Download tale Fundamental Groups

ÉTALE 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 étale
fundamental groups offered at Columbia University in Fall 2015 (MATH G4263: Topics in
Algebraic Geometry).
Due to my own lack of understanding of the materials, I have inevitably introduced
both mathematical and typographical errors in these notes. Please send corrections and
comments to [email protected].
WARNING: I am unable to commit to editing these notes outside of lecture time, so they
are likely riddled with mistakes and poorly formatted.
Contents
1. Lecture 1 (September 8, 2015)
2. Lecture 2 (September 10, 2015)
2.1. References
2.2. Galois Categories
3. Lecture 3 (September 15, 2015)
3.1. Reminders
3.2. Fundamental Groups of Schemes
4. Lecture 4 (September 17, 2015)
4.1. Fibre functor
5. Lecture 5 (September 22, 2015)
5.1. Complex Varieties
5.2. Applications
6. Lecture 6 (September 24, 2015)
6.1. Two more examples
6.2. Fundamental groups of normal schemes
6.3. Action of Galois groups on fundamental groups
7. Lecture 7 (September 29, 2015)
7.1. Short exact sequence of fundamental groups
8. Lecture 8 (October 1, 2015)
8.1. Ramification theory
8.2. Topological invariance of étale topology
9. Lecture 9 (October 6, 2015)
9.1. Special case of Theorem last time
9.2. Structure of proof
Last updated: December 10, 2015.
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10. Lecture 10 (October 8, 2015)
10.1. Purity of branch locus
11. Lecture 11 (October 13, 2015)
11.1. Local Lefschetz
12. Lecture 12 (October 15, 2015)
12.1. Specialization of the fundamental group
13. Lecture 13 (October 20, 2015)
13.1. Abhyankar’s lemma
13.2. End of proof of Theorem last time
13.3. Applications
14. Lecture 14 (October 22, 2015)
14.1. Quasi-unipotent monodromy over C
14.2. Étale cohomology version
15. Lecture 15 (October 27, 2015)
16. Lecture 16 (October 29, 2015)
16.1. Outline of proof of quasi-unipotent monodromy theorem
17. Lecture 17 (November 5, 2015)
17.1. Proof in the mixed characteristic case
17.2. Birational invariance of π1
18. Lecture 18 (November 17, 2015)
18.1. Semi-stable reduction theorem
19. Lecture 19 (November 19, 2015)
19.1. Resolution of singularities
20. Lecture 20 (November 24, 2015)
20.1. Method of Artin–Winters
20.2. Example applications
20.3. Idea of Artin–Winters
21. Lecture 21 (December 1, 2015)
22. Lecture 22 (December 3, 2015)
22.1. Abstract types of genus g
22.2. Part 2 of Artin–Winters’ argument
22.3. Saito
23. Lecture 23 (December 8, 2015)
23.1. Néron models
23.2. Group schemes over fields
23.3. Néron–Ogg–Shafarevich criterion
24. Lecture 24 (December 10, 2015)
24.1. Semi-abelian reduction
24.2. Proof of Theorem 24.1
24.3. Weight monodromy conjecture
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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. Morphisms
are 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∧ =
lim
U CG open, finite index
G/U.
It satisfies the universal property: if G → H is continuous and H is a profinite group, then
there 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 Q
explain the proof. A basis of the topology on Aut(F ) is
the 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 C G is open and finite index, then X = G/U belongs
to Finite-G-Sets. So
/ Aut(F )
G∧
G/U 
/
Aut(F (X))
T
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,··· ,n Xi , where Xi ∼
= G/Hi with Hi ⊂ G open subgroup of finite
T
T
−1
index. Take U = i=1,··· ,n g∈G gHi g 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
g0
Y
X
/
γY
g0
γX
/
/
/
Y
X
Think!
• Say γY (eU ) = g0 U where e ∈ G is the neutral element. Then
U CG
γY (gU ) → γY (Rg (eU ))
where Rg : G/U → G/U is right multiplication in Arrows(Finite-G-Sets). Since γ is
a transformation of functors, this is equal to
Rg (γY (eU )) = Rg (g0 U ) = g0 gU.
Done!
Lemma 2.2. Any exact functor F : Finite-G-Sets → Sets, with F (X) finite for all X, is
isomorphic 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 commutes
with them (⇔ F (∗A ) = ∗B and F (X ×Y Z) = F (X) ×F (Y ) F (Z)), then we say F is
left exact.
(2) If A has finite colimits (⇔ A has an initial object ∅A and pushouts) and F commutes
with them (⇔ F (∅A ) = ∅B and F (X tY Z) = F (X) tF (Y ) F (Z)), then we say F is
right exact.
(3) F exact ⇔ left and right exact.
In Grothendieck’s original exposition, things are more general but it is easier for us to
work with exactness properties.
If F : C → Sets is a functor with finite values, we get a functor C → Finite- Aut(F )-Sets
by 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;
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(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) is
isomorphic to either ∅ → X or X → X (this is the definition in the context of Galois
categories 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) ⊂ X. Since X is connected, Eq(a, b) = X and a = b.
subobject
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 X n = t∈T Zt . Then Sn acts in the same
way on F (X)n = F (X n ) = 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 (x01 , · · · , x0n ) ∈
F (Zt ) implies x0i pairwise distinct. If not, say x0i = x0j , then pri |Zt = prj |Zt by example fact
above. This contradicts pri (ξ) = prj (ξ).
The same argument can be used to construct Galois extensions of finite separable extensions (everything needs to be dualized, and X n 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 transitive
for all X ∈ Ob(C) connected.
Idea of proof. We need to introduce some notations. Let I be the set of isomorphism classes
of Galois objects. For i ∈ I, let Xi be a representative. Pick xi ∈ F (Xi ). We say i ≥ i0 if
there exists a morphism Xi → Xi0 . We may pick fii0 : Xi → Xi0 such that F (fii0 )(xi ) = xi0
(⇒ fii0 unique) (because Galois objects).
Claim. F is isomorphic to the functor F 0 : X 7→ colimi∈I MorC (Xi , X).
If this is true, then just as in the case of Galois theory, we set
H = lim Aut(Xi )
i∈I
and get H
opp
0
→ Aut(F ) = Aut(F ).
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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,··· ,n Xi , 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 T̃ has group Sn over T . For the
n-to-1 covering T 0 over T , T̃ has group Stab(1) over T 0 .
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 a
representative Xi .
• i ≥ i0 ⇔ there exists Xi → Xi0 .
• Pick γi ∈ F (Xi ), for i ≥ i0 pick fii0 : Xi → Xi0 such that F (fii0 )(γi ) = (γi0 ). (This
morphism is uniquely determined.)
• Ai = Aut(Xi ) acts transitively on F (Xi ).
Suppose
ai
/ Xi
Xi
fii0
∃!
fii0
/ Xi0
Xi0
This gives a map αii0 : Ai → Ai0 .
• The collection of Ai and transitive maps {αii0 : Ai → Ai0 } forms an inverse system of
finite groups over (I, ≥).
• A = lim Ai . 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 is
nonempty.
• There exists Aopp → Aut(F ) by proving that the functor F 0 is isomorphic to F where
F 0 (X) = colimI MorC (Xi , X).
• αii0 (ai ) is the unique map that makes
Xi
ai
/
Xi
α 0 (ai ) ii
/ Xi0
Xi0
commute. Details... Details...
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• F 0 → F is the map
fi
(if f ∈ F 0 (X) is given by Xi → X) ∈ F 0 (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 there
exists Z ⊂ X × Y which is a coproduct of connected components of X × Y such
that F (Z) = Γs . This implies pr1 |Z : Z → X is an isomorphism because F (pr1 |Z ) is
bijective. Then Z is the graph of a morphism X → Y
f
X
/
(pr1 |Z )−1
?Y
pr2 |Z
Z
with F (f ) = s.
(3) Essentially injective:
(a) Enough to construct X with F (X) ∼
= Aut(F )/H, for H ⊂ Aut(F ) an open
subgroup (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 0 ↔ (H/U )opp .
!
(e) Let X = Coeq
Y
h01
h0r
/
/
Y
, where H 0 = {h01 , · · · , h0r }.
(f) Since F commutes with colimits,
F (X) = Coeq
F (Y )
F (h01 )
F (h0r )

