Étale Fundamental Groups

ETALE´ FUNDAMENTAL GROUPS

JOHAN DE JONG

NOTES TAKEN BY PAK-HIN LEE

Abstract. Here are the notes I am taking for Johan de Jong’s ongoing course on ´etale 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) 3 2. Lecture 2 (September 10, 2015) 3 2.1. References 3 2.2. Galois Categories 3 3. Lecture 3 (September 15, 2015) 6 3.1. Reminders 6 3.2. Fundamental Groups of Schemes 8 4. Lecture 4 (September 17, 2015) 9 4.1. Fibre functor 10 5. Lecture 5 (September 22, 2015) 12 5.1. Complex Varieties 12 5.2. Applications 15 6. Lecture 6 (September 24, 2015) 15 6.1. Two more examples 15 6.2. Fundamental groups of normal schemes 16 6.3. Action of Galois groups on fundamental groups 17 7. Lecture 7 (September 29, 2015) 18 7.1. Short exact sequence of fundamental groups 18 8. Lecture 8 (October 1, 2015) 21 8.1. Ramiﬁcation theory 21 8.2. Topological invariance of ´etaletopology 23 9. Lecture 9 (October 6, 2015) 24 9.1. Special case of Theorem last time 24 9.2. Structure of proof 25

Last updated: December 10, 2015. 1 10. Lecture 10 (October 8, 2015) 28 10.1. Purity of branch locus 28 11. Lecture 11 (October 13, 2015) 30 11.1. Local Lefschetz 30 12. Lecture 12 (October 15, 2015) 34 12.1. Specialization of the fundamental group 34 13. Lecture 13 (October 20, 2015) 36 13.1. Abhyankar’s lemma 36 13.2. End of proof of Theorem last time 38 13.3. Applications 39 14. Lecture 14 (October 22, 2015) 39 14.1. Quasi-unipotent monodromy over C 39 14.2. Etale´ cohomology version 40 15. Lecture 15 (October 27, 2015) 41 16. Lecture 16 (October 29, 2015) 44 16.1. Outline of proof of quasi-unipotent monodromy theorem 46 17. Lecture 17 (November 5, 2015) 46 17.1. Proof in the mixed characteristic case 47 17.2. Birational invariance of π1 48 18. Lecture 18 (November 17, 2015) 49 18.1. Semi-stable reduction theorem 49 19. Lecture 19 (November 19, 2015) 51 19.1. Resolution of singularities 51 20. Lecture 20 (November 24, 2015) 55 20.1. Method of Artin–Winters 55 20.2. Example applications 57 20.3. Idea of Artin–Winters 58 21. Lecture 21 (December 1, 2015) 58 22. Lecture 22 (December 3, 2015) 58 22.1. Abstract types of genus g 58 22.2. Part 2 of Artin–Winters’ argument 60 22.3. Saito 61 23. Lecture 23 (December 8, 2015) 61 23.1. Néronmodels 61 23.2. Group schemes over fields 62 23.3. Néron–Ogg–Shafarevich criterion 62 24. Lecture 24 (December 10, 2015) 63 24.1. Semi-abelian reduction 63 24.2. Proof of Theorem 24.1 64 24.3. Weight monodromy conjecture 66

2 1. Lecture 1 (September 8, 2015) PH: I missed the lecture.

2. Lecture 2 (September 10, 2015) 2.1. References. • Stacks project, chapter on fundamental groups (tag 0BQ6). • Lenstra, Galois Theory for Schemes. • SGA I. • Murre, Lectures on an introduction to Grothendieck’s theory of the fundamental group. 2.2. Galois Categories. The idea is to consider Topological groups ↔ Categories C with F : C → Sets Those categories which go back to themselves after we go around will be the Galois categories. Notation. Let G be a topological group. Denote by G-Sets the category with objects (X, a) where X is a set (with the discrete topology) and a : G × X → X a continuous action. Because the topology is discrete, this means the stabilizer of any point is open. 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 G/U. UCG open, finite index 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 explain the proof. A basis of the topology on Aut(F ) is Qn the kernels of the maps Aut(F ) → i=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 G∧ / Aut(F )

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 gHig 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 g 0 / Y / Y γY

g0 / X / X γX Think!

