∗OY is a sheaf of flat OX -modules. Lemma 4.1.1. Let ϕ : Y → X be a finite étalemorphism. Then (a) If ψ : Z → Y is finite étale,then so is ϕ ◦ ψ : Z → X.

(b) If ψ : Z → Y is any morphism, then the base change Y ×X Z → Z is ﬁnite ´etale.

The category FétX is defined to be the full subcategory of SchX consisting of finite étale ϕ covers Y −→ X. To compare to the topological case of a covering space (see Section 0.1), we define: Definition. A geometric point of X is a morphism x¯ : Spec Ω → X for some algebraically closed field Ω. Concretely, the image ofx ¯ is some point x ∈ X for which κ(x) ⊆ Ω. Definition. Let ϕ : Y → X be a morphism and x¯ : Spec Ω → X a geometric point. Then the geometric fibre of x¯ is the fibre product Yx¯ := Y ×X Spec Ω. Proposition 4.1.2. For a morphism ϕ : Y → X, ϕ is étaleat each P ∈ X if and only if every geometric fibre Yx¯ of ϕ is of the form Spec(Ω × · · · × Ω), where Ω ⊇ κ(P ) is an algebraically closed field. Proof. This comes from the algebraic fact that for any field k, a finitely generated k-algebra ¯ ¯ A is finite étaleif and only if A ⊗k k is isomorphic to a finite direct sum of copies of k, where k¯ is an algebraic closure of k.

Example 4.1.3. The cover C r {0} → C r {0}, y 7→ yn in Example 4.0.2 is a finite étale cover. Example 4.1.4. Let X be a normal scheme of dimension 1 and let ϕ : Y → X be a finite morphism. In the affine case, with X = Spec A and Y = Spec B, this corresponds to an ∗ extension of Dedekind rings ϕ : A → B. Then points in the fibre YP of a point P ∈ X correspond to a prime factorization of ideals:

r Y ei PB = Qi i=1

74 4.1 Galois Theory for Schemes 4 Fundamental Groups of Schemes

where Q1,...,Qr are prime ideals of B. Then YP is a finite étalealgebra if and only if ei = 1, i.e. the prime P is unramified in the language of algebraic number theory. In the general case, a finite morphism of normal schemes of dimension 1 is étaleif and only if each affine piece corresponds to a finite, unramified extension of Dedekind rings.

Proposition 4.1.5. If ϕ : Y → X is a ﬁnite, locally free morphisim of schemes then the following are equivalent:

(1) ϕ is ´etale.

(2) ΩY/X = 0.

(3) The diagonal ∆ : Y → Y ×X Y induces an embedding Y,→ ∆(Y ) as an open and closed subscheme of Y ×X Y . Proof. (1) ⇐⇒ (2) When X and Y are affine, the equivalence is implied by Proposi- tion 3.4.17. Then the general case is obtained by base change and localization properties of differentials (Lemma 3.4.10). (2) =⇒ (3) Since ϕ is finite, it is separated and thus by definition the diagonal ∆ : Y → Y ×X Y is a closed immersion. By Lemma 3.4.14, ∆(Y ) corresponds to a sheaf of ideals I 2 on Y ×X Y such that ΩY/X is naturally identified with the pullback of I/I along ∆. By assumption ΩY/X = 0, so it follows that IP = 0 for all P ∈ ∆(Y ) and consequentially I = 0 on an open subset of Y ×X Y . Finally, this means ∆(Y ) is open and closed in Y ×X Y . (3) =⇒ (1) We will use the criterion in Proposition 4.1.2. Fix a geometric point x¯ : Spec Ω → X and consider the base change of the geometric fibre Yx¯:

Yx¯ ×X Y = (Spec Ω ×X Y ) ×Spec Ω Spec Ω ×X Y = Yx¯ ×Spec ω Yx¯.

This comes equipped with a morphism ∆x¯ : Yx¯ → Yx¯ ×X Y . Since (3) is consistent under base change, ∆x¯ is then an isomorphism from Yx¯ to an open and closed subscheme of Yx¯ ×Spec Ω Yx¯. Now Yx¯ is concretely the spectrum of a finite dimensional Ω-algebra so schematically it has finitely many points, each having residue field Ω. For any of these pointsy ¯ : Spec Ω → Yx¯, one more base change withy ¯ as above yields an open and closed immersion Spec Ω ,→ Yx¯. But since Spec Ω is connected, the image of this map is a connected component of Yx¯. Hence Yx¯ is a finite disjoint union of schematic points, and thus the criterion in Proposition 4.1.2 for ϕ to be étaleis satisfied. We next translate the ‘local triviality’ condition from covering space theory to schemes.

Proposition 4.1.6. Let X be a connected scheme and ϕ : Y → X an affine surjective morphism. Then ϕ is a finite étalecover if and only if there exist a locally free, surjective morphism ψ : Z → X such that Y ×X Z is a trivial cover of Z, i.e. a disjoint union of copies of Z with the morphism Y ×X Z → Z equal to the identity on each piece. Proof. ( =⇒ ) The assumption that X is connected means every fibre of ϕ has the same cardinality, say n. If n = 1, the property is trivial, so we may induct on n. By (3) of ` 0 Proposition 4.1.5, the diagonal Y → Y ×X Y induces a decomposition Y ×X Y = ∆(Y ) Y 0 0 for Y an open and closed subscheme of Y ×X Y . The maps Y ,→ Y ×X Y (obvious)

75 4.1 Galois Theory for Schemes 4 Fundamental Groups of Schemes

and Y ×X Y → Y (Lemma 4.1.1(b)) are finite étale. Thus Lemma 4.1.1(a) implies their composition Y 0 → Y is finite étale as well. But by construction the fibres of Y 0 → Y have cardinality n − 1 so by induction, there exists a finite, locally free, surjective morphism 0 0 ψ : Z → Y such that Y ×Y Z consists of n − 1 disjoint copies of Z. Then finally the composition ψ = ϕ ◦ ψ0 : Z → X is finite, locally free and surjective (it is the composition ∼ of two such maps) and (Y ×X Y ) ×Y Z = Y ×X Z is the disjoint union of n copies of Z. ( ⇒ = ) Since ψ is locally free, each P ∈ X has an open neighborhood U = Spec A such that ψ−1(U) = Spec C for some finitely generated free A-algebra C. Let ϕ−1(U) ⊆ Y be ϕ−1(U) = Spec B for B an A-module. Then the base change of ϕ to ψ over ϕ−1(U) is given by Spec(B ⊗A C) → Spec B. By linear algebra, B ⊗A C is a finitely generated, free C-module so it is also finitely generated and free as an A-module. Hence B must be finitely generated and free as an A-module, too. We next show ϕ is étale using Proposition 4.1.2. Letz ¯ : Spec Ω → Z be a geometric point of Z; thenx ¯ = ψ ◦ z¯ : Spec Ω → X is a geometric point of X. There is a natural isomorphism of fibres ∼ Yx¯ = Spec Ω ×X Y = Spec Ω ×Z (Y ×X Z) = (Y ×X Z)z¯

