Etale´ Cohomology

Andrew Kobin 2017 – 2020 Contents Contents

Contents

0 Introduction 1

1 Scheme Theory 2 1.1 Affine Schemes ...... 2 1.2 Schemes ...... 4 1.3 Properties of Schemes ...... 6 1.4 Sheaves of Modules ...... 12 1.5 Group Schemes ...... 17

2 Etale´ Fundamental Groups of Schemes 21 2.1 Galois Theory for Schemes ...... 22 2.2 The Etale´ Fundamental Group ...... 25 2.3 Properties of the Etale´ Fundamental Group ...... 28 2.4 Structure Theorems ...... 31

3 Sites 36 3.1 Grothendieck Topologies and Sites ...... 36 3.2 Sheaves on Sites ...... 38 3.3 The Etale´ Site ...... 42 3.4 A Word on Algebraic Spaces ...... 48

4 Cohomology 50 4.1 Direct and Inverse Image Functors ...... 50 4.2 Etale´ Cohomology ...... 53

i 0 Introduction

0 Introduction

These notes give an introduction to ´etalecohomology as part of my dissertation research in 2017 – 2020 at the University of Virginia. The main texts I use for reference are Milne’s Lectures on Etale Cohomology and Etale Cohomology, as well as Arapura’s Introduction to Etale Cohomology. The main topics covered are:

ˆ Review of scheme theory

ˆ The ´etalefundamental group for schemes

ˆ Grothendieck topologies

ˆ Sheaves and cohomology on the ´etalesite

ˆ Galois cohomology

ˆ Cohomology of curves

ˆ Etale´ versions of standard theorems in algebraic topology, including:

– The Gysin sequence – Finiteness theorems – The comparison theorems – K¨unnethformulas – Poincar´eduality – The Lefschetz fixed-point theorem

ˆ The Weil conjectures.

1 1 Scheme Theory

1 Scheme Theory

1.1 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 1.1.1 (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 1.1.2. 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}.

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 1.1.3. 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).

2 1.1 Affine Schemes 1 Scheme Theory

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 1.1.4. (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 1.1.5. For any field k, Spec k is a single point ∗ corresponding to the zero ideal, with sheaf O(∗) ∼= k.

Example 1.1.6. 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)

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 cor- responding 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 1.1.7. 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:

3 1.2 Schemes 1 Scheme Theory

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.

1.2 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 1.2.1. Any affine scheme Spec A is a locally ringed space by (1) of Theorem 1.1.4. We will sometimes denote the structure sheaf O by OA. 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 1.2.2. There is an isomorphism of categories AffSch −→∼ CommRingsop

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

4 1.2 Schemes 1 Scheme Theory

Proof. (Sketch) First suppose we have a homomorphism of rings f : A → B. By Lemma 1.1.2 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 1.1.4, ∼ ∼ 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 1.2.3. We saw in Example 1.1.5 that for any field k, Spec k = ∗ is a point with structure sheaf O(∗) = k. If A = L1 × · · · × Lr is a finite ´etale k-algebra, then ` ` Spec A = Spec L1 ··· Spec Lr is (schematically) a disjoint union of points.

Example 1.2.4. 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 1.2.5. 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.

Example 1.2.6. 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 1.3). 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 1.2.7. 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}

X

5 1.3 Properties of Schemes 1 Scheme Theory

1.3 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 1.3.1. 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 1.1.4, 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. 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 1.3.2. Let A be a noetherian ring. Then dim Spec A = dim A, the Krull dimension of A.

Be warned that the converse to Proposition 1.3.2 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

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

6 1.3 Properties of Schemes 1 Scheme Theory

ˆ An integral scheme X is normal if each stalk OX,P is integrally closed in its field of fractions. ˆ 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 1.3.3. For X = Spec A, let f ∈ A and recall the open set D(f) defined in Theorem 1.1.4. 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 mor- phism 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

X S ∼ Example 1.3.4. 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

p

{x} X

We next construct fibre products in the category SchX and use these to construct the alge- braic 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

7 1.3 Properties of Schemes 1 Scheme Theory

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

X

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.

Theorem 1.3.5. 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:

8 1.3 Properties of Schemes 1 Scheme Theory

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 1.2.2, 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 W

Yi Z

X

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.

9 1.3 Properties of Schemes 1 Scheme Theory

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 1.3.6. 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 1.3.7. Let A be a DVR and consider the affine scheme X = Spec A. We saw in Example 1.2.4 that X has a closed point m = mA and a generic point (0). For any morphism p : Y → X, there are two fibres:

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

ˆ 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

X

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 1.3.8. 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.

10 1.3 Properties of Schemes 1 Scheme Theory

Example 1.3.9. One can show that the affine line with the origin doubled (Example 1.2.7) 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 1.3.10. 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.

(c) If g : Z → Y is separated (resp. proper) then the composition f ◦ g : Z → X is separated (resp. 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 1.3.11. (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

A[t, t−1]

A[t] A[t−1]

A

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 .

11 1.4 Sheaves of Modules 1 Scheme Theory

n Example 1.3.12. 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 1.3.13. If f : Y → X is a projective morphism of noetherian schemes, then f is proper.

n The idea behind the proof of Theorem 1.3.13 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 1.3.10 to obtain the general result.

1.4 Sheaves of Modules

Through Proposition 1.2.2, 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 )

If F(U) ⊆ OX (U) is an ideal for each open set U, then we call F a sheaf of ideals on X. Example 1.4.1. 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,

ˆ 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.

12 1.4 Sheaves of Modules 1 Scheme Theory

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

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

0 00 0 ˆ 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.

ˆ 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 oper- ation.

∼ ⊕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 com- ponents. 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 1.1. Also, note that necessarily the s ∈ A in the definition above must lie outside of all q ∈ V .)

Proposition 1.4.2. 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 1.1.4; both can be found in Hartshorne.

13 1.4 Sheaves of Modules 1 Scheme Theory

Proposition 1.4.3. 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 1.2.2.

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 1.4.4. 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 1.4.5. QCohX and CohX are abelian categories. Example 1.4.6. 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 1.4.3. Theorem 1.4.7. 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.

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 1.4.6.

14 1.4 Sheaves of Modules 1 Scheme Theory

Lemma 1.4.8. 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 1.4.9. 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 1.4.10. 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 1.4.11. Let A → B be a ring homomorphism. Then

(a) ΩeB/A is a quasi-coherent sheaf on Spec B. ∼ (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 1.3.8. 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.

15 1.4 Sheaves of Modules 1 Scheme Theory

∼ 2 Lemma 1.4.12. ΩB/A = I/I . ∼ 2 Example 1.4.13. In Example 1.4.9, 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 1.4.14. 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. Theorem 1.4.15. 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.

16 1.5 Group Schemes 1 Scheme Theory

(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 1.4.16. 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 ´etaleif A = L1 × · · · × Lr for finite, separable extensions Li/k.

1 Proposition 1.4.17. A finitely generated k-algebra A is finite ´etaleif 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:

ˆ ωX is an invertible sheaf on X.

ˆ 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.

ˆ 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).

1.5 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.

17 1.5 Group Schemes 1 Scheme Theory

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 1.5.1. 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 1.5.2. 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 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 (´etale)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 1.5.3. Let X = Spec A be affine and recall the affine line AX = Spec A[t] con- 1 structed in Example 1.3.11. Then AX is an affine group scheme over X, denoted Ga, called

18 1.5 Group Schemes 1 Scheme Theory

the additive group scheme over X, with morphisms induced by the following ring homomor- phisms:

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 1.5.4. 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 1.5.5. 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 1.5.6. 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 1.5.7. 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.

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.

19 1.5 Group Schemes 1 Scheme Theory

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 1.5.8. Left multiplication defines a G-torsor structure on G itself.

Example 1.5.9. 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 fi- nite 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).

20 2 Etale´ Fundamental Groups of Schemes

2 Etale´ Fundamental Groups of Schemes

In this chapter we define the ´etalefundamental group of a scheme. This will 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 2.0.1. 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 2.0.2. 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

f

X

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.

