ETALE FUNDAMENTAL GROUPS

JOHAN DE JONG

NOTES TAKEN BY PAK-HIN LEE

Abstract. Here are the notes I am taking for Johan de Jong’s ongoing course on etalefundamental groups offered at Columbia University in Fall 2015 (MATH G4263: Topics inAlgebraic Geometry).

Due to my own lack of understanding of the materials, I have inevitably introducedboth mathematical and typographical errors in these notes. Please send corrections andcomments to phlee@math.columbia.edu.

WARNING: I am unable to commit to editing these notes outside of lecture time, so theyare likely riddled with mistakes and poorly formatted.

Contents

1. Lecture 1 (September 8, 2015) 32. Lecture 2 (September 10, 2015) 32.1. References 32.2. Galois Categories 33. Lecture 3 (September 15, 2015) 63.1. Reminders 63.2. Fundamental Groups of Schemes 84. Lecture 4 (September 17, 2015) 94.1. Fibre functor 105. Lecture 5 (September 22, 2015) 125.1. Complex Varieties 125.2. Applications 156. Lecture 6 (September 24, 2015) 156.1. Two more examples 156.2. Fundamental groups of normal schemes 166.3. Action of Galois groups on fundamental groups 177. Lecture 7 (September 29, 2015) 187.1. Short exact sequence of fundamental groups 188. Lecture 8 (October 1, 2015) 218.1. Ramification theory 218.2. Topological invariance of etale topology 239. Lecture 9 (October 6, 2015) 249.1. Special case of Theorem last time 249.2. Structure of proof 25

Last updated: December 10, 2015.1

10. Lecture 10 (October 8, 2015) 2810.1. Purity of branch locus 2811. Lecture 11 (October 13, 2015) 3011.1. Local Lefschetz 3012. Lecture 12 (October 15, 2015) 3412.1. Specialization of the fundamental group 3413. Lecture 13 (October 20, 2015) 3613.1. Abhyankar’s lemma 3613.2. End of proof of Theorem last time 3813.3. Applications 3914. Lecture 14 (October 22, 2015) 3914.1. Quasi-unipotent monodromy over C 3914.2. Etale cohomology version 4015. Lecture 15 (October 27, 2015) 4116. Lecture 16 (October 29, 2015) 4416.1. Outline of proof of quasi-unipotent monodromy theorem 4617. Lecture 17 (November 5, 2015) 4617.1. Proof in the mixed characteristic case 4717.2. Birational invariance of π1 4818. Lecture 18 (November 17, 2015) 4918.1. Semi-stable reduction theorem 4919. Lecture 19 (November 19, 2015) 5119.1. Resolution of singularities 5120. Lecture 20 (November 24, 2015) 5520.1. Method of Artin–Winters 5520.2. Example applications 5720.3. Idea of Artin–Winters 5821. Lecture 21 (December 1, 2015) 5822. Lecture 22 (December 3, 2015) 5822.1. Abstract types of genus g 5822.2. Part 2 of Artin–Winters’ argument 6022.3. Saito 6123. Lecture 23 (December 8, 2015) 6123.1. Neron models 6123.2. Group schemes over fields 6223.3. Neron–Ogg–Shafarevich criterion 6224. Lecture 24 (December 10, 2015) 6324.1. Semi-abelian reduction 6324.2. Proof of Theorem 24.1 6424.3. Weight monodromy conjecture 66

2

1. Lecture 1 (September 8, 2015)

PH: I missed the lecture.

2. Lecture 2 (September 10, 2015)

2.1. References.

• Stacks project, chapter on fundamental groups (tag 0BQ6).• Lenstra, Galois Theory for Schemes.• SGA I.• Murre, Lectures on an introduction to Grothendieck’s theory of the fundamental

group.

2.2. Galois Categories. The idea is to consider

Topological groups↔ Categories C with F : C → Sets

Those categories which go back to themselves after we go around will be the Galois categories.

Notation. Let G be a topological group. Denote by G-Sets the category with objects (X, a)where X is a set (with the discrete topology) and a : G × X → X a continuous action.Because the topology is discrete, this means the stabilizer of any point is open. Morphismsare the obvious ones. Finite-G-Sets is the full subcategory of G-Sets of (X, a) with #X <∞.

The profinite completion G∧ of G is

G∧ = limUCG open, finite index

G/U.

It satisfies the universal property: if G→ H is continuous and H is a profinite group, thenthere exists a unique factorization G→ G∧ → H.

The first interesting statement is

Proposition 2.1. Consider the forgetful functor F : Finite-G-Sets→ Sets. Then

G∧ ∼= Aut(F )

as topological groups.

This is not a triviality and I will explain the proof. A basis of the topology on Aut(F ) isthe kernels of the maps Aut(F )→

∏ni=1 Aut(F (Xi)).

Proof. The steps are

• There exists a map G→ Aut(F ).• This is continuous.• We get G∧ → Aut(F ) because Aut(F ) is profinite and universal property.• G∧ → Aut(F ). To see this, if U CG is open and finite index, then X = G/U belongs

to Finite-G-Sets. SoG∧ //

Aut(F )

G/U

// Aut(F (X))

so ker(G∧ → Aut(F )) ⊂⋂U U = e.

• Enough to show image is dense (by basic topology).3

• Pick X ∈ Finite-G-Sets, and γ ∈ Aut(F ). Enough to find g ∈ G such that

(γX : X → X) = (action of G on X)

(This uses the fact that the category has finite disjoint unions. Silly.)

Proof. Let X =∐

i=1,··· ,nXi, where Xi∼= G/Hi with Hi ⊂ G open subgroup of finite

index. Take U =⋂i=1,··· ,n

⋂g∈G gHig

−1 C G, which is open and finite index. Then

Y = G/U maps into Xi for i = 1, · · · , n.Enough to find g0 ∈ G such that γY is the action of g0. This is because

Yg0//

γY//

Y

Xg0//

γX// X

Think!

• Say γY (eU) = g0U where e ∈ G is the neutral element. Then

γY (gU)UCG→ γY (Rg(eU))

where Rg : G/U → G/U is right multiplication in Arrows(Finite-G-Sets). Since γ isa transformation of functors, this is equal to

Rg(γY (eU)) = Rg(g0U) = g0gU.

Done!

Lemma 2.2. Any exact functor F : Finite-G-Sets → Sets, with F (X) finite for all X, isisomorphic to the forgetful functor.

Proof. Omitted (exercise).

We need to define exactness.

Definition 2.3. Let F : A → B be a functor.

(1) If A has finite limits (⇔ A has a final object ∗A and fibre products) and F commuteswith them (⇔ F (∗A) = ∗B and F (X ×Y Z) = F (X)×F (Y ) F (Z)), then we say F isleft exact.

(2) If A has finite colimits (⇔ A has an initial object ∅A and pushouts) and F commuteswith them (⇔ F (∅A) = ∅B and F (X tY Z) = F (X) tF (Y ) F (Z)), then we say F isright exact.

(3) F exact ⇔ left and right exact.

In Grothendieck’s original exposition, things are more general but it is easier for us towork with exactness properties.

If F : C → Sets is a functor with finite values, we get a functor C → Finite- Aut(F )-Setsby sending X 7→ F (X).

Definition 2.4. Let F : C → Sets be a functor. We say (C, F ) is a Galois category if

(1) C has finite limits and colimits;(2) every object of C is a finite coproduct of connected objects;

4

(3) F has finite values;(4) F is exact and reflects isomorphisms ;

where

• X is connected ⇔ X is not initial and any subobject (Y → X monomorphism) isisomorphic to either ∅ → X or X → X (this is the definition in the context of Galoiscategories only);• F reflects P ⇔ if (F (f) has P ⇒ f has P ).

Warning. This definition is not the same as SGA I, but equivalent.

We will do a bunch of lemmas to see that nice things happen for Galois categories.

Lemma 2.5 (Example Fact). Suppose a, b : X → Y in C, X connected, F (a)(x) = F (b)(x)for some x ∈ F (X). Then a = b.

Proof. F commutes with limits, so Eq(F (a), F (b)) = F (Eq(a, b)) contains x. This implies∅C 6∼= Eq(a, b) ⊂

subobjectX. Since X is connected, Eq(a, b) = X and a = b.

Corollary 2.6. # AutC(X) ≤ #F (X) for X connected.

Definition 2.7. We say X ∈ Ob(C) is Galois if X is connected and equality holds.

Next we need to prove there are enough Galois objects.

Lemma 2.8. For any connected X, there exists a Galois object Y and Y → X.

Proof. Say F (X) = x1, · · · , xn. Sn acts on Xn =∐

t∈T Zt. Then Sn acts in the sameway on F (X)n = F (Xn) =

∐F (Zt). Pick t ∈ T such that ξ = (x1, · · · , xn) ∈ F (Zt). Set

G = σ ∈ Sn | σ(Zt) = Zt. This is the same as

σ ∈ Sn | F (σ)(ξ) = F (Zt)(argument omitted). But F (σ)(ξ) = σ(ξ) = (xσ(1), · · · , xσ(n)), so we win if (x′1, · · · , x′n) ∈F (Zt) implies x′i pairwise distinct. If not, say x′i = x′j, then pri |Zt = prj |Zt by example factabove. This contradicts pri(ξ) = prj(ξ).

The same argument can be used to construct Galois extensions of finite separable exten-sions (everything needs to be dualized, and Xn corresponds to the n-fold tensor product).Also applies to n-fold Galois covering of connected and locally connected spaces.

Lemma 2.9. Let (C, F ) be a Galois category. The action of Aut(F ) on F (X) is transitivefor all X ∈ Ob(C) connected.

Idea of proof. We need to introduce some notations. Let I be the set of isomorphism classesof Galois objects. For i ∈ I, let Xi be a representative. Pick xi ∈ F (Xi). We say i ≥ i′ ifthere exists a morphism Xi → Xi′ . We may pick fii′ : Xi → Xi′ such that F (fii′)(xi) = xi′(⇒ fii′ unique) (because Galois objects).

Claim. F is isomorphic to the functor F ′ : X 7→ colimi∈I MorC(Xi, X).

If this is true, then just as in the case of Galois theory, we set

H = limi∈I

Aut(Xi)

and get Hopp → Aut(F ′) = Aut(F ). 5

3. Lecture 3 (September 15, 2015)

3.1. Reminders. Galois category is a pair (C, F ) where

• C has all finite limits and colimits;• F : C → Sets is an exact functor;• F (X) is finite for all X ∈ Ob(C);• F reflects isomorphisms;• for all X ∈ Ob(C), X ∼=

∐i=1,··· ,nXi, Xi connected.

I was proving the

Lemma 3.1. If X ∈ Ob(C) is connected, then Aut(F ) acts on F (X) transitively.

For example, if π1(T ) = Sn, then the universal cover T has group Sn over T . For then-to-1 covering T ′ over T , T has group Stab(1) over T ′.

Definition 3.2. X is Galois if X is connected and Aut(X) acts transitively on F (X).

Sketch of proof.

• Let I be the set of isomorphism classes of Galois objects in C. For each i ∈ I, pick arepresentative Xi.• i ≥ i′ ⇔ there exists Xi → Xi′ .• Pick γi ∈ F (Xi), for i ≥ i′ pick fii′ : Xi → Xi′ such that F (fii′)(γi) = (γi′). (This

morphism is uniquely determined.)• Ai = Aut(Xi) acts transitively on F (Xi).

Suppose

Xiai//

fii′

Xi

fii′

Xi′∃!// Xi′

This gives a map αii′ : Ai → Ai′ .• The collection of Ai and transitive maps αii′ : Ai → Ai′ forms an inverse system of

finite groups over (I,≥).• A = limAi. I claim that A Ai.

To prove this claim, you show I is a directed partially ordered set (if i1, i2 ∈ I,pick a Galois object Y → Xi1 ×Xi2 , then Y ∼= Xi for some i with i ≥ i1 and i ≥ i2)and you use

Lemma 3.3 (Set Theory Lemma). Directed inverse limit of finite nonempty sets isnonempty.

• There exists Aopp → Aut(F ) by proving that the functor F ′ is isomorphic to F whereF ′(X) = colimI MorC(Xi, X).• αii′(ai) is the unique map that makes

Xiai//

Xi

Xi′αii′ (ai)

// Xi′

commute. Details... Details...6

• F ′ → F is the map

(if f ∈ F ′(X) is given by Xifi→ X) ∈ F ′(X) 7→ F (fi)(γi) ∈ F (X).

We have two more statements about general Galois categories.

Proposition 3.4. Suppose (C, F ) is a Galois category. Then the functor

F : C → Finite- Aut(F )-Sets

is an equivalence.

Proof.

(1) F is faithful: This is the first result on Galois categories we proved.(2) F is fully faithful: Let X, Y ∈ Ob(C) and s : F (X) → F (Y ) commuting with

Aut(F )-action. Then the graph Γs ⊂ F (X)×F (Y ) = F (X×Y ) is a union of orbits.This implies by the lemma (on action Aut(F ) transitive on F (connected)) that thereexists Z ⊂ X × Y which is a coproduct of connected components of X × Y suchthat F (Z) = Γs. This implies pr1 |Z : Z → X is an isomorphism because F (pr1 |Z) isbijective. Then Z is the graph of a morphism X → Y

Xf

//

(pr1 |Z)−1

Y

Zpr2 |Z

??

with F (f) = s.(3) Essentially injective:

(a) Enough to construct X with F (X) ∼= Aut(F )/H, for H ⊂ Aut(F ) an opensubgroup (automatically of finite index).

(b) Can find Y Galois with U = ker(Aut(F )→ Aut(F (Y ))) contained in H.(c) Then by fully faithfulness

Aut(Y ) ∼= AutAut(F )-Sets(F (Y ))∼= AutAut(F )-Sets(Aut(F )/U)∼= (Aut(F )/U)opp

which contains (H/U)opp.(d) Get H ′ ↔ (H/U)opp.

(e) Let X = Coeq

(Y

h′1//

h′r

// Y

), where H ′ = h′1, · · · , h′r.

(f) Since F commutes with colimits,

F (X) = Coeq

(F (Y )

F (h′1)//

F (h′r)// F (Y )

)

∼= Coeq

Aut(F )/URh′1//

Rh′r

// Aut(F )/U

∼= Aut(F )/H

7

where H = h1U t · · · t hrU .

Addendum: Let (C, F ) and (C ′, F ′) be two Galois categories, and H : C → C ′ an exactfunctor. Then

(1) There exists an isomorphism t : F → F ′ H.(2) The choice of t determines h : G′ := Aut(F ′) → G := Aut(F ) a continuous homo-

morphism of groups well-defined up to inner automorphisms of G.(3)

C H//

∼=

C ′

∼=

Finite-G-Setsh// Finite-G′-Sets

is 2-commutative (via t).

3.2. Fundamental Groups of Schemes. Let f : X → Y be a morphism of schemes.TFAE:

(1) f is etale.(2) f is smooth of relative dimension 0.(3) f is flat, lfp (locally of finite presentation), and fibres etale.(4) f lfp and infinitesimally lifting criterion:

X

T0oo

_

first order thickening

Y Taffineoo

∃!

bb

(formally etale).(5) f lfp, flat, unramified (⇔ f locally finite type and ΩX/Y = 0.)

Moreover,

(6) If Y is locally Noetherian, lfp = lft (locally of finite type)(7) If Y = Spec(k), then X → Y is etale if and only if there exists a set I and X ∼=∐

i∈I Spec(ki) with ki/k finite separable.

Formal properties:

(A) Being etale is preserved under base change:

Y ′ ×Y X //

X

f

Y ′ // Y

(B) Being etale is preserved under composition.(C)

Xf

//

h=gf

Y

g

Z

g, h etale ⇒ f etale.8

(D) A→ B etale ring map ⇔ B ∼= A[x1, · · · , xn]/(f1, · · · , fn) such that

∆ = det

(∂fi∂xj

)is invertible. (There is a structure theorem for etale algebras, which is a strongerstatement.)

We will be talking about finite etale morphisms.Let f : X → Y be a morphism. TFAE:

(1) f is finite, etale. (f : X → Y finite iff for all V ⊂ Y affine, U = f−1V is affine andO(V )→ O(U) is a finite ring map (module finite).)

(2) For all V ⊂ Y affine open, f−1V = U is affine and OY (V )→ OX(U) has (*).(3) Same as (2) for some affine open covering.

(*): A → B with B finite locally free as A-module and Q : B × B → A, (b1, b2) 7→TrB/A(b1b2) is nondegenerate.

If B ∼= A⊕n is free and multiplication by b1b2 has matrix (aij(b1b2)), then TrB/A(b1b2) isthe trace of this matrix.

Exercise. If K → B with (*) and K field, then B =∏Li where Li/K finite separable.

Next time: Let X be a scheme. Then

FEtX = Y → X finite etale Fx→ Sets

(fibre functor) is a Galois category, and π1(X, x) = Aut(Fx).

4. Lecture 4 (September 17, 2015)

Today we will talk about etale fundamental groups.

Notation. Let X be a scheme. FEtX is the category with

• objects: Yf→ X finite etale,

• morphisms:

MorFEtX(Y

f→ X, Y ′f ′→ X) = MorX(Y, Y ′) = g : Y → Y ′ | f ′ g = f.

Example 4.1.