but the hypothesis implies (Y ×X Z)z¯ is a disjoint union of copies of Spec Ω. Thus Yx¯ is a disjoint union of points, and since ψ is surjective, every geometric fibre of ϕ is as well. Hence Proposition 4.1.2 implies ϕ is étale. Recall from Section 3.3 that the codimension of a point P ∈ X is defined as the dimension of the local ring OX,P (or the height of the prime associated to P in any affine neighborhood). The powerful Zariski-Nagata purity theorem says that to show a map is étale,it suffices to check the étaleproperty at all codimension 1 points of the base scheme. Theorem 4.1.7 (Zariski-Nagata Purity Theorem). If ϕ : Y → X is a finite surjective morphism of integral schemes, with Y normal and X regular, such that the fibre YP of ϕ above each codimension 1 point P ∈ X is étaleover κ(P ), then ϕ itself is étale. Corollary 4.1.8. Let X be a regular integral scheme and U ⊆ X an open subscheme consisting of points of codimension at least 2. Then there is an equivalence of categories

F´etX −→ F´etU

Y 7−→ Y ×X U. Proposition 4.1.9. Let ϕ : Y → X be a finite étalecover, z¯ : Spec Ω → z a geometric point and Z a connected scheme over X. If f, g : Z → Y are two X-morphisms such that f ◦ z¯ = g ◦ z¯, then f = g. Proof. By Lemma 4.1.1(b), we may assume X = Z, so that f and g are two sections of ϕ : Y → X. It follows that f and g are finite étalemorphisms and each induces an isomorphism of Z = X with an open and closed subscheme of Y . Since Z is connected, the images of f and g are determined by the images of any geometric point, hence f ◦ z¯ = g ◦ z¯ implies f = g. Corollary 4.1.10. Let ϕ : Y → X be a finite étalecover. Then Aut(Y/X) is finite.

76 4.1 Galois Theory for Schemes 4 Fundamental Groups of Schemes

Proof. Take σ ∈ Aut(Y/X), σ 6= 1, and set f = ϕ and g = ϕ ◦ σ. Then by Proposition 4.1.9, f and g send some geometric point of Y to different points. In other words, the action of Aut(Y/X) on any geometric fibre is free, or the permutation representation of Aut(Y/X) on any of the geometric fibres is faithful. Now since the map is finite étale,each geometric fibre is finite as a set, so this implies Aut(Y/X) is itself finite. Now let ϕ : Y → X be a finite étalecover and let G be a group scheme over X such that Y is a left G-torsor (see Section 3.5). Let Y/G be the quotient space with projection map G π : Y → Y/G. We define a sheaf on Y/G by OY/G := (π∗OY ) , the subsheaf of G-invariants of the pushforward of OY to Y/G along π. This makes Y/G into a ringed space.

Proposition 4.1.11. The ringed space (Y/G, OY/G) is a scheme over X. Moreover, ϕ : Y → X factors through a ﬁnite morphism ψ : Y/G → X.

The following are analogues of the basic Galois theory of covering spaces (see Section 0.1, e.g.).

Proposition 4.1.12. If ϕ : Y → X is a connected, finite étalecover and G ≤ Aut(Y/X) is any finite subgroup of automorphisms, then π : Y → Y/G is a finite étalecover.

Definition. A connected, finite étalecover ϕ : Y → X is a Galois cover if Aut(Y/X) acts transitively on every geometric fibre of ϕ.

Theorem 4.1.13. Let ϕ : Y → X be a Galois cover and suppose ψ : Z → X is a connected, ﬁnite ´etalecover such that Z is a scheme over Y and the diagram

Y Z

ϕ ψ

commutes. Then

(1) Y → Z is a Galois cover and Z ∼= Y/G for some subgroup G ≤ Aut(Y/X). (2) There is a bijection

{subgroups G ≤ Aut(Y/X)} ←→ {intermediate covers Y → Z → X}.

(3) The correspondence is bijective on normal subgroups of Aut(Y/X) and Galois covers Z → X, and in this case Aut(Z/X) ∼= Aut(Y/X)/G as groups.

77 4.2 The Etale´ Fundamental Group 4 Fundamental Groups of Schemes

4.2 The Etale´ Fundamental Group

Let X be a scheme and FétX the category of finite étalecovers of X. Fix a geometric point x¯ : Spec Ω → X and let

Fibx¯ : F´etX −→ Sets

Y 7−→ Yx¯ = Y ×X Spec Ω be the fibre functor overx ¯. By Lemma 3.3.6(c), Fibx¯ is indeed a functor. Using the definition of the automorphism group of a functor from Section 0.1, we can now define the algebraic, or étale,fundamental group of a scheme.

Definition. The algebraic, or étalefundamental group of a scheme of X at a geometric point x¯ : Spec Ω → X is the automorphism group of the fibre functor over x¯:

π1(X, x¯) := Aut(Fibx¯).

By Theorem 0.1.3, this coincides with the topological fibre functor and fundamental group. We will also see below that this definition for schemes coincides with the definition given for curves in Section 1.5.

Theorem 4.2.1 (Grothendieck). Let X be a connected scheme and x¯ : Spec Ω → X a geometric point. Then

(1) π1(X, x¯) is a proﬁnite group and its action on Fibx¯(Y ) is continuous for all Y ∈ F´etX .

(2) Fibx¯ induces an equivalence of categories

∼ F´etX −→{ﬁnite, continuous, left π1(X, x¯)-sets}

Y 7−→ Fibx¯(Y ).

Example 4.2.2. Let k be a field and consider X = Spec k. Then finite étalecovers Y → Spec k are precisely Y = Spec A for A = L1 × · · · × Lr a finite étale k-algebra. Here, the fibre functor over anyx ¯ : Spec k¯ → Spec k (equivalent to a choice of algebraic closure k¯ of k) is exhibited by ¯ Fibx¯(Y ) = Spec(A ⊗k k) = Spec(Ω × · · · × Ω), and Spec(Ω×· · ·×Ω) is a finite set of r closed points indexed by the homomorphisms A → k¯. ∼ Indeed, as we saw in Section 0.2, Fibx¯(Y ) = Homk(A, ks), where ks is the separable closure of ¯ ¯ k in k via the embedding k ,→ k. The action of Aut(Homk(−, ks)) on any given Homk(A, ks) is given by T · σ = σ ◦ T , or precisely the action of Gk = Gal(ks/k) on Homk(A, ks) from Example 0.4.2. Therefore ∼ ∼ π1(X, x¯) = Aut(Fibx¯) = Aut(Homk(−, ks)) = Gal(ks/k).