∼
= Coeq  Aut(F )/U
/
/
!
F (Y )
1
Rh0
r
∼
= Aut(F )/H
7

R h0
/
/
Aut(F )/U 
where H = h1 U t · · · t hr U .
Addendum: Let (C, F ) and (C 0 , F 0 ) be two Galois categories, and H : C → C 0 an exact
functor. Then
(1) There exists an isomorphism t : F → F 0 ◦ H.
(2) The choice of t determines h : G0 := Aut(F 0 ) → G := Aut(F ) a continuous homomorphism of groups well-defined up to inner automorphisms of G.
(3)
∼
=
/
H
C
/
h
Finite-G-Sets
is 2-commutative (via t).
C0
∼
=
Finite-G0 -Sets
3.2. Fundamental Groups of Schemes. Let f : X → Y be a morphism of schemes.
TFAE:
(1) f is étale.
(2) f is smooth of relative dimension 0.
(3) f is flat, lfp (locally of finite presentation), and fibres étale.
(4) f lfp and infinitesimally lifting criterion:
X bo
T _ 0
∃!
Y o
first order thickening
Taffine
(formally étale).
(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 étale if and only if there exists a set I and X ∼
=
Spec(k
)
with
k
/k
finite
separable.
i
i
i∈I
Formal properties:
(A) Being étale is preserved under base change:
/
Y 0 ×Y X
X
/
Y0
f
Y
(B) Being étale is preserved under composition.
(C)
X
f
h=g◦f
Z
g, h étale ⇒ f étale.
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/
g
Y
(D) A → B étale ring map ⇔ B ∼
= A[x1 , · · · , xn ]/(f1 , · · · , fn ) such that
∂fi
∆ = det
∂xj
is invertible. (There is a structure theorem for étale algebras, which is a stronger
statement.)
We will be talking about finite étale morphisms.
Let f : X → Y be a morphism. TFAE:
(1) f is finite, étale. (f : X → Y finite iff for all V ⊂ Y affine, U = f −1 V is affine and
O(V ) → O(U ) is a finite ring map (module finite).)
(2) For all V ⊂ Y affine open, f −1 V = 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 (b1 b2 ) is nondegenerate.
If B ∼
= A⊕n is free and multiplication by b1 b2 has matrix (aij (b1 b2 )), then TrB/A (b1 b2 ) is
the trace of this matrix.
Q
Exercise. If K → B with (*) and K field, then B = Li where Li /K finite separable.
Next time: Let X be a scheme. Then
F
FÉtX = {Y → X finite étale} →x Sets
(fibre functor) is a Galois category, and π1 (X, x) = Aut(Fx ).
4. Lecture 4 (September 17, 2015)
Today we will talk about étale fundamental groups.
Notation. Let X be a scheme. FÉtX is the category with
f
• objects: Y → X finite étale,
• morphisms:
f
f0
MorFÉtX (Y → X, Y 0 → X) = MorX (Y, Y 0 ) = {g : Y → Y 0 | f 0 ◦ g = f }.
Example 4.1.
(1) Let X = Spec(k) where k is a field. Then
FÉtX = {finite separable k-algebras}opp .
(2) Let X = Spec(A). Then
FÉtX = {A → B which are finite locally free
opp
and Q : B × B → A, (b1 , b2 ) 7→ TrB/A (b1 b2 ) is nondegenerate
= {separable A-algebras}opp .
FÉtX will be a Galois category only when X is connected. In general,
Lemma 4.2. FÉtX has all finite limits and finite colimits and for a morphism of schemes
X 0 → X, then base change functor FÉtX → FÉtX 0 , Y /X 7→ X 0 ×X Y /X 0 is an exact functor.
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Proof.
• Limits: Check there exists final object and fibre products.
– Final object: id : X → X.
– Fibre products:
FÉtX 3 Y1
~
Y2
/
Y1 ×Y2 Y3
Y3
finite étale as
b.c. of Y1 → X
$
?X
finite étale
Y3
X
• Colimits: enough to check finite coproducts and coequalizers
– Coproducts: disjoint unions
– Coequalizers: Y1
a
b
/
/
Y2 in FÉtX .
Note Yi = SpecX (fi,∗ OYi ).
f1,∗ OY1 oo
a#
b#
f2,∗ OY2 o
A
where A is a quasicoherent sheaf of OX -algebras, and the last map is the equalizer
of a# and b# .
Claim. SpecX (A) → X is in FÉtX and is coequalizer of a and b.
Proof of claim (please find your own).
`
(*): Étale locally on X you can write Yi = j=1,··· ,mi X.
`
(**): Yi = j=1,··· ,mi X, then any morphism Y → Y 0 over X Zariski locally on
X comes from a map {1, · · · , m1 } → {1, · · · , m2 }.
(*) follows from the local structure theorem for finite unramified maps. Suppose
we have g : Y → X finite unramified and x ∈ X. Then there exists (U, u) →
(X, x) étale such that
a
Y ×X U =
Vj
j=1,··· ,m
where Vj are open and closed subsets, and Vj → U is a closed immersion for all
j.
Finish the proof:
(***): Descent theory “says” it is enough to prove claim after étale base change.
(****): Explicitly compute what happens when Y and Y 0 are map are as in
(**).
4.1. Fibre functor.
x
Definition 4.3. A geometric point x of a scheme is a morphism Spec(K) → X whre k is
an lagebraically closed field.
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Abusive notation: k = κ(x), x = Spec(κ(x)), x = Im(x).
Base change gives
Fx : FÉtX → FÉtx ∼
= category of finite sets → Sets
Y /X 7→ Fx (Y ) = Yx
where Yx is the set of y such that
?Y
y
x
x
X
commutes.
Theorem 4.4. If X is a connected scheme, then (FÉtX , 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 étale 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 étale. (Need to check: monos in FÉtX are open
immersions.)
g
• Lastly: Fx reflects isomorphisms. Suppose Y → Y 0 in FÉtX and Fx (g) bijective.
– by above reduce to Y 0 connected.
– then g is finite étale (general properties of morphisms)
– g is finite locally free, so degree of g is locally constant on Y 0 . Since Y 0 is
connected, degree of g is constant.
– degree of g is 1 by looking at the fibre over some y 0 ∈ Fx (Y 0 ).
– degree 1 implies g isomorphism.
Lemma 4.6. Let f : X 0 → X be a morphism of connected schemes. Let x0 be a geometric
point of X 0 . Set x = f (x0 ). Then we get a canonical continuous homomorphism of profinite
groups
f∗ : π1 (X 0 , x0 ) → π1 (X, x)
such that
FÉtX 0 o
base change
Fx0 ∼
=
Finite-π1 (X 0 , x0 )-Sets o
FÉtX
∼
= Fx
f∗
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Finite-π1 (X, x)-Sets
2-commutes.
Proof. Fx ∼
= Fx0 ◦ (base change) because x = f ◦ x0 .
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 ∼
= F x2 .
Example 4.8. Fix k ⊂ k sep ⊂ k. Then
π1 (Spec(k), Spec(k)) = Gal(k sep /k).
Proof. The functor
FÉtX → Finite- Gal(k sep /k)-Sets
Y 7→ MorSpec(k) (Spec(k), Y ) = MorSpec(k) (Spec(k sep ), Y )
(the right hand side has a left action of Gal(k sep /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)))) = Ẑ.
Example 4.12. π1 (P1C ) = {1}.
Proof. Every object in FÉtP1C is isomorphic to a disjoint union of copies of P1C . Let f : Y → P1C
be finite étale, with Y connected. Have to show Y ∼
= P1C .
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 X an the
usual 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 ) ⊂
AnC ,
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 : X an → Y an is
continuous.
• If f is an open/closed immersion, then f an is an open/closed immersion.
• If X = AnC , then X an = Cn with Euclidean topology.
Question/Remark: The properties of topology on C we need to do this construction are:
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(1) Multivariate polynomials give continuous maps. (This is implied by the fact that C
is 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 : X an →
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 a
projective surjection X 0 X such that the composition X 0 → 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 étale at some
x ∈ X(C), then f an is a local isomorphism at x: there exist x ∈ U ⊂ X an and f (x) ∈
open
V ⊂ Y an such that
open
f an |U : U → V
is a homeomorphism.
Proof. We may shrink X and Y and assume they are affine and that f is étale.
First Proof. Reduce to Y = AnC ; to do this pick
X

/
W affine

/
Y
étale
AnC
such that X = Y ×AnC W .
Lemma 5.3. Let A be a ring, I ⊂ A be an ideal and A/I → B étale. Then there exists
A → B étale such that B ∼
= B/IB as A/I-algebras.
∂fi
) invertible. Set
Proof. Write B = A/I[x1 , · · · , xn ]/(f1 , · · · , fn ) with ∆ = det( ∂x
j
−1
∂fi
B = A[x1 , · · · , xn ]/(f1 , · · · , fn ) det
.
∂xj
Then W → AnC is smooth of relative dimension 0 so we can write
W = Spec C[x1 , · · · , xn , z1 , · · · , zm ]/(F1 (z, x), · · · , Fm (z, x))
∂F
and det( ∂zkj )j,k=1,··· ,m invertible on W .
Implicit function theorem says
(W an , w) → (Cn , f (w))
is a local isomorphism.
Second Proof. Use
13
Theorem 5.4 (Structure Theorem). Let A → B be a ring map étale at a prime q ⊂ B.
Then there exists a g ∈ B and g ∈
/ q such that A → Bg is standard étale, i.e.,
Bg ∼
= (A[z]/(P ))Q
where P, Q ∈ A[z], P is monic and
dP
dz
is invertible in (A[z]/(P ))Q .
Suppose P (z) = z n + a1 z n−1 + · · · + an ∈ C[z] and α is a simple root. Then there exists
> 0 such that for |bi | < , i = 1, · · · , n, the polynomial
Pb1 ,··· ,bn (z) = z n + (a1 + b1 )z n−1 + · · · + (an + bn )
has a simple root α(b1 , · · · , bn ) depending continuously on b1 , · · · , bn and converging to α as
bi 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 étale) to
get a standard étale map. Then we can pick Py (z) ∈ OY (Y )[z].
Corollary 5.5. Given X lft over C, there is a functor
FÉtX → {finite covering spaces of X an }
f an
f
(Y → X) 7→ (Y an → X an ).
Theorem 5.6 (Riemann Existence Theorem, SGA 1, Exp XII). This functor is an equivalence.
Corollary 5.7. Let X lft over C be connected. Then X an is connected, and for x ∈ X(C),
there is an isomorphism
∼
=
π1top (X an , x)∧ → π1 (X, x),
where the left is the profinite completion of the topological fundamental group, and the right
is 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 X an
such that
• If f : X → Y is a morphism with X, Y smooth, then f an : X an → Y an is holomorphic.
• If f is étale, then f an is a local isomorphism of complex manifolds.
• If X = AnC , 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
? _U
Xo
f
So
étale
AnS
commutes.
14
Let X/C be a projective and smooth variety. To show
∼
=
FÉtX → finite topological covers of X an ,
let π : M → X an be a finite topological covering. This implies M has a unique structure of
complex manifolds such that π is a local isomorphism of complex manifolds.
Let L be a positive line bundle on X an coming from X ,→ PnC , so X an ,→ Pn (C) and O(r)
with Fubini–Study metric). This means L has a metric whose curvature is positive, so π ∗ L
is positive on M .
By the Kodaira Embedding Theorem, M is algebraic, say M = Y an . (Namely you prove
H 0 (M, L⊗k ) gives M ,→ Pn (C). By Chow’s theorem, M ,→ Pn (C) is cut out by polynomials
equations, which gives us Y .)
π
The problem left over: Y an = M → X an is equal to f an for some f : Y → X. We can
either do Chow’s theorem for the graph Γ ,→ M × X an , or use GAGA.
Remark. More generally, if X and Y are projective smooth over C, any holomorphic map
X an → Y an is f an for some f : X → Y algebraic.
5.2. Applications. Recall the profinite completion
Y
Ẑ = lim Z/nZ =
Z` .
n≥1
` prime
• π1 (PnC ) = {1}.
• π1 (AnC ) = {1}. (
Ẑ if n = 1, n
• π1 (AnC \{0}) =
(AC \{0} is homeomorphic to S 1 and S 2n−1 in these two
0 if n > 1.
cases respectively.)
• π1 (P1C \{0, 1, ∞}) = profinite completion of free group on 2 generators. This is something 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 , · · · , βg
subject to [α1 , β1 ][α2 , β2 ] · · · [αg , βg ] = 1.
• If A/C is an abelian variety of dimension g, then
π1 (A) ∼
= Ẑ⊕2g .
We don’t know how to prove the first one algebraically. Philosophically, it is interesting
to prove these without topological or analytic methods not because they are worse than
algebraic proofs, but because new proofs give new insights!
Today we will study the fundamental groups of normal schemes and their relationships to
Galois 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 finite
separable. 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 multiplying
by an element of A, we may assume βi ∈ B. Let β1∨ , · · · , βn∨ ∈ L be the dual basis with
respect 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 implies
B ⊂ Aβ1∨ + · · · Aβn∨
L
L
(as can be seen by taking dual on
Aβi ⊂ B, which gives B ∨ ⊂
Aβi∨ ). The right hand
side 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. Let
L/K be finite separable. There exists a finite dominant morphism Y → X with Y normal
and 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 not
necessarily finite.
Definition 6.5. We say X is unramified in L if the morphism Y → X of Corollary is
unramified.
Lemma 6.6. In situation of Corollary, we have Y → X unramified if and only if Y → X
is étale.
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 (étale) 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(K sep /K) = π1 (Spec(K), Spec(K)) → π1 (X, Spec(K))
is surjective and this identifies π1 (X, Spec(K)) with the quotient
Gal(K sep /K) → Gal(M/K),
16
where M is the compositum of all finite K ⊂ L ⊂ K sep such that X is unramified in L, and
M/K is Galois.
The key ingredient to the proof is:
Lemma 6.8. If Y → X is finite étale and Y is connected, then Y is normal and integral
and is the normalization of X in f.f.(Y ).
Fact (already used in Lemma 6.6). If f : Y → X is étale, 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 ramification
contribution 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 AF1 p → A1Fp given by x 7→ f (x) is unramified if
f 0 (x) 6= 0 for all x. An example is f (x) = xp − x. Look up Artin–Schreier coverings. In fact,
Hom π1 (A1Fp ), Z/pZ ∼
= Γ(A1Fp , O)/(subgroup of f p − 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 quasiseparated
scheme over k. Assume Xk is connected (i.e., X is geometrically connected over k). Pick
ξ ∈ 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(k sep /k).
The middle term is sometimes called the arithmetic fundamental group.
Example 6.13. X = P1Q \{0, 1, ∞}. Then
1 → Fˆ2 → 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
Aut(Fˆ2 )
Gal(Q/Q) ,→ Out(Fˆ2 ) =
.
Inn(Fˆ2 )
The method is to prove corresponding results for the functors
FÉtXk ← FÉtX ← FÉtSpec(k) .
17
7. Lecture 7 (September 29, 2015)
7.1. Short exact sequence of fundamental groups. Today we will consider the fundamental group of a scheme X over k.
/
Xk
/
Spec(k)
X
Spec(k)
If Xk is connected, then there is a short exact sequence of fundamental groups. But before
involving the geometric side, we will consider what this means for Galois categories.
Consider functors of Galois categories
H0
H
C → C 0 → C 00 ,
which give rise to the 2-commutative diagram
F
/
H
C
∼
=
H0
C0
/
=
F0 ∼
Finite-G-Sets
h
/
C 00
=
F 00 ∼
Finite-G0 -Sets
h0
/
Finite-G00 -Sets
We want to study when the sequence of profinite groups
h0
h
G ← G0 ← G00
is exact.
Properties of these two diagrams are related as follows:
h
h0
G ← G0 ← G00
h surjective
h0 injective
h ◦ h0 trivial
Im(h0 ) normal
h surjective + kernel
h is smallest normal
closed subgroup
containing Im(h0 ).
We can prove that:
H
H0
C → C 0 → C 00
(A) H fully faithful ⇔ X ∈ C connected and if there
exists ∗C 0 → H(X), then X = ∗C .
(B) For all X 00 ∈ C 00 connected, there exists
epi
mono
H 0 (X 0 ) ← Y 00 →`X 00 .
(C) H 0 (H(X)) ∼
= j=1,··· ,m ∗C 00 for all X ∈ C.
(D) For all X ∈ C 0 connected, `
if there exists
0
0
0
0
∼
∗C 00 → H (X ) then H (X ) = j=1,··· ,m ∗C 00 .
(E) H fully faithful + essential
`image of H is exactly
those X 0 such that H 0 (X 0 ) ∼
= j=1,··· ,m ∗C 00 .
`
Fact. (E) ⇔ (A) + (C) + (H 0 (X 0 ) ∼
= ∗C 00 ⇒ there exists X ∈ C and an epi H(X) X 0 ).
Now apply this to
H0
H
FÉtSpec(k) → FÉtX → FÉtXk
with X/k quasicompact, quasiseparated and geometrically connected (for example, a geometrically connected k-variety) and k perfect.
18
(A) Have to show: for k 0 /k finite separable field extension we have Spec(k 0 ) ×Spec(k) X =
Xk0 is connected. This is clear because Xk Xk0 and Xk is connected by assumption.
(B) First we have the following
Claim. For every Z → Xk finite étale there exists a k/k 0 /k finite and Y → Xk0 finite
étale such that Z ∼
= Y ×Xk0 Xk = Y ⊗k0 k.
If the claim is true, then the composition Y → Xk0 → X is finite étale and
Z = union of connected components of Y ×Spec(k) Spec(k)
The multiplication map k 0 ⊗k k → k gives a connected component
Spec(k) ,→ Spec(k 0 ) ×Spec(k) Spec(k),
and Z is the fibre product
Z
open+
closed