21 2.1 Galois Theory for Schemes 2 Etale´ Fundamental Groups of Schemes

2.1 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 ´etale κ(P )-algebra A, we say ϕ is ´etale at P . If this holds for each point P ∈ X, we call ϕ a finite ´etalemorphism. Finally, if ϕ is also surjective, it is called a finite ´etalecover 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 2.1.1. Let ϕ : Y → X be a finite ´etalemorphism. Then (a) If ψ : Z → Y is finite ´etale,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 F´etX is defined to be the full subcategory of SchX consisting of finite ´etale ϕ covers Y −→ X. To compare to the topological case of a covering space, 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 2.1.2. For a morphism ϕ : Y → X, ϕ is ´etaleat 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 ´etaleif 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 2.1.3. The cover C r {0} → C r {0}, y 7→ yn in Example 2.0.2 is a finite ´etale cover. Example 2.1.4. 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

22 2.1 Galois Theory for Schemes 2 Etale´ Fundamental Groups of Schemes

where Q1,...,Qr are prime ideals of B. Then YP is a finite ´etalealgebra 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 ´etaleif and only if each affine piece corresponds to a finite, unramified extension of Dedekind rings.

Proposition 2.1.5. 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 1.4.17. Then the general case is obtained by base change and localization properties of differentials (Lemma 1.4.10). (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 1.4.14, ∆(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 2.1.2. 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 2.1.2 for ϕ to be ´etaleis satisfied. We next translate the ‘local triviality’ condition from covering space theory to schemes.

Proposition 2.1.6. Let X be a connected scheme and ϕ : Y → X an affine surjective morphism. Then ϕ is a finite ´etalecover 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 2.1.5, 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)

23 2.1 Galois Theory for Schemes 2 Etale´ Fundamental Groups of Schemes

and Y ×X Y → Y (Lemma 2.1.1(b)) are finite ´etale. Thus Lemma 2.1.1(a) implies their composition Y 0 → Y is finite ´etale 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 ´etale using Proposition 2.1.2. 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 2.1.2 implies ϕ is ´etale. Recall from Section 1.3 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 ´etale,it suffices to check the ´etaleproperty at all codimension 1 points of the base scheme. Theorem 2.1.7 (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 ´etaleover κ(P ), then ϕ itself is ´etale. Corollary 2.1.8. Let X be a regular integral scheme and U ⊆ X an open subscheme con- sisting of points of codimension at least 2. Then there is an equivalence of categories

F´etX −→ F´etU

Y 7−→ Y ×X U. Proposition 2.1.9. Let ϕ : Y → X be a finite ´etalecover, 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 2.1.1(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 ´etalemorphisms 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 2.1.10. Let ϕ : Y → X be a finite ´etalecover. Then Aut(Y/X) is finite.

24 2.2 The Etale´ Fundamental Group 2 Etale´ Fundamental Groups of Schemes

Proof. Take σ ∈ Aut(Y/X), σ 6= 1, and set f = ϕ and g = ϕ ◦ σ. Then by Proposition 2.1.9, 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 ´etale,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 ´etalecover and let G be a group scheme over X such that Y is a left G-torsor (see Section 1.5). 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 2.1.11. 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. Proposition 2.1.12. If ϕ : Y → X is a connected, finite ´etalecover and G ≤ Aut(Y/X) is any finite subgroup of automorphisms, then π : Y → Y/G is a finite ´etalecover. Definition. A connected, finite ´etalecover ϕ : Y → X is a Galois cover if Aut(Y/X) acts transitively on every geometric fibre of ϕ. Theorem 2.1.13. 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

X 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.

2.2 The Etale´ Fundamental Group

Let X be a scheme and F´etX the category of finite ´etalecovers 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 1.3.6(c), Fibx¯ is indeed a functor. We can now define the algebraic, or ´etale,fundamental group of a scheme in analogy with the topological case.

25 2.2 The Etale´ Fundamental Group 2 Etale´ Fundamental Groups of Schemes

Definition. The algebraic, or ´etalefundamental 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¯).

Theorem 2.2.1 (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 2.2.2. Let k be a field and consider X = Spec k. Then finite ´etalecovers Y → Spec k are precisely Y = Spec A for A = L1 × · · · × Lr a finite ´etale 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, 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) given by σ ·f = σ ◦f for any f : A → ksep and σ ∈ Gk. Therefore ∼ ∼ π1(X, x¯) = Aut(Fibx¯) = Aut(Homk(−, ks)) = Gal(ks/k).

Moreover, there is an identification (via Prop. 1.2.2) Homk(A, ks) = HomSpec k(Spec ks, Spec k). The above shows that although Fibx¯ is not a representable functor – Spec ks is not a finite ´etale k-scheme – it is pro-representable in F´etSpec 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 para- graph.

Proposition 2.2.3. For a connected scheme X and any geometric point x¯ : Spec Ω → X, the fibre functor Fibx¯ is pro-representable. Explicitly, for any finite ´etalecover Y → X,

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

26 2.2 The Etale´ Fundamental Group 2 Etale´ Fundamental Groups of Schemes

Lemma 2.2.4. 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 2.2.1.

0 Proof. (1) By Proposition 2.2.3, 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 2.2.4 implies

0 Aut(Fibx¯) = lim Aut(Y /X) ←− where again Y 0 ranges over all finite Galois covers. Since Corollary 2.1.10 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 ´etalecover 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 2.2.3, 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.

Theorem 2.2.5. Let X be an integral normal scheme with function field K and fix a sepa- rable 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 ´etaleover X. Then

(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 straightforward. Then (2) follows from (2) of Theorem 2.2.1.

Example 2.2.6. Let X = Y = C r {0} and define the finite ´etalecover f : Y −→ X y 7−→ πyn.

(Here, we use π to ensure there are no algebraic relations on this map.) As in Example 2.1.3, ∼ f is a finite ´etaleGalois 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 2.2.7. If a scheme X/C is defined over Q, then any finite ´etalecover of X is also defined over Q.

27 2.3 Properties of the Etale´ Fundamental Group2 Etale´ Fundamental Groups of Schemes

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 2.2.8. If G is an abelian group, then the unique maximal field of definition for Y → X is precisely M. Note that Proposition 2.2.8 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: ˆ (Geometric) Every Galois cover of schemes has a group of automorphisms and a cor- responding field extension for which this is the Galois group.

ˆ (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. 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. The sequence splits if and only if X contains a k-rational point.

2.3 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, there is a nice notion of a path between two points. We now describe an analogue for schemes. Proposition 2.3.1. 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 2.2.3 that each fibre functor is pro-representable by an inverse system (Pα) of finite ´etaleGalois 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 2.2.3, for any ∼ Y ∈ F´etX , 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):

28 2.3 Properties of the Etale´ Fundamental Group2 Etale´ Fundamental Groups of Schemes

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 2.1.9 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 2.3.2. 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 2.3.1 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. 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 2.1.6 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 2.3.3. ϕ∗ : π1(X , x¯ ) → π1(X, x¯) is a continuous homomorphism of profinite groups. Proposition 2.3.4. 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.

29 2.3 Properties of the Etale´ Fundamental Group2 Etale´ Fundamental Groups of Schemes

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 ∈ F´etX , 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 ∈ F´etX (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 2.1.13 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 2.3.5. Let U ⊆ π1(X, x¯) be any open subgroup and let Y → X be the corre- sponding 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 . 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 2.3.6. 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).

30 2.4 Structure Theorems 2 Etale´ Fundamental Groups of Schemes

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 2.3.7. 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 2.3.8. 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 2.3.9. 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:

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∗

2.4 Structure Theorems

In this section we prove several fundamental results about π1(X) for schemes over C, and then generalize. 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 2.4.1 (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.

31 2.4 Structure Theorems 2 Etale´ Fundamental Groups of Schemes

Proof. Hartshorne, II.8.1.8.

n Lemma 2.4.2. 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. Using Riemann’s existence theorem in complex geometry, one obtains the following result for curves:

Theorem 2.4.3. 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 topo- logical 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).

Theorem 2.4.4. 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 2.4.2, 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 2.1.1(b), Y×(X ∩H) → X ∩H is a (connected) finite ´etalecover, so Proposition 2.3.4 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 2.4.3, 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.

Theorem 2.4.5 (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.