Moreover, there is an identiﬁcation (via Prop. 3.2.2) Homk(A, ks) = HomSpec k(Spec ks, Spec k).

78 4.2 The Etale´ Fundamental Group 4 Fundamental Groups of Schemes

The above shows that although Fibx¯ is not a representable functor – Spec ks is not a finite étale k-scheme – it is pro-representable in FétSpec k. Explicitly,

0 Fibx¯(Spec A) = lim HomSpec k(Y , Spec A) −→ where the direct limit is over all ﬁnite ´etaleGalois covers Y 0 → Spec k ordered by the existence of a Spec k-morphism Y 0 → Y 00.

In fact, there was nothing special about X being Spec k or even aﬃne in the last paragraph.

Proposition 4.2.3. For a connected scheme X and any geometric point x¯ : Spec Ω → X, the fibre functor Fibx¯ is pro-representable. Explicitly, for any finite étalecover Y → X,

0 Fibx¯(Y ) = lim HomX (Y ,Y ) −→ where the direct limit is over all ﬁnite ´etaleGalois covers Y 0 → Y .

Lemma 4.2.4. Every automorphism of Fibx¯ is determined by a unique automorphism of the direct system (Y 0 → X) of ﬁnite Galois covers of X.

We can now give the proof of Theorem 4.2.1.

0 Proof. (1) By Proposition 4.2.3, Fibx¯ is pro-representable by the direct system of HomX (Y ,Y ) where Y 0 ranges over all ﬁnite Galois covers of X. Now Lemma 4.2.4 implies

0 Aut(Fibx¯) = lim Aut(Y /X) ←− where again Y 0 ranges over all finite Galois covers. Since Corollary 4.1.10 says that each 0 Aut(Y /X) is a finite group, we see that π1(X, x¯) = Aut(Fibx¯) is a profinite group. Note that π1(X, x¯) has a natural action on each Fibx¯(Y ) for any finite étalecover Y → X. It remains to show this action is continuous. Take a geometric pointy ¯ ∈ Fibx¯(Y ) lying 0 overx ¯ : Spec Ω → X. Ify ¯ comes from an element of HomX (Y ,Y ) by the direct limit 0 in Proposition 4.2.3, then the action of π1(X, x¯) factors through Aut(Y /X). This implies continuity. (2) Take a finite, continuous, left π1(X, x¯)-set S. The action of π1(X, x¯) is transitive on each orbit of S, so we may assume it is transitive on S to begin with. Let H be the stabilizer of any point in S. Then H is an open subgroup of π1(X, x¯), so it contains the open normal 0 subgroup corresponding to the kernel of π1(X, x¯) → Aut(Y /X) for some finite Galois cover Y 0 → X. Let H be the image of H in Aut(Y 0/X). Then one proves S ∼= Y 0/H. This proves essential surjectivity; fully faithfulness is routine. The following shows that our definition of the fundamental group for schemes properly captures the notion we described for curves in Section 1.5.

Theorem 4.2.5. Let X be an integral normal scheme with function field K and fix a separable closure Ks/K. Let KX be the compositum of all finite subextensions L/K in Ks such that the normalization XL of X in L is étaleover X. Then

79 4.2 The Etale´ Fundamental Group 4 Fundamental Groups of Schemes

(1) KX /K is Galois. ∼ (2) For any geometric point x¯ : Spec K → X, we have π1(X, x¯) = Gal(KX /K). Proof. The proof of (1) is basically the same as the proof of Proposition 1.5.1. Then (2) follows from (2) of Theorem 4.2.1.

Example 4.2.6. Let X = Y = C r {0} and define the finite étalecover f : Y −→ X y 7−→ πyn.

(Here, we use π to ensure there are no algebraic relations on this map.) As in Example 4.1.3, ∼ f is a finite étaleGalois cover of degree n, with Galois group G = hσi = Z/nZ, where σ is the automorphism σ : y 7→ ζny for ζn a primitive nth root of unity. Notice that by setting z = π1/ny, we may define f over Q. We make use of the following important result.

Theorem 4.2.7. If a scheme X/C is defined over Q, then any finite étalecover of X is also defined over Q.

In particular, the finite cover f : y 7→ zn is defined over some number field K/Q.A classic question asks for a description of K in terms of the topology of this cover.

Set GQ = Gal(Q/Q) and fix an automorphism ω ∈ GQ. Let G be any group and suppose Y → X is a G-Galois cover which is defined over a number field K. Then Gal(Q/K) ≤ GQ and if ω lies in Gal(Q/K), then ω carries this cover Y → X to itself (the action on Y being the action of ω on the coefficients of the map). The field of moduli for the cover Y → X is defined to be the subfield M of Q/Q fixed by all ω taking Y → X to itself. Proposition 4.2.8. If G is an abelian group, then the unique maximal field of definition for Y → X is precisely M.

Note that Proposition 4.2.8 is completely open when G is nonabelian. In any case, the consequences for abelian covers is that we can see the Galois theory of ﬁnite ´etalecovers in two parallel ways:

(Geometric) Every Galois cover of schemes has a group of automorphisms and a corresponding ﬁeld extension for which this is the Galois group.

(Arithmetic) Every cover of schemes has a field of definition over Q, and in the abelian case there is even a unique minimal such field of definition.

As in Section 1.6, these related pieces are governed by a short exact sequence ¯ 1 → π1(X) → π1(X) → Gal(k/k) → 1

(here, the base point can be omitted and k can be any ﬁeld). Accordingly, π1(X) is called ¯ the geometric fundamental group of X, π1(X) the algebraic fundamental group and Gal(k/k) the arithmetic fundamental group. As in Corollary 1.6.4, the sequence splits if and only if X contains a k-rational point.

80 4.3 Properties of the Etale´ Fundamental Group 4 Fundamental Groups of Schemes

4.3 Properties of the Etale´ Fundamental Group

Let X be a connected scheme and take two geometric pointsx ¯ : Spec Ω → X andx ¯0 : Spec Ω0 → X. In the topological case (Section 0.1), there was a nice notion of a path between two points. We now describe an analogue for schemes.