Spec(k) open+
 closed /
/
Y ×Spec(k) Spec(k)
Spec(k 0 ) ×Spec(k) Spec(k)
i.e., Z = Y ×Spec(k0 ) Spec(k).
About the claim: k = colimk/k0 /k k 0 , so
Spec(k) = lim Spec(k 0 )
k/k0 /k
and
Xk = lim Xk0
in the category of schemes.
In general, if (I, ≥) is a directed partially ordered set, and (Xi , ϕii0 ) is an inverse
system of schemes over I such that ϕii0 are affine, then X = limi∈I Xi exists in
schemes.
Lemma 7.1. In this situation, if Xi is quasicompact and quasiseparated for all i, then
FÉtX = colim FÉtXi , and if Xi is connected for all i 0, then π1 (X) = lim π1 (Xi ).
Affine case. Suppose A = colim Ai filtered. Then
category of A-algebras
category of Ai -algebras
= colim
.
of finite presentation
of finite presentation
An A-algebra of finite presentation looks like A[x1 , · · · , xn ]/(f1 , · · · , fm ), where fj =
P
I
finite aj,I x . Pick i large enough such that aj,I ∈ Im(Ai → A).
The same holds for categories of étale, smooth, finite+fp, flat+fp, and combinations
of these.
The lemma implies the claim by above.
19
(C) This is clear from
/
Xk
/
Spec(k)
X
Spec(k)
and the fact that Spec(k) has trivial π1 .
(D) Suppose U → X finite étale, U connected, and there is a section s : Xk → Uk .
/
U
F k
s
/
Xk
U
X
Then you can consider
[
T =
sσ (Xk ) ⊂ Uk
σ∈Gal(k/k)
which is open and closed, and Gal(k/k)-invariant. With some work, T is the inverse
image of an open and closed T ⊂ U . Since
` U is connected, this implies T =0 U .
∼
(E) Suppose U → X is finite étale and Uk = i=1,··· ,n Xk . Then there exists k/k /k (with
k 0 /k finite) such that
a
Uk 0 ∼
Xk 0 .
=
i=1,··· ,n
Then
!
U ← Uk0 ∼
=
a
Xk0 = X ×Spec(k)
i=1,··· ,n
a
Spec k 0
i=1,··· ,n
where the last coproduct is in FÉtk .
Remark. We’ve proved the exact sequence
1 → π1 (Xksep ) → π1 (X) → Gal(k sep /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
(·)n
Xn = Gm,k → Gm,k = Xk
x 7→ xn .
Using Riemann–Hurwitz, you can show these are cofinal in FÉtXk . This implies
π1 (Xk ) = lim µ(k) ∼
= Ẑ(1).
n
20
So we have
/
0
/
π1 (Xk )
/
0
∼
=
/
Ẑ
/
π1 (X)
∼
=
/
?
/
π1 (Spec(k))
1
∼
=
/
Gal(k/k)
1
and get
Gal(k/k) → Out(Z) = Ẑ× .
This is the cyclotomic character.
Example 7.3. If X → Spec(k) is an elliptic curve with identity O : Spec(k) → X and
char(k) = 0, then
π1 (Xk ) ∼
= Ẑ⊕2 .
Again multiplying by n gives a cofinal system in FÉtXk . The exact sequence is
0
/
Ẑ⊕2
/
t
π1 (X)
OX
/
Gal(k/k)
/
1.
Then
Gal(k/k) → Aut(Ẑ⊕2 ) =
Y
GL2 (Z` ).
`
8. Lecture 8 (October 1, 2015)
Today we will talk about ramification theory and topological invariance of the étale topology.
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 previous
lemma, we know B is finite over A. Ramification theory will take some work to prove, and
the 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
Y
ei fi .
i=1
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 and
κ(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 . Set
{1} ⊂ 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 C D and D/I → Aut(κ(m)/κA ).
• P C D 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(K sep /K)
S
by passing to the limit. (Write K sep = L where L runs over K sep /L/K Galois, then for
each L we get
PL ⊂ IL ⊂ DL = Gal(L/K)
and then
P = lim PL → I = lim IL → Gal(K sep /K) = lim Gal(L/K) = lim DL .)
Both P and I are normal in G and
sep
(1) G/I ∼
= Gal(κQ
A /κA ).
(2) It = I/P ∼
= `6=char(κA ) Z` .
(*)
(3) The short exact sequence
1 → It → G/P → Gal(κsep
A /κA ) → 1
gives an action of Gal(κsep
A /κA ) on It which is via the cyclotomic character.
(4) P = {1} if char(κA ) = 0.
The isomorphism (*) is noncanonical. Canonically,
It = lim µe (κsep
A ).
e
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 ) = ζσ t1/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
g · π mod π 2 Bm
σ(π) = θ(σ)
g ∈ Bm is any lift of θ(σ) ∈ κ(m). The reason this
where π ∈ Bm is a uniformizer and θ(σ)
e
works is that π = (unit)πA .
We are interested in studying the fundamental groups of curves using this. We will see
how 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 }.
bX,x (completion) or Ai = Oh (henselization). We get
Apply with Ai = O
i
X,xi
ai : Spec(f.f.(Ai )) → X.
Hence
π1 (ai )
Ẑ ∼
= π1 (Spec(f.f.(Ai ))) → π1 (X),
well-defined up to inner conjugation on the right hand side, measures the ramification of
finite étale cover above xi . It is possible to choose base-points to make this well-defined, but
there 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
Ẑ⊕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) ∼
= Ẑ (see earlier), then
∼
=
∼
=
inertia at 0 → π1 (X) ← inertial at ∞.
The isomorphism between the two inertia is via inverse. This is compatible with the topological picture.
In characteristic p, we can do a similar thing but we need to use the tame inertia group
instead.
8.2. Topological invariance of étale topology.
Theorem 8.8. Let f : X → Y be a universal homeomorphism of schemes. Then the base
change functor
schemes étale over Y → schemes étale over X
∼
=
is an equivalence. Same for FÉtY → FÉtX 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 0 → Y , X ×Y Y 0 → Y 0
is a homeomorphism.
Fact. f is a universal homeomorphism if and only if f is integral, surjective and universally
injective.
Example 8.10.
(1) Thickenings: i : X → X 0 is called a thickening if and only if i is a closed immersion
and bijective on points.
Example 8.11. Spec(A/I) ,→ Spec(A) if I ⊂ A is an ideal and:
• I 2 = 0 (square zero ideals), or
• I n = 0 for some n (nilpotent ideal), or
• for all x ∈ I, there exists n = n(x) such that xn = 0 (locally nilpotent ideal).
Étale 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 the
quasi-inverse functor.
Example 8.12. Let A = Fp [x1 , · · · , xn ]/I and B = Fp [x1 , · · · , xn ]/I 39 . Clearly there
m
is a surjection B → A. To get a map A → B, we send xi 7→ xpi where m is such
that pm > 39. We can always use this if f : X → Y is a finite morphism of varieties
over 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 schemes
inducing a homeomorphism |X0 | → |X| on the underlying topological spaces, then the
functor
FÉtX → FÉtX0
Y 7→ Y0 = X0 ×X Y
is an equivalence.
∼
=
Remark. Even Xét → X0,ét (small étale sites).
We often call this the topological equivalence of étale topology. For example, we see that
the étale site of a scheme only depends on the reduction of the scheme.
Today we will prove the following
Theorem 9.1. Let f : X → Spec A be a proper morphism of schemes. If (A, m, κ) is a
henselian local ring and X0 = X ×Spec(A) Spec(κ), then the base change functor
FÉtX → FÉtX0
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 starting
from 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 the
fundamental 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 really
Grothendieck) will look at
Thm
π1 (Xη ) → π1 (Xη ) → π1 (X) = π1 (X0 ).
This composition is called the specialization map. This allows us to compare the
fundamental group of the generic fibre with that of the special fibre. In characteristic
p, this map has a kernel.
9.2. Structure of proof. I will focus on “essential surjectivity”. Assume Y0 → X0 finite
étale is given and we try to construct Y → X finite étale with Y0 = X0 ×X X.
The argument is standard, and the other parts of this proof (such as full faithfulness) will
involve 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 étale scheme Yn → Xn and isomorphisms
∼
Xn ×Xn+1 Yn+1 → Yn over Xn .
Remark. Can put these together to get a formal scheme
Y = “ colim Yn ”.
25
9.2.2. Algebraization. We will use the
Theorem 9.2 (Grothendieck’s existence theorem). Suppose given X → Spec(A) proper, A
Noetherian complete local and
? _ Z1 o
? _ Z2 o
··· o
h2
? _ X2 o
··· o
? _ Z0
h1
? _ X1 o
h0
? _ X0
with cartesian squares and hn finite, then there exist h : Z → X finite and isomorphisms
Zn = 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 compatible with transition maps.
Back to our Theorem in case (A, m, κ) is complete local Noetherian. Let Y → X be the
finite morphism such that Y ×X Xn ∼
= Yn for all n where Yn is as in deformation step. Then
Y → X is étale by:
Lemma 9.4. Let f : Y → X be lft with X locally Noetherian. Let X0 ,→ X be a closed
subscheme with n-th infinitesimal neighborhood Xn ,→ X. If Xn ×X Y → Xn are smooth,
resp. étale for all n 0, then f is smooth, resp. étale at all points of f −1 (X0 ).
Finally: X → Spec(A) proper and A local. Then any open neighborhood of X0 is all of
X.
9.2.3. Approximation (Artin, ..., Popescu).
Lemma 9.5. Let (A, m, κ) be a local ring. The henselization is
Ah = colim B = colim Bq
(A→B,q)
(A→B,q)
where the colimit is over the category of A → B étale and q ⊂ B prime lying over m such
that 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 α
mod 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 ring
of a scheme of finite type over a field or Z, or over any Dedekind domain of characteristic
0), then approximation holds for Ah ⊂ A∧ = (Ah )∧ .
26
Deformation theory combined with algebraization produces objects over the completion
A∧ , and now we want to descend to Ah .
Definition 9.9. Let R be a Noetherian ring, I ⊂ R an ideal and R∧ = lim R/I n be the
completion. 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 for
j = 1, · · · , m, there exist a1 , · · · , an ∈ R such that
fj (a1 , · · · , an ) = 0
for j = 1, · · · , m, and ai ≡ bi mod I N R∧ .
With approximation, we see that bad properties of A∧ are inherited from Ah , for examples
zero divisors and nilpotence.
9.2.4. Apply. Theorem holds if A = C h where C is a Noetherian local ring essentially of
finite type over Z (more generally if approximation holds for A → A∧ ).
Proof. By deformation and algebraization, we have Y 0 → Spec(A∧ ) ×Spec(A) X recovering Y0
finite étale. Say A∧ = colim Ai 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 étale whose base
change is Y 0 . Write (for the map coming from A∧ = colim A)
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 ) o
Y0
Spec(Ai ) o
YO i
Y
Spec(ρ)
Spec(A)
Spec(A) ×Spec(ρ),Spec(Ai ) Yi = Y
Since ai ≡ bi mod mA∧ , we get that the special fibre of Y is the same as the special fibre
of Y 0 , 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 of
branch 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.
def
(2) x is isolated in Xy (⇔ {x} is open in Xy ).
(3) x is closed in Xy and no x0
x (specialization) in Xy except x0 = x.
(4) For some (any) affine opens
Spec(A) Spec(B) 
/