32 2.4 Structure Theorems 2 Etale´ Fundamental Groups of Schemes

top To compare π1 and π1 , we associate to any C-scheme X of finite type a complex analytic an S ∼ 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 each fj may be n n regarded as a holomorphic function on C , so the zero set of {f1, . . . , fr} 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, OYi ) as the basic model of a complex analytic space. Now S an an glue the Yi together using the same gluing as for Spec Ai to get a 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 2.4.6. 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 1.3) 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 Theorem 2.4.3 for curves, Grothendieck proved the following equivalence of cate- gories.

Theorem 2.4.7 (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. 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?

33 2.4 Structure Theorems 2 Etale´ Fundamental Groups of Schemes

To extend Theorem 2.4.7 to any algebraically closed field k of characteristic 0, we need the following. Proposition 2.4.8. 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 2.4.9. 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 2.4.8, 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 2.4.10. 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 2.3.4, 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 ´etalecover Y → X ×k T . Fix a geometric ∼ point t¯ : Spec k → T . By Corollary 2.4.9, there is an isomorphism π1(X ×k T, (¯x, t¯)) = π1(X, x¯) × π1(T, t¯), so Corollary 2.3.7 implies there exist connected covers Z → X and 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 ´etalecover of X and there is a 0 0 morphism Z ×T T → Y of schemes over XK . Thus by Corollary 2.3.7, π1(XK , x¯ ) → π1(X, x¯) is injective. Corollary 2.4.11. 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.

34 2.4 Structure Theorems 2 Etale´ Fundamental Groups of Schemes

Proof. We know the result for k = C by Theorem 2.4.4, so applying Corollary 2.4.10 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 2.4.10 finishes the proof. Theorem 2.4.3 generalizes to curves over an algebraically closed field k of characteristic 0 in the following sense.

Corollary 2.4.12. 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 2.4.13. If k is an algebraically closed field of characteristic 0 and X → Pk is a finite ´etalecover, 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 2.4.14. 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 2.4.3 and Corollary 2.4.12.

1 Example 2.4.15. Theorem 2.4.3 and Corollary 2.4.12 also imply that π1(Pk r {0, ∞}) = 1 ∼ π1(Ak r {0}) = Zb, the profinite completion of the integers. Thus we get a finite ´etalecover 1 Xn → Ak r {0} for each n ≥ 1 having Galois group Z/nZ.

1 Example 2.4.16. By Theorem 2.4.3 and Corollary 2.4.12, π1(Pk r {0, 1, ∞}) is the free profinite group on two generators.

35 3 Sites

3 Sites

This chapter covers the basic definitions and results in Grothendieck’s theory of sites, a useful generalization of a topological space. The main motivation is to develop a working sheaf theory on schemes that can detect the features of ´etalemorphisms and more general properties.

3.1 Grothendieck Topologies and Sites

To every topological space X, we can associate a category Top(X) consisting of the open subsets U ⊆ X with morphisms given by inclusions of open sets U,→ V .A presheaf on X is a functor F : Top(X)op → Set, i.e. a contravariant functor on the category Top(X). The conditions for F to be a sheaf on X can be summarized by saying that for every open set S U ∈ Top(X) and every open covering U = Ui, the set F (U) is an equalizer in the following diagram: Y Y F (U) F (Ui) F (Ui ∩ Uj) i i,j

This generalizes as follows. Definition. A Grothendieck topology on a category C is a set of collections of morphisms Cov(X) = {{Xi → X}i} for every objects X ∈ C, called coverings, satisfying: (i) Every isomorphism X0 → X defines a covering {X0 → X} in Cov(X).

(ii) For any covering {Xi → X} of X and any morphism Y → X in C, the fibre products Xi ×X Y exist and the induced maps {Xi ×X Y → Y } are a covering of Y .

(iii) If {Xi → X}i is a covering of X and {Yij → Xi}j is a covering of Xi for each i, then the compositions {Yij → Xi → X}i,j are a covering of X. A category equipped with a Grothendieck topology is called a site. Example 3.1.1. For a topological space X, the category Top(X) is a site with coverings ( ) [ Cov(U) = {Ui ,→ U} : Ui ⊆ U are open and U = Ui . i When X is a scheme with the Zariski topology, Top(X) is called the (small) Zariski site on X. Example 3.1.2. The category Top of all topological spaces with continuous maps between them is a site, called the big topological site, whose coverings are defined by ( ) [ Cov(X) = {fi : Xi ,→ X} : fi is an open embedding and X = Xi . i

36 3.1 Grothendieck Topologies and Sites 3 Sites

Example 3.1.3. Similarly, for a scheme X, let SchX be the category of X-schemes (the category Sch of all schemes can be viewed in this framework by setting X = Spec Z since this is a terminal object in Sch). Then SchX is a site, called the big Zariski site on X, with coverings ( ) [ Cov(Y ) = {ϕi : Yi → Y } : ϕi is an open embedding and Y = Yi . i Example 3.1.4. Let C be a site and X ∈ C be an object. Define the localized site (or the slice category) C/X to be the category with objects Y → X ∈ HomC(Y,X), morphisms Y → Z in C such that Y Z

X

commutes. Then C/X can be equipped with a Grothendieck topology by defining

Cov(Y → X) = {{Yi → Y } : Yi → Y ∈ HomX (Yi,Y ), {Yi → Y } ∈ CovC(Y )}.

Example 3.1.5. Let X be a scheme and define the (small) ´etalesite on X to be the category ´ Et(X) of X-schemes with ´etale morphisms Y → X and covers {Yi → Y } ∈ Cov(Y ) such ` that Yi → Y is surjective. Example 3.1.6. In contrast, we can equip the slice category Sch/X with a Grothendieck topology by declaring {Yi → Y } to be a covering of Y → X if each Yi → Y is ´etaleand ` Yi → Y is surjective. The resulting site is referred to as the big ´etalesite on X. Example 3.1.7. Similar constructions can be made by replacing “´etale”with other prop- erties, such as:

ˆ The fppf site is the category Sch/X with coverings {Yi → Y } ∈ Cov(Y ) such that ` Yi → Y are flat and locally of finite presentation and Yi → Y is surjective. ˆ The lisse-´etalesite LisEt(´ X) is the category of X-schemes with smooth morphisms between them, whose coverings are {Yi → Y } ∈ Cov(Y ) such that the Yi → Y are ` ´etale and Yi → Y is surjective. ˆ The smooth site Sm(X) is the category of X-schemes with smooth morphisms between them and surjective families of smooth coverings.

ˆ Most generally, the flat site is Sch/X with surjective families of flat morphisms of finite type as coverings.

Definition. A continuous map between sites f : C1 → C2 is a functor F : C2 → C1 that preserves fibre products and takes coverings in C2 to coverings in C1.

37 3.2 Sheaves on Sites 3 Sites

Remark. Notice that a continuous map between sites is a functor in the opposite direction. This is in analogy with the topological notion: a continuous map f : X → Y between topological spaces induces a functor F : Top(Y ) → Top(X) given by V 7→ f −1(V ). Example 3.1.8. When X is a scheme, there are continuous maps between the various sites we have defined on Sch/X. We collect some of these sites in the following table, along with their relevant features. (The arrows between sites represent continuous maps between sites, so the functors on the underlying categories go in the opposite direction. Note that when we define sheaves in the next section, sheaves will pull back in the same direction as these arrows.)

Xflat → Xfppf → Xsmooth → X´et → XNis → XZar

name flat fppf smooth ´etale Nisnevich Zariski ´etale, with flat, maps all smooth ´etale residue field all locally f.p. isomorphisms

Example 3.1.9. Let G be a profinite group and let CG be the category of all finite, discrete ` G-sets. Then the collections of G-homomorphisms {Xi → X} such that i Xi → X is ¯ surjective form a Grothendieck topology on CG. When G = Gal(k/k) for some field k, the category CG is equivalent to X´et for X = Spec k.

3.2 Sheaves on Sites

In this section we generalize the notions of presheaf and sheaf to an arbitrary category with a Grothendieck topology.

Definition. A presheaf on a category C is a functor F : Cop → Set, that is, a contravari- ant functor from C to the category of sets. The category of presheaves on C (with natural transformations between them) will be denoted PreShC.