Proposition 4.3.1. For any two geometric points of X, x¯ : Spec Ω → X and x¯0 : Spec Ω0 → ∼ X, there is a natural isomorphism Fibx¯ −→ Fibx¯0 . Proof. We saw in Proposition 4.2.3 that each fibre functor is pro-representable by an inverse system (Pα) of finite étaleGalois covers of X, but the choice of X-morphisms in each case 0 depends on the choices of basepoints in each Pα lying overx ¯ orx ¯ . Let the covering morphisms 0 be denoted ϕαβ, ψαβ : Pβ → Pα forx, ¯ x¯ , respectively. Then by Proposition 4.2.3, for any ∼ Y ∈ FétX , Fibx¯(Y ) = lim Hom(Pα,Y ) where the connecting homomorphisms in the direct −→ ∼ limit are those induced by the ϕαβ; and similarly Fibx¯0 (Y ) = lim Hom(Pα,Y ) with the maps −→ induced by the ψαβ. Therefore is suffices to exhibit an isomorphism of inverse systems ∼ (Pα, ϕαβ) = (Pα, ψαβ). Let α ≤ β and take λβ ∈ AutX (Pβ). Let xα ∈ Pα and xβ ∈ Pβ be the specified points 0 overx ¯ and set xα = ψαβ ◦ λβ(xβ):

xβ λ (x ) λβ β β Pβ Pβ

ϕαβ ψαβ

Pα Pα λα 0 xα xα

Since Pα is a Galois cover of X, there exist a (unique) automorphism λα ∈ AutX (Pα) 0 such that λ(xα) = xα. It follows from Proposition 4.1.9 that this λα makes the diagram above commute. Define ραβ : AutX (Pβ) → AutX (Pα) by sending λβ ∈ AutX (Pβ) to the λα described above. This gives an inverse system (AutX (Pα, ραβ) in which each AutX (Pα) is finite and nonempty, so the corresponding inverse limit lim AutX (Pα) is nonempty. That is, ←− there is at least one system of isomorphisms of (Pα) compatible with the ϕαβ and ψαβ. Corollary 4.3.2. For any pair of geometric points x¯ : Spec Ω → X and x¯0 : Spec Ω0 → X, 0 there is a continuous isomorphism of profinite groups π1(X, x¯ ) → π1(X, x¯). ∼ Proof. Any natural isomorphism λ : Fibx¯ −→ Fibx¯0 from Proposition 4.3.1 determines an ∗ 0 ∗ −1 isomorphism λ : π1(X, x¯ ) → π1(X, x¯) by λ (ϕ) = λ ◦ ϕ ◦ λ.

0 Deﬁnition. A natural isomorphism λ : Fibx¯ → Fibx¯0 is called a path from x¯ to x¯ .

∗ 0 Note that the isomorphism λ : π1(X, x¯ ) → π1(X, x¯) depends on the choice of path λ, but only up to inner automorphism.

81 4.3 Properties of the Etale´ Fundamental Group 4 Fundamental Groups of Schemes

Now let X and X0 be two connected schemes, take a geometric point in each,x ¯ : Spec Ω → X andx ¯0 : Spec Ω → X0, and suppose there is a covering morphism ϕ : X0 → X such that ϕ ◦ x¯0 =x ¯. Then ϕ induces a base change functor

Bx,¯ x¯0 : F´etX −→ F´etX0 0 Y 7−→ Y ×X X . ∼ Recall from the proof of Proposition 4.1.6 that there is a natural isomorphism Fibx¯ = Fibx¯0 ◦Bx,¯ x¯0 . This induces a map

0 0 ϕ∗ : π1(X , x¯ ) −→ π1(X, x¯).

0 0 Lemma 4.3.3. ϕ∗ : π1(X , x¯ ) → π1(X, x¯) is a continuous homomorphism of proﬁnite groups.

Proposition 4.3.4. Let ϕ : X0 → X be a cover of connected schemes. Then

0 0 (1) ϕ∗ is trivial if and only if for all connected Y ∈ F´etX , the base change Y ×X X → X is a trivial cover.

0 (2) ϕ∗ is surjective if and only if for all connected Y ∈ F´etX , the base change Y ×X X → X0 is a connected cover.

0 0 0 Proof. (1) First note that an X -cover is trivial if and only if π1(X , x¯ ) acts trivially on its 0 0 0 fibre overx ¯ . Thus if, for any λ ∈ π1(X , x¯ ) and Y ∈ FétX , λ acts trivially on Fibx¯0 (Y ×X 0 ∼ 0 X ) = Fibx¯(Y ), then ϕ∗(λ) must be trivial. Conversely, if some Y ×X X is a nontrivial 0 0 ∼ 0 cover, there exists λ ∈ π1(X , x¯ ) acting nontrivially on Fibx¯(Y ) = Fibx¯0 (Y ×X X ). Hence ϕ∗(λ) is nontrivial. 0 (2) Suppose Y ∈ FétX (connected) is such that Y ×X X is not connected. Then there 0 0 exist x1, x2 ∈ Fibx¯(Y ) such that ϕ∗(λ)(x1) 6= x2 for any λ ∈ π1(X , x¯ ). However, the fact that Y is connected implies there exists µ ∈ π1(X, x¯) such that µ(x1) = x2. Thus µ is not in the image of ϕ∗, so ϕ∗ is not surjective. Going the other direction, if ϕ∗ is not surjective then im ϕ∗ is a proper, closed subgroup of the profinite group π1(X, x¯), so there exists a 0 0 proper, open subgroup U ⊆ π1(X, x¯) such that U ⊇ im ϕ∗. Thus π1(X , x¯ ) acts trivially on the coset space π1(X, x¯)/U, or in other words, the connected cover Y → X corresponding 0 0 to π1(X, x¯)/U from Theorem 4.1.13 base changes to a trivial cover Y ×X X → X . But if 0 0 U 6= π1(X, x¯), then Y ×X X 6= X and is therefore a disconnected cover.

Remark. Suppose Y → X is connected,y ¯ is a geometric point overx ¯ and U = Stabπ1(X,x¯)(¯y). ∼ Then U is an open subgroup of π1(X, x¯) and Fibx¯(Y ) = π1(X, x¯)/U as left π1(X, x¯)-sets. As above, let ϕ : X0 → X be a cover of connected schemes such that ϕ ◦ x¯0 =x ¯.

Proposition 4.3.5. Let U ⊆ π1(X, x¯) be any open subgroup and let Y → X be the corresponding connected cover; let y¯ ∈ Fibx¯(Y ) be the geometric point such that U = Stab(¯y). 0 0 Then U ⊇ im ϕ∗ if and only if the base change Y ×X X → X has a section such that 0 0 0 0 y :=x ¯ (Spec Ω) ⊆ X maps to y :=y ¯(Y ×X Spec Ω) ⊂ Y ×X X .

82 4.3 Properties of the Etale´ Fundamental Group 4 Fundamental Groups of Schemes

Proof. Note that

U ⊇ im ϕ∗ ⇐⇒ every element of π1(X, x¯) ﬁxesy ¯

⇐⇒ the connected component of y is ﬁxed under the π1(X, x¯)-action ⇐⇒ the connected component of y is isomorphic to X0.