(1) Let X = Spec(k) where k is a field. Then

FEtX = finite separable k-algebrasopp.

(2) Let X = Spec(A). Then

FEtX = A→ B which are finite locally free

and Q : B ×B → A, (b1, b2) 7→ TrB/A(b1b2) is nondegenerateopp

= separable A-algebrasopp.

FEtX will be a Galois category only when X is connected. In general,

Lemma 4.2. FEtX has all finite limits and finite colimits and for a morphism of schemesX ′ → X, then base change functor FEtX → FEtX′, Y/X 7→ X ′×X Y/X ′ is an exact functor.

9

Proof.

• Limits: Check there exists final object and fibre products.– Final object: id : X → X.– Fibre products:

FEtX 3 Y1

Y3

~~

Y2

X

Y1 ×Y2 Y3//

finite etale asb.c. of Y1 → X $$

X

Y3

finite etale

??

• Colimits: enough to check finite coproducts and coequalizers– Coproducts: disjoint unions

– Coequalizers: Y1

a//

b// Y2 in FEtX .

Note Yi = SpecX

(fi,∗OYi).

f1,∗OY1 f2,∗OY2b#oo

a#oo Aoo

whereA is a quasicoherent sheaf ofOX-algebras, and the last map is the equalizerof a# and b#.

Claim. SpecX

(A)→ X is in FEtX and is coequalizer of a and b.

Proof of claim (please find your own).

(*): Etale locally on X you can write Yi =∐

j=1,··· ,mi X.

(**): Yi =∐

j=1,··· ,mi X, then any morphism Y → Y ′ over X Zariski locally on

X comes from a map 1, · · · ,m1 → 1, · · · ,m2.(*) follows from the local structure theorem for finite unramified maps. Supposewe have g : Y → X finite unramified and x ∈ X. Then there exists (U, u) →(X, x) etale such that

Y ×X U =∐

j=1,··· ,m

Vj

where Vj are open and closed subsets, and Vj → U is a closed immersion for allj.Finish the proof:(***): Descent theory “says” it is enough to prove claim after etale base change.(****): Explicitly compute what happens when Y and Y ′ are map are as in(**).

4.1. Fibre functor.

Definition 4.3. A geometric point x of a scheme is a morphism Spec(K)x→ X whre k is

an lagebraically closed field.10

Abusive notation: k = κ(x), x = Spec(κ(x)), x = Im(x).Base change gives

Fx : FEtX → FEtx ∼= category of finite sets→ Sets

Y/X 7→ Fx(Y ) = Yx

where Yx is the set of y such that

Y

x

y??

x

Xcommutes.

Theorem 4.4. If X is a connected scheme, then (FEtX , Fx) is a Galois category!

Warning. Not true if X is not connected.

Definition 4.5 (Grothendieck’s Fundamental Group). For X connected,

π1(X, x) := Aut(Fx).

Proof. Already know

• There exist finite (co)limits.• Fx exact.• Decomposition into connected components

– Fact: A finite morphism is closed (hence proper and integral).– Fact: An etale morphism is open (hence smooth and flat+lfp).– If X is Noetherian, then Y is Noetherian and Y =

∐i=1,··· ,n Yi into connected

components, so Yi → X is finite etale. (Need to check: monos in FEtX are openimmersions.)

• Lastly: Fx reflects isomorphisms. Suppose Yg→ Y ′ in FEtX and Fx(g) bijective.

– by above reduce to Y ′ connected.– then g is finite etale (general properties of morphisms)– g is finite locally free, so degree of g is locally constant on Y ′. Since Y ′ is

connected, degree of g is constant.– degree of g is 1 by looking at the fibre over some y′ ∈ Fx(Y ′).– degree 1 implies g isomorphism.

Lemma 4.6. Let f : X ′ → X be a morphism of connected schemes. Let x′ be a geometricpoint of X ′. Set x = f(x′). Then we get a canonical continuous homomorphism of profinitegroups

f∗ : π1(X ′, x′)→ π1(X, x)

such that

FEtX′

Fx′ ∼=

FEtXbase change

oo

∼= Fx

Finite-π1(X ′, x′)-Sets Finite-π1(X, x)-Setsf∗oo

11

2-commutes.

Proof. Fx ∼= Fx′ (base change) because x = f x′.

Lemma 4.7 (Change of base point). If x1, x2 are geometric points of connected X, then

π1(X, x1) ∼= π1(X, x2)

well-defined up to inner automorphism.

Proof. Choose isomorphism Fx1∼= Fx2 .

Example 4.8. Fix k ⊂ ksep ⊂ k. Then

π1(Spec(k), Spec(k)) = Gal(ksep/k).

Proof. The functor

FEtX → Finite- Gal(ksep/k)-Sets

Y 7→ MorSpec(k)(Spec(k), Y ) = MorSpec(k)(Spec(ksep), Y )

(the right hand side has a left action of Gal(ksep/k)) is an equivalence.

Example 4.9. π1(Spec(C)) = 1.

Example 4.10. π1(Spec(R)) = Z/2Z.

Example 4.11. π1(Spec(C((t)))) = Z.

Example 4.12. π1(P1C) = 1.

Proof. Every object in FEtP1C

is isomorphic to a disjoint union of copies of P1C. Let f : Y → P1

Cbe finite etale, with Y connected. Have to show Y ∼= P1

C.By Hurwitz,

2gY − 2 = (2gP1C− 2) deg(f) + deg(R) = −2 deg(f).

Since gY ≥ 0, this implies deg(f) = 1.

5. Lecture 5 (September 22, 2015)

5.1. Complex Varieties.

Notation. Given a scheme X that is lft (locally of finite type) over C we denote Xan theusual topological space whose underlying set of points is X(C).

Recall X(C) is the set of maps Spec(C)→ X. For an affine variety X = V (f1, · · · , ft) ⊂An

C,X(C) = (a1, · · · , an) ∈ Cn | fi(a1, · · · , an) = 0 for i = 1, · · · , t.

Properties:

• If f : X → Y is a morphism of schemes lft over C, then f an : Xan → Y an iscontinuous.• If f is an open/closed immersion, then f an is an open/closed immersion.• If X = An

C, then Xan = Cn with Euclidean topology.

Question/Remark: The properties of topology on C we need to do this construction are:12

(1) Multivariate polynomials give continuous maps. (This is implied by the fact that Cis a topological field.)

(2) 0 ⊂ C is closed.

Examples of other fields are Qp, R, Cp, ...

Lemma 5.1. If f : X → Y is a proper morphism of schemes lft over C, then f an : Xan →Y an is proper (⇔ f closed and fibres are compact).

Proof idea.

• Reduce to projective case by Chow’s lemma: given X → Y proper, there exists aprojective surjection X ′ X such that the composition X ′ → Y is projective.• Reduce to PnY → Y by second axiom.• (Pn)an = Pn(C) is compact.

Lemma 5.2. If f : X → Y is a morphism of schemes lft over C, and f is etale at somex ∈ X(C), then f an is a local isomorphism at x: there exist x ∈ U ⊂

openXan and f(x) ∈

V ⊂open

Y an such that

f an|U : U → V

is a homeomorphism.

Proof. We may shrink X and Y and assume they are affine and that f is etale.

First Proof. Reduce to Y = AnC; to do this pick

X

// W affine

etale

Y

// AnC

such that X = Y ×AnC W .

Lemma 5.3. Let A be a ring, I ⊂ A be an ideal and A/I → B etale. Then there existsA→ B etale such that B ∼= B/IB as A/I-algebras.

Proof. Write B = A/I[x1, · · · , xn]/(f1, · · · , fn) with ∆ = det( ∂fi∂xj

) invertible. Set

B = A[x1, · · · , xn]/(f1, · · · , fn)

[det

(∂fi∂xj

)]−1

.

Then W → AnC is smooth of relative dimension 0 so we can write

W = SpecC[x1, · · · , xn, z1, · · · , zm]/(F1(z, x), · · · , Fm(z, x))

and det(∂Fj∂zk

)j,k=1,··· ,m invertible on W .Implicit function theorem says

(W an, w)→ (Cn, f(w))

is a local isomorphism.

Second Proof. Use13

Theorem 5.4 (Structure Theorem). Let A → B be a ring map etale at a prime q ⊂ B.Then there exists a g ∈ B and g /∈ q such that A→ Bg is standard etale, i.e.,

Bg∼= (A[z]/(P ))Q

where P,Q ∈ A[z], P is monic and dPdz

is invertible in (A[z]/(P ))Q.

Suppose P (z) = zn + a1zn−1 + · · · + an ∈ C[z] and α is a simple root. Then there exists

ε > 0 such that for |bi| < ε, i = 1, · · · , n, the polynomial

Pb1,··· ,bn(z) = zn + (a1 + b1)zn−1 + · · ·+ (an + bn)

has a simple root α(b1, · · · , bn) depending continuously on b1, · · · , bn and converging to α asbi 7→ 0. This can be proved by Newton’s method.

Returning to the proof, we can shrink the map X → Y near x 7→ y (where it is etale) toget a standard etale map. Then we can pick Py(z) ∈ OY (Y )[z].

Corollary 5.5. Given X lft over C, there is a functor

FEtX → finite covering spaces of Xan

(Yf→ X) 7→ (Y an fan→ Xan).

Theorem 5.6 (Riemann Existence Theorem, SGA 1, Exp XII). This functor is an equiva-lence.

Corollary 5.7. Let X lft over C be connected. Then Xan is connected, and for x ∈ X(C),there is an isomorphism

πtop1 (Xan, x)∧

∼=→ π1(X, x),

where the left is the profinite completion of the topological fundamental group, and the rightis the Grothendieck fundamental group.

Warning. Weird things can happen. For example, the exponential map

exp : (A1C)an = C→ Gan

m,C = C×

is not the analytification of any f : A1C → Gm,C.

Sketch of proof for X smooth and projective.

Claim. For any X/C smooth there exists a unique structure of a complex manifold on Xan

such that

• If f : X → Y is a morphism with X, Y smooth, then f an : Xan → Y an is holomorphic.• If f is etale, then f an is a local isomorphism of complex manifolds.• If X = An

C, then get usual complex structure on Cn.

Characterization of smooth morphism: Let f : X → S be a morphism of schemes. TFAE:

(1) f is smooth.(2) For all x ∈ X, there exists x ∈ U such that

X

f

U? _oo

etale

S AnS

oo

commutes.14

Let X/C be a projective and smooth variety. To show

FEtX∼=→ finite topological covers of Xan,

let π : M → Xan be a finite topological covering. This implies M has a unique structure ofcomplex manifolds such that π is a local isomorphism of complex manifolds.

Let L be a positive line bundle on Xan coming from X → PnC, so Xan → Pn(C) and O(r)with Fubini–Study metric). This means L has a metric whose curvature is positive, so π∗Lis positive on M .

By the Kodaira Embedding Theorem, M is algebraic, say M = Y an. (Namely you proveH0(M,L⊗k) gives M → Pn(C). By Chow’s theorem, M → Pn(C) is cut out by polynomialsequations, which gives us Y .)

The problem left over: Y an = Mπ→ Xan is equal to f an for some f : Y → X. We can

either do Chow’s theorem for the graph Γ →M ×Xan, or use GAGA.

Remark. More generally, if X and Y are projective smooth over C, any holomorphic mapXan → Y an is f an for some f : X → Y algebraic.

5.2. Applications. Recall the profinite completion

Z = limn≥1

Z/nZ =∏

` prime

Z`.

• π1(PnC) = 1.• π1(An

C) = 1.

• π1(AnC\0) =

Z if n = 1,

0 if n > 1.(An

C\0 is homeomorphic to S1 and S2n−1 in these two

cases respectively.)• π1(P1

C\0, 1,∞) = profinite completion of free group on 2 generators. This is some-thing we don’t know how to prove without using topology!

6. Lecture 6 (September 24, 2015)

6.1. Two more examples.

• If X is a smooth projective genus g curve over C, then

π1(X) ∼= profinite completion of the free group on α1, · · · , αg, β1, · · · , βgsubject to [α1, β1][α2, β2] · · · [αg, βg] = 1.

• If A/C is an abelian variety of dimension g, then

π1(A) ∼= Z⊕2g.

We don’t know how to prove the first one algebraically. Philosophically, it is interestingto prove these without topological or analytic methods not because they are worse thanalgebraic proofs, but because new proofs give new insights!

Today we will study the fundamental groups of normal schemes and their relationships toGalois groups.

15

6.2. Fundamental groups of normal schemes.

Lemma 6.1. Let A be a normal Noetherian domain with fraction field K. Let L/K be finiteseparable. Then the integral closure B of A in L is finite over A.

Example 6.2. A = Z ⊂ Q ⊂ K a number field. Then OK is finite over Z.

There are counterexamples if A is not assumed to be normal.

Proof. L/K is separable if and only if the trace pairing QL/K : L × L → K given by(x, y) 7→ TrL/K(xy) is nondegenerate. Choose a K-basis β1, · · · , βn for L. After multiplyingby an element of A, we may assume βi ∈ B. Let β∨1 , · · · , β∨n ∈ L be the dual basis withrespect to QL/K .

Fact. TrL/K(B) ⊂ A.

Reason. The minimal polynomial of an element of B has coefficients in A (as A is normal).Then TrL/K(−) is expressible in terms of coefficients of the minimal polynomial.

This impliesB ⊂ Aβ∨1 + · · ·Aβ∨n

(as can be seen by taking dual on⊕

Aβi ⊂ B, which gives B∨ ⊂⊕

Aβ∨i ). The right handside is a finite A-module. Since A is Noetherian, B is a finite A-module as well.

In particular, this lemma applies to geometric rings.

Corollary 6.3. Let X be a normal Noetherian integral scheme with function field K. LetL/K be finite separable. There exists a finite dominant morphism Y → X with Y normaland integral such that the function field f.f.(Y ) is L as extension of K = f.f.(X).

Proof. Lemma over affine opens, and glue.

Definition 6.4. The Y of Corollary is called the normalization of X in L.

More generally, without the Noetherian assumption, we get a normalization that is notnecessarily finite.

Definition 6.5. We say X is unramified in L if the morphism Y → X of Corollary isunramified.

Lemma 6.6. In situation of Corollary, we have Y → X unramified if and only if Y → Xis etale.

Proof. See Lenstra or Stacks project. For an easy proof, use the following facts:

• Structure theorem of finite unramified morphisms (earlier in lectures).• Normalization commutes with smooth (etale) base change.• Closed immersion is never a normalization.

Proposition 6.7. Let X be a Noetherian normal integral scheme with function field K.Then

Gal(Ksep/K) = π1(Spec(K), Spec(K))→ π1(X, Spec(K))

is surjective and this identifies π1(X, Spec(K)) with the quotient

Gal(Ksep/K)→ Gal(M/K),16

where M is the compositum of all finite K ⊂ L ⊂ Ksep such that X is unramified in L, andM/K is Galois.

The key ingredient to the proof is:

Lemma 6.8. If Y → X is finite etale and Y is connected, then Y is normal and integraland is the normalization of X in f.f.(Y ).

Fact (already used in Lemma 6.6). If f : Y → X is etale, then

X normal⇒ Y normal.

(The converse is true if f is surjective.)

Example 6.9. π1(Spec(Z)) = 1, since the maximal unramified extension Q/M/Q is Q.

Example 6.10. π1(A1C) = 1. We saw this last time using topology (C is a contractible

space), but now we can use algebra. Suppose there is an n-to-1 cover. The ramificationcontribution is at most n− 1, so

2gC − 2 ≤ −2 · n+ (n− 1).

This gives a contradiction if n > 1.

Example 6.11. π1(A1Fp

) 6= 1. A map A1Fp→ A1

Fpgiven by x 7→ f(x) is unramified if

f ′(x) 6= 0 for all x. An example is f(x) = xp−x. Look up Artin–Schreier coverings. In fact,

Hom(π1(A1

Fp),Z/pZ)∼= Γ(A1

Fp ,O)/(subgroup of fp − f).

6.3. Action of Galois groups on fundamental groups.

Proposition 6.12. Let k be a perfect field. Let X be a quasicompact and quasiseparatedscheme over k. Assume Xk is connected (i.e., X is geometrically connected over k). Pickξ ∈ Xk. Then there exists a short exact sequence

1→ π1(Xk, ξ)→ π1(X, ξ)→ π1(Spec(k), ξ)→ 1.

The first term is the geometric fundamental group of X, and the last term is Gal(ksep/k).The middle term is sometimes called the arithmetic fundamental group.

Example 6.13. X = P1Q\0, 1,∞. Then

1→ F2 → π1(X)→ Gal(Q/Q)→ 1,

where F2 is the free group on 2 generators. Here we have used:

Fact. π1(P1C\0, 1,∞)

∼=→ π1(PQ\0, 1,∞).

The exact sequence gives a continuous group homomorphism

Gal(Q/Q) → Out(F2) =Aut(F2)

Inn(F2).

The method is to prove corresponding results for the functors

FEtXk ← FEtX ← FEtSpec(k).

17

7. Lecture 7 (September 29, 2015)

7.1. Short exact sequence of fundamental groups. Today we will consider the funda-mental group of a scheme X over k.

Xk//

X

Spec(k) // Spec(k)

If Xk is connected, then there is a short exact sequence of fundamental groups. But beforeinvolving the geometric side, we will consider what this means for Galois categories.