Definition. We say F is separated if for every collection of maps {Xi → X}, the map ` F (X) → i F (Xi) is injective. Definition. Let C be a site. A sheaf on C is a presheaf F : Cop → Set such that for every object X ∈ C and every covering {Xi → X} ∈ Cov(X), the sequence of based sets Y Y F (X) F (Xi) F (Xi ×X Xj) i i,j

is exact, or equivalently, F (X) is an equalizer in the diagram. The category of sheaves on C will be denoted ShC. As in topology, we can consider sheaves on C with values in set categories with further

structure, e.g. Group, Ring,R−Mod, Algk.

38 3.2 Sheaves on Sites 3 Sites

Theorem 3.2.1 (Sheafification). The forgetful functor ShC → PreShC has a left adjoint F 7→ F a.

Proof. First consider the forgetful functor SepC → PreShC defined on the subcategory of separated presheaves on C. For a presheaf F on C, let F sep be the presheaf X 7−→ F sep(X) := F (X)/ ∼

where, for a, b ∈ F (X), a ∼ b if there is a covering {Xi → X} of X such that a and b have the same image under the map a F (X) → F (Xi). i By construction, F sep is a separated presheaf on C and for any other separated presheaf F 0, any morphism of presheaves F → F 0 factors through F sep uniquely. Hence F 7→ F sep is left adjoint to the forgetful functor SepC → PreShC so it remains to construct a sheafification of every separated presheaf on C. For a separated presheaf F , define F a to be the presheaf

a X 7−→ F (X) := ({Xi → X}, {αi})/ ∼ where {Xi → X} ∈ CovC(X), {αi} is a collection of elements in the equalizer   Y Y Eq  F (Xi) F (Xi ×X Xj)  , i i,j

and ({Xi → X}, {αi}) ∼ ({Yj → Y }, {βj}) if αi and βj have the same image in F (Xi ×X Yj) for all i, j. Then as above, F a is a sheaf which is universal with respect to all morphisms of 0 a sheaves F → F . Thus F 7→ F defines a left adjoint to the forgetful functor ShC → SepC and composition with the first construction proves the theorem.

Definition. A topos is a category T which is equivalent to ShC for some site C. The plural of ‘topos’ is ‘topoi’. Remark. Grothendieck held the view that the topos of a site is more important than the site itself. In practice, many naturally occurring sites induce the same topos (that is, their sheaf theories are equivalent). For example, the Zariski sites on Sch and AffSch (the categories of schemes and affine schemes, respectively) induce the same topos. In general, AffSch is easier to study since it is equivalent to the opposite category of commutative rings.

0 ∗ Definition. A morphism of topoi f : T → T is an adjoint pair f = (f , f∗) of functors 0 ∗ 0 ∗ f∗ : T → T and f : T → T such that the functors HomT (f (−), −) and HomT 0 (−, f∗(−)) are naturally isomorphic. Definition. Let T be a topos and A ∈ T an object. Then A is called a group object (or a group in T ) if it possesses distinguished morphisms m : A × A → A, e : ∗ → A, i : A → A satisfying the following axioms:

39 3.2 Sheaves on Sites 3 Sites

(i) (Associativity) The diagram

1A × m A × A × A A × A

m × 1A m

A × A m A

commutes.

(ii) (Identity) The composition m ◦ (1A × e): A → A × A → A is equal to 1A. (iii) (Inverses) The diagram

(i, 1A) A A × A

m

∗ e A

commutes. In addition, A is called abelian if the diagram

A × A m p A m A × A commutes, where p flips the factors of A × A. In this case, we will often write the map m as a to denote addition. Definition. An abelian group A in a topos T is called a ring if, in addition to a : A × A → A, e : ∗ → A and i : A → A, it possesses morphisms

m : A × A → A and 1 : ∗ → A satisfying the usual axioms of associativity, identity and left and right distributivity. Further, if m satisfies the axiom of commutativity, A will be called a commutative ring in T . Definition. A ringed topos is a pair (T ,A) where T is a topos and A is a ring in T .A morphism of ringed topoi is a pair (f, f #):(T ,A) → (T 0,A0) consisting of a morphismi ∗ 0 # 0 of topoi f = (f , f∗): T → T and a morphism of ring objects f : A → f∗A, or equivalently by adjointness, f ∗A0 → A.

40 3.2 Sheaves on Sites 3 Sites

Proposition 3.2.2. For every continuous map of sites f : C0 → C, where C and C0 are small ∗ categories, there exists a morphism of topoi f∗ : ShC → ShC0 with left adjoint f : SchC0 → SchC. 0 0 Proof. Let T = SchC, T = SchC0 and define the morphism f∗ : T → T for each object X0 ∈ C0 and sheaf F ∈ T by 0 0 (f∗F )(X ) = F (f(X )). 0 0 0 If {Xi → X } is a covering in C , we have a commutative diagram

Y 0 Y 0 0 0 (f∗F )(X ) (f∗F )(X ×X0 X ) (f∗F )(X ) i i j i i,j = = ∼=

Y 0 Y 0 0 F (f(X0)) F (f(Xi)) F (f(Xi) ×f(X0) f(Xj)) i i,j

Here, the bottom row is exact since F is a sheaf and f is continuous; the right vertical arrow is an isomorphism since f is continuous; and the other vertical arrows are equalities 0 by definition. Hence by the Five Lemma, the top row is exact, so f∗F defines a sheaf on C . ∗ To define the left adjoint f : ShC0 → ShC, take an object X ∈ C and define a category 0 0 0 0 IX with objects (X , ρ) where X ∈ C and ρ ∈ HomC(X, f(X )), and with morphisms (X0, ρ) → (Y 0, σ) given by a morphism g : X0 → Y 0 in C0 making the following diagram commute: f(X0) ρ X f(g) σ f(Y 0)

Now define f ∗F for a sheaf F ∈ T 0 on an object X ∈ C by (f ∗F )(X) = lim F (X0) −→ 0 op where the limit is over all objects (X , ρ) in the opposite category IX . If h : X → Y is a morphism in C, then there is an induced map (f ∗F )(Y ) → (f ∗F )(X) given by the functor

IY −→ IX (Y 0, ρ) 7−→ (Y 0, ρ ◦ h). This shows that f ∗F is a presheaf on C. Moreover, the maps ∗ 0 (f f∗F )(X) = lim F (f(X )) → F (X) −→ for each F ∈ T ,X ∈ C give a natural transformation ∗ HomT 0 (F, f∗G) −→ HomT (f F,G). ∗ One can show that it is an isomorphism, which establishes that (f , f∗) is an adjoint pair. Hence we have a morphism of topoi T → T 0.

41 3.3 The Etale´ Site 3 Sites

Example 3.2.3. Let f : C0 → C be a continuous map of sites. For an object X0 ∈ C0, consider the presheaf “ represented by X0”:

0 op hX0 :(C ) −→ Set 0 0 0 Y 7−→ HomC0 (Y ,X ). ∗ ∼ It’s easy to see that f hX0 = hf(X0) as functors, that is, the induced morphism of topoi ∗ (f , f∗) commutes with representable functors.

3.3 The Etale´ Site

Let X´et denote the ´etalesite on a scheme X. Fix a faithfully flat morphism ϕ : Y → X and a group G acting on the morphism on the right via α : G → AutX (Y ). Definition. We say ϕ : Y → X is a Galois cover with Galois group G, or a G-cover for short, if the morphism

Y × G −→ Y ×X Y (y, g) 7−→ (y, yg) is an isomorphism. Lemma 3.3.1. ϕ : Y → X is a G-cover if and only if ϕ is surjective, finite, ´etaleand deg ϕ = |G|. Definition. A Galois cover ϕ : Y → X is said to be generically Galois if k(Y )/k(X) is a Galois extension of fields. Example 3.3.2. Let A be a ring, B an A-algebra and consider the corresponding morphism of affine schemes ϕ : Spec B −→ Spec A. Then ϕ is a Galois cover with Galois group G if and only if A → B is faithfully flat and G acts on B such that Y B ⊗A A −→ B × G = B g∈G 0 0 b ⊗ b 7−→ (bgb )g is an isomorphism. Example 3.3.3. Let k be a field and f ∈ k[t] a monic irreducible polynomial. Set K = e1 er k[t]/(f). Then f = f1 ··· fr in K[t] and by the Chinese remainder theorem, r ∼ ∼ Y ei K ⊗k K = K[t]/(f) = K[t]/(fi ). i=1

Thus Spec K → Spec k is a Galois cover if and only if each fi is linear, fi 6= fj for any i 6= j and ei = 1 for all i. That is, Spec K → Spec k is Galois if and only if f is separable with splitting field K, just as in classical Galois theory.