0 0 0 Therefore under Y ×X X → X , the component of y maps isomorphically to X , so we get 0 0 0 0 a section X ,→ Y ×X X . Conversely, any such section determines a π1(X , x¯ )-equivariant 0 map Spec Ω → Fibx¯0 (Y ×X X ) whose image must be ﬁxed. Hence U ⊇ im ϕ∗.

0 0 0 0 0 Proposition 4.3.6. Let U ⊇ π1(X , x¯ ) be an open subgroup, Y → X the corresponding 0 0 0 0 0 0 connected cover and y¯ the geometric point such that Stabπ1(X ,x¯ )(¯y ) = U . Then U ⊇ ker ϕ∗ 0 0 if and only if there exists a connected cover Y → X and an X -morphism Y0 → Y such that 0 0 Y0 is a connected component of Y ×X X → X .

0 0 Proof. ( =⇒ ) Suppose U ⊇ ker ϕ∗. We know im ϕ∗ is a subgroup in π1(X, x¯) and V := 0 ϕ∗(U ) is open in im ϕ∗, so by proﬁnite group theory, there exists an open subgroup V ⊆ 0 π1(X, x¯) such that V ∩ im ϕ∗ = V . This V corresponds to a cover Y → X. Let Y0 be the 0 0 0 connected component of Y ×X X → X containing y =y ¯(Y ×X Spec Ω) ⊆ Y ×X X . Then 0 0 y¯ ∈ Fibx¯(Y0) and there is an isomorphism of π1(X , x¯ )-sets ∼ 0 0 00 00 Fibx¯(Y0) = π1(X , x¯ )/U where U = Stab(¯y).

0 0 0 0 00 Now an X -morphism Y0 → Y exists if and only if there is an equivariant map π1(X , x¯ )/U → 0 0 0 00 0 π1(X , x¯ )/U . This in turn occurs if and only if U ⊆ U . If λ ∈ ker ϕ∗, then λ fixesy ¯ in 0 00 Fibx¯(Y ) = Fibx¯0 (Y ×X X ), hence λ fixesy ¯ in Fibx¯0 (Y0). This means λ ∈ U and so we see 00 0 that U ⊇ ker ϕ∗. By hypothesis, U ⊇ ker ϕ∗ as well, so by the correspondence theorem, 00 0 00 0 U ⊆ U if and only if ϕ∗(U ) ⊆ ϕ∗(U ), but this holds by construction. Therefore an 0 0 X -morphism Y0 → Y exists. 0 ( ⇒ = ) We must show U ⊇ ker ϕ∗. Applying the fibre functor Fibx¯0 to the morphism 0 0 0 0 Y0 → Y gives a map Fibx¯0 (Y0) → Fibx¯0 (Y ). Note thaty ¯ ∈ Fibx¯0 (Y ); choose any lift 0 00 0 0 y¯ ∈ Fibx¯0 (Y0) and set U = Stab(¯y). Identifying Y0 → Y with a subgroup U ⊆ π1(X , x¯ ), 00 0 00 0 we get U ⊆ U by the above. Then once again, ker ϕ∗ has to fixy ¯, so ker ϕ∗ ⊆ U ⊆ U .

0 0 Corollary 4.3.7. ϕ∗ is injective if and only if for all connected covers Y → X , there exists 0 0 a cover Y → X and a morphism Y0 → Y over X such that Y0 is isomorphic to a connected 0 component of Y ×X X .

0 0 0 0 Corollary 4.3.8. If every connected cover Y → X arises as a base change Y = Y ×X X of some Y → X, then ϕ∗ is injective.

ψ ϕ Corollary 4.3.9. Let X00 −→ X0 −→ X be a pair of morphisms of connected schemes and let x¯ : Spec Ω → X, x¯0 : Spec Ω → X0 and x¯00 : Spec Ω → X00 be geometric points such that ϕ ◦ x¯0 =x ¯ and ψ ◦ x¯00 =x ¯0. Then the sequence of proﬁnite groups

00 00 ψ∗ 0 0 ϕ∗ π1(X , x¯ ) −→ π1(X , x¯ ) −→ π1(X, x¯) is exact if and only if both of the following conditions hold:

83 4.4 Structure Theorems 4 Fundamental Groups of Schemes

00 00 (1) For all covers Y → X, the base change Y ×X X → X is a trivial cover.

0 0 0 00 00 (2) For all connected covers Y → X such that Y ×X0 X → X admits a section 00 0 00 0 0 X → Y ×X0 X , there exists a connected cover Y → X and an X -morphism Y0 → Y 0 for Y0 a connected component of Y ×X X . Remark. It suﬃces to check condition (2) on Galois covers, since \ ker ϕ∗ = U. open normal U⊇ker ϕ∗

4.4 Structure Theorems

In this section we prove several fundamental results about π1(X) for schemes over C, and then generalize. In particular, we will state an analogue of Theorem 1.5.5. First, let us show that for any smooth, projective, integral scheme X over C, π1(X) is ﬁnitely generated. To do this, we need:

n Theorem 4.4.1 (Bertini). Let k be an algebraically closed ﬁeld and let X ⊆ Pk be a smooth, n closed subscheme. Then there exists a hyperplane H ⊂ Pk not containing X such that X ∩H is regular.

Proof. Hartshorne, II.8.1.8.

n Lemma 4.4.2. Let k be algebraically closed and let X ⊆ Pk be a smooth, connected, closed subscheme of dimension dim X ≥ 2 and let Y → X be a connected, ﬁnite ´etalecover. Then n there exists a hyperplane H ⊆ Pk not containing X so that X ∩ H is smooth and connected and Y ×X (X ∩ H) is connected. Proof. Let H be as in Bertini’s theorem. When dim X ≥ 2, one can use sheaf cohomology (cf. Hartshorne, III.7.9.1) to show that X ∩ H is not just regular but also smooth and connected. Then the same proof applies to the closed immersion Y ×X (X ∩ H) ,→ Y to show that Y ×X (X ∩ H) is connected.

Theorem 4.4.3. For any smooth, projective, integral scheme X over C and every geometric point x¯ : Spec C → X, π1(X, x¯) is finitely generated. Proof. Suppose dim X ≥ 2. Then by Lemma 4.4.2, for any connected Y → X there exists a hyperplane H ⊆ n such that X∩H is smooth, connected and Y × (X∩H) is connected. By PC X Lemma 4.1.1(b), Y×(X ∩H) → X ∩H is a (connected) finite étalecover, so Proposition 4.3.4 implies π1(X ∩H, x¯) → π1(X, x¯) is surjective. Now note that X ∩H is smooth, projective and dim X ∩ H < dim X, so repeating this process, we eventually obtain a surjection π1(C, x¯) → π1(X, x¯) for some smooth projective curve C over C. By Theorem 1.5.5, we know π1(C, x¯) is finitely generated so this implies π1(X, x¯) is as well.