Consider functors of Galois categories

C H→ C ′ H′→ C ′′,

which give rise to the 2-commutative diagram

C H//

F ∼=

C ′ H′//

F ′ ∼=

C ′′

F ′′ ∼=

Finite-G-Setsh// Finite-G′-Sets

h′// Finite-G′′-Sets

We want to study when the sequence of profinite groups

Gh← G′

h′← G′′

is exact.Properties of these two diagrams are related as follows:

Gh← G′

h′← G′′ C H→ C ′ H′→ C ′′

h surjective(A) H fully faithful ⇔ X ∈ C connected and if thereexists ∗C′ → H(X), then X = ∗C.

h′ injective(B) For all X ′′ ∈ C ′′ connected, there exists

H ′(X ′)mono← Y ′′

epi→ X ′′.h h′ trivial (C) H ′(H(X)) ∼=

∐j=1,··· ,m ∗C′′ for all X ∈ C.

Im(h′) normal(D) For all X ∈ C ′ connected, if there exists∗C′′ → H ′(X ′) then H ′(X ′) ∼=

∐j=1,··· ,m ∗C′′ .

h surjective + kernelh is smallest normal

closed subgroupcontaining Im(h′).

(E) H fully faithful + essential image of H is exactlythose X ′ such that H ′(X ′) ∼=

∐j=1,··· ,m ∗C′′ .

We can prove that:

Fact. (E) ⇔ (A) + (C) + (H ′(X ′) ∼=∐∗C′′ ⇒ there exists X ∈ C and an epi H(X) X ′).

Now apply this to

FEtSpec(k)H→ FEtX

H′→ FEtXkwith X/k quasicompact, quasiseparated and geometrically connected (for example, a geo-metrically connected k-variety) and k perfect.

18

(A) Have to show: for k′/k finite separable field extension we have Spec(k′)×Spec(k) X =Xk′ is connected. This is clear because Xk Xk′ and Xk is connected by assumption.

(B) First we have the following

Claim. For every Z → Xk finite etale there exists a k/k′/k finite and Y → Xk′ finiteetale such that Z ∼= Y ×Xk′ Xk = Y ⊗k′ k.

If the claim is true, then the composition Y → Xk′ → X is finite etale and

Z = union of connected components of Y ×Spec(k) Spec(k)

The multiplication map k′ ⊗k k → k gives a connected component

Spec(k) → Spec(k′)×Spec(k) Spec(k),

and Z is the fibre product

Z

open+closed

//

Y ×Spec(k) Spec(k)

Spec(k) open+closed

// Spec(k′)×Spec(k) Spec(k)

i.e., Z = Y ×Spec(k′) Spec(k).

About the claim: k = colimk/k′/k k′, so

Spec(k) = limk/k′/k

Spec(k′)

and

Xk = limXk′

in the category of schemes.In general, if (I,≥) is a directed partially ordered set, and (Xi, ϕii′) is an inverse

system of schemes over I such that ϕii′ are affine, then X = limi∈I Xi exists inschemes.

Lemma 7.1. In this situation, if Xi is quasicompact and quasiseparated for all i, thenFEtX = colim FEtXi, and if Xi is connected for all i 0, then π1(X) = lim π1(Xi).

Affine case. Suppose A = colimAi filtered. Thencategory of A-algebrasof finite presentation

= colim

category of Ai-algebrasof finite presentation

.

An A-algebra of finite presentation looks like A[x1, · · · , xn]/(f1, · · · , fm), where fj =∑finite aj,Ix

I . Pick i large enough such that aj,I ∈ Im(Ai → A).The same holds for categories of etale, smooth, finite+fp, flat+fp, and combinations

of these.

The lemma implies the claim by above.19

(C) This is clear from

Xk//

X

Spec(k) // Spec(k)

and the fact that Spec(k) has trivial π1.(D) Suppose U → X finite etale, U connected, and there is a section s : Xk → Uk.

Uk//

U

Xk//

s

FF

X

Then you can consider

T =⋃

σ∈Gal(k/k)

sσ(Xk) ⊂ Uk

which is open and closed, and Gal(k/k)-invariant. With some work, T is the inverseimage of an open and closed T ⊂ U . Since U is connected, this implies T = U .

(E) Suppose U → X is finite etale and Uk∼=∐

i=1,··· ,nXk. Then there exists k/k′/k (with

k′/k finite) such that

Uk′ ∼=∐

i=1,··· ,n

Xk′ .

Then

U ← Uk′ ∼=∐

i=1,··· ,n

Xk′ = X ×Spec(k)

( ∐i=1,··· ,n

Spec k′

)where the last coproduct is in FEtk.

Remark. We’ve proved the exact sequence

1→ π1(Xksep)→ π1(X)→ Gal(ksep/k)→ 1

for X/k as before and k not necessarily perfect.

Fact. π1(Xk) = π1(Xksep).

Example 7.2. Let X = Gm,k = A1k\0 with char(k) = 0. Then Xk = Gm,k and consider

Xn = Gm,k

(·)n→ Gm,k = Xk

x 7→ xn.

Using Riemann–Hurwitz, you can show these are cofinal in FEtXk . This implies

π1(Xk) = limnµ(k) ∼= Z(1).

20

So we have

0 // π1(Xk)//

∼=

π1(X) //

∼=

π1(Spec(k)) //

∼=

1

0 // Z // ? // Gal(k/k) // 1

and get

Gal(k/k)→ Out(Z) = Z×.This is the cyclotomic character.

Example 7.3. If X → Spec(k) is an elliptic curve with identity O : Spec(k) → X andchar(k) = 0, then

π1(Xk)∼= Z⊕2.

Again multiplying by n gives a cofinal system in FEtXk . The exact sequence is

0 // Z⊕2 // π1(X) // Gal(k/k) //

OXtt

1.

Then

Gal(k/k)→ Aut(Z⊕2) =∏`

GL2(Z`).

8. Lecture 8 (October 1, 2015)

Today we will talk about ramification theory and topological invariance of the etale topol-ogy.

8.1. Ramification theory. A will be a discrete valuation ring with fraction field K =f.f.(A). Let L/K be finite separable, and B the integral closure of A in L. By a previouslemma, we know B is finite over A. Ramification theory will take some work to prove, andthe proofs will be omitted.

Fact. B is a Dedekind domain with a finite number of maximal ideals m1, · · · ,mn.

Throughout today’s lecture n will stand for the number of maximal ideals.A ⊂ Bmi is an extension of DVRs; let ei be the ramification index and fi = [κ(mi), κA].

Fact. [L : K] =n∏i=1

eifi.

Fact. If A is complete or more generally henselian, then n = 1.

Definition 8.1. We say L/K is:

• unramified with respect to A ⇔ all ei = 1 and κ(mi)/κA is separable.• tamely ramified with respect to A ⇔ all ei are prime to the characteristic of κA andκ(mi)/κA is separable.• totally ramified with respect to A ⇔ n = 1 and f1 = 1.

Assume L/K Galois with G = Gal(L/K).21

Fact. G acts on transitively on m1, · · · ,mn ⇒ e1 = · · · = en = e, f1 = · · · = fn = f and[L : K] = nef .

Pick m = m1. Set1 ⊂ P ⊂ I ⊂ D ⊂ G

where

• D = σ ∈ G | σ(m) = m is the deccomposition group of m.• I = σ ∈ D | σ mod m = idκ(m) is the inertia group of m.• P = σ ∈ D | σ mod m2 = id is the wild inertia group of m.

Fact.

• [G : D] = n• κ(m)/κA is normal.

Warning. Not necessarily separable.

• I CD and D/I∼→ Aut(κ(m)/κA).

• P CD and P = 1 if char(κA) = 0; P is a p-group if char(κA) = p > 0.• I/P is cyclic of order the prime-to-p part of e. (There is a canonical isomorphism

I/P∼→ µe(κ(m)).)

Definition 8.2. It = I/P is the tame inertia group of m.

Here is an application. If A is henselian (so n = 1), then

P ⊂ I ⊂ D = G = Gal(Ksep/K)

by passing to the limit. (Write Ksep =⋃L where L runs over Ksep/L/K Galois, then for

each L we getPL ⊂ IL ⊂ DL = Gal(L/K)

and then

P = limPL → I = lim IL → Gal(Ksep/K) = lim Gal(L/K) = limDL.)

Both P and I are normal in G and

(1) G/I ∼= Gal(κsepA /κA).

(2) It = I/P ∼=(*)

∏` 6=char(κA) Z`.

(3) The short exact sequence

1→ It → G/P → Gal(κsepA /κA)→ 1

gives an action of Gal(κsepA /κA) on It which is via the cyclotomic character.

(4) P = 1 if char(κA) = 0.

The isomorphism (*) is noncanonical. Canonically,

It = limeµe(κ

sepA ).

We will soon explain why IL/PL ∼= µeL(κL) at the finite level.

Example 8.3. Let A = κ[[t]] with κ = κ of characteristic 0. The extensions are Le = K(t1/e)(e ≥ 1) with Gal(Le/K) = µe(κ) via (t1/e 7→ ζt1/e) ↔ ζ. (Note A = κ[[t]] ⊂ Be = κ[[t1/e]].)Given σ ∈ Gal(Le/K), the corresponding root of unity is the ζσ such that σ(t1/e) = ζσt

1/e.22

Remark. Back in the finite case, the map

θ : I/P → µe(residue field κ(m) of m ⊂ B)

is given by the rule that

σ(π) = θ(σ) · π mod π2Bm

where π ∈ Bm is a uniformizer and θ(σ) ∈ Bm is any lift of θ(σ) ∈ κ(m). The reason thisworks is that πe = (unit)πA.

We are interested in studying the fundamental groups of curves using this. We will seehow little we get and this is disappointing!

Example 8.4. Let X be a smooth curve over k = k, char(k) = 0. Let X = X\x1, · · · , xm.Apply with Ai = OX,xi (completion) or Ai = OhX,xi (henselization). We get

ai : Spec(f.f.(Ai))→ X.

Hence

Z ∼= π1(Spec(f.f.(Ai)))π1(ai)→ π1(X),

well-defined up to inner conjugation on the right hand side, measures the ramification offinite etale cover above xi. It is possible to choose base-points to make this well-defined, butthere is no canonical choice.

Lemma 8.5. The kernel of the surjective map

π1(X)→ π1(X)

is the smallest closed normal subgroup of π1(X) containing the images of π1(ai).

Corollary 8.6. There exists a short exact sequence

Z⊕m → π1(X)ab → π1(X)ab → 0.

Example 8.7. Let X = A1k and X = Gm,k, so m = 1 and x1 = 0. Suppose you knew the

lemma and π(A1k) = 1. Then π1(X) is topologically generated by 1 conjugacy class. This

is rather weak.If we use that π1(X) ∼= Z (see earlier), then

inertia at 0∼=→ π1(X)

∼=← inertial at ∞.The isomorphism between the two inertia is via inverse. This is compatible with the topo-logical picture.

In characteristic p, we can do a similar thing but we need to use the tame inertia groupinstead.

8.2. Topological invariance of etale topology.

Theorem 8.8. Let f : X → Y be a universal homeomorphism of schemes. Then the basechange functor

schemes etale over Y → schemes etale over X

is an equivalence. Same for FEtY → FEtX and if X and Y are connected then π1(X)∼=→

π1(Y ).23

Definition 8.9. f : X → Y is a universal homeomorphism if for all Y ′ → Y , X×Y Y ′ → Y ′

is a homeomorphism.

Fact. f is a universal homeomorphism if and only if f is integral, surjective and universallyinjective.

Example 8.10.

(1) Thickenings: i : X → X ′ is called a thickening if and only if i is a closed immersionand bijective on points.

Example 8.11. Spec(A/I) → Spec(A) if I ⊂ A is an ideal and:• I2 = 0 (square zero ideals), or• In = 0 for some n (nilpotent ideal), or• for all x ∈ I, there exists n = n(x) such that xn = 0 (locally nilpotent ideal).

Etale cohomology doesn’t see multiplicities.(2) Purely inseparable field extensions.(3) Suppose f : Spec(A) → Spec(B) where A,B are Fp-algebras, and there exists a

g : Spec(B)→ Spec(A) such that

f g = Spec(B → B, b 7→ bq),

g f = Spec(A→ A, a 7→ aq),

where q = pn. Here you get the theorem because base change by g will be thequasi-inverse functor.

Example 8.12. Let A = Fp[x1, · · · , xn]/I and B = Fp[x1, · · · , xn]/I39. Clearly there

is a surjection B → A. To get a map A → B, we send xi 7→ xpm

i where m is suchthat pm > 39. We can always use this if f : X → Y is a finite morphism of varietiesover Fq inducing a purely inseparable function field extension.

9. Lecture 9 (October 6, 2015)

9.1. Special case of Theorem last time. If X0 → X is a closed immersion of schemesinducing a homeomorphism |X0| → |X| on the underlying topological spaces, then thefunctor

FEtX → FEtX0

Y 7→ Y0 = X0 ×X Yis an equivalence.

Remark. Even Xet

∼=→ X0,et (small etale sites).

We often call this the topological equivalence of etale topology. For example, we see thatthe etale site of a scheme only depends on the reduction of the scheme.

Today we will prove the following

Theorem 9.1. Let f : X → SpecA be a proper morphism of schemes. If (A,m, κ) is ahenselian local ring and X0 = X ×Spec(A) Spec(κ), then the base change functor

FEtX → FEtX0

24

Y 7→ Y0 = X0 ×X Y

is an equivalence.

This is an amazing theorem! I will spend some time discussing what this theorem means,how we will use it, and finally sketch the proof. If we want to prove this theorem startingfrom basic commutative algebra, it will take a year!

Remark.

(1) What does it mean? The fundamental group of the total space X is the same as thefundamental group of the special fibre:

π1(X) ∼= π1(X0).

This suggests that a small neighborhood of the special fibre X0 contracts onto X0.This is true in the C-world.

(2) How will we use it? If η ∈ Spec(A) is a generic point then we (this “we” is reallyGrothendieck) will look at

π1(Xη)→ π1(Xη)→ π1(X)Thm= π1(X0).

This composition is called the specialization map. This allows us to compare thefundamental group of the generic fibre with that of the special fibre. In characteristicp, this map has a kernel.

9.2. Structure of proof. I will focus on “essential surjectivity”. Assume Y0 → X0 finiteetale is given and we try to construct Y → X finite etale with Y0 = X0 ×X X.

The argument is standard, and the other parts of this proof (such as full faithfulness) willinvolve similar ingredients:

• deformation (A will be complete local Noetherian),• algebraization (A will be complete local Noetherian),• approximation (A will be excellent and henselian),• limits (A will be arbitrary henselian).

9.2.1. Deformation. Assume (A,m, κ) is complete local Noetherian. Set

Xn = Spec(A/mn+1)×Spec(A) X.

Then we have a sequence of first-order thickenings

· · · ← X2 ← X1 ← X0.

By topological invariance, we get for each n a finite etale scheme Yn → Xn and isomorphismsXn ×Xn+1 Yn+1

∼→ Yn over Xn.

Remark. Can put these together to get a formal scheme

Y = “ colimYn”.25

9.2.2. Algebraization. We will use the

Theorem 9.2 (Grothendieck’s existence theorem). Suppose given X → Spec(A) proper, ANoetherian complete local and

· · · Z2? _oo

h2

Z1? _oo

h1

Z0? _oo

h0

· · · X2? _oo X1

? _oo X0? _oo

with cartesian squares and hn finite, then there exist h : Z → X finite and isomorphismsZn = Xn ×X Z compatible with h, hn and the squares.

This can be deduced from:

Theorem 9.3 (Sheaf version). Given Fn coherent OXn-modules and isomorphisms Fn+1⊗OXn+1

OXn∼→ Fn, there exist a coherent OX-module F and isomorphisms F ⊗OX OXn

∼→ Fn com-patible with transition maps.

Back to our Theorem in case (A,m, κ) is complete local Noetherian. Let Y → X be thefinite morphism such that Y ×X Xn

∼= Yn for all n where Yn is as in deformation step. ThenY → X is etale by:

Lemma 9.4. Let f : Y → X be lft with X locally Noetherian. Let X0 → X be a closedsubscheme with n-th infinitesimal neighborhood Xn → X. If Xn ×X Y → Xn are smooth,resp. etale for all n 0, then f is smooth, resp. etale at all points of f−1(X0).

Finally: X → Spec(A) proper and A local. Then any open neighborhood of X0 is all ofX.

9.2.3. Approximation (Artin, ..., Popescu).

Lemma 9.5. Let (A,m, κ) be a local ring. The henselization is

Ah = colim(A→B,q)

B = colim(A→B,q)

Bq

where the colimit is over the category of A → B etale and q ⊂ B prime lying over m suchthat κ = κ(m) = κ(q). The index category is filtered.

Note that a filtered colimit of local rings is a local ring.

Definition 9.6. A local ring (A,m, κ) is henselian if Hensel’s lemma holds: for all P ∈ A[x]monic, and α0 ∈ κ a simple root of P ∈ κ[x], there exists a root α ∈ A of P such that αmod m = α0.

Lemma 9.7. Ah is henselian.

Lemma 9.8. If A is Noetherian, then

A ⊂ Ah ⊂ A∧ = (Ah)∧

are Noetherian and both inclusions are flat (hence faithfully flat).