42 3.3 The Etale´ Site 3 Sites

Proposition 3.3.4. Suppose ϕ : Y → X is a G-cover and F is a presheaf on the ´etale site X´et taking disjoint unions to products. Then F is a sheaf on X´et if and only if F (ϕ): F (X) → F (Y ) is an isomorphism onto the fixed set F (Y )G ⊆ F (Y ).

Proof. Consider the two maps Y × G Y given by (y, g) 7→ y and (y, g) 7→ yg. These fit into a commutative diagram with the two coordinate projections : Y ×X Y Y

Y × G Y X

∼= id id

Y ×X Y Y X

Applying F to the diagram, we obtain a commutative diagram of sets

F (X) F (Y ) F (Y ×X Y ) ∼ id id = Y F (X) F (Y ) F (Y ) g∈G

where the maps Q are given by s 7→ (s) and s 7→ (gs) . Then these F (Y ) g∈G F (Y ) g g maps agree precisely when gs = s for all g ∈ G, i.e. F (X) is the equalizer in the top row if and only if it identifies with F (Y )G in the bottom row.

Proposition 3.3.5. Suppose F is a presheaf on X´et which satisfies the condition that Y Y F (U) F (Ui) F (Ui ×U Uj) i i,j

is an equalizer diagram for all covers {Ui → U} of U ∈ XZar in the Zariski site on X and for all ´etaleaffine covers {V → U} of U ∈ X´et consisting of a single morphism in the ´etale site. Then F is a sheaf on X´et. ` Proof. If U = i Ui for schemes Ui ∈ X´et, then the first condition implies that F (U) = Q 0 i F (Ui). Thus for a covering {Ui → U } in X´et, the sequence Y Y F (U 0) F (Ui) F (Ui ×U 0 Uj) i i,j

is isomorphic to the sequence

0 F (U ) F (U) F (U ×U 0 U)

43 3.3 The Etale´ Site 3 Sites

0 ` ` ` for the covering {U → U }, using the fact that ( i Ui) ×U 0 ( i Ui) = i,j(Ui ×U 0 Uj). Since the equalizer condition is assumed to hold for all ´etaleaffine covers, this argument shows the condition holds for all {Ui → U}i∈I with I finite and each Ui affine. 0 ` 0 S Let {Uj → U}j be a covering and set U = j Uj and f : U → U. Write U = i Vi −1 S for open affine subschemes Vi ⊆ U and for each i, write f (Vi) = k Wik for open affine 0 subschemes Wik ⊆ U . Fix one of the Vi. Then each f(Wik) is open in Vi, so by quasi- K compactness, we may reduce to a finite cover {Wik → Vi}k=1. Now consider the diagram 0 0 0 F (U) F (U ) F (U ×U U )

Y Y Y Y Y F (Vi) F (Wik) F (Wik ×U Wi`) i i k i k,`

Y Y Y F (Vi ×U Vj) F (Wik ×Vi Wj`) i,j i,j k,`

0 The two columns correspond to the coverings {Vi → U}i and {Wik → U }i,k which are all coverings in the Zariski site on X and hence these columns are exact by hypothesis. Moreover, the middle row corresponds to the coverings {Wik → f(Wik) ⊆ Vi}k which for each i is finite and affine, so by the above paragraph this row is exact. An easy diagram chase then implies the top row of the diagram is also exact, which is what we want. Example 3.3.6. Let A → B be a ring homomorphism such that Spec B → Spec A is surjective and ´etale.In particular, A → B is faithfully flat and unramified. We claim that the sequence

0 → A → B → B ⊗A B (*) is exact, where the second map is b 7→ 1 ⊗ b − b ⊗ 1. First note that the map g : B → 0 0 B ⊗A B, b 7→ b ⊗ 1 has a section s : b ⊗ b 7→ bb . Then consider

h :(B ⊗A B) ⊗B (B ⊗A B) −→ B ⊗A B, x ⊗ y 7−→ xgs(y). We have h(1 ⊗ x − x ⊗ 1) = gs(x) − x, so if 1 ⊗ x − x ⊗ 1 = 0, we get x = gs(x) ∈ im g and the sequence g 0 → B −→ B ⊗A B → (B ⊗A B) ⊗B (B ⊗A B) is exact. Tensoring (∗) with B induces the vertical arrows in the following diagram:

0 A B B ⊗A B

0 B B ⊗A B (B ⊗A B) ⊗B (B ⊗A B)

Therefore the top row is exact as claimed.

44 3.3 The Etale´ Site 3 Sites

Example 3.3.7. Let X be a scheme and Y → X an ´etalemorphism. Set OX´et (Y ) =

Γ(Y, OY ), which defines a sheaf OX´et for the Zariski topology by standard calculations. To

check that OX´et is a sheaf for the ´etalesite, it suffices to check the conditions of Proposi- tion 3.3.5, but the Zariski condition was just seen to hold. If {Y → Z} is an ´etaleaffine covering in X´et, with Y = Spec B and Z = Spec A, then the corresponding ring map A → B satisfies the condition of Example 3.3.6, meaning

0 → A → B → B ⊗A B is exact and hence so is the sequence

OX´et (Z) → OX´et (Y ) → OX´et (Y ×Z Y ).

(Note that this is precisely the same as the equalizer condition since the map B → B ⊗A B is b 7→ 1 ⊗ b − b ⊗ 1 and Γ(Y, OY ) are abelian groups.) Example 3.3.8. Any scheme Z → X defines a presheaf

FZ : X´et −→ Set

Y 7−→ HomX (Y,Z).

Then FZ is a sheaf for XZar so once again, to show it is a sheaf for the ´etaletopology, by Proposition 3.3.5 it will suffice to check the sheaf condition for single ´etaleaffine covers. For such a cover {Spec B → Spec A}, the map A → B is faithfully flat with

0 → A → B → B ⊗A B

exact. Take an open affine subscheme U = Spec C,→ Z. Applying HomA(C, −) to the above sequence gives

HomA(C,A) → HomA(C,B) → HomA(C,B ⊗A B) which is exact since HomA(C, −) is a left-exact functor. Generalizing to HomX (−,Z) is straightforward using patching, so ultimately we conclude that FZ is a sheaf on X´et. As usual, by Yoneda’s Lemma the assignment Z 7→ FZ is injective so we will often write FZ simply by Z.

Example 3.3.9. Let n ≥ 1 be an integer and let µn be the group scheme defined (locally) n by t − 1 = 0. Then for any Y ∈ X´et, µn(Y ) coincides with the set of nth roots of unity in Γ(Y, OY ).

Example 3.3.10. Let X be a k-scheme and let Ga be the affine group scheme defined by the additive group of k. Then for each Y ∈ X´et, Ga(Y ) = Γ(Y, OY ).

Example 3.3.11. Similarly, when Gm is the affine multiplicative group scheme defined by × × k , then for each Y ∈ X´et, Gm(Y ) = Γ(Y, OY ) . p Example 3.3.12. When char k = p > 0, let αp denote the group scheme defined by t = 0. Note that αp is not an ´etalegroup scheme (but it is flat). For Y ∈ X´et, αp(Y ) corresponds to the set of nilpotent elements in Γ(Y, OY ).

45 3.3 The Etale´ Site 3 Sites

Example 3.3.13. Consider the ring k[ε] = k[t]/(t2). Write T = Spec k[ε]. The functor T = Homk(−,T ) is called the (´etale) tangent space functor since for any Y ∈ X´et, T (Y ) is 2 the tangent space to Y , which is locally given by T (Y )x = TxY := (mx/mx).

Example 3.3.14. Let R be a set and let FR be the sheaf on X´et defined by Y FR : Y 7−→ FR(Y ) := R

π0(Y )

where π0(Y ) denotes the set of connected components of Y . Then FR is called the constant sheaf on X´et associated to R.