/
X
f
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 ramification locus. Roughly speaking, “purity” means “if it happens, then it happens in codimension
1”.
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 x0
x with dim(OX,x0 ) = 1 we have f unramified at x0 .
Then f is étale at x.
The flatness assumption is what makes the proof easy.
Now we will translate this into algebra, which requires lots of technical garbage:
• Étale 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 Noetherian
local ring. If
• dim(A) ≥ 2,
• Spec(B) → Spec(A) is étale over the punctured spectrum Spec(A)\{mA },
then A → B is étale.
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 (b1 b2 ) (trace of the matrix of multiplication by b1 b2 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 étale 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 over
this prime).
If d = discB/A ∈ m, then dim V (d) ≥ 1, so there exists p ⊂ A, p 6= m, p ∈ V (d). This
contradicts the assumption. Therefore d ∈ A× , which proves that f is étale.
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 x0
x with dim(OX,x ) = 1, f is unramified at x0 .
Then f is étale 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 ring
of dim ≥ 2 and B normal. If Spec(B) → Spec(A) is étale over the punctured spectrum, then
A → B is étale.
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, dim Y = 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 ) and
√
ϕ(x2 ) form a regular sequence in B. Then use mB = mA B.
• Case II: dim(A) ≥ 3. To prove this Grothendieck uses a local Lefschetz theorem.
Pick f ∈ mA \m2A . Contemplate
? _ X0
Spec(A) = X o
O
?
Spec(A)\{m} = U o
? _ U0
29
= Spec(A/f A)
O
?
= Spec(A/f A)\{m}
Note A/f A is regular local of dimension 1 less, so by induction we have the result
for A/f A. Reformulated, then result is the statement that FÉtX0 → FÉtU0 is an
equivalence.
If A is henselian, then we have
FÉtX ∼
= FÉtX ∼
= FÉtSpec(κ ) .
0
A
So we’re done if we can show
FÉtX → FÉtU0
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/f A) and U0 the punctured spectrum
of A/f A. Let V be finite étale over U . Assume
(1) f is a nonzerodivisor,
(2) Hm1 (A) is a finite A-module,
(3) a power of f annihilates Hm2 (A),
(4) V0 = V ×U U0 is equal to Y0 ×X0 U0 for some Y0 → X0 finite étale.
Then V = Y ×X U for some Y → X finite étale.
Remark.
depth(A) ≥ t ⇔ Hmi (A) = 0 for i = 0, · · · , t − 1.
So if dim A ≥ 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 complete
intersection.
Roughly speaking, complete intersection means that A is a regular local ring modulo a
regular sequence. This result is sharp because it doesn’t work when dim A = 2.
Example 10.9. A = C[x2 , xy, y 2 ] ⊂ B = C[x, y]. This looks like the plane mapping to the
cone via quotienting out by {±1}. Then B is not finite étale 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.
Consider
Spec(A)\{m}
=U o

_
? _ U0
= Spec(A/f
A)\{m}

? _ X0
Spec(A) = X o
1PH:
_
= Spec(A/f A)
This is a corrected version of the theorem stated in lecture (see here), as pointed out to me by
Johan. To quote him: “It should not be an equivalence between the categories of finite étale schemes from U
to U0 . This theorem is enough to prove purity in the regular local case (the implication direction being that
if the thing extends after restriction to the hypersurface, then it extends). But, it is not enough to get the
application to Grothendieck’s thing about complete intersection rings. The result you need there is this. But
this 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 ) o
π1 (U0 )
π1 (X) o
π1 (X0 )
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} and
π1 (X0 ) = {1}.
Purity says that the map on the left is an isomorphism, hence in the strictly henselian
case π1 (U ) = {1} and any finite étale cover of U is trivial. Lefschetz is about the top map
π1 (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, Hm1 (A) is finite, and f N kills Hm2 (A) for some N .
Then
FÉtU → FÉtU0
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 for
A/f A, i.e., π1 (U0 ) = {1}, then local purity holds for A, i.e., π1 (U ) = {1}. (This is an
example of going from knowing local purity for lower dimension to knowing local purity for
higher dimension.)
We have to show
FÉtU → FÉtU0
V 7→ V0 = V ×U U0
is fully faithful.
Set Vn = V ×U Un where Un = V (f n+1 ) ⊂ U .
V o
U _ o
Spec(A) o
? _· · · o
? _ V2 o
? _ V1 o
? _ V0
? _· · · o
? _ U2 o
_
? _ U1 o
_
? _ U0
_
? _· · · o
? _ Spec(A/f 3 A) o
? _ Spec(A/f 2 A) o
open
? _ Spec(A/f A)
bij.
Lemma 11.2. The functor is fully faithful if and only if for all V ∈ FÉtU , we have π0 (V ) →
π0 (V0 ).
31
Looking at the diagram above, we get
∼
=
∼
=
π0 (V ) → · · · → π0 (V2 ) → π0 (V1 ) → π0 (V0 ).
Note that
bij.
subsets of π0 (Vi ) → set of idempotents of Γ(Vi , OVi )
for each i, so we have
/
subsets of π0 (V )
/
···
subsets of π0 (V2 )
bij.
set of idempotents
of Γ(V, OV )
/
/
···
∼
=
/
subsets of π0 (V1 )
bij.
set of idempotents
of Γ(V2 , OV2 )
∼
=
/
∼
=
/
subsets of π0 (V0 )
bij.
set of idempotents
of Γ(V1 , OV1 )
∼
=
/
set of idempotents
of 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 H p and short
exact sequences
0 → R1 lim H p−1 (OVn ) → H p → lim H p (OVn ) → 0
and
0 → H 0 (H p (OV )∧ ) → H p → Tf H p+1 (OV ) → 0,
where
• H P (OV )∧ is the derived completion of H p (OV ) with respect to F , and for any Amodule M one has a short exact sequence,
c→0
0 → R1 lim M [f n ] → H 0 (M ∧ ) M
fn
c = lim M/f n M is the usual completion, and M ∧ = R lim(M → M ) (at
where M
degrees −1 and 0 respectively) is the derived completion.
• Tf (M ) = limn M [f n ] is the f -Tate module of M :
←−
×f
· · · → M [f 2 ] → M [f ] = {x ∈ M | f x = 0}.
We will take p = 0:
• H 0 = lim H 0 (OVn ) by the first exact sequence.
• If H 0 (OV ) has bounded f -power torsion (e.g. when V is Noetherian), then
0 (O ).
H 0 (H 0 (OV )∧ ) = H\
V
We get
0 → H 0\
(V, OV ) → lim H 0 (OVn ) → Tf H 1 (OV ) → 0.
Corollary 11.4. To show FÉtU → FÉtU0 is fully faithful it suffices to show for all V → U
finite étale that:
(1) H 0 (OV ) is f -adically complete.
(2) f -power torsion on H 1 (OV ) is bounded.
32
bij.
Proof of (1) when A is complete. Hm1 (A) finite and V → U finite étale imply that H 0 (V, OV )
is finite over A. This uses finiteness theorems in coherent cohomology.
Example 11.5.
• If dim A = 1, then H 0 (U, OU ) is never finite: pick x ∈ m nonzerodivisor, and then
1
∈ H 0 (U, OU ) for all n ≥ 1.
xn
• If A is a normal domain of dim ≥ 2, then H 0 (U, OU ) = A is finite. We have
Hm1 (A) = 0.
In general, there is an exact sequence
0 → Hm0 (A) → A → H 0 (U, OU ) → Hm1 (A) → 0
and
∼
=
H i (U, OU ) → Hmi+1 (A)
if i ≥ 1. (Look in Hartshorne for
0 → HZ0 (X, F) → H 0 (X, F) → H 0 (U, F) → HZ1 (X, F) → · · · .)
Proof of (2). We have to show π : V → U étale finite implies
H 1 (V, OV ) = H 1 (U, π∗ OV ) = H 1 (U, some vector bundle on U )
has bounded f -torsion. This follows from the next lemma.
Lemma 11.6. If H 1 (U, OU ) ∼
= Hm2 (A) is annihilated by f N for some N , then for any finite
locally free sheaf E on U there is an M = M (E) such that f M annihilates H 1 (U, E).
Proof. For all u ∈ U there exists u ∈ U 0 ⊂ U such that
open
∼
=
E|U 0 ← OU⊕r |U 0 .
ϕ
We can pick U 0 = D(g) for some g ∈ A.
There exists a map α : OU⊕r → E such that α|U 0 = g N1 ϕ. Dually, there exists a map
β : E → OU⊕r such that β|U 0 = g N2 ϕ−1 . Thus the composition
β
α
E → OU⊕r → E
is multiplication by g N1 +N2 (maybe after increasing N1 and N2 ).
Now for all u ∈ U , there exists g ∈ A such that u ∈ D(g) and g N1 +N2 f N ∈ Ann(H 1 (U, E)).
The ideal generated by all these g N1 +N2 is m-primary, so
mN101 · f N ⊂ Ann H 1 (U, E)
and hence
f N101 +N ∈ Ann H 1 (U, E).
Upshot: We are done with Theorem 11.1 in the complete case. Then you still have to
reduce 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 specialization
of the fundamental group in the smooth proper case.
Theorem 12.1. Let k ⊂ k 0 be an extension of algebraically closed fields. Let X/k be a
proper connected scheme. Then Xk0 is connected and π1 (X) = π1 (Xk0 ).
Sketch of proof. Recall “cohomology and base change”: if X/R is quasicompact and quasiseparated and R → R0 is flat, then
H i (X, OX ) ⊗R R0 = H i (XR0 , OXR0 ).
To prove connectedness, we use
H 0 (X, OX ) ⊗k k 0 = H 0 (Xk0 , OXk0 ).
Note H 0 (X, OX ) is a finite-dimensional k-algebra, hence artinian. It is a product of local
rings. Since k is algebraically closed, the number of local components remains the same after
base change to k 0 .
The proof of π1 (X) = π1 (Xk0 ) uses our previous theorem about π1 (X) where X is proper
over 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 (AF1 p ) × 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 smooth
morphism with geometrically connected fibres. Let s
s0 be a specialization of points of S.
Then there is a map
sp : π1 (Xs ) → π1 (Xs0 )
which is surjective and
• an isomorphism if char(κ(s0 )) = 0.
• an isomorphism on prime-to-p quotients if char(κ(s0 )) = p.
Remark. If X is connected, then
sp
π1 (Xs )
$
/
π1 (Xs0 )
z
π1 (X)
commutes.
Remark (Missing material). Let f : X → S be a flat proper morphism with geometrically
connected and reduced fibres. If S is Noetherian and connected and s ∈ S, then there is an
exact 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.
s0 in S Noetherian, we can find
Proof. Use that given a specialization s
Spec(A) → S
η 7→ s,
0 7→ s0 ,
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 the
exceptional 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, s0 = 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 previous
theorem.
Step 5: sp is surjective.
Proof. Let K = f.f.(A). Let Y0 → X0 be connected finite étale. Let Y → X be the
corresponding finite étale 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/K L
finite
and ...). Let B ⊂ L be the integral closure of A in L. Note A/mA = B/mB and B is local.
YL 
Spec(L) /