Fact (Artin–Popescu). If A is an excellent Noetherian local ring (for example the local ringof a scheme of finite type over a field or Z, or over any Dedekind domain of characteristic0), then approximation holds for Ah ⊂ A∧ = (Ah)∧.

26

Deformation theory combined with algebraization produces objects over the completionA∧, and now we want to descend to Ah.

Definition 9.9. Let R be a Noetherian ring, I ⊂ R an ideal and R∧ = limR/In be thecompletion. We say approximation holds for R ⊂ R∧ if and only if for all N ≥ 1, all n,m,all f1, · · · , fm ∈ R[x1, · · · , xn], and all b1, · · · , bn ∈ R∧ such that fj(b1, · · · , bn) = 0 forj = 1, · · · ,m, there exist a1, · · · , an ∈ R such that

fj(a1, · · · , an) = 0

for j = 1, · · · ,m, and ai ≡ bi mod INR∧.

With approximation, we see that bad properties of A∧ are inherited from Ah, for exampleszero divisors and nilpotence.

9.2.4. Apply. Theorem holds if A = Ch where C is a Noetherian local ring essentially offinite type over Z (more generally if approximation holds for A→ A∧).

Proof. By deformation and algebraization, we have Y ′ → Spec(A∧)×Spec(A) X recovering Y0

finite etale. Say A∧ = colimAi with A→ Ai of finite type. Then

Spec(A∧)×Spec(A) X = lim Spec(Ai)×Spec(A) X.

By a previous result, there exist i and Yi → Spec(Ai) ×Spec(A) X finite etale whose basechange is Y ′. Write (for the map coming from A∧ = colimA)

Ai = A[x1, · · · , xn]/(f1, · · · , fm)→ A∧

xi 7→ bi ∈ A∧

and (b1, · · · , bn) is a solution!By approximation, we get (a1, · · · , an) ∈ A with fj(a1, · · · , an) = 0 for j = 1, · · · ,m and

ai ≡ bi mod mA∧, so an A-algebra homomorphism

ρ : Ai = A[x1, · · · , xn]/(fj)→ A

xi 7→ ai.

Take

Y = Spec(A)×Spec(ρ),Spec(Ai) Yi.

Spec(An)

Y ′oo

Spec(Ai)

Yioo

Spec(A)

Spec(ρ)

YY

Spec(A)×Spec(ρ),Spec(Ai) Yi = Y

OO

Since ai ≡ bi mod mA∧, we get that the special fibre of Y is the same as the special fibreof Y ′, so equal to our given Y0.

9.2.5. Limits. Any A henselian is a filtered colimit of A’s as above.27

10. Lecture 10 (October 8, 2015)

10.1. Purity of branch locus. Suppose X is a curve over Zp, with special fibre X0 (which

is a curve in characteristic p), generic fibre Xη over Qp, and geometric fibre Xη over Qp.We want to analyze how close π1(Xη) is to π1(X0). This will have to do with the purity ofbranch locus, which is one of Grothendieck’s big ideas.

Lemma 10.1. Let f : X → Y be lft. Let x ∈ X with image y = f(x) ∈ Y . TFAE:

(1) f is quasifinite at x.

(2) x is isolated in Xy (def⇔ x is open in Xy).

(3) x is closed in Xy and no x′ x (specialization) in Xy except x′ = x.(4) For some (any) affine opens

Spec(A)

//

X

f

Spec(B)

// Y

and x ∈ Spec(A) corresponding to p ⊂ A, the ring map B → A is quasifinite at p.

Definition 10.2.

• f : X → Y is locally quasifinite (lqf) if f is quasifinite at all x.• f : X → Y is quasifinite if f is locally quasifinite and quasicompact.

Let me state the first version of purity of branch locus, which is really about the ramifica-tion locus. Roughly speaking, “purity” means “if it happens, then it happens in codimension1”.

Lemma 10.3 (Easy case). Let f : X → Y and x ∈ X. Assume

(1) X and Y are locally Noetherian,(2) f is lft and quasifinite at x,(3) f is flat,(4) x is not a generic point,(5) for all x′ x with dim(OX,x′) = 1 we have f unramified at x′.

Then f is etale at x.

The flatness assumption is what makes the proof easy.Now we will translate this into algebra, which requires lots of technical garbage:

• Etale locally a quasifinite morphism becomes finite.• Zariski’s main theorem.

Lemma 10.4 (Algebra problem). Let A→ B be a finite flat ring map, with A a Noetherianlocal ring. If

• dim(A) ≥ 2,• Spec(B)→ Spec(A) is etale over the punctured spectrum Spec(A)\mA,

then A→ B is etale.

We can see how this is related to the above. We can look at OY,y → OX,x. It is not finite,but maybe after some completion we get a finite map.

28

Proof. Flatness implies B ∼= A⊕n for some n, so we can define the trace pairing QB/A :B × B → A by (b1, b2) 7→ TrB/A(b1b2) (trace of the matrix of multiplication by b1b2 on B).Let

discB/A = det(the matrix of QB/A with respect to some basis).

We know that the

branch locus := p ∈ Spec(A) | Spec(B)→ Spec(A) not etale at all points over p

is equal to

p ∈ Spec(A) | discB/A ∈ p = V (discB/A)

(i.e., a prime of A is in the branch locus precisely when the trace pairing is degenerate overthis prime).

If d = discB/A ∈ m, then dimV (d) ≥ 1, so there exists p ⊂ A, p 6= m, p ∈ V (d). Thiscontradicts the assumption. Therefore d ∈ A×, which proves that f is etale.

Lemma 10.5 (Difficult case). Let f : X → Y , x ∈ X, y = f(x). Assume

(1) X and Y are locally Noetherian,(2) OX,x is normal,(3) OY,y is regular,(4) f is qf at x,(5) dim(OX,x) = dim(OY,y) ≥ 1,(6) for all x′ x with dim(OX,x) = 1, f is unramified at x′.

Then f is etale at x.

Again it takes lots of technical garbage to translate this into the following

Lemma 10.6 (Algebra problem). Let A ⊂ B be a finite extension, with A regular local ringof dim ≥ 2 and B normal. If Spec(B)→ Spec(A) is etale over the punctured spectrum, thenA→ B is etale.

Sketch of proof.

• Case I: dim(A) = 2. In this case A→ B will be flat. This is because of the

Fact.(1) Let f : X → Y be a qf and dominant morphism of integral schemes, with Y

regular, dimY = 2 and X normal. Then f is flat.(2) Let f : X → Y be a qf and dominant morphism of integral schemes, with Y

regular and X Cohen–Macauley. Then f is flat.

The idea is that if x1, x2 are a regular system of parameters of A, then ϕ(x1) andϕ(x2) form a regular sequence in B. Then use mB =

√mAB.

• Case II: dim(A) ≥ 3. To prove this Grothendieck uses a local Lefschetz theorem.Pick f ∈ mA\m2

A. Contemplate

Spec(A) = X X0 = Spec(A/fA)? _oo

Spec(A)\m = U?

OO

U0 = Spec(A/fA)\m? _oo?

OO

29

Note A/fA is regular local of dimension 1 less, so by induction we have the result

for A/fA. Reformulated, then result is the statement that FEtX0 → FEtU0 is anequivalence.

If A is henselian, then we have

FEtX ∼= FEtX0∼= FEtSpec(κA).

So we’re done if we can show

FEtX → FEtU0

is an equivalence when A is henselian or something.

Theorem 10.7. 1 Let (A,m) be a local ring and f ∈ m a nonzerodivisor. Let X = Spec(A)and U the punctured spectrum of A. Let X0 = Spec(A/fA) and U0 the punctured spectrumof A/fA. Let V be finite etale over U . Assume

(1) f is a nonzerodivisor,(2) H1

m(A) is a finite A-module,(3) a power of f annihilates H2

m(A),(4) V0 = V ×U U0 is equal to Y0 ×X0 U0 for some Y0 → X0 finite etale.

Then V = Y ×X U for some Y → X finite etale.

Remark.depth(A) ≥ t⇔ H i

m(A) = 0 for i = 0, · · · , t− 1.

So if dimA ≥ s and Cohen–Macauley (e.g. regular), then the assumptions are satisfied.

Corollary 10.8. Local purity of branch locus holds for A local, dim(A) ≥ 3 and A completeintersection.

Roughly speaking, complete intersection means that A is a regular local ring modulo aregular sequence. This result is sharp because it doesn’t work when dimA = 2.

Example 10.9. A = C[x2, xy, y2] ⊂ B = C[x, y]. This looks like the plane mapping to thecone via quotienting out by ±1. Then B is not finite etale over A.

11. Lecture 11 (October 13, 2015)

11.1. Local Lefschetz.

Notation. (A,m, κ) will be a Noetherian local ring, and f ∈ m will be a nonzerodivisor.

ConsiderSpec(A)\m = U

_

U0 = Spec(A/fA)\m _

? _oo

Spec(A) = X X0 = Spec(A/fA)? _oo

1PH: This is a corrected version of the theorem stated in lecture (see here), as pointed out to me byJohan. To quote him: “It should not be an equivalence between the categories of finite etale schemes from Uto U0. This theorem is enough to prove purity in the regular local case (the implication direction being thatif the thing extends after restriction to the hypersurface, then it extends). But, it is not enough to get theapplication to Grothendieck’s thing about complete intersection rings. The result you need there is this. Butthis is a bit harder to state... I may come back to this in the next lecture, but probably not.”

30

If U and U0 are connected, we get a diagram of fundamental groups

π1(U)

π1(U0)oo

π1(X) π1(X0)oo

Observation: If A is strictly henselian (i.e., A is henselian and κ is separably closed,for example A is complete with algebraically closed residue field), then π1(X) = 1 andπ1(X0) = 1.

Purity says that the map on the left is an isomorphism, hence in the strictly henseliancase π1(U) = 1 and any finite etale cover of U is trivial. Lefschetz is about the top mapπ1(U)← π1(U0).

Local Lefschetz type questions: Find conditions on A such that

• π1(U0) π1(U), or

• π1(U0)∼=→ π1(U), or

• Pic(U) → Pic(U0), or

• Pic(U)∼=→ Pic(U0), ...

The misstated theorem from last time has been corrected in the notes.

Theorem 11.1. Assume A is henselian, H1m(A) is finite, and fN kills H2

m(A) for some N .Then

FEtU → FEtU0

is fully faithful, i.e., π1(U0) π1(U) if U and U0 are connected.

In the situation of Theorem 11.1, if A is strictly henselian and if local purity holds forA/fA, i.e., π1(U0) = 1, then local purity holds for A, i.e., π1(U) = 1. (This is anexample of going from knowing local purity for lower dimension to knowing local purity forhigher dimension.)

We have to show

FEtU → FEtU0

V 7→ V0 = V ×U U0

is fully faithful.Set Vn = V ×U Un where Un = V (fn+1) ⊂ U .

V

· · ·? _oo V2? _oo

V1? _oo

V0? _oo

U _

· · ·? _oo U2? _oo _

U1? _oo _

U0? _oo _

open

Spec(A) · · ·? _oo Spec(A/f 3A)? _oo Spec(A/f 2A)? _oo Spec(A/fA)? _oo

Lemma 11.2. The functor is fully faithful if and only if for all V ∈ FEtU , we have π0(V )bij.→

π0(V0).31

Looking at the diagram above, we get

π0(V )→ · · · → π0(V2)∼=→ π0(V1)

∼=→ π0(V0).

Note that

subsets of π0(Vi)bij.→ set of idempotents of Γ(Vi,OVi)

for each i, so we have

subsets of π0(V ) //

bij.

· · · // subsets of π0(V2)∼=//

bij.

subsets of π0(V1)∼=//

bij.

subsets of π0(V0)

bij.

set of idempotentsof Γ(V,OV )

// · · · //set of idempotentsof Γ(V2,OV2)

∼=//set of idempotentsof Γ(V1,OV1)

∼=//set of idempotentsof Γ(V0,OV0)

Therefore it is enough to show

Γ(V,OV )∼=→ lim Γ(Vn,OVn).

Here completeness is crucial.

Lemma 11.3 (Tag 0BLD). In the setting above there exist mysterious modules Hp and shortexact sequences

0→ R1 limHp−1(OVn)→ Hp → limHp(OVn)→ 0

and0→ H0(Hp(OV )∧)→ Hp → TfH

p+1(OV )→ 0,

where

• HP (OV )∧ is the derived completion of Hp(OV ) with respect to F , and for any A-module M one has a short exact sequence,

0→ R1 limM [fn]→ H0(M∧) M → 0

where M = limM/fnM is the usual completion, and M∧ = R lim(Mfn→ M) (at

degrees −1 and 0 respectively) is the derived completion.• Tf (M) = lim←−nM [fn] is the f -Tate module of M :

· · · →M [f 2]×f→M [f ] = x ∈M | fx = 0.

We will take p = 0:

• H0 = limH0(OVn) by the first exact sequence.• If H0(OV ) has bounded f -power torsion (e.g. when V is Noetherian), then

H0(H0(OV )∧) = H0(OV ).

We get

0→ H0(V,OV )→ limH0(OVn)→ TfH1(OV )→ 0.

Corollary 11.4. To show FEtU → FEtU0 is fully faithful it suffices to show for all V → Ufinite etale that:

(1) H0(OV ) is f -adically complete.(2) f -power torsion on H1(OV ) is bounded.

32

Proof of (1) when A is complete. H1m(A) finite and V → U finite etale imply that H0(V,OV )

is finite over A. This uses finiteness theorems in coherent cohomology.

Example 11.5.

• If dimA = 1, then H0(U,OU) is never finite: pick x ∈ m nonzerodivisor, and then1xn∈ H0(U,OU) for all n ≥ 1.

• If A is a normal domain of dim ≥ 2, then H0(U,OU) = A is finite. We haveH1

m(A) = 0.

In general, there is an exact sequence

0→ H0m(A)→ A→ H0(U,OU)→ H1

m(A)→ 0

and

H i(U,OU)∼=→ H i+1

m (A)

if i ≥ 1. (Look in Hartshorne for

0→ H0Z(X,F)→ H0(X,F)→ H0(U,F)→ H1

Z(X,F)→ · · · .)

Proof of (2). We have to show π : V → U etale finite implies

H1(V,OV ) = H1(U, π∗OV ) = H1(U, some vector bundle on U)

has bounded f -torsion. This follows from the next lemma.

Lemma 11.6. If H1(U,OU) ∼= H2m(A) is annihilated by fN for some N , then for any finite

locally free sheaf E on U there is an M = M(E) such that fM annihilates H1(U, E).

Proof. For all u ∈ U there exists u ∈ U ′ ⊂open

U such that

E|U ′∼=←ϕO⊕rU |U ′ .

We can pick U ′ = D(g) for some g ∈ A.There exists a map α : O⊕rU → E such that α|U ′ = gN1ϕ. Dually, there exists a map

β : E → O⊕rU such that β|U ′ = gN2ϕ−1. Thus the composition

E β→ O⊕rUα→ E

is multiplication by gN1+N2 (maybe after increasing N1 and N2).Now for all u ∈ U , there exists g ∈ A such that u ∈ D(g) and gN1+N2fN ∈ Ann(H1(U, E)).

The ideal generated by all these gN1+N2 is m-primary, so

mN101 · fN ⊂ AnnH1(U, E)

and hence

fN101+N ∈ AnnH1(U, E).

Upshot: We are done with Theorem 11.1 in the complete case. Then you still have toreduce to the complete case.

Next time we will talk about specialization for fundamental groups.33

12. Lecture 12 (October 15, 2015)

12.1. Specialization of the fundamental group. We will talk about the specializationof the fundamental group in the smooth proper case.

Theorem 12.1. Let k ⊂ k′ be an extension of algebraically closed fields. Let X/k be aproper connected scheme. Then Xk′ is connected and π1(X) = π1(Xk′).

Sketch of proof. Recall “cohomology and base change”: if X/R is quasicompact and qua-siseparated and R→ R′ is flat, then

H i(X,OX)⊗R R′ = H i(XR′ ,OXR′ ).To prove connectedness, we use

H0(X,OX)⊗k k′ = H0(Xk′ ,OXk′ ).Note H0(X,OX) is a finite-dimensional k-algebra, hence artinian. It is a product of localrings. Since k is algebraically closed, the number of local components remains the same afterbase change to k′.

The proof of π1(X) = π1(Xk′) uses our previous theorem about π1(X) where X is properover a henselian local ring.

Remark. It would be easy to prove the theorem if you knew that π1(X×Y ) = π1(X)×π1(Y )when X and Y are varieties over k = k. But this is not true:

π1(A2Fp) 6= π1(A1

Fp)× π1(A1Fp).

In characteristic 0, this is true but I don’t know a truly simple proof.

Theorem 12.2 (Specialization of fundamental group). Let X → S be a proper smoothmorphism with geometrically connected fibres. Let s s′ be a specialization of points of S.Then there is a map

sp : π1(Xs)→ π1(Xs′)

which is surjective and

• an isomorphism if char(κ(s′)) = 0.• an isomorphism on prime-to-p quotients if char(κ(s′)) = p.

Remark. If X is connected, then

π1(Xs)sp

//

$$

π1(Xs′)

zz

π1(X)

commutes.