Even stronger than the surjections π1(X ∩ H) → π1(X) in the above proof, we have the following result of Lefschetz.

84 4.4 Structure Theorems 4 Fundamental Groups of Schemes

Theorem 4.4.4 (Lefschetz’s Hyperplane Theorem). Let X be a smooth, closed projective scheme over and H ⊆ n a hyperplane such that X ∩ H is smooth. Then C PC

Hk(X ∩ H) −→ Hk(X) Hk(X ∩ H) −→ Hk(X)

πk(X ∩ H) −→ πk(X) are all isomorphisms for k < dim X − 1 and are surjective for k = dim X − 1. top To compare π1 and π1 as we did in Chapter 2, we associate to any C-scheme X of finite an S type a complex analytic space X as follows. Write X = Spec Ai for C-algebras Ai of finite ∼ type. Then Ai = C[x1, . . . , xn]/(f1, . . . , fr) for polynomials f1, . . . , fr ∈ C[x1, . . . , xn]. Then n each fj may be regarded as a holomorphic function on C , so the zero set of {f1, . . . , fr} n in C is a ringed space (Yi, OYi ), where OYi is the ring of holomorphic functions on Yi (in the sense of complex geometry). We regard this (Yi, OY ) as the basic model of a complex i S analytic space. Now glue the Yi together using the same gluing as for Spec Ai to get a an an ringed space (X , OX ). Note that for any morphism of schemes Y → X, there is a natural induced morphism of complex analytic spaces Y an → Xan. The complex analytic space Xan has the following properties relative to X. Proposition 4.4.5. Let X/C be a scheme of finite type. Then (1) X is separated if and only if Xan is Hausdorff. (2) X is connected if and only if Xan is connected. (3) X is reduced if and only if Xan is reduced. (4) X is smooth if and only if Xan is a complex manifold. (5) A morphism Y → X is proper (i.e. of finite type and the base change is closed, as in Section 3.3) if and only if Y an → Xan is proper (i.e. compact sets pull back to compact sets).

(6) In particular, X is proper as a C-scheme if and only if Xan is compact. # an an (7) There is a morphism of locally ringed spaces (ϕ, ϕ ):(X , OX ) → (X, OX ) sending Xan bijectively to the closed points of X. As in Chapter 2 for curves, Grothendieck proved the following equivalence of categories. Theorem 4.4.6 (Grothendieck). Let X be a connected scheme over C of ﬁnite type. Then there is an equivalence of categories

F´etX −→ FCovXan (ﬁnite-sheeted topological covers) (Y → X) 7−→ (Y an → Xan).

Therefore, for any geometric point x¯ : Spec C → X with image x =x ¯(Spec C), the induced map top\an π1 (X , x) −→ π1(X, x¯) is an isomorphism.

85 4.4 Structure Theorems 4 Fundamental Groups of Schemes

Just like in Chapter 2, the diﬃcult part of this proof is establishing the essential surjectivity of the functor Y 7→ Y an. This is essentially the main idea behind Serre’s GAGA principle.

Question (Serre). Does there exist a scheme X over C of ﬁnite type such that π1(X) = 1 top an but π1 (X ) 6= 1? To extend Theorem 4.4.6 to any algebraically closed ﬁeld k of characteristic 0, we need the following generalization of Proposition 1.6.1.

Proposition 4.4.7. Let X be a noetherian integral scheme, k an algebraically closed field, ϕ : Y → X a proper flat morphism with geometrically integral fibres and suppose x¯ : Spec k → X is a geometric point such that k is the algebraic closure of the residue field κ(x), where x = imx ¯. Fix y¯ ∈ Yx¯. Then there is an exact sequence

π1(Yx¯, y¯) → π1(Y, y¯) → π1(X, x¯) → 1.

Corollary 4.4.8. Let k be algebraically closed and suppose X,Y are noetherian, connected schemes over k, with X proper and geometrically integral. Then for any geometric points x¯ : Spec k → X, y¯ : Spec k → Y , the natural map π1(X ×k Y, (¯x, y¯)) → π1(X, x¯) × π1(Y, y¯) is an isomorphism.

Proof. From Proposition 4.4.7, we get a diagram (with basepoints suppressed):

1 π1(X) π1(X ×k Y ) π1(Y ) 1

id id

1 π1(X) π1(X) × π1(Y ) π1(Y ) 1

with the 1 on the left in the top row coming from the natural section X,→ X ×k Y of the projection morphism X ×k Y → X, and the entire bottom row representing the direct product of proﬁnite groups. Since the left and right vertical arrows are identities, the Five Lemma implies the middle vertical arrow is an isomorphism.

Corollary 4.4.9. Let K ⊇ k be an extension of algebraically closed ﬁelds, X a proper integral scheme over k and XK its base change. Then for any geometric points x¯ : Spec k → 0 0 X, x¯ : Spec K → XK , the induced map π1(XK , x¯ ) → π1(X, x¯) is an isomorphism.

0 Proof. We first show that π1(XK , x¯ ) → π1(X, x¯) is surjective. Suppose Y → X is connected. Then since k is algebraically closed, the K-algebra k(Y )⊗k K is a field, but since this is equal to the function field of YK , we see that YK is connected. Therefore by Proposition 4.3.4, the induced map on fundamental groups is surjective. For injectivity, suppose Y → XK is a connected cover. It is possible (see Szamuely) to find a subfield k0 ⊆ K such that k0/k is finitely generated, as well as an integral affine k-scheme T 0 0 such that k(T ) = k and there is a connected finite étalecover Y → X ×k T . Fix a geometric ∼ point t¯ : Spec k → T . By Corollary 4.4.8, there is an isomorphism π1(X ×k T, (¯x, t¯)) = π1(X, x¯) × π1(T, t¯), so Corollary 4.3.7 implies there exist connected covers Z → X and

86 4.5 Specialization and Characteristic p Results 4 Fundamental Groups of Schemes

0 0 0 T → T and Z ×T T → Y a morphism of T -schemes. Take a k-point t ∈ T ; then the 0 0 fibre (Z ×T T )t of the morphism Z ×T T → T is a finite étalecover of X and there is a 0 0 morphism Z ×T T → Y of schemes over XK . Thus by Corollary 4.3.7, π1(XK , x¯ ) → π1(X, x¯) is injective.

Corollary 4.4.10. Let k be an algebraically closed ﬁeld of characteristic 0, X a smooth, connected, projective scheme over k and x¯ : Spec k → X a geometric point. Then π1(X, x¯) is ﬁnitely generated.