/
/
YB
/
Y
/
Spec(B)
#
X
Spec(A)
Now YB → Spec(B) is smooth proper with disconnected generic fibre. Two components
have to meet in YB , but this cannot happen because YB is normal (even regular). The next
lemma 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 locally
constant 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.
sp
π1 (Xη )
#
G
35
{
/
π1 (X0 )
Proof. Let G acting on Y 0 → Xη be the corresponding finite étale cover. By a limit argument
(see Step 5), there exists κ(η)/L/K finite such that Y 0 is defined over L. After replacing A
by B ⊂ L (see Step 5), we may assume we have G acting on Y 0 → Xη .
Let Y → X be the normalization of X in f.f.(Y 0 ). Then we get

Y 0
πη
/
Xη

/
Y
π
X
where πη is finite étale, π is finite and Y is normal. As X is a regular scheme (smooth
over dvr), by purity of branch locus, if π is not unramified then π is not unramified at a
codimension 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 #G
prime 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) 
/
O
?
OX,ξ 
f.f.(Y )
/
O
?
C
where f.f.(X) ⊂ f.f.(Y ) is a Galois extension with group G, and C is the integral closure of
OX,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 base
change 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 Abhyankar’s lemma, which is about lowering ramification index by base change. Today we will
lower this down to 1 so that there is no ramification. In fact the general version allows us
to 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 that
A ,→ B be a local ring map, i.e., mA ⊂ mB . Let L = f.f.(B) and K = f.f.(A). Then L/K is
an extension (not necessarily finite). We have the invariants:
• e = ramification index e: πA = (unit)πBe ,
• κB /κA extension of residue fields (not necessarily finite).
Reminder: An extension of fields κ0 /κ is called separable if and only if every finitely
generated subextension κ0 /`/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 is
separable!
Proposition 13.1. TFAE:
36
•
•
•
•
•
•
κ0 /κ is separable.
κ0 is geometrically reduced over κ.
κ0 ⊗κ Ωκ/Z ,→ Ωκ0 /Z .
Any derivation of κ extends to a derivation of κ0 .
κ0 is formally smooth over κ.
H1 (Lκ0 /κ ) = 0.
If κ0 /κ is a finitely generated field extension, then these are also equivalent to:
• κ0 /κ is separably generated.
• κ0 = κ(X) where X/κ is a smooth variety.
• dimκ0 (Ωκ0 /κ ) = Tr(κ0 /κ).
S
n
Example 13.2. Fp ⊂ n Fp (t1/p ) is separable: the only derivation of Fp is trivial!
Example 13.3. Let κ = Fp (t, s, r). Then the variety X : txp + sy p + rz p = 0 is nowhere
smooth over κ, so κ(X)/κ is not separable. Such a variety does arise in nature: it is the
generic fibre of a smooth morphism X → A3Fp over Spec(Fp (t, s, r)).
Going back to A ⊂ B, let K1 /K be a finite separable extension.
BO 1 r
oO BO
e
L
K
?

oO A
?
ei
eij
A1 r

$
L ⊗K K1
$
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 and
that κB /κA is separable. If e | ei , then eij = 1 for all j = 1, · · · , mi , and κ(mij )/κ(mi ) is
separable.
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
/
/
A[x]/(x5 − t)
O
A = k[[t]]
/
(z, · · · , z)
Q5
(z, ζ5 z, ζ52 z, ζ53 z, ζ54 z)
i=1 O k[[z]]
O
_
A[y]/(y 5 − t)
y
where ζ5 ∈ k is a primitive 5-th root of unity. Here
A[x, y]/(x5 − t, y 5 − t).
Q5
i=1
k[[z]] is the normalization of
13.2. End of proof of Theorem last time. Recall that A is a complete dvr with κA = κA
and char(κA ) = p. X → Spec(A) is a smooth proper geometrically connected fibre. Y → X
is the normalization of X in a Galois extension, with f.f.(X) ⊂ f.f.(Y ) with group G of order
prime to p and Yη → Xη étale.
Y → X → Spec(A) 3 η
G
Yη → X η
Let ξ ∈ X0 be the generic point of the special fibre, and y1 , · · · , yn ∈ Y0 are points lying
over 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 to
OX,ξ , we see that e(OY,yi /OX,ξ ) is prime to p. Note e(OX,ξ /A) = 1 as X → Spec(A) is
smooth. 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.
1/e
Pick A1 = A[πA ] where e ∈ N is sufficiently divisible (e.g. #G | e). (This was called B
in the previous lecture.) Then
Y o
Y1
Xo
XA 1
Spec(A) o
Spec(A1 )
where Y1 is the normalization of YA1 . By Abhyankar’s lemma, Y1 → XA1 has ramification
index 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 unramified in codimension 1. By purity, Y1 → XA1 is étale. 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 example
of 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, so
we 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 a
surjection
∧
hα1 , · · · , αg , β1 , · · · , βg i
π1 (X)
[α1 , β1 ] · · · [αg , βg ]
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 g
Isomorphism classes of
→
smooth projective curves over Fp
profinite groups
given by
X 7→ π1 (X)
is finite to 1.
Example 13.7. If (E, O) is an elliptic curve over k = k, then