Remark (Missing material). Let f : X → S be a flat proper morphism with geometricallyconnected and reduced fibres. If S is Noetherian and connected and s ∈ S, then there is anexact sequence

π1(Xs)→ π1(X)→ π1(S)→ 1.

This is the “first homotopy sequence”. See Murre.

Remark. sp exists only assuming X → S proper.34

Step 0: Reduce to S Noetherian (technical).Step 1: Reduce to S = Spec(A) where A is a dvr.

Proof. Use that given a specialization s s′ in S Noetherian, we can find

Spec(A)→ S

η 7→ s,

0 7→ s′,

where A is a dvr, η ∈ Spec(A) is the generic point, and 0 ∈ Spec(A) is the closed point.(Roughly, this follows by blowing-up to get codimension 1, taking the generic point of theexceptional fibre, and normalizing to get a dvr.)

By Theorem 12.1, the residue field extensions don’t matter.Step 2: We may assume A is a complete dvr with algebraically closed residue field (same

arguments).Step 3+4: Now s = η, s′ = 0, S = Spec(A), A as in Step 2, and we get

π1(Xη)→ π1(Xη)→ π1(X) ∼= π1(X0)

whose composition is the specialization map sp. The isomorphism is given by a previoustheorem.

Step 5: sp is surjective.

Proof. Let K = f.f.(A). Let Y0 → X0 be connected finite etale. Let Y → X be thecorresponding finite etale cover. Then Y is connected (which follows from π1(X) = π1(X0)).We have to show Yη is connected.

If not, then there exists L/K finite such that YL is disconnected (because κ(η) = colimκ(η)/L/Kfinite

L

and ...). Let B ⊂ L be the integral closure of A in L. Note A/mA = B/mB and B is local.

YL //

YB //

Y //

##

X

Spec(L)

// Spec(B) // Spec(A)

Now YB → Spec(B) is smooth proper with disconnected generic fibre. Two componentshave to meet in YB, but this cannot happen because YB is normal (even regular). The nextlemma implies Y0 = YB/mB is disconnected, a contradiction.

Lemma 12.3. If Z → T is smooth and proper. Then the function t 7→ #π0(Zt) is locallyconstant on T .

Step 6: The kernel of sp?

Claim. Let γ : π1(Xη)→ G be a continuous finite quotient with #G prime to char(A/mA).Then γ factors through sp.

π1(Xη)sp

//

##

π1(X0)

G

35

Proof. Let G acting on Y ′ → Xη be the corresponding finite etale cover. By a limit argument(see Step 5), there exists κ(η)/L/K finite such that Y ′ is defined over L. After replacing Aby B ⊂ L (see Step 5), we may assume we have G acting on Y ′ → Xη.

Let Y → X be the normalization of X in f.f.(Y ′). Then we get

Y ′

//

πη

Y

π

Xη // X

where πη is finite etale, π is finite and Y is normal. As X is a regular scheme (smoothover dvr), by purity of branch locus, if π is not unramified then π is not unramified at acodimension 1 point. Such a point always lies over ξ ∈ X0, the generic point of X0.

Let y1, · · · , yn ∈ Y0 be the generic points of Y0. Then π−1(ξ) = y1, · · · , yn. Now #Gprime to char(κ(ξ)) implies that the ramification of OY,yi/OX,ξ is tame.

(We will continue the proof next time.)

To elucidate, here we have exactly the situation as in the section on ramification theory:

f.f.(X)

// f.f.(Y )

OX,ξ?

OO

// C?

OO

where f.f.(X) ⊂ f.f.(Y ) is a Galois extension with group G, and C is the integral closure ofOX,ξ in f.f.(Y ). We have maximal ideals m1, · · ·mn with Cmi = OY,yi .

Next time: Abhyankar’s lemma guarantees that we can lower the e of this after basechange with a suitable B/A finite as before.

13. Lecture 13 (October 20, 2015)

13.1. Abhyankar’s lemma. The Stacks Project (Tag 0BRM) has a general version of Ab-hyankar’s lemma, which is about lowering ramification index by base change. Today we willlower this down to 1 so that there is no ramification. In fact the general version allows usto lower the ramification index to any specified divisor.

The situation is as follows. Let A ⊂ B be an extension of dvr’s; we only require thatA → B be a local ring map, i.e., mA ⊂ mB. Let L = f.f.(B) and K = f.f.(A). Then L/K isan extension (not necessarily finite). We have the invariants:

• e = ramification index e: πA = (unit)πeB,• κB/κA extension of residue fields (not necessarily finite).

Reminder: An extension of fields κ′/κ is called separable if and only if every finitelygenerated subextension κ′/`/κ is separably generated : there exists x1, · · · , xr ∈ ` such that

` ⊃ κ(x1, · · · , xr) ⊃ κ

where the first extension is finite separable and the second extension is purely transcendental.Under this definition, it is true (but not obvious) that a separably generated extension isseparable!

Proposition 13.1. TFAE:36

• κ′/κ is separable.• κ′ is geometrically reduced over κ.• κ′ ⊗κ Ωκ/Z → Ωκ′/Z.• Any derivation of κ extends to a derivation of κ′.• κ′ is formally smooth over κ.• H1(Lκ′/κ) = 0.

If κ′/κ is a finitely generated field extension, then these are also equivalent to:

• κ′/κ is separably generated.• κ′ = κ(X) where X/κ is a smooth variety.• dimκ′(Ωκ′/κ) = Tr(κ′/κ).

Example 13.2. Fp ⊂⋃n Fp(t1/p

n) is separable: the only derivation of Fp is trivial!

Example 13.3. Let κ = Fp(t, s, r). Then the variety X : txp + syp + rzp = 0 is nowheresmooth over κ, so κ(X)/κ is not separable. Such a variety does arise in nature: it is thegeneric fibre of a smooth morphism X→ A3

Fp over Spec(Fp(t, s, r)).

Going back to A ⊂ B, let K1/K be a finite separable extension.

BOo

B1 r

$$

L A?

e

OO

Oo

eiA1

?

eij

OO

r

$$

L⊗K K1

K K1

where:

• L⊗K K1 is a finite product of finite separable extensions of L,• A1 is the integral closure of A in K1,• B1 is the integral closure of B in L⊗K K1,• m1, · · · ,mn are the maximal ideals of A1, and• mi1, · · · ,mimi are the maximal ideals of B1 lying over mi.

We have new invariants:

• ei = ramification of A→ (A1)mi ,• eij = ramification of (A1)mi → (B1)mij ,• κ(mij)/κ(mi).

Theorem 13.4 (Abhyankar’s lemma). Assume e is prime to the characteristic of κA andthat κB/κA is separable. If e | ei, then eij = 1 for all j = 1, · · · ,mi, and κ(mij)/κ(mi) isseparable.

Remark. In fact under some hypothesis you get

eij =e

gcd(e, ei).

This is not yet in the Stacks Project.37

Example 13.5.

x // (z, · · · , z)

A[x]/(x5 − t) //∏5

i=1 k[[z]] (z, ζ5z, ζ25z, ζ

35z, ζ

45z)

A = k[[t]]

OO

// A[y]/(y5 − t)

OO

y_

OO

where ζ5 ∈ k is a primitive 5-th root of unity. Here∏5

i=1 k[[z]] is the normalization ofA[x, y]/(x5 − t, y5 − t).

13.2. End of proof of Theorem last time. Recall that A is a complete dvr with κA = κAand char(κA) = p. X → Spec(A) is a smooth proper geometrically connected fibre. Y → Xis the normalization of X in a Galois extension, with f.f.(X) ⊂ f.f.(Y ) with group G of orderprime to p and Yη → Xη etale.

Y → X → Spec(A) 3 η

YηG→ Xη

Let ξ ∈ X0 be the generic point of the special fibre, and y1, · · · , yn ∈ Y0 are points lyingover ξ.

Claim. The hypotheses of Abhyankar’s lemma apply to A ⊂ OY,yi for i = 1, · · · , n.

Proof. Look at A ⊂ OX,ξ ⊂ OY,yi . Since f.f.(Y )/f.f.(X) is tamely ramified with respect toOX,ξ, we see that e(OY,yi/OX,ξ) is prime to p. Note e(OX,ξ/A) = 1 as X → Spec(A) issmooth. Hence

e(OY,yi/A) = e(OY,yi/OX,ξ) · e(OX,ξ/A)

is prime to p. Since κA = κA, we get that the residue field extension is separable.

Pick A1 = A[π1/eA ] where e ∈ N is sufficiently divisible (e.g. #G | e). (This was called B

in the previous lecture.) Then

Y

Y1oo

X

XA1oo

Spec(A) Spec(A1)oo

where Y1 is the normalization of YA1 . By Abhyankar’s lemma, Y1 → XA1 has ramificationindex 1 at all points of closed fibre of Y1.

By tameness of f.f.(Y )/f.f.(XA1) with respect to OXA1,ξ, we see that Y1 → XB1 is unram-

ified in codimension 1. By purity, Y1 → XA1 is etale. This means G is a quotient of π1(X0)as desired.

38

13.3. Applications. Let us now apply this famous theorem of Grothendieck. An exampleof a variety in characteristic p that lifts to characterisitc 0 is complete intersection.

13.3.1. Complete intersections. Let X ⊂ Pnk be a smooth complete intersection and k = k.If dim(X) ≥ 2, then π1(X) = 1.

Proof. Over k = C, we use Lefschetz. Then we get it over any k = k of characteristic 0, sowe can lift (using the Witt ring) and apply the theorem.

13.3.2. Curves. Let X be a smooth projective curve of genus g over k = k. Then there is asurjection (

〈α1, · · · , αg, β1, · · · , βg〉[α1, β1] · · · [αg, βg]

)∧ π1(X)

which induces an isomorphism on maximal prime-to-p quotients.This is one of the victories of Grothendieck’s school.

Theorem 13.6 (Tamagawa, Raynaud, ...). Fix p and g > 1. Then the association

Isomorphism classes of genus gsmooth projective curves over Fp

→ Isomorphism classes ofprofinite groups

given byX 7→ π1(X)

is finite to 1.

Example 13.7. If (E,O) is an elliptic curve over k = k, then

π1(E) =

Z⊕2 if char(k) = 0,∏6=p Z

⊕2` × Zp if char(k) = 0, E ordinary,∏

6=p Z⊕2` × 0 if char(k) = 0, E supersingular.

Example 13.8. If A is an abelian variety, then

π1(A) =∏`6=p

Z⊕2g` × Zfp

where f ∈ 0, · · · , g is the p-rank of A.

14. Lecture 14 (October 22, 2015)

First we will discuss quasi-unipotent monodromy over C, but this will be so much easierwith algebraic geometry. This will also fit well with the Remynar.

14.1. Quasi-unipotent monodromy over C. Let f : X → S be a smooth proper mor-phism of schemes l.f.t. over C.

Fact. f an : Xan → San is a fibre bundle: for all s ∈ San = S(C), there exists s ∈ U ⊂ San

open such that (f an)−1(U) ∼= Xs × U as homeomorphism of spaces over U .

Corollary 14.1. The sheaves Rif an∗ (Z) are locally constant on San (such a thing is often

called a local system) with stalk H i(Xs,Z) at s. In particular, there is a monodromy repre-sentation

ρi : π1(San, s)→ AutZ(H i(Xs,Z)).39

Theorem 14.2. The representation ρi sends “loops around∞” to quasi-unipotent operators.

Proof. Hodge theory; but see later.

Definition 14.3. An element g ∈ GLn(Field) is quasi-unipotent if some power of g isunipotent (⇔ the eigenvalues of g are roots of unity).

What do we mean by “loops around ∞”?

(1) If S is a smooth curve, choose a smooth projective compactification S ⊂ S. Then foreach x ∈ S\S there is a well-defined conjugacy class in π1(San, s) consisting of loopsaround x (counterclockwise).

(2) If dim(S) is arbitrary, we consider finite morphisms C → S for C smooth, and lookat the images of loops around ∞ in π1(C) in π1(S).

14.2. Etale cohomology version. Let f : X → S be a smooth proper morphism ofNoetherian schemes. Let ` be a prime number invertible on S. Then we get Rifet,∗(Z`)locally constant (technically we haven’t discusses etale sites yet, but fet is the etale analogueof the analytification for the complex topology), with

Rifet,∗(Z`)s = H iet(Xs,Z`)

is a finitely generated Z`-module. There is a monodromy representation

ρi` : π1(S, s)→ AutZ`(Hiet(Xs,Z`)).

Theorem 14.4 (Grothendieck). ρi` have quasi-unipotent local monodromy, i.e., for everymorphism Spec(K)→ S where K = f.f.(A) and A is a dvr, the action

ρiXK/K,` : Gal(Ksep/K)→ AutZ`(Hiet(XK ,Z`))

when restricted to an inertia subgroup has the following properties, provided ` is invertiblein κA:

(1) the image of wild inertia is finite: #ρiXK/K,`(P ) >∞.

(2) if τ ∈ I is an element mapping a topological generator of It, then ρiXK/K,`(τ) isquasi-unipotent.

Recall 1 ⊂ P ⊂ I ⊂ D ⊂ Gal(Ksep/K), where P is the wild inertia, I is the inertia, Dis the decomposition, PI = I/P ∼=

∏`′ 6=char(κA) Z`′ is the tame inertia.

The argument of Grothendieck shows that this is a consequence of the structure of inertiagroups (which can be thought of as fundamental groups), without knowing about etalecohomology and the local systems Rifet,∗(Z`)! This theorem really gets used in geometricsituations when we are given a smooth proper morphism, even though the proof goes throughthe inertia groups.

Remark. You can deduce the theorem on quasi-unipotent monodromy over C from Grothendieck’sversion by using comparison theorems.

Remark. Immediate reduction to S = Spec(K), where K = f.f.(A).

I am going to prove part (1).

Lemma 14.5. Let `, p be two distinct primes. Let H,G be profinite groups with H pro-` andG pro-p. Then there is no nontrivial continuous group homomorphism G→ H.

40

Proof. The same is true for finite groups.

Lemma 14.6. Let M be a finitely generated Z`-module. Endow M with the `-adic topology.Then

AutZ`(M) = Autcont(M) = lim Aut(M/`nM)

is a profinite group and the kernel of the continuous

AutZ`(M)→ Aut(M/`M)

is a pro-` group.

Proof. Omitted. (This is clear from the description lim Aut(M/`nM).)

Lemma 14.7. If X/K is a variety, then ρiX/K,` : Gal(Ksep/K) → AutZ`(Hiet(XK ,Z`)) is

continuous.

Proof. By definition,H i

et(XK ,Z`) = limnH i

et(XK ,Z/`nZ)

and the action of Gal(Ksep/K) on these is continuous.

Remark. Even thoughH iet(XK ,Z/`nZ) is finite, continuity is not automatic because Gal(Ksep/K)

is a profinite group!

Proof of part (1) of Theorem 14.4. Done by combining the lemmas and using that the wildinertia is a pro-p group where p = char(κA). This is why ` 6= char(κA) is needed.

15. Lecture 15 (October 27, 2015)

Let A be a dvr with fraction field K and residue field κ, X/ Spec(K) be a smooth andproper variety, and ` 6= char(κ) be a prime number. We have the picture

1 ⊂ P ⊂ I ⊂ D ⊂ Gal(Ksep/K)

where P is the wild inertia, I is the inertia and D is the decomposition (if K is henselianlocal, then D = Gal(Ksep/K)), together with representations

ρiX/K,` : Gal(Ksep/K)→ Aut(H iet(XK ,Z`)).

Last time we showed that ρiX/K,`(P ) is finite, using ` 6= char(κ) and the continuity of ρiX/K,`.

Theorem 15.1. If τ ∈ I maps to a topological generator of It = I/P , then ρiX/K,`(τ) isquasi-unipotent.

We will do this by mapping Aut(H iet(XK ,Z`)) into

AutQ`(Hiet(XK ,Z`)⊗Z` Q`) ∼= GLBiX (Qp),

where BiX is the i-th Betti number of X.

Remark. This just means every τ ∈ I maps to a quasi-unipotent element of GL.

Today we will consider the case when κ has a finite number of `-power roots of 1, and saya few words about why the general case does not follow by simply taking limits.

Lemma 15.2. Assume κ has finitely many `-power roots of 1. Then there exist σ ∈ D andτ ∈ I such that

41

• τ maps to a topological generator of It,

• στσ−1 and τα map to the same element of It where α ∈ Z× is α = (α2, α3, α5, α7, · · · )with α` ≡ 1 (mod `) but α` 6= 1.

Aside: Suppose G is a profinite group, τ ∈ G, α ∈ Z. Then τα is defined.

Proof. We may assume G is finite. Then τn = 1 for some n. Pick a ∈ Z such that a ≡ α(mod n). Then set τα = τa in G.

More conceptually, there is a map Z→ G given by a 7→ τa. By the universal property of

profinite completion, we get Z→ G by α 7→ τα.

Proof of lemma. There are canonical maps D → Gal(κsep/κ) and

θcan : I → limn prime to p

µn(κsep)

where p = char(κ) (or 1 when char(κ) = 0) (and the limit on the right is noncanonicallyisomorphic to

∏`′ 6=p Z`′), such that

θcan(στσ−1) = σ(θcan(τ))

for τ ∈ I and σ ∈ D. Note that θcan factors through It, and any σ ∈ D acts on It ∼=limn prime to p µn(κsep) '

∏`′ Z`′ by multiplication by some ασ ∈

∏`′ 6=p Z

×`′ . So we just pick a

σ ∈ D such that

(1) σ(ζ`) = ζ`,(2) σ(ζ`n) 6= ζ`n for some n > 1,

where ζ` and ζ`n are primitive `- and `n-th roots of 1 in κsep. This is possible by ourassumptions on κ.