Proof. We know the result for k = C by Theorem 4.4.3, so applying Corollary 4.4.9 to the extension C/Q shows that any Q-scheme has finitely generated fundamental group. Finally, any algebraically closed field of characteristic 0 contains Q, so one more application of Corollary 4.4.9 finishes the proof.

4.5 Specialization and Characteristic p Results

To study fundamental groups of schemes in characteristic p > 0, we give a brief account of Grothendieck’s theory of specialization of the fundamental group. Suppose A is a complete DVR with fraction ﬁeld K and residue ﬁeld k of characteristic p > 0. Set X = Spec A and let η : Spec K,→ X and x : Spec k ,→ X be the generic and closed points of X, respectively. Also letη ¯ : Spec K,→ Spec K,→ X andx ¯ : Spec k¯ ,→ Spec k ,→ X be the corresponding geometric points. For any X-scheme Y → X, set Yη = Y ×X Spec K,Yx = Y ×X Spec k, Yη¯ = ¯ Y ×X Spec K and Yx¯ = Y ×X Spec k. Grothendieck proves: Theorem 4.5.1. If ϕ : Y → X is a proper map, then

0 (1) For any geometric point y¯ ∈ Yx, the natural map

0 0 π1(Yx, y¯ ) −→ π1(Y, y¯ )

is an isomorphism.

(2) Further, if k is algebraically closed, ϕ is ﬂat and Yη¯ and Yx¯ are geometrically reduced, then for any geometric point y¯ ∈ Yη¯, the natural map

π1(Yη¯, y¯) −→ π1(Y, y¯)

is an isomorphism.

For the moment, assume k is algebraically closed, ϕ : Y → X is a morphism of schemes 0 andy ¯ ∈ Yη¯ andy ¯ ∈ Yx are geometric points as above. By Corollary 4.3.2, there is a ∼ 0 noncanonical isomorphism π1(Y, y¯) −→ π1(Y, y¯ ) determined by picking a path λ : Fiby¯ → Fiby¯0 . The isomorphisms in Theorem 4.5.1 allow us to make the following deﬁnition. Deﬁnition. The specialization map from y¯ to y¯0 (associated to ϕ : Y → X) is the composite ∼ 0 ∼ 0 sp : π1(Yη¯, y¯) → π1(Y, y¯) −→ π1(Y, y¯ ) −→ π1(Yx¯, y¯ ).

87 4.5 Specialization and Characteristic p Results 4 Fundamental Groups of Schemes

For any profinite group G and prime integer p, we denote by G(p0) the maximal quotient of G whose order (as a profinite group) is relatively prime to p. This is typically called the prime-to-p part of G. We will prove an important result of Grothendieck which says that specialization induces an isomorphism on prime-to-p parts of the fundamental groups of Yη¯ and Yx¯. First, we need: Lemma 4.5.2 (Abhyankar). Let (A, m, k) be a DVR with fraction field K and L/K and M/K finite Galois extensions in which A has integral closures AL and AM , respectively, and integral closure B in the compositum LM. Suppose pL ⊂ AL and pM ⊂ AM are maximal ideals lying over m such that the inertia degrees e(pL | m) and e(pM | m) are relatively prime to char k and e(pL | m) | e(pM | m). Then Spec B → Spec AM is a finite étalemorphism of affine schemes.

Proof. Set p = char k. Fix a maximal ideal P ⊂ B lying over m. We may assume P∩AL = pL and P ∩ AM = pM . Then the injection Gal(LM/K) ,→ Gal(L/K) × Gal(M/K) restricts to

an injection of inertia groups IP ,→ IpL × IpM . Moreover, the induced projections IP → IpL and IP → IpM are surjective. Since |Ip| = e(p | m) for any prime p lying over m, then by hypothesis the inertia groups IP,IpL and IpM all have prime-to-p order and hence are cyclic. Further, since |IpL | divides |IpM |, every element of IP ⊆ IpL × IpM has order dividing e(pM | m), but since there is a surjection IP → IpM and the target has order e(pM | m), this ∼ implies the map is really an isomorphism IP = IpM . By transitivity of inertia degrees, that is e(P | m) = e(P | pM )e(pM | m), we get e(P | pM ) = 1, so P is unramified over M. Since P was arbitrary, it follows from Proposition 1.4.6 that Spec B → Spec AM is étale. Theorem 4.5.3 (Grothendieck). If k is algebraically closed of characteristic p > 0, ϕ : Y → X = Spec A is smooth, proper and has geometrically connected fibres, then for any geometric 0 points y¯ ∈ Yη¯ and y¯ ∈ Yx, the specialization map descends to an isomorphism of prime-to-p parts (p0) ∼ 0 (p0) π1(Yη¯, y¯) −→ π1(Yx¯, y¯ ) .

(p0) (p0) Proof. By definition of sp, it’s enough to show π1(Yη¯, y¯) → π1(Y, y¯) is an isomorphism of prime-to-p profinite groups. To begin, suppose Z → Y is a connected finite étalecover with base change Zη¯ = Z ×X Spec K over the generic point. Then

Zη¯ = lim Z ×X Spec L ←− where the inverse limit is over all finite extensions L/K contained in a fixed algebraic closure K corresponding toη ¯. Then each Z ×X Spec L is an open subscheme of the base change ZL = Z ×X XL, where XL = Spec AL and AL is the integral closure of A in L. We claim ZL is connected. If ZL = Z1 ∪ Z2 for two closed subsets Z1,Z2, then the special fibre Zx = Z ×X Spec k lies in one of them, say Z1. Under the map ZL → XL, Zx is taken to Yx, so Z2 must be open, contradicting the assumption that Y → X, and therefore also ZL → XL, is proper and in particular closed. Thus ZL is connected, so Z ×X Spec L is as well. Since L was arbitrary and the inverse limit of connected spaces is connected, we have shown that Zη¯ is connected. Now (2) of Proposition 4.3.4 implies that π1(Yη¯, y¯) → π1(Y, y¯) is surjective and in particular it is surjective on prime-to-p parts.