b ⊕2

if char(k) = 0,
Z
Q
⊕2
π1 (E) =
if char(k) = 0, E ordinary,
`6=p Z` × Zp

Q Z⊕2 × {0} if char(k) = 0, E supersingular.
`6=p `
Example 13.8. If A is an abelian variety, then
Y ⊕2g
π1 (A) =
Z` × Zfp
`6=p
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 easier
with 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 morphism of schemes l.f.t. over C.
Fact. f an : X an → S an is a fibre bundle: for all s ∈ S an = S(C), there exists s ∈ U ⊂ S an
open such that (f an )−1 (U ) ∼
= Xs × U as homeomorphism of spaces over U .
Corollary 14.1. The sheaves Ri f∗an (Z) are locally constant on S an (such a thing is often
called a local system) with stalk H i (Xs , Z) at s. In particular, there is a monodromy representation
ρi : π1 (S an , 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 is
unipotent (⇔ 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 for
each x ∈ S\S there is a well-defined conjugacy class in π1 (S an , s) consisting of loops
around x (counterclockwise).
(2) If dim(S) is arbitrary, we consider finite morphisms C → S for C smooth, and look
at the images of loops around ∞ in π1 (C) in π1 (S).
14.2. Étale cohomology version. Let f : X → S be a smooth proper morphism of
Noetherian schemes. Let ` be a prime number invertible on S. Then we get Ri fét,∗ (Z` )
locally constant (technically we haven’t discusses étale sites yet, but fét is the étale analogue
of the analytification for the complex topology), with
i
Ri fét,∗ (Z` )s = Hét
(Xs , Z` )
is a finitely generated Z` -module. There is a monodromy representation
i
(Xs , Z` )).
ρi` : π1 (S, s) → AutZ` (Hét
Theorem 14.4 (Grothendieck). ρi` have quasi-unipotent local monodromy, i.e., for every
morphism Spec(K) → S where K = f.f.(A) and A is a dvr, the action
i
ρiXK /K,` : Gal(K sep /K) → AutZ` (Hét
(XK , Z` ))
when restricted to an inertia subgroup has the following properties, provided ` is invertible
in 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 ) is
quasi-unipotent.
sep
Recall {1} ⊂ P ⊂ I ⊂ D ⊂ Gal(K
Q /K), where P is the wild inertia, I is the inertia, D
∼
is the decomposition, PI = I/P = `0 6=char(κA ) Z`0 is the tame inertia.
The argument of Grothendieck shows that this is a consequence of the structure of inertia
groups (which can be thought of as fundamental groups), without knowing about étale
cohomology and the local systems Ri fét,∗ (Z` )! This theorem really gets used in geometric
situations when we are given a smooth proper morphism, even though the proof goes through
the inertia groups.
Remark. You can deduce the theorem on quasi-unipotent monodromy over C from Grothendieck’s
version 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-` and
G 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/`n M )
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/`n M ).)
i
(XK , Z` )) is
Lemma 14.7. If X/K is a variety, then ρiX/K,` : Gal(K sep /K) → AutZ` (Hét
continuous.
Proof. By definition,
i
i
Hét
(XK , Z` ) = lim Hét
(XK , Z/`n Z)
n
and the action of Gal(K sep /K) on these is continuous.
i
Remark. Even though Hét
(XK , Z/`n Z) is finite, continuity is not automatic because Gal(K sep /K)
is a profinite group!
Proof of part (1) of Theorem 14.4. Done by combining the lemmas and using that the wild
inertia 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 and
proper variety, and ` 6= char(κ) be a prime number. We have the picture
1 ⊂ P ⊂ I ⊂ D ⊂ Gal(K sep /K)
where P is the wild inertia, I is the inertia and D is the decomposition (if K is henselian
local, then D = Gal(K sep /K)), together with representations
i
ρiX/K,` : Gal(K sep /K) → Aut(Hét
(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 ) is
quasi-unipotent.
i
We will do this by mapping Aut(Hét
(XK , Z` )) into
i
AutQ` (Hét (XK , Z` ) ⊗Z` Q` ) ∼
= GLBXi (Qp ),
i
where BX
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 say
a 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 and
τ ∈ I such that
41
• τ maps to a topological generator of It ,
b × is α = (α2 , α3 , α5 , α7 , · · · )
• στ σ −1 and τ α map to the same element of It where α ∈ Z
with α` ≡ 1 (mod `) but α` 6= 1.
b Then τ α is defined.
Aside: Suppose G is a profinite group, τ ∈ G, α ∈ Z.
Proof. We may assume G is finite. Then τ n = 1 for some n. Pick a ∈ Z such that a ≡ α
(mod n). Then set τ α = τ a in G.
More conceptually, there is a map Z → G given by a 7→ τ a . By the universal property of
b → G by α 7→ τ α .
profinite completion, we get Z
Proof of lemma. There are canonical maps D → Gal(κsep /κ) and
θcan : I →
lim
n prime to p
µn (κsep )
where p = char(κ)
Q (or 1 when char(κ) = 0) (and the limit on the right is noncanonically
isomorphic to `0 6=p Z`0 ), such that
θcan (στ σ −1 ) = σ(θcan (τ ))
for τ ∈ I and σ ∈ D. QNote that θcan factors through It , andQany σ ∈ D acts on It ∼
=
×
sep
limn prime to p µn (κ ) ' `0 Z`0 by multiplication by some ασ ∈ `0 6=p Z`0 . 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 our
assumptions 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 ) where
M 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(στ σ −1 ) (here we need
the fact that ρ is defined on D, not just I). Since #ρ(P ) < ∞ and στ σ −1 and τ α have the
same image mod P , this implies by thinking plus group theory that
ρ((στ σ −1 )N ) = ρ((τ α )N )
for some N > 0. Then ρ(τ )N is conjugate to (ρ(τ )α )N = (ρ(τ )N )α . Finish by the following
lemma.
Lemma 15.4. If g : M → M is an automorphism of a finite Z` -module such that g is
conjugate to g α 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 exists
r
r
s
j such that λαi = λj . Then for all i, there exists 0 ≤ r < s such that λαi = (λαi )α , so
r
(λαi )α
s −1
= 1.
Note αr 6= 0 and αs − 1 6= 0, by looking at the `-component of α: there are no roots of 1 in
1 + `Z` if ` > 2, or in 1 + 4Z2 if ` = 2.
42
×
b and β 6= 0, then λ is a root
We still have to prove that if λ ∈ Z` and λβ = 1 with β ∈ Z
of 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
⊕2g
T` X = lim X[`n ](K) ∼
=Z
`
sep
with a continuous action ρ of Gal(K /K). Then we have shown, after replacing K by
a finite separable extension, we have ρ(P ) = {1} and ρ of the tame inertia is given by a
unipotent operator.
Fact (I don’t know an easy proof). Each Jordan block in our unipotent thing is 1 × 1 or
2 × 2.
Remark. If X has good reduction (i.e., X extends to an abelian scheme over Spec(A)), then
ρ(I) = {1}. The converse is true, but a lot harder to prove.
The picture is

1
...







ρ(τ ) = 








1
...
0



1
1 


1

.

..
0
0


1


1


.
..
0
0

1
0
0
of size 2g × 2g, with the middle block of size 2g × 2g . Namely, it will turn out that X
extends 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 0 ,
and T is a torus over κ:
g−g 0
T ⊗κ κ ∼
= Gm,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(K sep /K)
ρiX/K,`
i
Aut(Hét
(XK , Q` ))
∼
/
Gal(K0sep /K0 )
ρiX
0 /K0 ,`
i
Aut(Hét
(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 for
the previous argument to work.
P n
Example 15.6. Consider K0 = Q(x, y) → K = C[[t]] given by x 7→ t and y 7→
ζn t ,
S
2πi
where ζn = e n . Then κ = n Q(ζn ).
To prove the general case, we will write A = colim A0 as a colimit of regular rings, and
apply 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 when
A 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 of
dvr’s A ⊂ A0 with residue field of A0 algebraically closed. The tame inertia of the Galois
group of the fraction field of A0 will map onto that of A.
Néron desingularization is the same process needed for constructing Néron models. The
references are Tag 0BJ7 of the Stacks Project and Artin’s paper on approximation.
Theorem 16.2 (Néron 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 = colim Ai
is a filtered colimit of smooth R-algebras.
This theorem is amazing, but Néron’s procedure of proving it is even more amazing. This
is proved using Néron 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 we
consider the affine blowup at σ(0) to get X1 → Spec(R) with a new section σ1 . The point
σ1 (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 Néron
blowups, 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 A
contains a field. Then (Cohen structure theorem) A ∼
= κ[[t]] with κ = κ. Say char(κ) = p > 0.
Then we may apply Néron 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 = colim Ai is a filtered colimit with Ai ’s smooth over Fp [[t]] or Q[[t]] and
t maps to a uniformizer π of A.
Remark. In particular, K = f.f.(A) = A[ π1 ] = colimi Ai [ 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 ∈
/ m2Ai (because Fp [[t]] → Ai or Q[[t]] → Ai is smooth).
Important question: What is the structure of
π1 (Spec(Ai )\V (t)) = π1
1
?
Spec Ai
t
Theorem 16.4 (Generalized Abhyankar’s lemma). Let (A, m, κ) be a regular henselian local
ring. Let t1 , · · · , td be a regular system of parameters. Let 0 ≤ r ≤ d. Then there exists a
quotient
1
π1 Spec Ai
π1t
t1 · · · tr
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
r Y
sep
0→
lim µn (κ ) → π1t → Gal(κsep /κ) = π1 (Spec(A)) → 1
i=1
n prime to p
where limn prime to p µn (κsep ) is non-canonically isomorphic to
action of Gal(κsep /κ) on the group on the left is as indicated.
Q
`6=p
Z` , such that the
Remark. π1t is the tame fundamental group of (Spec(A), D = V (t1 · · · tr )). There is a general
definition of tame fundamental groups for pairs (S, D) where D ⊂ S is a normal crossings
divisor.
Idea of proof. The map comes from considering the Galois closure of
[
√
√
f.f.(A[ n t1 , · · · , n tr ])
n prime to p
and combining with ramification theory (at generic points of ti = 0), purity and Abhyankar’s
lemma.
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
1
ρ : π1 Spec Ai
→ GLn (Q` )
t1 · · · tr
whose image is pro-`, factors through π1t and maps elements of ker(π1t → Gal(κsep /κ)) to
quasi-unipotent elements.
Proof. Exactly the same as last lecture except we need to assume our image is pro-` so that
pro-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,
i
(XK , Z/`Z)
• the action ρiX/K,` is trivial mod `, i.e., the action of Gal(K sep /K) on Hét
is trivial.
16.1. Outline of proof of quasi-unipotent monodromy theorem. We combine the
above to prove the general case of the quasi-unipotent monodromy theorem.
Step 1: Reduce to A complete dvr, κ = κ (Lemma 16.1).
i
(XK , Z/`Z) trivial (Lemma 16.6).
Step 2: Reduce to action on Hét
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.
1
i
i
→ Aut(Hét
(XK , Z` )).
ρXi /Ai [ 1 ],` : π1 Spec Ai
t
t
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) with
ramification index e. A uniformizer π ∈ A satisfies an Eisenstein equation
π e + λ1 π e−1 + · · · + λe = 0.
To finish, consider the diagram
/
RO
?
/
W (κ0 )
AO
?
W (κ)
where
• κ0 ⊂ κ is the perfection of a finitely field extension of Fp ,
• R∼
= W (κ0 )[X]/(X e + λ01 X e−1 + · · · + λ0e ) with λ0i 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 field
κi ∈ Ai /mAi a purely inseparable extension of a finitely generated extension of its prime
field.
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 mixed
characteristic case.
Step 1: For every perfect field κ of characteristic p > 0 there exists a canonical complete
dvr 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
π e + λ1 π e−1 + · · · λe = 0
with λi ∈ W (κ).
Lemma 17.1 (Krasner’s lemma). There exists N > 0 such that if λ01 , · · · , λ0e ∈ W (κ) satisfy
λi − λ0i ∈ pN W (κ), then
P (x) = xe + λ01 xe−1 + · · · + λ0e
has a root π 0 ∈ A with π 0 ≡ π (mod m2A ).
Last lecture we considered Néron desingularization, which corresponds to the case e = 1
and 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 of
a finitely generated extension of Fp , and λ0 ∈ W (κ0 ) such that λ − λ0 ∈ pN W (κ). Here we
think of W (κ0 ) ⊂ W (κ) via the map corresponding to κ0 → κ.
Step 4: We get
W (κ)[x]/(P (x))
AO 0
Krasner
?
W (κ0 )
/
/
AO
?
W (κ)
•
•
•
•
Pick N as in Krasner’s lemma.
Pick κ0 , λ01 , · · · , λ0e using Step 3 (repeatedly).
Set A0 = W (κ0 )[x]/(xe + λ01 xe−1 + · · · + λ0e ).
Because we started with an Eisenstein polynomial we see that A0 is a complete dvr
with uniformizer the class x of x.
• We get A0 → A by mapping x to π 0 .
Step 5: Néron desingularization applies to A0 → A. Thus
(A, π 0 ) = colim(Ai , t)
with Ai the henselization of smooth algebras over A0 at a prime. Then κi = Ai /mAi has the
desired property.
Remark. This argument does not reduce the difficult case of Grothendieck’s quasi-unipotent
monodromy theorem back to the easy case, but it uses the generalized Abhyankar lemma
instead.
47
17.2. Birational invariance of π1 .
Lemma 17.2. Let f : X → Y be a birational proper morphism of varieties with X normal
and 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 that
f (U ) ⊂ Y is the largest open over which f −1 lives. By purity, the top map is an isomorphism
in the diagram
π1 (Y ) o
∼
=
O
π1 (f (U ))
π1 (X) o o
π1 (U )
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 0 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 whose
geometric 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 = P1k and X be a smooth projective surface over k = k. Then
π1 (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 case
if g(Xη ) = 0.
Example 17.6. Suppose g(Xη ) = 1 and all fibres are at worst nodal. Then Xη is an elliptic
curve 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
b and π1 (X) is procyclic. We
cases occur when there exists a bad fibre Xt . Then π1 (Xt ) ∼
=Z
have the correspondence
n
o
n
o
connected cyclic finite étale covers
L invertible on X with L⊗n ∼
= OX
↔
with group Z/nZ
but L⊗m ∼
6= OX for 0 < m < n
0
(g : X → X) 7→ Lχ
×
by choosing a primitive n-th root of 1 in k and the corresponding
L χ : Z/nZ → k . More
precisely, the action of Z/nZ gives the decomposition (g∗ OX 0 ) = 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
0
K-point of E = PicXη /η (an elliptic curve over K) whch has order n.
By Lang–Néron, E(K) is a finitely generated abelian group, because j(E/K) ∈
/ k. This
implies 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 about
the semi-stable reduction theorem for curves.
18.1. Semi-stable reduction theorem. This is what the theorem says in the complexanalytic category. Suppose we have a family of curves over the punctured disk D∗ , and we
want 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→ z n , we can fill in a nodal curve.
This complex version is not that hard to prove. I will formulate this in the correct generality in algebraic geometry, with dvr’s.
Definition 18.1. Let X be a locally Noetherian scheme. A strict normal crossings divisor
on X is an effective Cartier divisor D ⊂ X such that for all p ∈ D, the local ring OX,p is
regular 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;P
• 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 normal
crossings divisor if there exists an étale covering {Uk → X}k∈K such that D ×X Uk ⊂ Uk is
a strict normal crossings divisor for all k ∈ K.
Remark. Normal crossings divisors “can have self-intersections”.
Example 18.5. The curve Y 2 + X 3 + X 2 = 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 and
closed 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
P
(c2) this implies Xs = i∈I Xi .)
(c4) T
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
bX,x ∼
O
= 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 = κ0 [[t]] with κ0 /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 étale locally on X we have X is strictly
semi-stable (s.s.-s.) over S.
We have the following diagram of implications:
κ(s) is perfect
(Xs )red n.c.d. ks
KS
'/
ks
Xs n.c.d.
KS
X s.s./S
KS
κ(s) is perfect
(Xs )red s.n.c.d. ks
Xs s.n.c.d. ks
'/
X s.s.s./S.
Definition 18.9. Let S be a scheme. A semi-stable curve over S is a morphism X → S
which is flat, proper, of finite presentation, such that all geometric fibres are connected, and
of 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
bX,x ∼
if and only if O
= k[[u, v]]/(uv).
Definition 18.10. A split semi-stable curve X over a field k is a semi-stable curve over
k whose irreducible components are all geometrically irreducible, and whose nodes are all
k-rational. A split semi-stable curve over S is a semi-stable curve over S such that all fibres
are split.
Lemma 18.11. Let R be a dvr, S = Spec(R), and X → S a proper S-variety of relative
dimension 1 with geometrically connected fibres. If X/S is semi-stable as an S-variety, then
X 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 over
K by lifting Spec(K) → Mg to Spec(R) → Mg
Spec(K)
Mg