Corollary 15.3. Theorem 15.1 holds if κ has only a finite number of `-power roots of 1,and in fact it holds for any continuous ρ : Gal(κsep/κ)→ AutZ`(M) and → AutQ`(V ) whereM is a finite generated Z`-module and V is a finite-dimensional Q`-vector space.

Proof. Pick σ, τ as in the previous lemma. Then ρ(τ) is conjugate to ρ(στσ−1) (here we needthe fact that ρ is defined on D, not just I). Since #ρ(P ) < ∞ and στσ−1 and τα have thesame image mod P , this implies by thinking plus group theory that

ρ((στσ−1)N) = ρ((τα)N)

for some N > 0. Then ρ(τ)N is conjugate to (ρ(τ)α)N = (ρ(τ)N)α. Finish by the followinglemma.

Lemma 15.4. If g : M → M is an automorphism of a finite Z`-module such that g isconjugate to gα with α as in the previous lemma, then g is quasi-unipotent.

To prove this for Q`-vector spaces, pick a stable lattice, but we will not do this in detail.

Proof. Look at eigenvalues λ1, · · · , λt; these will be in Z`×

. Then we get: for all i there existsj such that λαi = λj. Then for all i, there exists 0 ≤ r < s such that λα

r

i = (λαr

i )αs, so

(λαr

i )αs−1 = 1.

Note αr 6= 0 and αs − 1 6= 0, by looking at the `-component of α: there are no roots of 1 in1 + `Z` if ` > 2, or in 1 + 4Z2 if ` = 2.

42

We still have to prove that if λ ∈ Z`×

and λβ = 1 with β ∈ Z and β 6= 0, then λ is a rootof 1. This will be omitted.

Example 15.5. Let K be a local field, A = OK be the ring of integers and X → Spec(K)be an abelian variety of dimension g. Consider the Tate module

T`X = limX[`n](K) ∼= Z⊕2g`

with a continuous action ρ of Gal(Ksep/K). Then we have shown, after replacing K bya finite separable extension, we have ρ(P ) = 1 and ρ of the tame inertia is given by aunipotent operator.

Fact (I don’t know an easy proof). Each Jordan block in our unipotent thing is 1 × 1 or2× 2.

Remark. If X has good reduction (i.e., X extends to an abelian scheme over Spec(A)), thenρ(I) = 1. The converse is true, but a lot harder to prove.

The picture is

ρ(τ) =

1. . .

1

0

1. . .

1

0

1. . .

1

0

0 0

1. . .

1

of size 2g × 2g, with the middle block of size 2g′ × 2g′. Namely, it will turn out that Xextends to a semi-abelian scheme G over Spec(OK) with special fibre

0→ T → G0 → Y → 0

(extension of commutative group schemes) where Y is an abelian variety of dimension g′,and T is a torus over κ:

T ⊗κ κ ∼= Gg−g′m,κ .

Now we consider the general case, when κ has lots of roots of 1, e.g. κ = κ or A = C[[t]].Here is a naive idea: there is a K0 ⊂ K which is finitely generated over Q or Fp and X0/K0

is smooth proper such that X ∼= X0 ×Spec(K0) Spec(K).

Proof. See SGA.

We have a commutative diagram

Gal(Ksep/K) //

ρiX/K,`

Gal(Ksep0 /K0)

ρiX0/K0,`

Aut(H iet(XK ,Q`))

∼// Aut(H i

et(X0,K0,Q`))

Now the discrete valuation on K induces a discrete valuation v on K0.43

Warning. This idea FAILS because the residue field OK0,v may have too many roots of 1 forthe previous argument to work.

Example 15.6. Consider K0 = Q(x, y) → K = C[[t]] given by x 7→ t and y 7→∑ζnt

n,

where ζn = e2πin . Then κ =

⋃nQ(ζn).

To prove the general case, we will write A = colimA0 as a colimit of regular rings, andapply Abhyankar’s lemma.

16. Lecture 16 (October 29, 2015)

Recall that we are trying to prove the quasi-unipotent monodromy theorem.

Lemma 16.1. To prove the quasi-unipotent monodromy theorem, it suffices to prove it whenA is a complete dvr with κ = κ.

This is counterintuitive: in the previous lecture we needed κ to be small!

Proof. Omitted. Reason: any A can be completed and we can always find an extension ofdvr’s A ⊂ A′ with residue field of A′ algebraically closed. The tame inertia of the Galoisgroup of the fraction field of A′ will map onto that of A.

Neron desingularization is the same process needed for constructing Neron models. Thereferences are Tag 0BJ7 of the Stacks Project and Artin’s paper on approximation.

Theorem 16.2 (Neron desingularization). Let R ⊂ Λ be an extension of dvr’s with e = 1,f.f.(Λ)/f.f.(R) separable, and κΛ/κR separable. Then

Λ = colimAi

is a filtered colimit of smooth R-algebras.

This theorem is amazing, but Neron’s procedure of proving it is even more amazing. Thisis proved using Neron blowups, which I claim can be proved on a napkin!

Suppose X → Spec(R) is a morphism of finite type with a section σ. Let η ∈ Spec(R)be the generic point and 0 ∈ Spec(R) be a closed point. σ(0) may not be smooth, so weconsider the affine blowup at σ(0) to get X1 → Spec(R) with a new section σ1. The pointσ1(0) may still not be smooth, so we do it again.

Fact. If σ(η) is in the smooth locus of f , then after a finite number (n) of these Neronblowups, σn(0) is in the smooth locus of Xn → Spec(R).

Application (in the equal characteristic case): Suppose A is a complete dvr, κ = κ and Acontains a field. Then (Cohen structure theorem)A ∼= κ[[t]] with κ = κ. Say char(κ) = p > 0.Then we may apply Neron desingularization to Fp[[t]] ⊂ κ[[t]].

We need to check the hypotheses are satisfied: indeed Ω1Fp[[t]]/Fp = Fp[[t]] dt shows the

extension of fraction fields is separable, and the other parts are clear.

Corollary 16.3. A = colimAi is a filtered colimit with Ai’s smooth over Fp[[t]] or Q[[t]] andt maps to a uniformizer π of A.

Remark. In particular, K = f.f.(A) = A[ 1π] = colimiAi[

1t].

44

Remark. We can replace Ai by the henselization of (Ai)Ai∩mA and then we see

(A, π) = colim(Ai, t)

is filtered, where

• Ai is a henselian regular local ring,• Ai/mAi is a finitely generated field extension of Fp or Q,• t ∈ mAi , t /∈ m2

Ai(because Fp[[t]]→ Ai or Q[[t]]→ Ai is smooth).

Important question: What is the structure of

π1(Spec(Ai)\V (t)) = π1

(Spec

(Ai

[1

t

]))?

Theorem 16.4 (Generalized Abhyankar’s lemma). Let (A,m, κ) be a regular henselian localring. Let t1, · · · , td be a regular system of parameters. Let 0 ≤ r ≤ d. Then there exists aquotient

π1

(Spec

(Ai

[1

t1 · · · tr

])) πt1

with the following properties:

(1) The kernel of this map is topologically generated by pro-p-groups, where p = char(κ)(a pro-0-group is 1).

(2) There is a short exact sequence

0→r∏i=1

(lim

n prime to pµn(κsep)

)→ πt1 → Gal(κsep/κ) = π1(Spec(A))→ 1

where limn prime to p µn(κsep) is non-canonically isomorphic to∏

`6=p Z`, such that the

action of Gal(κsep/κ) on the group on the left is as indicated.

Remark. πt1 is the tame fundamental group of (Spec(A), D = V (t1 · · · tr)). There is a generaldefinition of tame fundamental groups for pairs (S,D) where D ⊂ S is a normal crossingsdivisor.

Idea of proof. The map comes from considering the Galois closure of⋃n prime to p

f.f.(A[ n√t1, · · · , n

√tr])

and combining with ramification theory (at generic points of ti = 0), purity and Abhyankar’slemma.

Corollary 16.5. In the situation of generalized Abhyankar’s lemma, assume ` 6= char(κ)and κ has a finite number of `-power roots of 1. Then any continuous

ρ : π1

(Spec

(Ai

[1

t1 · · · tr

]))→ GLn(Q`)

whose image is pro-`, factors through πt1 and maps elements of ker(πt1 → Gal(κsep/κ)) toquasi-unipotent elements.

Proof. Exactly the same as last lecture except we need to assume our image is pro-` so thatpro-p-groups map to 1.

45

Lemma 16.6. To prove the quasi-unipotent monodromy theorem, it suffices to prove it for(A,X/K, `, i) when

• A is a complete dvr,• the action ρiX/K,` is trivial mod `, i.e., the action of Gal(Ksep/K) on H i

et(XK ,Z/`Z)is trivial.

16.1. Outline of proof of quasi-unipotent monodromy theorem. We combine theabove to prove the general case of the quasi-unipotent monodromy theorem.

Step 1: Reduce to A complete dvr, κ = κ (Lemma 16.1).Step 2: Reduce to action on H i

et(XK ,Z/`Z) trivial (Lemma 16.6).Step 3: Write (A, π) = colim(Ai, t) with Ai henselian regular local, κAi finitely generated

over the prime field, and regular parameters (Remark).Step 4 (limit): Get Xi → Spec(Ai[

1t]) proper smooth whose base change is X.

ρiXi/Ai[

1t],`

: π1

(Spec

(Ai

[1

t

]))→ Aut(H i

et(XK ,Z`)).

Step 5 (limit): After possibly increasing i we may assume ρiXi/Ai,` mod ` is trivial. By

the previous lemma, Im(ρiXi/Ai,`) is pro-`.Step 6: Conclude by Corollary 16.5.Left over: mixed characteristic complete dvr A with κ = κ. Consider Zp ⊂ A, with

absolute ramification index e. Then the Cohen structure theorem gives A ⊃ W (κ) withramification index e. A uniformizer π ∈ A satisfies an Eisenstein equation

πe + λ1πe−1 + · · ·+ λe = 0.

To finish, consider the diagram

R // A

W (κ0) //?

OO

W (κ)?

OO

where

• κ0 ⊂ κ is the perfection of a finitely field extension of Fp,• R ∼= W (κ0)[X]/(Xe + λ′1X

e−1 + · · ·+ λ′e) with λ′i close to λi.

This uses Krasner’s lemma.

17. Lecture 17 (November 5, 2015)

Next week: no lectures.Last lecture we have seen that Grothendieck’s quasi-unipotent monodromy theorem follows

from a statement of the form

Fact. Let A be a complete dvr with algebraically closed residue field κ and uniformizer π.Then we can write

(A, π) = colim(Ai, t)

as a filtered colimit, where each Ai is regular local henselian, t ∈ mAi/m2Ai

, with residue fieldκi ∈ Ai/mAi a purely inseparable extension of a finitely generated extension of its primefield.

46

Last time we saw that the Fact is true in the equicharacteristic case.

17.1. Proof in the mixed characteristic case. We will prove the Fact in the mixedcharacteristic case.

Step 1: For every perfect field κ of characteristic p > 0 there exists a canonical completedvr W (κ) with uniformizer p such that (universal property): for any complete local ring(B,mB) and κ→ B/mB there exists a unique lift W (κ)→ B.

(References for the Witt ring include Lang, Serre and Zink.)Step 2: Apply to our complete dvr A with residue field κ = κ to get

W (κ)→ A.

Then A will be finite flat over W (κ) with some ramification index e. Pick a minimal equation

πe + λ1πe−1 + · · ·λe = 0

with λi ∈ W (κ).

Lemma 17.1 (Krasner’s lemma). There exists N > 0 such that if λ′1, · · · , λ′e ∈ W (κ) satisfyλi − λ′i ∈ pNW (κ), then

P (x) = xe + λ′1xe−1 + · · ·+ λ′e

has a root π′ ∈ A with π′ ≡ π (mod m2A).

Last lecture we considered Neron desingularization, which corresponds to the case e = 1and gives A as a colimit of nice Zp-algebras. We want to work with general e.

Step 3: Given λ ∈ W (κ) and N > 0 we can find κ0 ⊂ κ, where κ0 is the perfection ofa finitely generated extension of Fp, and λ′ ∈ W (κ0) such that λ − λ′ ∈ pNW (κ). Here wethink of W (κ0) ⊂ W (κ) via the map corresponding to κ0 → κ.

Step 4: We get

W (κ)[x]/(P (x)) A0Krasner

// A

W (κ0) //?

OO

W (κ)?

OO

• Pick N as in Krasner’s lemma.• Pick κ0, λ

′1, · · · , λ′e using Step 3 (repeatedly).

• Set A0 = W (κ0)[x]/(xe + λ′1xe−1 + · · ·+ λ′e).

• Because we started with an Eisenstein polynomial we see that A0 is a complete dvrwith uniformizer the class x of x.• We get A0 → A by mapping x to π′.

Step 5: Neron desingularization applies to A0 → A. Thus

(A, π′) = colim(Ai, t)

with Ai the henselization of smooth algebras over A0 at a prime. Then κi = Ai/mAi has thedesired property.

Remark. This argument does not reduce the difficult case of Grothendieck’s quasi-unipotentmonodromy theorem back to the easy case, but it uses the generalized Abhyankar lemmainstead.

47

17.2. Birational invariance of π1.

Lemma 17.2. Let f : X → Y be a birational proper morphism of varieties with X normaland Y nonsingular. Then

π1(X) ∼= π1(Y ).

Proof. Let U ⊂ X be the largest open such that f |U : U → f(U) is an isomorphism.Then codim(Y \f(U), Y ) ≥ 2 by the valuative criterion of properness for f and the fact thatf(U) ⊂ Y is the largest open over which f−1 lives. By purity, the top map is an isomorphismin the diagram

π1(Y ) π1(f(U))∼=

oo

π1(X)

OO

π1(U)oooo

The bottom map is surjective because X is normal. This proves that π1(X) ∼= π1(Y ).

Corollary 17.3. “π1 is a birational invariant.”

Idea: Given K/C finitely generated, Pick X smooth projective over C with C(X) ∼= K.Then π1(X) is independent of the choice of X (only depends on K).

Corollary 17.4. If X is a rational smooth projective variety over k = k, then π1(X) = 1.Proof. Choose a birational map ϕ : Pnk 99K X, and let X ′ be the normalization of Γϕ ⊂Pnk ×X. Apply birational invariance twice.

Recall (not discussed): Let f : X → Y be a proper flat morphism of varieties all of whosegeometric fibres are connected and reduced. Then there exists an exact sequence

π1(Xt)→ π1(X)→ π1(Y )→ 1

where Xt is a geometric fibre of f .As a special case, let Y = P1

k and X be a smooth projective surface over k = k. Thenπ1(X) is a quotient of π1(Xt) for all t ∈ P1.

Example 17.5. If one fibre is a tree of P1’s then π1(X) = 1. For example, this is the caseif g(Xη) = 0.

Example 17.6. Suppose g(Xη) = 1 and all fibres are at worst nodal. Then Xη is an ellipticcurve E, and (with a grain of salt) either

• X = E × P1 and π1(X) ∼= π1(E), or• π1(X) is finite cyclic.

Proof. No singular fibres ⇔ X = E × P1 ⇔ j-invariant of fibres is constant. The rest of the

cases occur when there exists a bad fibre Xt. Then π1(Xt) ∼= Z and π1(X) is procyclic. Wehave the correspondence

connected cyclic finite etale coverswith group Z/nZ

↔L invertible on X with L⊗n ∼= OX

but L⊗m 6∼= OX for 0 < m < n

(g : X ′ → X) 7→ Lχ

by choosing a primitive n-th root of 1 in k and the corresponding χ : Z/nZ → k×. Moreprecisely, the action of Z/nZ gives the decomposition (g∗OX′) =

⊕χ:Z/nZ→k× Lχ.

48

Let η ∈ Y = P1 be the generic point, and K = κ(η) = k(P1) ∼= k(t). Then L|Xη gives a

K-point of E = Pic0Xη/η (an elliptic curve over K) whch has order n.

By Lang–Neron, E(K) is a finitely generated abelian group, because j(E/K) /∈ k. Thisimplies E(K) has a finite amount of torsion.

Warning. Prof. de Jong is worried that something went wrong!!

18. Lecture 18 (November 17, 2015)

Last time we finished our discussion on the monodromy theorem. Today we will talk aboutthe semi-stable reduction theorem for curves.

18.1. Semi-stable reduction theorem. This is what the theorem says in the complex-analytic category. Suppose we have a family of curves over the punctured disk D∗, and wewant to fill this in with a smooth proper curve at the center. In general this is not possible,but after base change by z 7→ zn, we can fill in a nodal curve.

This complex version is not that hard to prove. I will formulate this in the correct gener-ality in algebraic geometry, with dvr’s.

Definition 18.1. Let X be a locally Noetherian scheme. A strict normal crossings divisoron X is an effective Cartier divisor D ⊂ X such that for all p ∈ D, the local ring OX,p isregular and there exists a system of parameters x1, · · · , xd ∈ mp and 1 ≤ r ≤ d such that

D ×X Spec(OX,p) = V (x1 · · · xr).