Example 3.3.15. Let M be a sheaf of coherent OX -modules on the Zariski site XZar. This gives us an ´etale sheaf M´et as follows. If ϕ : Y → X is an ´etalemorphism, then ϕ∗M is a coherent OY -module on YZar which on affine patches U = Spec A ⊆ X,V = Spec B ⊆ Y takes the form

M ⊗A B M

B A

´et ∗ Let M be the presheaf Y 7→ Γ(Y, ϕ M) on X´et. By a similar proof to the one in Exam- ´et ple 3.3.7 for OX´et , one can show that M is then a sheaf on X´et. As a special case, note ´et that (OXZar ) = OX´et . Example 3.3.16. Let X be a k-scheme, ϕ : Y → X a morphism and consider the exact sequence of sheaves ∗ 1 1 1 ϕ ΩX/k → ΩY/k → ΩY/X → 0. 1 ∗ 1 1 If ϕ is an ´etalemorphism, then by Proposition 1.4.17, ΩY/X = 0 so ϕ ΩX/k → ΩY/k is ∗ 1 ∼ 1 surjective. In fact, both are locally free sheaves of the same rank, so ϕ ΩX/k = ΩY/k. It 1 ´et 1 follows that (ΩX/k) |YZar = ΩY/k.

d Example 3.3.17. Let k be a field, X = Spec k and consider the ´etalesite X´et. If G-Mod denotes the category of discrete G-modules, then there is an equivalence of categories

d Sh(X´et) ←→ G-Mod

F 7−→ MF

FM 7−→ M

where MF = lim F (L) is the direct limit over all finite, Galois extensions L/k and FM is the −→ sep sheaf A 7→ HomG(Homk(A, k ),M).

We next describe the category Sh(X´et). To begin, first consider the category Presh(X´et) of presheaves on the ´etalesite.

46 3.3 The Etale´ Site 3 Sites

Lemma 3.3.18. Presh(X´et) is an abelian category.

0 00 Let F → F → F be a sequence of presheaves on X´et. Then this sequence is exact if and only if the sequence F 0(Y ) → F (Y ) → F 00(Y ) is exact for all ´etalemorphisms Y → X. Let Sh(X´et) be the full subcategory of Presh(X´et) of sheaves of abelian groups on the ´etalesite X´et. Then Sh(X´et) is an additive category; we will prove that it is abelian.

0 Definition. A morphism of sheaves T : F → F on X´et is locally surjective if for every 0 Y ∈ X´et and s ∈ F (Y ), there exists a covering {Yi → Y } in the ´etaletopology such that for 0 each i, s|Yi lies in the image of F (Yi) → F (Yi). Proposition 3.3.19. For a morphism of sheaves T : F → F 0 on the ´etalesite, the following are equivalent:

(a) F −→T F 0 → 0 is exact.

(b) T is locally surjective.

¯ 0 (c) For every geometric point x¯ : Spec k → X, the map on stalks Tx¯ : Fx¯ → Fx¯ is surjective.

0 00 Proposition 3.3.20. For a sequence of sheaves 0 → F → F → F → 0 in Sh(X´et), the following are equivalent: (a) The sequence is exact.

0 00 (b) For all Y ∈ X´et, the sequence 0 → F (Y ) → F (Y ) → F (Y ) → 0 is exact.

¯ 0 00 (c) For all geometric points x¯ : Spec k → X, the sequence of stalks 0 → Fx¯ → Fx¯ → Fx¯ → 0 is exact.

Corollary 3.3.21. For any scheme X, Sh(X´et) is an abelian category.

Example 3.3.22. Let n ∈ N and assume n is invertible on X, i.e. char k - n for any residue fields k of X. Consider the following sequence of sheaves, called the Kummer sequence for X: n 0 → µn −→ Gm −→ Gm → 0 n n where Gm −→ Gm is the morphism induced by t 7→ t . We claim the Kummer sequence is exact. By Proposition 3.3.20, it suffices to show exactness on stalks at geometric points. × For such a pointx ¯, set A = OX,x¯. Then µn,x¯ = µn(A) and Gm,x¯ = A and moreover, × n 0 → µn(A) → A is clearly exact. For the right map, notice that t − a splits in A[t] for × d n n−1 every a ∈ A , since dt (t − a) = nt 6= 0 when n is invertible on X. Thus every a is an nth root in A×, and it follows that the sequence

× n × 0 → µn(A) −→ A −→ A → 0 is exact as required.

47 3.4 A Word on Algebraic Spaces 3 Sites

Example 3.3.23. When char k | n for some residue field k of X, the above example fails since ´etalelocally, we have d (tp − a) = ptp−1 = 0 in characteristic p > 0. dt

p Note however that the equation t 7→ t does define a (locally) flat covering Gm → Gm, so the sequence p 0 → µp −→ Gm −→ Gm → 0 is exact in the flat topology on X. On the ´etalesite, the appropriate characteristic p replacement for the Kummer sequence is the Artin-Schreier sequence

℘ 0 → Z/pZ −→ Ga −→ Ga → 0

where Z/pZ is the constant group scheme defined by the same group and ℘ : Ga → Ga is induced by the map t 7→ tp −t. To see that the AS sequence is exact, it suffices once again to check exactness on stalks at geometric points. For a geometric pointx ¯, again let A = OX,x¯ so that Z/pZx¯ = Z/pZ(A) and Ga,x¯ = A. As before, 0 → Z/pZ(A) → A is exact and the kernel of t 7→ tp − t : A → A is precisely Z/pZ(A). For surjectivity, note that for any a ∈ A, d (tp − t − a) = ptp−1 − 1 = −1 6= 0 dt so tp − t − a splits in A[t] for all a ∈ A. Thus the end of the sequence

℘ 0 → Z/pZ(A) −→ A −→ A → 0 is exact and we are done.

3.4 A Word on Algebraic Spaces

The idea behind algebraic spaces is to generalize to other Grothendieck topologies the prop- erty of the Zariski topology that allows affine varieties and schemes to be glued together to form more general varieties and schemes. In particular, we would like an analogous gluing construction for the ´etaletopology. We define them for the main case that will interest us. Let X = Spec k and consider the category Affk of affine k-schemes equipped with the ´etaletopology, which we will write Affk,´et.

Definition. A sheaf A on Affk,´et is an algebraic space provided it satisfies: (1) There exists an affine k-scheme X equipped with an ´etaleequivalence relation, i.e. a subscheme R ⊂ X×kX such that for each A ∈ Affk, R(A) is an ordinary equivalence p1,p2 relation on X(A) and each composition R,→ X ×k X −−−→ X is surjective and ´etale. (2) There is a locally surjective morphism of sheaves T : X → A giving A = X/R in the sense that for any A ∈ Affk and s1, s2 ∈ X(A), T (A)(s1) = T (A)(s2) in A(A) if and only if (s1, s2) ∈ R(A).

48 3.4 A Word on Algebraic Spaces 3 Sites

Remark. In the above, and from now on, we are denoting the sheaf HomAffk (−,X) by X, so that for example, X(A) means HomAffk (A, X). Example 3.4.1. Any affine scheme X/k is an algebraic space when viewed as the sheaf

HomAffk (−,X), with trivial ´etaleequivalence relation. The main utility of algebraic spaces is that one can take quotients by any free action of a finite group and still have an algebraic space. This is not true in general with schemes.

49 4 Cohomology

4 Cohomology

The ´etalesite on a scheme has a sheaf cohomology theory which mimics the ordinary (sin- gular) cohomology for schemes X/C with their natural complex structure. To define coho- mology, we must first show that Sh(X´et) has enough injectives. Along the way, we develop a few useful algebraic constructions on sheaf categories that will play a role in our study of cohomology and beyond. We will assume all sheaves have values in abelian groups.

4.1 Direct and Inverse Image Functors

Given a morphism of schemes f : Y → X, we have several ways of mapping sheaves on X´et to sheaves on Y´et and vice versa. The simplest to define (although not always the simplest to understand) is the pushfoward functor. Definition. For a morphism f : Y → X and a presheaf F on Et´ (Y ) (the category of ´etale X-schemes), define the direct image presheaf (or pushforward presheaf) of F along ´ op f to be the functor f∗F : Et(X) → Ab sending U 7→ f∗F (U) := F (Y ×X U) for all U → X ´etale.