/
Spec(R)
/
Mg
/
&
Spec(Z).
As stated this is wrong, and the issue is that Mg is an algebraic stack. The valuative
criterion for stacks should allow for finite extensions. With this modification, such a lifting
becomes possible.
Theorem 18.12 (Semi-stable reduction of curves). Given R dvr with fraction field K, and
C/K smooth, proper, geometrically connected curve there exist:
• R ⊂ R0 extension of dvr’s,
• X 0 → S 0 = Spec(R0 ) an s.s.s. S-variety,
• C ⊗K K 0 ∼
= X 0 ⊗R0 K 0 .
Additional properties we would like:
• K 0 /K finite separable,
• Y → Spec(B) where B is the integral closure of R in K 0 such that Y ⊗B Bm0 is an
s.s.s. variety over Bm0 for all maximal ideals m0 ⊂ B. (A standard argument allows
one to work with one m0 at a time; we will not discuss further.)
Here is the strategy.
Step 1: Pick some K sep /K 0 /K finite separable such that some property holds for the
GalK 0 -action on Pic0 (CK sep )tors .
Step 2: Let R0 = Bm0 with B ⊃ m0 as before. We will show there exists X 0 → Spec(R0 )
(relatively minimal model) which is proper flat, X 0 regular, (Xs )red s.n.c.d. and without
(−1)-curves, with X 0 ⊗ K 0 ∼
= C ⊗ K 0.
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 is
the 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 a
modification 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 singularities. For example, take the spectrum of a Noetherian domain of dimension 1 whose
completion is not reduced. This motivated Grothendieck to introduce excellent rings and
excellent schemes, which characterize in terms of commutative algebra when resolutions of
singularities 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 OY∧ 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 a
complete 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 Silverman.
Notation. Fix R a dvr with K = f.f.(R), and C/K a proper smooth geometrically connected
curve.
Proposition 19.4. There exists
/
X
C
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 is
regular 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 ⊂ Y
open
such that f
−1
(V ) → V is finite.)
10
Example 19.5. Consider Zp → Zp [x, y]/(xy(x + y) − p). This is regular at the maximal
ideal (x, y, p).
Theorem 19.6 (Embedded resolutions). Let Y be a regular 2-dimensional scheme. Let
Z ⊂ Y be a closed subscheme all of whose irreducible components Zi are either points or 1dimensional schemes whose normalization is finite. Then there exists a sequence of blowups
f : 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
/
Spec(K)
52
X
f
Spec(R)
such that f is projective and flat, X is regular, (Xs )red is a s.n.c.d.
Notation. Let X → Spec(R)P
= S be as in the proposition, C1 , · · · , Cn the irreducible components of Xs . Write Xs =
ri Ci 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 ), R0 = R[π 0 ] with π 0 = π 1/r , and X 0 the
normalization of X ×Spec(R) Spec(R0 ). Then
X 0 → Spec(R0 ) = S 0
is a semi-stable curve over R0 .
This is not the same thing as saying X 0 is a semi-stable S 0 -variety! However, we have the
Lemma 19.8 (Addendum to last time). If X is a semi-stable curve over a dvr R with
smooth 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 0 -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 = Spec κ[x, y]/(y 2 − xp − t) is a regular
non-smooth curve over κ, as follows. One can check that
Sing(C → Spec(κ)) = V (y, y 2 − 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 as
mQ = (y). But over the algebraic closure,
C ×Spec(κ) Spec(κ) = Spec(κ[x, y]/(y 2 − (x + t1/p )p )) ∼
= Spec(κ[x, y]/(y 2 − xp ))
is not smooth.
Example 19.10 (Worse). Let κ0 = κ[x]/(xp − t). Take C = Spec(κ0 [y]) → Spec(κ).
Sketch of proof of Claim. The idea is to describe étale local structure of X and use that
normalization commutes with étale localization.
We will not define “étale local structure”, but the picture is
···
(u1 , U1 )
étale
y
(x ∈ X)
étale
···
%
étale
{
étale
···
(u2 , U2 )
where (uk , Uk ) should be in a standard form.
Example 19.11. A smooth morphism is étale locally like AnS → S.
53
#
(uk , Uk )
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 and
m = rj . Then
π = u · an bm
×
where u ∈ OX,x
. Because char(κ(s)) = char(κ(x)) does not divide n, after replacing X by
an étale cover we may assume π = an bm (adjoin n-th root of u to OX,x ).
Now κ(x)/κ(s) is separable, so
OX,x ← R[u, v]/(un v m − π)
a ←[ u
b ←[ v
defines an étale morphism (X, x) → Spec(R[u, v]/(un v m − π)).
In characteristic 0, consider An = R[u, v]/(un − π) and An,m = R[u, v]/(un v m − π).
Final step: Compute the étale local structure of
A0n,m,d = (R0 [u, v]/(un v m − (π 0 )d ))norm
where R0 = R[π] and (π 0 )d = π, n | d, m | d.
Step 1: If e = gcd(n, m) > 1 and ζe ∈ R0 , then
A0n,m,d ∼
=
e
Y
A0n/e,m/e,d/e .
i=1
Why? (un/e v m/e )e = (π 0 d/e )e in A0n,m,d .
Step 2: If gcd(n, m) = 1, then
un v m = (π 0 )d
in A0n,m,d , i.e.,
(π 0 )d
u = m =
v
n
π 0 d/m
v
!m
,
so there exists u0 ∈ A0n,m,d such that
(u0 )m = u and (u0 )n =
(π 0 )d/m
.
v
Similarly, there exists v 0 ∈ A0n,m,d such that
(π 0 )d/n
(v ) = v and (v ) =
.
u
0 n
0 m
Check that
A0n,m,d ≡ R0 [u0 , v 0 ]/(u0 v 0 − (π 0 )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 characteristic 0 we have semi-stable reduction of curves (in either of the two senses). Today I will
say 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 ](K sep ).
←−
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
Spec(K) 
/
X
f
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
X
Xs =
ri Ci
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 H 0 (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 P1Q(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.
(
degree over κ(s) of the scheme Ci ∩ Cj
Ci · Cj = Cj · Ci =
deg(NCi /X )
if i 6= j,
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
2
NZ/X = HomOZ (IZ/X /IZ/X
, OZ )
2
where IZ/X /IZ/X
is the conormal sheaf.)
55
Lemma 20.2.
X
ri Ci · Cj = 0 for all j.
P
∼
=
Proof. OX ( ri Ci ) = OX (Xs ) ∼
= OX , where the isomorphism OX → OX (Xs ) is given by
multiplication by a uniformizer π ∈ R.
Ln
P
Set Λ = i=1 ZCi 3 F =
ri Ci . Define a pairing
D=
X
Λ×Λ→Z
X
X
ai C i , D 0 =
bi Ci 7→ D · D0 =
ai b j C i C j
which is a symmetric bilinear form.
Lemma 20.3. The pairing Λ × Λ → Z is semi-negative definite and if Z ∈ Λ with Z 2 = 0,
then Z ∈ Z( 1r F ).
P
Proof. Say Z =
si Ci . Then
!2
!
X
X
X
Z2 =
si C i
=
si C i
sj C j
i
=
X
i
si C i
i
=
X
si C i
i
=
X si
ri
i6=j
=
X
i<j
−
j
si X
sj C j −
rj Cj
ri j
j
!
X
si rj
sj −
Cj
r
i
j6=i
!
X
(ri sj − rj si )Ci Cj
(ri sj − rj si )2
Ci Cj .
ri rj
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
α
0 → Z → Λ → Λ∗ → G → 0
1
1 7→ F
r
where
α(D) = linear form D0 7→ D · D0 .
This implies
G∼
= Gtors ⊕ Z
with #Gtors < ∞.
Lemma 20.5. There is an exact sequence
α−1 (nΛ∗ )
.
nΛ + ZF
Proof. ker(Pic(X) → Pic(C)) = Λ/ZF . The torsion in this is Z( 1r F )/ZF ∼
= Z/rZ. Hence
(Z/rZ)[n] ∼
= Z/ gcd(r, n)Z.
0 → Z/ gcd(n, r)Z → Pic(X)[n] → Pic(C)[n] →
Suppose LC ∈ Pic(C)[n]. Pick L ∈ Pic(X) with LC ∼
= L|C . Then
X
L⊗n ∼
ai C i
= OC
P
for some
ai Ci ∈ Λ well-defined modulo ZF . Moreover
X
ai Ci · Cj = L⊗n · Cj = n(L · Cj ) ∈ nZ.
P
Hence Pai Ci ∈ α−1 (nΛ∗ )P
is well-defined modulo
P ZF . On the other
P hand wePcan replace
L by L( bi Ci ) for some
bi Ci ∈ Λ. Then
ai Ci changes to
ai Ci + n( bi Ci ). We
α−1 (nΛ∗ )
see that we get a well-defined class in nΛ+ZF which if zero means we can choose L to be
n-torsion.
Lemma 20.6. There is an exact sequence
α−1 (nΛ∗ )
→ Λ/nΛ → Λ∗ /nΛ∗ → G/nG → 0
0→
nΛ
and hence
# Pic(C)[n]
# Pic(X)[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(K sep /K) = I acts
trivially on J[`m ](K sep ) 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
57
P
gi = g.
2g
Example 20.8. Again say ` prime to char(κ(s)) and Pic(C)[`] ∼
= F` . 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 types
of 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
there exists a subgroup H ⊂ G of index dividing c
ρc (G) = min r |
.
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
ρcc0 (G2 ) ≤ ρc (G1 ) + ρc0 (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 an
abstract type of genus g, then
ρc (G) ≤ 1 + β
where
• β is the 1st Betti number of the graph T ,
L
(mij )
• G = coker(Λ → 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 no
subgraphs that are chains of (−2) with the same multiplicity r of length 4 (from last time):
−2
−2
−2
−2
r
r
r
r
◦
◦
◦
◦
OK for these with in fact ρc (G) ≤ 1 for a suitable c.
If the abstract type T does contain such a chain:
−2
−2
−2
−2
C1
r
C2
r
C3
r
C4
r
◦
◦
◦
···
Rest
0
then we remove the middle edge to form T :
◦
···
−2
−1
−1
−2
C1
r
C2
r
C3
r
C4
r
◦
◦
···
◦
Rest
◦
···
Recall that
1X
ki ri .
2
Since k1 = k2 = k3 = k4 = 0 in T but k20 = k30 = −1 in T 0 , we have g 0 < g. If T 0 is connected,
then it is an abstract type of genus g 0 . 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
!
X
mij
M = Im(Λ → Λ∗ ) = Span
mij xj , i 6= 2, 3; x1 − 2x2 + x3 ; x2 − 2x3 + x4
g =1+
j
and
m0ij
!
M 0 = Im(Λ → Λ∗ ) = Span
X
mij xj , i 6= 2, 3; x1 − x2 ; −x3 + x4 .
j
We see that
M 0 ⊂ M + Z(x2 − x3 ),
so there is a surjection
G = Λ∗ /M Λ∗ /M + Z(x2 − x3 ) = G/hx2 − x3 i
with cyclic kernel, and similarly a surjection
G0 = Λ∗ /M 0 G/hx2 − x3 i.
Hence
ρc (G) ≤ ρc (G/hx2 − x3 i) + 1 ≤ ρc (G0 ) + 1 ≤ 1 + (β − 1) + 1 = 1 + β,
where the last inequality follows by induction hypothesis if T 0 is connected and c works for
all 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 not
dividing c(g) and prime to char(κ) where g = g(C). Let K 0 be a finite separable extension
of K such that
Pic(CK 0 )[`] ∼
= (Z/`Z)2g
and
C(K 0 ) 6= ∅.
Replace R, K, C by R0 , K 0 , CK 0 . We will show C has semi-stable reduction.
Pick