Lemma 18.2. This is equivalent to:

• D ⊂ X effective Cartier divisor;• each irreducible component Di of D is regular;• for J = i1, · · · , it where t = #J , DJ = Di1 ∩ · · · ∩Dit is regular of codimension t

in X;• D =

∑i∈I Di (i.e., D is reduced).

Example 18.3.

• Two lines intersecting at a point: Yes.• Three lines intersecting at point: No.• Two curves meeting tangentially at a point: No.• A self-intersecting curve: No.• Cone: No.• Two plane curves intersecting transversally at two points: Yes.

Definition 18.4. For X locally Noetherian, an effective Cartier divisor D ⊂ X is a normalcrossings divisor if there exists an etale covering Uk → Xk∈K such that D ×X Uk ⊂ Uk isa strict normal crossings divisor for all k ∈ K.

Remark. Normal crossings divisors “can have self-intersections”.

Example 18.5. The curve Y 2 +X3 +X2 = 0 in A2R is a normal crossings divisor.

Definition 18.6. Let R be a dvr, and S = Spec(R) be the trait2 with generic point η andclosed point s.

2This is French!49

(a) An S-variety is an integral scheme, separated and of finite type over S with Xη 6= ∅.(b) Xs = f−1(s) = X ⊗ κ(s) = V (π) ⊂ X is the special fibre.(c) We say X is strictly semi-stable over S if (c1) to (c4) hold.

(c1) Xη is smooth over κ(η).(c2) Xs is reduced.(c3) Each irreducible component Xi of Xs is an effective Cartier divisor on X. (By

(c2) this implies Xs =∑

i∈I Xi.)(c4) For all nonempty J ⊂ I we have that the scheme-theoretic intersection XJ =⋂

i∈J Xi is smooth over κ(s) and has codimension #J in X.(In particular Xs is a strict normal crossings divisor on X.)

Fact. If x ∈ Xs, then

OX,x ∼= B[[t1, · · · , tr]]/(t1 · · · tr − π)

where B is a complete local R-algebra which is formally smooth over R.

Fact. If κ(s) is perfect, then

(c1), (c2), (c3) ⇔ Xs is an s.n.c.d. in X.

Example 18.7. Let R = κ[[t]] ⊂ A = κ′[[t]] with κ′/κ not separable. Then for X = Spec(A),Xs is an s.n.c.d. but (c4) is violated.

Definition 18.8. We say X is semi-stable over S if etale locally on X we have X is strictlysemi-stable (s.s.-s.) over S.

We have the following diagram of implications:

(Xs)red n.c.d. Xs n.c.d.ks

κ(s) is perfect'/X s.s./Sks

(Xs)red s.n.c.d.

KS

Xs s.n.c.d.ks

KS

κ(s) is perfect'/X s.s.s./S.ks

KS

Definition 18.9. Let S be a scheme. A semi-stable curve over S is a morphism X → Swhich is flat, proper, of finite presentation, such that all geometric fibres are connected, andof dimension 1 with singularities at worst nodes.

Recall that for X finite type over k = k and dim(X) = 1, a closed point x ∈ X is a node

if and only if OX,x ∼= k[[u, v]]/(uv).

Definition 18.10. A split semi-stable curve X over a field k is a semi-stable curve overk whose irreducible components are all geometrically irreducible, and whose nodes are allk-rational. A split semi-stable curve over S is a semi-stable curve over S such that all fibresare split.

Lemma 18.11. Let R be a dvr, S = Spec(R), and X → S a proper S-variety of relativedimension 1 with geometrically connected fibres. If X/S is semi-stable as an S-variety, thenX is a semi-stable curve over S.

Warning. The converse is not true.50

Remark. For the converse it is true that a blowup of an X on the RHS will be in the LHS.(Grain of salt!)

In terms of moduli theory, we want to produce a semi-stable model for every curve overK by lifting Spec(K)→Mg to Spec(R)→Mg

Spec(K) //

Spec(R)

&&

Mg //Mg

// Spec(Z).

As stated this is wrong, and the issue is that Mg is an algebraic stack. The valuativecriterion for stacks should allow for finite extensions. With this modification, such a liftingbecomes possible.

Theorem 18.12 (Semi-stable reduction of curves). Given R dvr with fraction field K, andC/K smooth, proper, geometrically connected curve there exist:

• R ⊂ R′ extension of dvr’s,• X ′ → S ′ = Spec(R′) an s.s.s. S-variety,• C ⊗K K ′ ∼= X ′ ⊗R′ K ′.

Additional properties we would like:

• K ′/K finite separable,• Y → Spec(B) where B is the integral closure of R in K ′ such that Y ⊗B Bm′ is an

s.s.s. variety over Bm′ for all maximal ideals m′ ⊂ B. (A standard argument allowsone to work with one m′ at a time; we will not discuss further.)

Here is the strategy.Step 1: Pick some Ksep/K ′/K finite separable such that some property holds for the

GalK′-action on Pic0(CKsep)tors.Step 2: Let R′ = Bm′ with B ⊃ m′ as before. We will show there exists X ′ → Spec(R′)

(relatively minimal model) which is proper flat, X ′ regular, (Xs)red s.n.c.d. and without(−1)-curves, with X ′ ⊗K ′ ∼= C ⊗K ′.

Step 3: Show that property in Step 1 implies the model of Step 2 is s.s.s.Artin–Winters picks ` g and trivializes the action on `-torsion. Saito makes the action

on T`(Pic(CK)) unipotent.

19. Lecture 19 (November 19, 2015)

The proof of the semistable reduction of curves involves many ingredients, one of which isthe resolution of singularities of surfaces.

19.1. Resolution of singularities.

Definition 19.1. Let Y be a Noetherian integral scheme. A resolution of singularities is amodification X → Y such that X is regular.

Definition 19.2. A modification is a proper birational morphism of integral schemes.51

In general, Noetherian schemes are horrible and might not admit resolutions of singu-larities. For example, take the spectrum of a Noetherian domain of dimension 1 whosecompletion is not reduced. This motivated Grothendieck to introduce excellent rings andexcellent schemes, which characterize in terms of commutative algebra when resolutions ofsingularities exist. There is the celebrated

Theorem 19.3 (Lipman). Let Y be an integral Noetherian 2-dimensional scheme such that:

(1) the normalization morphism Y ν → Y is finite;(2) Y ν has finitely many singular points y1, · · · , yn and O∧Y ν ,yi is normal.

Then there exists a resolution of singularities of Y . The converse is also true.

Remark. If Y is of finite type over a field or Z or a characteristic 0 Dedekind domain or acomplete Noetherian local ring, then (1) and (2) hold (in fact, Y is a quasi-excellent scheme).

A nice exposition is Artin’s article in Arithmetic Geometry, edited by Cornell and Silver-man.

Notation. Fix R a dvr with K = f.f.(R), and C/K a proper smooth geometrically connectedcurve.

Proposition 19.4. There exists

C //

X

f

Spec(K) // Spec(R)

with f flat proper and X regular.

Proof. Choose C → PnK and let Y ⊂ PnR be the Zariski closure.

Fact (Technical mumbo-jumbo). Y satisfies (1) and (2) of Lipman’s theorem.

Then we get X → Y a resolution of singularities. Note, since C ⊂ Y is open and C isregular of dimension 1, we see that X → Y is an isomorphism over C. Therefore, XK

∼= C.(Hint (alteration): If f : X → Y is a proper, generically finite, dominant morphism of

Noetherian integral schemes and if y ∈ Y has codimension≤ 1, then there exists y ∈ V ⊂open

Y

such that f−1(V )→ V is finite.)

Example 19.5. Consider Zp → Zp[x, y]/(xy(x + y)10 − p). This is regular at the maximalideal (x, y, p).

Theorem 19.6 (Embedded resolutions). Let Y be a regular 2-dimensional scheme. LetZ ⊂ Y be a closed subscheme all of whose irreducible components Zi are either points or 1-dimensional schemes whose normalization is finite. Then there exists a sequence of blowupsf : X → Y such that f−1(Z)red is a strict normal crossings divisor (s.n.c.d.).

This is a lot easier to show than Lipman’s theorem.

Proposition 19.7. There exists

C //

X

f

Spec(K) // Spec(R)52

such that f is projective and flat, X is regular, (Xs)red is a s.n.c.d.

Notation. Let X → Spec(R) = S be as in the proposition, C1, · · · , Cn the irreducible com-ponents of Xs. Write Xs =

∑riCi as Cartier divisors, where ri is the multiplicity of Ci in

Xs. Then Xs is reduced if and only if ri = 1 for all 1 ≤ i ≤ r.

Claim. If char(κ(s)) = 0, then set r = lcm(ri), R′ = R[π′] with π′ = π1/r, and X ′ the

normalization of X ×Spec(R) Spec(R′). Then

X ′ → Spec(R′) = S ′

is a semi-stable curve over R′.

This is not the same thing as saying X ′ is a semi-stable S ′-variety! However, we have the

Lemma 19.8 (Addendum to last time). If X is a semi-stable curve over a dvr R withsmooth generic fibre, then

• the singularities of X as a surface are (Ak), k ≥ 1;• there exists a repeated blowup Xn → Xn−1 → · · · → X1 → X0 = X such that Xn is

a semi-stable S-variety.

Applying the Claim followed by the addendum, we get a semi-stable S ′-variety.

Remark. Claim is also true if:

• char(κ(s)) = p > 0,• p - ri for all i = 1, · · · , n,• Ci → Spec(κ(s)) is smooth,• if x ∈ Ci ∩ Cj (i 6= j) then κ(x)/κ(s) is separable.

Example 19.9. Let κ = Fp(t) where p 6= 2. Then C = Specκ[x, y]/(y2−xp− t) is a regularnon-smooth curve over κ, as follows. One can check that

Sing(C → Spec(κ)) = V (y, y2 − xp − t) = (y, xp + t).Denote Q = (y, xp + t) ∈ C. Then κ(Q) = κ[x]/(xp + t), and OC,Q is a regular local ring asmQ = (y). But over the algebraic closure,

C ×Spec(κ) Spec(κ) = Spec(κ[x, y]/(y2 − (x+ t1/p)p)) ∼= Spec(κ[x, y]/(y2 − xp))is not smooth.

Example 19.10 (Worse). Let κ′ = κ[x]/(xp − t). Take C = Spec(κ′[y])→ Spec(κ).

Sketch of proof of Claim. The idea is to describe etale local structure of X and use thatnormalization commutes with etale localization.

We will not define “etale local structure”, but the picture is

(u1, U1)etale

yy

etale

%%

· · ·etale

· · ·etale

##

(x ∈ X) (u2, U2) · · · (uk, Uk)

where (uk, Uk) should be in a standard form.

Example 19.11. A smooth morphism is etale locally like AnS → S.

53

Let x ∈ Ci ∩ Cj (i 6= j) with

(1) κ(x)/κ(s) separable,(2) ri, rj prime to char(κ(s)).

We have R → OX,x ⊃ mx. Let a, b ∈ mx be local equations for Ci, Cj, and n = ri andm = rj. Then

π = u · anbm

where u ∈ O×X,x. Because char(κ(s)) = char(κ(x)) does not divide n, after replacing X byan etale cover we may assume π = anbm (adjoin n-th root of u to OX,x).

Now κ(x)/κ(s) is separable, so

OX,x ← R[u, v]/(unvm − π)

a←[ ub←[ v

defines an etale morphism (X, x)→ Spec(R[u, v]/(unvm − π)).In characteristic 0, consider An = R[u, v]/(un − π) and An,m = R[u, v]/(unvm − π).Final step: Compute the etale local structure of

A′n,m,d = (R′[u, v]/(unvm − (π′)d))norm

where R′ = R[π] and (π′)d = π, n | d, m | d.Step 1: If e = gcd(n,m) > 1 and ζe ∈ R′, then

A′n,m,d∼=

e∏i=1

A′n/e,m/e,d/e.

Why? (un/evm/e)e = (π′d/e)e in A′n,m,d.Step 2: If gcd(n,m) = 1, then

unvm = (π′)d

in A′n,m,d, i.e.,

un =(π′)d

vm=

(π′d/m

v

)m

,

so there exists u′ ∈ A′n,m,d such that

(u′)m = u and (u′)n =(π′)d/m

v.

Similarly, there exists v′ ∈ A′n,m,d such that

(v′)n = v and (v′)m =(π′)d/n

u.

Check that

A′n,m,d ≡ R′[u′, v′]/(u′v′ − (π′)d/nm).

This ring has singularities that are at worst nodes. 54

20. Lecture 20 (November 24, 2015)

Last time we showed that, conditional on resolution of singularities of surfaces, in charac-teristic 0 we have semi-stable reduction of curves (in either of the two senses). Today I willsay what problems we run into if we try to work in arbitrary characteristic.

Let C be a curve and J = Jac(C). There is an action ρ of I on

T`J = lim←− J [`n](Ksep).

If C has semi-stable reduction, then the action ρ is unipotent.

20.1. Method of Artin–Winters. Suppose we have a Cartesian diagram

C //

X

f

Spec(K)

// Spec(R)

where

• R is a dvr with fraction field K,• C is smooth, proper, and geometrically connected over K,• f is proper and flat,• X is regular.

Let C1, · · · , Cm be the irreducible components of Xs. Then

Xs =∑

riCi

as Cartier divisors. Let r = gcd(r1, · · · , rn).

Warning. r need not be 1.

Lemma 20.1. Xs is geometrically connected over κ(s).

Proof. By the conditions on C, we have H0(C,OC) = K and so f∗OX = OS where S =Spec(R). By the Zariski main theorem, all fibres are geometrically connected.

For L ∈ Pic(X) and i ∈ 1, · · · , n, we define

L · Ci = deg(L|Ci)where the degree is computed over κ(s). (For example, O(1) on P1

Q(i), as a variety over Q,

has degree 2.) If D ⊂ X is an (effective) Cartier divisor, we define

D · Ci = OX(D) · Ci.

Fact.

Ci · Cj = Cj · Ci =

degree over κ(s) of the scheme Ci ∩ Cj if i 6= j,

deg(NCi/X) if i = j.

Remark. Because X is regular, Ci ⊂ X is Cartier and I = ICi/X is invertible and OX(Ci) ∼=I−1 so OX(Ci)|Ci ∼= NCi/X . (It is always true that the normal sheaf

NZ/X = HomOZ (IZ/X/I2Z/X ,OZ)

where IZ/X/I2Z/X is the conormal sheaf.)

55

Lemma 20.2.(∑

riCi

)· Cj = 0 for all j.

Proof. OX(∑riCi) = OX(Xs) ∼= OX , where the isomorphism OX

∼=→ OX(Xs) is given bymultiplication by a uniformizer π ∈ R.

Set Λ =⊕n

i=1 ZCi 3 F =∑riCi. Define a pairing

Λ× Λ→ Z(D =

∑aiCi, D

′ =∑

biCi

)7→ D ·D′ =

∑aibjCiCj

which is a symmetric bilinear form.

Lemma 20.3. The pairing Λ× Λ→ Z is semi-negative definite and if Z ∈ Λ with Z2 = 0,then Z ∈ Z(1

rF ).

Proof. Say Z =∑siCi. Then

Z2 =

(∑i

siCi

)2

=∑i

siCi

(∑j

sjCj

)

=∑i

siCi

(∑j

sjCj −siri

∑j

rjCj

)

=∑i

siCi

(∑j 6=i

(sj −

sirjri

)Cj

)=∑i 6=j

siri

(risj − rjsi)CiCj

=∑i<j

−(risj − rjsi)2

rirjCiCj.

Now use the fact that Xs is connected.

There is an exact sequence

0→ Z→ Λ→ Pic(X)→ Pic(C)→ 0

1 7→ F.

There is also the restriction map

Pic(X)→ Pic(Xs)

L 7→ LXs .We want to relate Pic(C) (for the generic fibre) with Pic(Xs) (for the special fibre).

Lemma 20.4. If n is prime to char(κ(s)), then

Pic(X)[n]→ Pic(Xs)[n]

is injective.

Proof. Skipped. 56

Set Λ∗ = Hom(Λ,Z). Then we get an exact sequence

0→ Z→ Λα→ Λ∗ → G→ 0

1 7→ 1

rF

whereα(D) = linear form D′ 7→ D ·D′.

This impliesG ∼= Gtors ⊕ Z

with #Gtors <∞.

Lemma 20.5. There is an exact sequence

0→ Z/ gcd(n, r)Z→ Pic(X)[n]→ Pic(C)[n]→ α−1(nΛ∗)

nΛ + ZF.

Proof. ker(Pic(X)→ Pic(C)) = Λ/ZF . The torsion in this is Z(1rF )/ZF ∼= Z/rZ. Hence

(Z/rZ)[n] ∼= Z/ gcd(r, n)Z.Suppose LC ∈ Pic(C)[n]. Pick L ∈ Pic(X) with LC ∼= L|C . Then

L⊗n ∼= OC(∑

aiCi

)for some

∑aiCi ∈ Λ well-defined modulo ZF . Moreover(∑

aiCi

)· Cj = L⊗n · Cj = n(L · Cj) ∈ nZ.