Lemma 4.1.1. Direct image restricts to a functor f∗ : Sh(Y´et) → Sh(X´et).

Proof. This follows from the axiom (see Section 3.1) that if {Ui → U} is a covering in X´et then {Y ×X Ui → Y ×X U} is a covering in Y´et. Example 4.1.2. Ifx ¯ : ∗ → X is a geometric point of X, then any sheaf on ∗ is an abelian group, say A, andx ¯∗A = Ax¯ is a skyscraper sheaf supported at the image ofx ¯ with stalk A. This explains the usual notationx ¯∗A for such a skyscraper sheaf. Proposition 4.1.3. Let f : Y → X be a morphism and x¯ : ∗ → X a geometric point of X with image x ∈ X. Then for any sheaf F on Y´et,

(1) If f is an open immersion, then (f∗F )x¯ = Fx¯ if x ∈ Y . (2) If f is a closed immersion, then ( Fx¯, it x ∈ Y (f∗F )x¯ = 0, if x 6∈ Y.

(3) If f is finite, then M ⊕[k(z):k(x)]s (f∗F )x¯ = Fz¯ z7→x

where the sum is over all points z ∈ Y mapping to x and [k(z): k(x)]s denotes the separable degree of the residue field extension at z. In particular, if f is finite ´etaleof ⊕n degree n, then (f∗F )x¯ = Fy¯ .

Corollary 4.1.4. If f : Y → X is finite or a closed immersion, then f∗ : Sh(Y´et) → Sh(X´et) is an exact functor.

50 4.1 Direct and Inverse Image Functors 4 Cohomology

g f Proposition 4.1.5. Suppose Z −→ Y −→ X are morphisms of schemes. Then (f◦g)∗ = f∗◦g∗.

The direct image functor on presheaves is exact, so f∗ : Sh(Y´et) → Sh(X´et) is left exact – in general, it is not right exact. Thus we can define:

Definition. Let f : Y → X be a morphism with pushfoward functor f∗ : Sh(Y´et) → Sh(X´et). ∗ The inverse image functor (or pullback) along f is the left adjoint f : Sh(X´et) → Sh(Y´et) to f∗. ∗ Lemma 4.1.6. For a morphism f : Y → X and any sheaf F on X´et, f F is the sheafification −1 of the presheaf f F on Y´et defined by f −1F (V ) := lim F (U) −→ where V → Y is ´etaleand the direct limit is over all U → X ´etalesuch that V Y

f

U X

commutes.

−1 Proof. It is easy to show that f is left adjoint to f∗ on presheaves. Explicitly, for any presheaf Q on Y´et, there are bijections

−1 HomPresh(Y )(f F,Q) ←→ lim HomPresh(X )(F (U),Q(U ×X Y )) ←→ HomPresh(X )(F, f∗Q) ´et −→ ´et ´et which are functorial in F and Q. Therefore, passing to the sheafification yields −1 sh ∼ HomSh(Y´et)((f F ) , Q) = HomSh(X´et)(F, f∗Q) −1 sh ∗ for any sheaf Q on Y´et. Since left adjoints are unique, this proves (f F ) = f F .

g f Proposition 4.1.7. For any morphisms Z −→ Y −→ X, (f ◦ g)∗ = g∗ ◦ f ∗.

∗ ∗ ∗ Proof. Each of (f ◦ g) and g ◦ f is a left adjoint of (f ◦ g)∗ = f∗ ◦ g∗. ∗ Example 4.1.8. When f : Y → X is an ´etalemorphism, the inverse image f : Sh(X´et) →

Sh(Y´et) is just the restriction functor F 7→ F |Y´et . ∗ Proposition 4.1.9. For all morphisms f : Y → X, f is exact and f∗ preserves injectives. Proof. Lety ¯ : Spec k → Y be a geometric point of Y , letx ¯ = f ◦ y¯ and denote their images respectively by y ∈ Y and x ∈ X. Then for any sheaf F ∈ Sh(X´et), Example 4.1.2 and Proposition 4.1.7 give us

∗ ∗ ∗ ∗ ∗ (f F )y =y ¯ (f F ) = (f ◦ y¯) =x ¯ F = Fx¯. Therefore f ∗ is exact on stalks at geometric points, so it is exact on stalks and therefore exact by Proposition 3.3.20. The statement for f∗ follows from the more general statement that the right adjoint of an exact functor preserves injectives.

51 4.1 Direct and Inverse Image Functors 4 Cohomology

Theorem 4.1.10. For any scheme X, the category Sh(X´et) has enough injectives. Proof. Take a geometric pointx ¯ : Spec k → X with image x ∈ X and let F be a sheaf on X´et. Since Fx is an abelian group, there is some injective Ex such that Fx ,→ Ex. Now Ex defines a sheaf on k, necessarily injective, so that by Proposition 4.1.9,x ¯∗Ex is injective. Q Therefore E = x∈X x¯∗Ex is injective and by construction there is a monomorphism of sheaves F → E. A slight annoyance in Proposition 4.1.3 is that the direct image of a sheaf along an open immersion is not necessarily supported on the image of the immersion. Indeed, consider the following example.

Example 4.1.11. Let X = A1 be the affine line over an algebraically closed field k and consider the open immersion j : A1 r {0} ,→ A1. Recall (Example 2.4.15) that G := ´et 1 ∼ π1 (A r {0}, y¯) = Zb for any geometric pointy ¯. Let S be a module over this fundamental G group, which determines a locally constant sheaf F = FS. Then (j∗F )0 = S , the submodule of G-invariants of S. This shows that in (1) of Proposition 4.1.3, (f∗F )x need not be 0. However, we can define a version of the direct image which plays nicely with open im- mersions.

Definition. Let X be a scheme and j : Y,→ X an open immersion. For a sheaf F on Y´et, define the extension along j of F to be the sheafification of the presheaf P on X´et which sends an ´etalescheme U → X to ( F (U), if im(U) ⊆ Y P (U) = 0, otherwise.

This defines a functor j! : Sh(Y´et) → Sh(X´et).

∗ Proposition 4.1.12. For any open immersion j : Y,→ X, j! is exact and j preserves injectives.

∗ Proof. We first observe that (j!, j ) is an adjoint pair: for all F ∈ Sh(Y´et) and G ∈ Sh(X´et), ∼ HomSh(X´et)(j!F,G) = HomPresh(X´et)(P,G) where P is as above ∼ = HomSh(Y´et)(F,G|Y´et ) ∗ = HomSh(Y´et)(F, j G) by Example 4.1.8.

Now take a geometric pointx ¯ : Spec k → X with image x and note that ( Fx, if x ∈ Y (j!F )x = 0, if x 6∈ Y by definition of j!F . Then the first statement follows from Proposition 3.3.20 and the second follows from a similar categorical statement as in the proof of Proposition 4.1.9.

52 4.2 Etale´ Cohomology 4 Cohomology

Proposition 4.1.13. Let j : Y,→ X be an open immersion, Z = X r j(Y ) and i : Z,→ X the corresponding closed immersion. Then for any sheaf F on X´et, the sequence

∗ ∗ 0 → j!j F → F → i∗i F → 0

is exact. Proof. By Proposition 3.3.20, it suffices to check this on stalks, where for x ∈ X we have

( id 0 → Fx −→ Fx −→ 0 → 0, x ∈ Y id 0 → 0 −→ Fx −→ Fx → 0, x ∈ Z.

4.2 Etale´ Cohomology

Let X be a scheme and Sh(X´et) the abelian category of sheaves on the ´etalesite, which has enough injectives by Theorem 4.1.10. For a sheaf F on X´et, let Γ(X´et,F ) := F (X) be the abelian group of global sections of F . Then Γ(X´et, −): Sh(X´et) → Ab is a left exact functor, so we can take its right derived functors to define a cohomology theory on X´et. Definition. For i ≥ 0, the ith ´etalecohomology of X with coefficients in a sheaf F is

i i H´et(X,F ) := R Γ(X´et,F ).