/X
C
/
Spec(K)
Spec(R)
a regular flat proper model without (−1)-curves (omitted: can contract (−1)-curves). We
will show this model is semi-stable.
The existence of rational point implies:
• gcd(ri ) = 1 (there is an i with ri = 1).
• dimκ H 1 (Xs , O) = g.
• dimκ H 1 (Xs , O) ≥ dim H 1 ((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), material
of 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
n
n
Y
Y
Y
0,0
∗
∗
∗
0 → Γ(Y, OY ) →
Γ(Ci , OCi ) →
OCi ∩Cj ,x → Pic (Y ) →
Pic0 (Ci ) → 0.
x∈Ci ∩Cj
i6=j
i=1
i=1
Fact (Oort). dimF` Pic0 (Ci )[`] ≤ g(Ci ) + pa (Ci ), with equality if and only if Ci is nodal.
Then
dimF` (OC∗ i ∩Cj ,x )[`] = 1
and equal to dimκ (OCi ∩Cj ,x ) if and only if OCi ∩Cj ,x ∼
= κ.
Putting everything together, we get
X
2g − β ≤ dimF` Pic(Y )[`] ≤
(g(Ci ) + pa (Ci )) + β
which implies
2g ≤
X
(g(Ci ) + pa (Ci )) + 2β.
Now we consider the same long exact sequence as before without ∗’s:
n
n
Y
Y
M
0
1
0 → H (Y, OY ) →
Γ(Ci , OCi ) →
OCi ∩Cj ,x → H (Y, OY ) →
H 1 (Ci , O) → 0.
i=1
x∈Ci ∩Cj
i6=j
i=1
60
This gives
g = dimκ H 1 (Xs , O) ≥ dimκ H 1 (Y, OY ) =
X
pa (Ci ) + β.
Therefore,
2
X
pa (Ci ) + 2β ≤
X
(g(Ci ) + pa (Ci )) + 2β
so
X
pa (Ci ) ≤
X
g(Ci ).
This shows there are no δ-invariants, i.e., all Ci ’s are smooth and we have equality everywhere. This finishes the proof.
22.3. Saito. We have been following Artin–Winters, but there is a paper by Saito which
proves the theorem (same as in Deligne–Mumford): if the inertia acts unipotently, then we
have semi-stable reduction of curves. Saito’s proof uses étale cohomology and vanishing cycle
sheaves.
23. Lecture 23 (December 8, 2015)
23.1. Néron models. We will discuss the Néron–Ogg–Shafarevich criterion. First we need
to talk about Néron models. A reference is Artin’s article in Chapter VIII in Arithmetic
Geometry (edited by Cornell and Silverman).
Let R be a Dedekind domain (dvr) with fraction field K, and A an abelian variety over
K.
Definition 23.1. A Néron model for A is a smooth group scheme G → Spec(R) with
GK ∼
= A and such that the following universal property holds:
If X is smooth over Spec(R) then any rational map X 99K G extends
to a morphism X → G.
An alternative weaker condition is:
If X is smooth over Spec(R) then any morphism XK → A extends to
a morphism X → G.
(*)
(†)
It is clear that (*) implies (†).
Remark. In both cases we have, if R is a dvr, then
sp
A(K sep )I = A(f.f.(Rsh )) = G(Rsh ) → Gs (κsep ),
where I ⊂ Gal(K sep /K) is the inertia with (K sep )I = f.f.(Rsh ), the second equality follows
from (*) or (†), and sp is the specialization map.
Theorem 23.2 (Weil). Finite type Néron models for abelian varieties exist over Dedekind
domains.
Theorem 23.3 (Raynaud). Locally of finite type smooth models with (†) exist for semiabelian varieties. These models are also called Néron models.
Example 23.4. Consider Gm,K with R dvr. Pick π ∈ R uniformizer. Set
[
“G =
π n Gm,R ”
n∈Z
61
(construct by glueing). Then we have
K × = Gm,K (K) = G(R) =
[
π n R× .
n∈Z
Example 23.5. Consider elliptic curves with multiplicative reduction and v(∆) = 1. Assume we have a Weierstrass equation over R with discriminant ∆ = π · unit. Then
singular point of
closed subscheme of P2R defined
G=
\
the special fibre
by the Weierstrass equation
(over Spec(R)) is the Néron 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 exact
sequence
1→L→G→A→1
of group schemes over K, where L is a connected linear algebraic group, and A is an abelian
variety.
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
..

 ⊂ GLn .
.
0

1
Torus T means
T ⊗K K ∼
= G⊕r
m,K
for some r.
Definition 23.8. A group scheme G over a field K is semi-abelian if G is abelian and an
extension of an abelian variety by a torus. A group scheme G over a base S is (semi-)abelian
if G → S is smooth and all fibres are (semi-)abelian.
23.3. Néron–Ogg–Shafarevich criterion.
Definition 23.9. Let R be a dvr with fraction field K, and A/K an abelian variety. We
say A has good reduction if its Néron model is an abelian scheme, or equivalently if there
exists an abelian scheme G → Spec(R) with A ∼
= GK .
Theorem 23.10 (Néron–Ogg–Shafarevich). Let R be a dvr with fraction field K, and A/K
an 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(K sep )[`∞ ] is trivial.
62
“Proof ” of ⇐= . First we show that although
sp
A(K sep )I = A(f.f.(Rsh )) = G(Rsh ) → Gs (κsep )
is not injective, writing f.f.(Rsh ) = K sh we have
A(K sh )[`n ] ,→ Gs (κsep )[`n ].
Note that I acts trivially by assumption, so
A(K sh )[`n ] ∼
= (Z/`Z)2g
where g = dim(A).
Since G → Spec(R) is of finite type, [Gs : G0s ] < ∞ and hence
#G0s (κsep )[`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 = dim U + dim T + dim B = u + t + b. Then
#B(κ)[`n ] = `2bn ,
#T (κ)[`n ] = `tn ,
#U (κ)[`n ] = 1.
Combining everything we get
(
2g ≤ 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 a
dvr with residue field κ.
Theorem 24.1 (SGA7). A has semi-abelian reduction if and only if there exists ` different
from 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 if
and only if C has semi-stable reduction.
63
This is not very hard to show. We have to relate the Picard scheme of the model to the
Néron model of the Jacobian, and then using techniques similar to those in Artin–Winters
we can prove this theorem. We would in particular get the fact that if the inertia acts
unipotently 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 reduction
theorem for curves.
24.2. Proof of Theorem 24.1. We will proof the implication ⇐= following Deligne in
SGA7-I, Exp. I App.
For simplicity assume R complete with finite κ. The Tate module
sep
n
V` (A) = T` A ⊗Z` Q` = lim A(K )[` ] ⊗Z` Q`
n
is a Q` -vector space of dimension 2g endowed with a continuous action of
b → 0.
0 → I → Gal(K sep /K) Galκ = Z
By assumption, the action of I is unipotent.
Q
Let σ ∈ I be a topological generator, and recall the tame inertia It = I/P ∼
= `0 6=char(κ) Z`0 .
b 3 1, i.e.,
Suppose τ ∈ Gal(K sep /K) maps to the arithmetic Frobenius in Galκ = Z
τ στ −1 = σ q
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
O
Q = V` / ker((σ − 1)r )
∼
=
?
/
Im((σ − 1)r ) =: S
r
(σ−1)
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 q r on Q` (r).
Proof.
τ (σ − 1)r = (τ στ −1 − 1)r τ
= (σ q − 1)r τ
= q(σ − 1)r
by looking at



σ=






0 · · · 0 qr
∗
1 q
∗
.
.
..




. 0
1 ..  q  1 .. 
0


q
r
,
σ
=
,
(σ
−
1)
=




.
..
.
.
. . q
.. 0 


. 1
1
1
0
1 1
Definition 24.3. A q-Weil number of weight w is α ∈ Z such that |α0 | = q w/2 for all
conjugates α0 ∈ C of α.
64
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` (G0s ), where G is the Néron model and Gs is the special fibre. G0s has
two parts. On the the torus part, τ acts on V` (Gm,κ ) by the cyclotomic character, so the
eigenvalue of τ −1 is q −1 . On the abelian variety part, we get weight −1 by the Weil conjectures
for 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 gives
0, −1.
Conclusion: r ≤ 1.
0−2r
−1−2r
(V` )I
(V` )I (r)
O
∼
=
Q(r)
/
−2
−1
?
S
If r = 0, Néron–Ogg–Shafarevich says A has
goodreduction.
1 1
If r = 1, the Jordan blocks of σ are 1 or
, and
0 1
1 1
.
2g = #(1 × 1 blocks) + 2 · #
0 1
Looking at weights we see
dim(torus part of
G0s )
1 1
≥#
,
0 1
using V`I = V` (G0s ) and the fact that weight −2 eigenvalues come from the torus part. The
relation V`I = V` (G0s ) also shows
1 1
0
0
2 dim(abelian part of Gs ) + dim(torus part of Gs ) = 2g − #
0 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 H 1 . Scholze proved this for complete intersections using perfectoid spaces.
66