Hence∑aiCi ∈ α−1(nΛ∗) is well-defined modulo ZF . On the other hand we can replace

L by L(∑biCi) for some

∑biCi ∈ Λ. Then

∑aiCi changes to

∑aiCi + n(

∑biCi). We

see that we get a well-defined class in α−1(nΛ∗)nΛ+ZF which if zero means we can choose L to be

n-torsion.

Lemma 20.6. There is an exact sequence

0→ α−1(nΛ∗)

nΛ→ Λ/nΛ→ Λ∗/nΛ∗ → G/nG→ 0

and hence

# Pic(X)[n] ≥ # Pic(C)[n]

#(Gtor/nGtor).

Proof. Chase diagrams.

20.2. Example applications.

Example 20.7. If R is strictly henselian, ` ∈ κ(s)× is a prime and Gal(Ksep/K) = I actstrivially on J [`m](Ksep) for all m ≥ `, then we can conclude

# Pic(Xs)[`m] ≥ (`m)2g/fixed constant

where J is the Jacobian of C/K and g is the genus of C. This implies

dimF` Pic(Xs)[`] ≥ 2g.

We will see later this implies Xs is a tree of smooth curves with∑gi = g.

57

Example 20.8. Again say ` prime to char(κ(s)) and Pic(C)[`] ∼= F2g` . Then we get

dimF` Pic(Xs)[`] ≥ 2g − dimF`(Gtor/`Gtor).

Warning. We cannot pick ` after choosing model X.

20.3. Idea of Artin–Winters. The idea is to work with X such that (Xs)red n.c.d.

(1) Find an a priori bound on the “types” of graphs we can get for fixed genus g.(2) Show that dimF`(Pic(Xs)[`]) ≥ 2g − β where β = β(graph) with equality if and only

if semi-stable.

21. Lecture 21 (December 1, 2015)

PH: I missed the lecture.

22. Lecture 22 (December 3, 2015)

22.1. Abstract types of genus g. Last time we showed that: for g ≥ 2, the abstract typesof genus g with no (−1)-curves is finite up to equivalence.

Definition 22.1. Let G be a finitely generated abelian group. Let c ≥ 1. Then

ρc(G) = min

r | there exists a subgroup H ⊂ G of index dividing c

such that H can be generated by r elements

.

Example 22.2. ρ1(G) is the minimal number of generators of G.

Lemma 22.3. If 0→ G1 → G2 → G3 → 0 is exact, then

ρcc′(G2) ≤ ρc(G1) + ρc′(G3)

and

ρc(G2) ≥ ρc(G3).

Example 22.4. If ` - c is prime, then

dimF`(G/`G) ≤ ρc(G)

and

dimF`(G[`]) ≤ ρc(G).

Theorem 22.5 (Artin–Winters). For any g ≥ 0 there exists a c = c(g) such that if T is anabstract type of genus g, then

ρc(G) ≤ 1 + β

where

• β is the 1st Betti number of the graph T ,

• G = coker(Λ(mij)→ Λ∗) with Λ =

⊕ni=1 ZCi.

Proof. The steps are:Step 1: Argue that (−1)-curves can be “contracted”.Step 2: Do g = 0, 1 separately.Step 3: Use boundedness of last lecture to do g ≥ 2 by induction on g.

58

I will explain Step 3. First, there are finitely many abstract types of genus g with nosubgraphs that are chains of (−2) with the same multiplicity r of length 4 (from last time):

−2r

−2r

−2r

−2r

OK for these with in fact ρc(G) ≤ 1 for a suitable c.If the abstract type T does contain such a chain:

−2C1

r

−2C2

r

−2C3

r

−2C4

r

· · · Rest · · ·then we remove the middle edge to form T ′:

−2C1

r

−1C2

r

−1C3

r

−2C4

r

· · · Rest · · ·Recall that

g = 1 +1

2

∑kiri.

Since k1 = k2 = k3 = k4 = 0 in T but k′2 = k′3 = −1 in T ′, we have g′ < g. If T ′ is connected,then it is an abstract type of genus g′. Otherwise it breaks into two connected components,but we will omit this case.

Letting x1, · · · , xn be the basis of Λ∗ dual to C1, · · · , Cn in Λ, we see

M = Im(Λmij→ Λ∗) = Span

(∑j

mijxj, i 6= 2, 3;x1 − 2x2 + x3;x2 − 2x3 + x4

)and

M ′ = Im(Λm′ij→ Λ∗) = Span

(∑j

mijxj, i 6= 2, 3;x1 − x2;−x3 + x4

).

We see thatM ′ ⊂M + Z(x2 − x3),

so there is a surjection

G = Λ∗/M Λ∗/M + Z(x2 − x3) = G/〈x2 − x3〉with cyclic kernel, and similarly a surjection

G′ = Λ∗/M ′ G/〈x2 − x3〉.Hence

ρc(G) ≤ ρc(G/〈x2 − x3〉) + 1 ≤ ρc(G′) + 1 ≤ 1 + (β − 1) + 1 = 1 + β,

where the last inequality follows by induction hypothesis if T ′ is connected and c works forall lower genera.

The disconnected case is similar (but trickier). 59

22.2. Part 2 of Artin–Winters’ argument. Given C/K, R and κ = κ. Pick ` prime notdividing c(g) and prime to char(κ) where g = g(C). Let K ′ be a finite separable extensionof K such that

Pic(CK′)[`] ∼= (Z/`Z)2g

andC(K ′) 6= ∅.

Replace R,K,C by R′, K ′, CK′ . We will show C has semi-stable reduction.Pick

C

//

X

Spec(K) // Spec(R)

a regular flat proper model without (−1)-curves (omitted: can contract (−1)-curves). Wewill show this model is semi-stable.

The existence of rational point implies:

• gcd(ri) = 1 (there is an i with ri = 1).• dimκH

1(Xs,O) = g.• dimκH

1(Xs,O) ≥ dimH1((Xs)red,O) with equality if and only if Xs = (Xs)red.

By our choice of ` we have

2g − β ≤ dimF` Pic(Xs)[`].

This uses ρc(g)(G) ≤ 1 + β (where β is the 1st Betti number of our graph), ` - c(g), materialof two lectures ago, plus the example of today. It is easy to see that

dimF` Pic(Xs)[`] = dimF` Pic((Xs)red)[`].

Set Y = (Xs)red. Then we get a long exact sequence

0→ Γ(Y,O∗Y )→n∏i=1

Γ(Ci,O∗Ci)→∏

x∈Ci∩Cji 6=j

O∗Ci∩Cj ,x → Pic0,0(Y )→n∏i=1

Pic0(Ci)→ 0.

Fact (Oort). dimF` Pic0(Ci)[`] ≤ g(Ci) + pa(Ci), with equality if and only if Ci is nodal.

ThendimF`(O∗Ci∩Cj ,x)[`] = 1

and equal to dimκ(OCi∩Cj ,x) if and only if OCi∩Cj ,x ∼= κ.Putting everything together, we get

2g − β ≤ dimF` Pic(Y )[`] ≤∑

(g(Ci) + pa(Ci)) + β

which implies

2g ≤∑

(g(Ci) + pa(Ci)) + 2β.

Now we consider the same long exact sequence as before without ∗’s:

0→ H0(Y,OY )→n∏i=1

Γ(Ci,OCi)→∏

x∈Ci∩Cji 6=j

OCi∩Cj ,x → H1(Y,OY )→n⊕i=1

H1(Ci,O)→ 0.

60

This gives

g = dimκH1(Xs,O) ≥ dimκH

1(Y,OY ) =∑

pa(Ci) + β.

Therefore,

2∑

pa(Ci) + 2β ≤∑

(g(Ci) + pa(Ci)) + 2β

so ∑pa(Ci) ≤

∑g(Ci).

This shows there are no δ-invariants, i.e., all Ci’s are smooth and we have equality every-where. This finishes the proof.

22.3. Saito. We have been following Artin–Winters, but there is a paper by Saito whichproves the theorem (same as in Deligne–Mumford): if the inertia acts unipotently, then wehave semi-stable reduction of curves. Saito’s proof uses etale cohomology and vanishing cyclesheaves.

23. Lecture 23 (December 8, 2015)

23.1. Neron models. We will discuss the Neron–Ogg–Shafarevich criterion. First we needto talk about Neron models. A reference is Artin’s article in Chapter VIII in ArithmeticGeometry (edited by Cornell and Silverman).

Let R be a Dedekind domain (dvr) with fraction field K, and A an abelian variety overK.

Definition 23.1. A Neron model for A is a smooth group scheme G → Spec(R) withGK∼= A and such that the following universal property holds:

If X is smooth over Spec(R) then any rational map X 99K G extendsto a morphism X → G.

(*)

An alternative weaker condition is:

If X is smooth over Spec(R) then any morphism XK → A extends toa morphism X → G.

(†)

It is clear that (*) implies (†).

Remark. In both cases we have, if R is a dvr, then

A(Ksep)I = A(f.f.(Rsh)) = G(Rsh)sp→ Gs(κ

sep),

where I ⊂ Gal(Ksep/K) is the inertia with (Ksep)I = f.f.(Rsh), the second equality followsfrom (*) or (†), and sp is the specialization map.

Theorem 23.2 (Weil). Finite type Neron models for abelian varieties exist over Dedekinddomains.

Theorem 23.3 (Raynaud). Locally of finite type smooth models with (†) exist for semi-abelian varieties. These models are also called Neron models.

Example 23.4. Consider Gm,K with R dvr. Pick π ∈ R uniformizer. Set

“G =⋃n∈Z

πnGm,R”

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(construct by glueing). Then we have

K× = Gm,K(K) = G(R) =⋃n∈Z

πnR×.

Example 23.5. Consider elliptic curves with multiplicative reduction and v(∆) = 1. As-sume we have a Weierstrass equation over R with discriminant ∆ = π · unit. Then

G =

(closed subscheme of P2

R definedby the Weierstrass equation

)\(

singular point ofthe special fibre

)(over Spec(R)) is the Neron model of its generic fibre.

23.2. Group schemes over fields.

Theorem 23.6 (Chevalley). If G/K is smooth and connected, then there exists a short exactsequence

1→ L→ G→ A→ 1

of group schemes over K, where L is a connected linear algebraic group, and A is an abelianvariety.

Remark. If K is perfect then L is smooth.

Theorem 23.7. If L/K is a commutative, smooth, connected linear algebraic group, then

0→ U → L→ T → 0

with U unipotent and T a torus.

Remark. If K is perfect then L = U × T canonically.

Unipotent means it fits in 1 ∗. . .

0 1

⊂ GLn .

Torus T means

T ⊗K K ∼= G⊕rm,K

for some r.

Definition 23.8. A group scheme G over a field K is semi-abelian if G is abelian and anextension of an abelian variety by a torus. A group scheme G over a base S is (semi-)abelianif G→ S is smooth and all fibres are (semi-)abelian.

23.3. Neron–Ogg–Shafarevich criterion.

Definition 23.9. Let R be a dvr with fraction field K, and A/K an abelian variety. Wesay A has good reduction if its Neron model is an abelian scheme, or equivalently if thereexists an abelian scheme G→ Spec(R) with A ∼= GK .

Theorem 23.10 (Neron–Ogg–Shafarevich). Let R be a dvr with fraction field K, and A/Kan abelian variety. Then A has good reduction if and only if there exists ` prime to char(κ)such that the action of I on A(Ksep)[`∞] is trivial.

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“Proof” of ⇐= . First we show that although

A(Ksep)I = A(f.f.(Rsh)) = G(Rsh)sp→ Gs(κ

sep)

is not injective, writing f.f.(Rsh) = Ksh we have

A(Ksh)[`n] → Gs(κsep)[`n].

Note that I acts trivially by assumption, so

A(Ksh)[`n] ∼= (Z/`Z)2g

where g = dim(A).Since G→ Spec(R) is of finite type, [Gs : G0

s] <∞ and hence

#G0s(κ

sep)[`n] ≥ `2gn−constant.

The structure of group schemes over κ gives

0→ L→ Gs ⊗ κ→ B 0

where B is an abelian variety, and

0→ U → L→ T → 0.

Write g = dimU + dimT + dimB = u+ t+ b. Then

#B(κ)[`n] = `2bn,

#T (κ)[`n] = `tn,

#U(κ)[`n] = 1.

Combining everything we get2g ≤ t+ 2b,

g = u+ t+ b=⇒ t = u = 0.

Much harder is the following

Theorem 23.11. I acts unipotently on T`(A) if and only if A has semi-abelian reduction(SGA7 calls this stable reduction).

24. Lecture 24 (December 10, 2015)

24.1. Semi-abelian reduction. Recall the situation: A/K is an abelian variety, K ⊃ R advr with residue field κ.

Theorem 24.1 (SGA7). A has semi-abelian reduction if and only if there exists ` differentfrom char(κ) such that the action of I on T`(A) is unipotent.

The original proof was due to Grothendieck, but I will explain a simpler proof by Deligne.

Theorem 24.2 (Deligne–Mumford). If A = Jac(C), then A has semi-abelian reduction ifand only if C has semi-stable reduction.

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This is not very hard to show. We have to relate the Picard scheme of the model to theNeron model of the Jacobian, and then using techniques similar to those in Artin–Winterswe can prove this theorem. We would in particular get the fact that if the inertia actsunipotently on the Tate module of A = Jac(C), then C has semi-stable reduction.

Combining these Deligne–Mumford proved (for the first time) the semi-stable reductiontheorem for curves.

24.2. Proof of Theorem 24.1. We will proof the implication ⇐= following Deligne inSGA7-I, Exp. I App.

For simplicity assume R complete with finite κ. The Tate module

V`(A) = T`A⊗Z` Q` =(

limnA(Ksep)[`n]

)⊗Z` Q`

is a Q`-vector space of dimension 2g endowed with a continuous action of

0→ I → Gal(Ksep/K) Galκ = Z→ 0.

By assumption, the action of I is unipotent.Let σ ∈ I be a topological generator, and recall the tame inertia It = I/P ∼=

∏`′ 6=char(κ) Z`′ .

Suppose τ ∈ Gal(Ksep/K) maps to the arithmetic Frobenius in Galκ = Z 3 1, i.e.,

τστ−1 = σq mod P.

Let r be the greatest integer such that (σ − 1)r 6= 0 on V`(A).Consider the coinvariants and invariants

(V`)I

(V`)I

Q = V`/ ker((σ − 1)r)∼=

(σ−1)r// Im((σ − 1)r) =: S

?

OO

This map Q→ S is not Galκ-invariant but it is if we twist

Q⊗Q` Q`(r)∼→ S

as Galκ-representations. Note τ acts as q on Q`(1) and as qr on Q`(r).

Proof.

τ(σ − 1)r = (τστ−1 − 1)rτ

= (σq − 1)rτ

= q(σ − 1)r

by looking at

σ =

1 1 ∗

1. . .. . . 1

1

, σq =

1 q ∗

1. . .. . . q

1

, (σq − 1)r =

0 · · · 0 qr

0. . . 0. . . 0

0

.

Definition 24.3. A q-Weil number of weight w is α ∈ Z such that |α′| = qw/2 for allconjugates α′ ∈ C of α.

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Lemma 24.4. The eigenvalues of τ−1 on (V`)I are either:

• q-Weil numbers of weight −2 or• q-Weil numbers of weight −1.

Proof. (V`)I = V`(G

0s), where G is the Neron model and Gs is the special fibre. G0

s hastwo parts. On the the torus part, τ acts on V`(Gm,κ) by the cyclotomic character, so theeigenvalue of τ−1 is q−1. On the abelian variety part, we get weight−1 by the Weil conjecturesfor abelian varieties over κ.

Lemma 24.5. The eigenvalues of τ−1 on (V`)I are either:

• q-Weil numbers of weight 0, or• q-Weil numbers of weight −1.

Proof. Let At = Pic0A be the dual abelian variety. Then there exists a nondegenerate Galκ-

equivariant bilinear pairing

V`(A)× V`(At)→ Q`(1).

Note

(V`(A))I ∼= ((V`(At))I)∗(1).

By the previous lemma for At (with weights −2,−1), dualizing gives 2, 1 and twisting gives0,−1.

Conclusion: r ≤ 1.0−2r−1−2r (V`)I(r)

(V`)I −2

−1

Q(r)∼=

// S?

OO

If r = 0, Neron–Ogg–Shafarevich says A has good reduction.

If r = 1, the Jordan blocks of σ are(1)

or

(1 10 1

), and

2g = #(1× 1 blocks) + 2 ·#(

1 10 1

).

Looking at weights we see

dim(torus part of G0s) ≥ #

(1 10 1

),

using V I` = V`(G

0s) and the fact that weight −2 eigenvalues come from the torus part. The

relation V I` = V`(G

0s) also shows

2 dim(abelian part of G0s) + dim(torus part of G0

s) = 2g −#

(1 10 1

)≥ 2g − dim(torus part),

which must be an equality because

g = dim(unipotent part) + dim(torus part) + dim(abelian part).

So we are done!65

24.3. Weight monodromy conjecture. Let X/K be a variety over a local field, and

V = H i(XK ,Q`).

There are two filtrations on V :

• filtration coming from the nilpotent operator N = σ − 1,• weight filtration.

Conjecture 24.6. The two filtrations agree.

We just proved this for abelian varieties on H1. Scholze proved this for complete intersec-tions using perfectoid spaces.

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