88 4.5 Specialization and Characteristic p Results 4 Fundamental Groups of Schemes

For injectivity (on prime-to-p parts), Corollary 4.3.7 says that it’s enough to show that 0 every finite étalecover Z → Yη¯ of prime-to-p degree is the base change of some cover Z → Y of prime-to-p degree. In fact, the remark at the end of Section 4.3 implies that we can check 0 the condition for Galois covers Z → Yη¯. Applying Theorem 4.5.1, we have an isomorphism ∼ 0 π1(Y ×X XL, z¯) = π1(Y, z¯) so replacing X with XL, we may assume Z is obtained by base change from a finite Galois cover Zη → Yη. Let Y be the normalization of Y in the function field K(Zη) of Zη. Then K(Zη)/K(Y ) is a finite Galois extension, say of degree d, and by the Zariski-Nagata purity theorem (4.1.7), it’s enough to find a finite Galois extension K0/K such that Y ×X XK0 → Y ×X XK0 is étaleover all codimension 1 points. Further, since all codimension 1 points of Y lie in Yη except for the generic point ξ of Yx, we need only check that Y ×X XK0 is étaleover every point of Y ×X XK0 lying over ξ. If A has maximal ideal m = (π), then π also generates the√ maximal ideal of the local ring OY,ξ. It follows from Kummer theory that K0 = K(Y )( d π) is a finite Galois extension of K(Y ) of degree d and 0 since A is a DVR, ramification theory implies that there is a unique point ξ ∈ YK0 lying 0 ∼ 0 above ξ, with inertia group I(ξ | ξ) = Z/dZ. Now Abhyankar’s lemma shows that YK(Zη)K 0 0 ∼ 0 is étaleover ξ . Moreover, YK(Zη)K = Y ×X XK by construction. Finally, this shows that 0 the base change of Z := Y ×X XK0 → Y to Yη¯ is precisely Z , completing the proof. Now let X be any locally noetherian scheme and Y → X a proper morphism with connected geometric fibres. Suppose x0, x1 ∈ X such that x0 is a specialization of x1, i.e. x0 ∈ {x1}. Letx ¯0, x¯1 be geometric points of X with images x0, x1, respectively, and fix geometric pointsy ¯0, y¯1 ∈ Y lying overx ¯0, x¯1, respectively. Set Z = {x1}, consider the local ring OZ,x0 and note that the geometric fibres Yx¯0 and Yx¯1 over X remain the same over

Spec OZ,x0 . Denote by A the completion of the localization at a height 1 prime ideal of

OZ,x0 . Then A is a complete DVR and we have a specialization map

spy1 : π1(Yη¯, y¯1) −→ π1(Yx¯1 , y¯1) for the generic pointη ¯ of Spec A. On the other hand, by Corollary 4.4.9, the morphism

Yη¯ → Yx¯0 induces an isomorphism

∼ π1(Yη¯, y¯0) −→ π1(Yx¯0 , y¯0).

Deﬁnition. The specialization map from y¯0 to y¯1 is the composition ∼ π1(Yx¯1 , y¯1) = π1(Yη¯, y¯0) −→ π1(Yx¯0 , y¯0).

Theorem 4.5.4. Suppose ϕ : Y → X is a proper ﬂat morphism over a locally noetherian scheme X with connected geometric ﬁbres. Then for any geometric points y¯0, y¯1 of Y lying over x0, x1 ∈ X with x0 ∈ {x1}, the specialization map from y¯0 to y¯1 descends to an isomorphism of prime-to-p parts

∼ π1(Yx¯1 , y¯1) −→ π1(Yx¯0 , y¯0).

Grothendieck used this specialization result to prove the following analogue of Theo- rem 4.4.6.

89 4.5 Specialization and Characteristic p Results 4 Fundamental Groups of Schemes

Theorem 4.5.5. Let X be an integral, proper, normal curve of genus g over an algebraically closed ﬁeld k of characteristic p > 0. Then for any geometric point x¯ : Spec k → X,

(1) π1(X, x¯) is ﬁnitely generated as a proﬁnite group.

(p0) (2) π1(X, x¯) is isomorphic to the proﬁnite prime-to-p completion of the topological fundamental group

Πg = ha1, b1, . . . , ag, bg | [a1, b1] ··· [ag, bg] = 1i.

Corollary 4.5.6. Let k be an algebraically closed field of characteristic p > 0, X a smooth, connected, projective scheme over k and x¯ : Spec k → X a geometric point. Then π1(X, x¯) is finitely generated. Proof. Same as the proof in characteristic 0, using the same reduction to the case of curves as in Theorem 4.4.3. Example 4.5.7. This result is false in characteristic p when X is not assumed to proper. 1 1 For example, Raynaud proved that for X = Ak = Pk r {∞}, π1(X, x¯) is infinitely generated. (Compare this to Example 1.5.10.) Any Artin-Schreier equation yp − y = f(x), where 1 f(x) ∈ k[x] has prime-to-p degree, defines a nontrivial Galois cover Yf → Pk that is étale 1 over Ak and has Galois group Z/pZ. If k is algebraically closed, this gives an infinitely family 1 of finite Galois covers of Ak. More generally, π1(X, x¯) is infinitely generated for any affine curve X over k. Definition. Let ϕ : Y → X be a finite separable morphism of normal integral schemes. Then ϕ is tamely ramified at a codimension 1 point P ∈ X if ϕ is ramified at P and for each Q ∈ ϕ−1(P ), the ramification index e(Q | P ) of ϕ at Q is relatively prime to the characteristic of κ(P ). We say ϕ is tamely ramified, or simply tame, if it is tamely ramified at every codimension 1 point of X. Let X be an integral normal scheme, let U ⊆ X be an open subscheme and denote by t FétU the full subcategory of FétU consisting of finite separable covers ϕ : Y → X which are t tamely ramified over X r U. Note that the fibre functor Fibx¯ restricts to a functor on FétU . Definition. For a geometric point x¯ : Spec Ω ,→ X with image x ∈ U, the tame fundamental group of U at x¯ is

t π (U, x¯) := Aut(Fib | t ). 1 x¯ F´etU Theorem 4.5.8. Let x¯ be a geometric point in X and U ⊆ X an open subscheme containing x¯. Then

t (1) π1(U, x¯) is a quotient of π1(U, x¯) and in particular a profinite group and its action on t Fibx¯(Y ) is continuous for all Y ∈ FétU . t (2) The restriction of Fibx¯ to FétU induces an equivalence of categories

t ∼ t F´etU −→{ﬁnite, continuous, left π1(U, x¯)-sets}

Y 7−→ Fibx¯(Y ).

90 4.5 Specialization and Characteristic p Results 4 Fundamental Groups of Schemes

t (3) Let KU be the compositum inside a fixed algebraic closure of K = k(U) of all finite extensions L/K such that the normalization XL of X in L is étaleover U and tamely ramified over X r U.

t (4) The tame fundamental group π1(U, x¯) is finitely generated as a profinite group. Statements (1) – (3) can be derived from similar arguments as for the full étalefunda- mental group. The proof of (4) relies on the deep fact that tame covers in characteristic p lift to p-tame covers in characteristic 0, where a p-tame cover is one defined over a field of characteristic 0 in which p does not divide the order of any inertia groups of the cover. However, this more or less allows one to compute the tame fundamental group by lifting to characteristic 0. In contrast, the pro-p-part of the fundamental group in characteristic p is not completely understood and is the subject of much current research. But we will end the discussion here.