• Explicitly, if F → E is an injective resolution of F in Sh(X´et), then

i i • H´et(X,F ) = H (Γ(X´et,E )). Note the following properties which are immediate from the definition/the theory of derived functors: ˆ 0 For all sheaves F on X´et, H´et(X,F ) = Γ(X´et,F ). ˆ i If E is an injective sheaf, then H´et(X,E) = 0 for all i > 0. ˆ Every short exact sequence of sheaves 0 → F 00 → F → F 0 → 0 determines a long exact sequence in ´etalecohomology:

0 00 0 0 0 1 00 0 → H´et(X,F ) → H´et(X,F ) → H´et(X,F ) → H´et(X,F ) → · · ·

ˆ For an ´etalemorphism f : Y → X, the right derived functors of the composite functor ∗ f Γ(Y´et,−) i Γ(Y, −): Sh(X´et) −→ Sh(Y´et) −−−−−→ Ab are H´et(Y, (−)|Y ), which we will simply write i H´et(Y, −) when the context is clear.

• We now verify that the Eilenberg-Steenrod axioms hold for H´et(X, −), showing it is a • cohomology theory. Note that additivity is guaranteed since H´et is constructed as a right derived functor.

53 4.2 Etale´ Cohomology 4 Cohomology

Theorem 4.2.1 (Dimension). Let x = Spec k be a geometric point, i.e. k is separably closed. i Then for any sheaf F on x, H´et(x, F ) = 0 for all i > 0. Proof. If k is an arbitrary field, then by Example 3.3.17 there is an equivalence of categories

∼ d Sh(Spec k´et) −→ G-Mod

F 7−→ Fx

sep i ∼ i where G = Gal(k /k). Therefore H´et(x, F ) = H (G, Fx), the group cohomology of G with coefficients in Fx, so when k is separably closed, these are all trivial for i > 0.

Fix a sheaf F on X´et and consider the right derived functors of the left exact functor i HomSh(X´et)(F, −): Sh(X´et) → Ab, which are denoted Ext (F, −).

Example 4.2.2. For the constant sheaf Z on X, there are isomorphisms ∼ HomSh(X´et)(Z,G) −→ G(X) = Γ(X´et,G) α 7−→ α(1)

i ∼ i for all G ∈ Sh(X´et). Therefore Ext (Z, −) = H´et(X, −).

Theorem 4.2.3 (Exactness). Let Z ⊆ X be a closed subscheme and F any sheaf on X´et. Then there is a long exact sequence

i−1 i i i · · · → H´et (X r Z,F ) → HZ (X,F ) → H´et(X,F ) → H´et(X r Z,F ) → · · ·

i where HZ (X, −) denotes the ith right derived functor of F 7→ ΓZ (X,F ) := ker(Γ(X´et,F ) → Γ((X r Z)´et,F )). Moreover, the sequence is functorial in the pair (X,X r Z) and in F .

Proof. Let i : Z,→ X and j : X r Z,→ X be the (closed and open, resp.) immersions. For the constant sheaf Z on X, Proposition 4.1.13 gives us a short exact sequence

∗ ∗ 0 → j!j Z → Z → i∗i Z → 0.

For F ∈ Sh(X´et), applying HomSh(X´et)(−,F ), we get an exact sequence

∗ ∗ 0 → Hom(i∗i Z,F ) → Hom(Z,F ) → Hom(j!j Z,F ).

∗ The adjointness of (j!, j ) from Proposition 4.1.12 implies that

∗ ∗ ∗ HomSh(X´et)(j!j Z,F ) = HomSh((XrZ)´et)(j Z, j F ) = F (X r Z) = Γ((X r Z)´et,F ).

i i ∗ Taking derived functors gives H´et(X r Z,F ) = Ext (j!j Z,F ) for all i ≥ 0. Therefore ∗ i ∗ i Hom(i∗i Z,F ) = ΓZ (X,F ), Ext (i∗i Z,F ) = HZ (X,F ) for all i ≥ 0, and the long exact sequence in Ext for the above Hom sequence is precisely the long exact sequence for the pair (X,X r Z) that we are looking for. Theorem 4.2.4 (Excision). Let π : Y → X be an ´etalemorphism and W ⊆ Y a closed subscheme such that

54 4.2 Etale´ Cohomology 4 Cohomology

(1) Z = π(W ) is closed in X and π|W : W → Z is an isomorphism.

(2) π(Y r W ) ⊆ X r Z.

Then for any sheaf F on X´et, the induced map

i i HZ (X,F ) −→ HW (Y,F |Y )

is an isomorphism for all i ≥ 0.

Proof. The maps in the commutative diagram

Y r W Y W

π ∼=

X r Z X Z induce a commutative diagram

∗ ∗ ∗ 0 ΓW (Y, π F ) Γ(Y, π F ) Γ(Y r W, π F )

α

0 ΓZ (X,F ) Γ(X,F ) Γ(X r Z,F )

By Proposition 4.1.9 and Example 4.1.8, π∗ is exact and preserves injectives, so it’s enough to prove the isomorphism for i = 0, that is, to show α is an isomorphism.

Take s ∈ ΓZ (X,F ) and suppose α(s) = 0. Then s|XrZ = 0 by exactness of the bottom row and α(s) = 0 in Γ(Y, π∗F ), but since {Y → X,X r Z → X} is an ´etalecovering of X, the sheaf axioms imply s = 0. Therefore α is injective. ∗ ∗ On the other hand, if t ∈ ΓW (Y, π F ), we may view it as an element of Γ(Y, π F ). Let ∗ −1 t0 ∈ Γ(Y, π F ) be the zero section. Note that (X r Z) ×X Y = π (X r Z) ⊆ Y r W and we have t|Y rW = 0 by exactness of the top row and t0|Y rW = 0 because it’s the zero section. ∗ Thus the sheaf condition on π F says that both t and t0 come from a section s ∈ Γ(X,F )

which must satisfy s|XrZ = 0 by commutativity. Hence by exactness of the bottom row, s ∈ ΓZ (X,F ) and α(s) = t, proving surjectivity.

Corollary 4.2.5. For any closed point x ∈ X and any sheaf F on X´et, there is an isomor- phism i ∼ i h H{x}(X,F ) −→ H{x}(Spec OX,x,F ) h for all i ≥ 0, where OX,x is the henselization of OX,x. Proof. Let π : U → X be an ´etaleneighborhood of X with u = π−1(x). Taking Z = {x} and W = {u}, Theorem 4.2.4 gives us isomorphisms

i ∼ i H{x}(X,F ) −→ H{u}(U, F |U )

55 4.2 Etale´ Cohomology 4 Cohomology

h for all i ≥ 0. Since OX,x = lim Γ(U, OU ) where the limit is over all ´etaleneighborhoods of x, −→ and since cohomology commutes with direct limits, we get

i h ∼ i ∼ i H{x}(Spec OX,x,F ) = lim H{u}(U, F |U ) = H{x}(X,F ) −→ as desired. Finally, to prove the homotopy axiom, we need a suitable analogue of “homotopic maps” in the category of schemes. There are various concepts of homotopy that can be introduced in algebraic geometry, some giving quite different theories, but we will focus on one that gives an analogue of the homotopy axiom for ´etalecohomology. Let k be a field (perhaps algebraically closed), X a scheme over k and suppose Z ⊆ X×P1 is a closed subscheme whose image under the projection π : X × P1 → P1 is dense. For each 1 closed point t ∈ P , set Z(t) = Z ×X (X × {t}), so that each Z(t) may be viewed as an algebraic cycle on X. If Z and Z0 are algebraic cycles on X, we say that they are rationally equivalent, denoted Z ∼ Z0, if there are finitely many such families of algebraic cycles Z1(t),Z2(t),...,Zr(t) interpolating Z1 and Z2 in the following sense: for each 1 ≤ i ≤ r − 1, 0 1 0 there are closed points ti, ti ∈ P such that Zi(ti) = Zi+1(ti), and in addition there are 0 1 0 0 t, t ∈ P such that Z = Z1(t) and Z = Zr(t ). We say two morphisms f, g : Y → X are rationally equivalent if their graphs Γf ∼ Γg as algebraic cycles in X ×k Y . Theorem 4.2.6 (Homotopy). If f, g : Y → X are rationally equivalent, then f ∗ = g∗ : • • H´et(X,F ) → H´et(Y,F ) for any sheaf F ∈ Sh(X´et). Proof. See Milne’s LEC, Ch. 9.

56