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

UCG 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(g0U) = g0gU. 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; 4 (3) F has finite values; (4) F is exact and reflects isomorphisms; where • X is connected ⇔ X is not initial and any subobject (Y → X monomorphism) 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. n ` Proof. Say F (X) = {x1, ··· , xn}. Sn acts on X = t∈T Zt. Then Sn acts in the same n n ` way on F (X) = F (X ) = 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)} 0 0 (argument omitted). But F (σ)(ξ) = σ(ξ) = (xσ(1), ··· , xσ(n)), so we win if (x1, ··· , xn) ∈ 0 0 0 F (Zt) implies xi pairwise distinct. If not, say xi = xj, 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 Xn corresponds to the n-fold tensor product). Also applies to n-fold Galois covering of connected and locally connected spaces. Lemma 2.9. Let (C,F ) be a Galois category. The action of Aut(F ) on F (X) is transitive for all X ∈ Ob(C) connected. Idea of proof. We need to introduce some notations. Let I be the set of isomorphism classes 0 of Galois objects. For i ∈ I, let Xi be a representative. Pick xi ∈ F (Xi). We say i ≥ i if there exists a morphism Xi → Xi0 . We may pick fii0 : Xi → Xi0 such that F (fii0 )(xi) = xi0 (⇒ fii0 unique) (because Galois objects). 0 Claim. F is isomorphic to the functor F : X 7→ colimi∈I MorC(Xi,X). If this is true, then just as in the case of Galois theory, we set

H = lim Aut(Xi) i∈I opp 0 and get H → Aut(F ) = Aut(F ). 5 3. Lecture 3 (September 15, 2015) 3.1. Reminders. Galois category is a pair (C,F ) where •C has all finite limits and colimits; • F : C → Sets is an exact functor; • F (X) is finite for all X ∈ Ob(C); • F reflects isomorphisms; ∼ ` • for all X ∈ Ob(C), X = i=1,··· ,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 T has group Sn over T . For the n-to-1 covering T 0 over T , T˜ 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. 0 • i ≥ i ⇔ there exists Xi → Xi0 . 0 • Pick γi ∈ F (Xi), for i ≥ i 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 0 F (X) = colimI MorC(Xi,X). • αii0 (ai) is the unique map that makes

ai Xi / Xi

αii0 (ai) Xi0 / Xi0 commute. Details... Details... 6 • F 0 → F is the map

0 fi 0 (if f ∈ F (X) is given by Xi → X) ∈ F (X) 7→ F (fi)(γi) ∈ F (X). We have two more statements about general Galois categories. Proposition 3.4. Suppose (C,F ) is a Galois category. Then the functor F : C → Finite- Aut(F )-Sets is an equivalence. Proof. (1) F is faithful: This is the ﬁrst result on Galois categories we proved. (2) F is fully faithful: Let X,Y ∈ Ob(C) and s : F (X) → F (Y ) commuting with Aut(F )-action. Then the graph Γs ⊂ F (X) × F (Y ) = F (X × Y ) is a union of orbits. This implies by the lemma (on action Aut(F ) transitive on F (connected)) that 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 ? Y

−1 pr | (pr1 |Z ) 2 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 H0 ↔ (H/U)opp. 0 ! h1 / 0 0 0 (e) Let X = Coeq Y / Y , where H = {h1, ··· , hr}. 0 hr (f) Since F commutes with colimits,

F (h0 ) ! 1 / F (X) = Coeq F (Y ) / F (Y ) 0 F (hr)   Rh0 1 / ∼= Coeq  Aut(F )/U / Aut(F )/U  R 0 hr ∼= Aut(F )/H 7 where H = h1U t · · · t hrU. Addendum: Let (C,F ) and (C0,F 0) be two Galois categories, and H : C → C0 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) C H / C0

=∼ =∼ Finite-G-Sets h / Finite-G0-Sets is 2-commutative (via t). 3.2. Fundamental Groups of Schemes. Let f : X → Y be a morphism of schemes. TFAE: (1) f is 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 o T b _ 0 first order thickening ∃! YTo affine (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 étaleif and only if there exists a set I and X ∼= ` i∈I Spec(ki) with ki/k finite separable. Formal properties: (A) Being étaleis preserved under base change:

0 Y ×Y X / X

f Y 0 / Y (B) Being ´etaleis preserved under composition. (C) f X / Y

g h=g◦f Z g, h étale ⇒ f étale. 8 ∼ (D) A → B étalering map ⇔ B = A[x1, ··· , xn]/(f1, ··· , fn) such that ∂f ∆ = det i ∂xj is invertible. (There is a structure theorem for étalealgebras, which is a stronger statement.) We will be talking about finite étalemorphisms. 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 −1V is affine and O(V ) → O(U) is a finite ring map (module finite).) −1 (2) For all V ⊂ Y affine open, f 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(b1b2) is nondegenerate. ∼ ⊕n If B = A is free and multiplication by b1b2 has matrix (aij(b1b2)), then TrB/A(b1b2) 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 ´ Fx FEtX = {Y → X finite étale} → Sets

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

4. Lecture 4 (September 17, 2015) Today we will talk about étalefundamental groups. ´ Notation. Let X be a scheme. FEtX is the category with f • objects: Y → X finite étale, • morphisms: f f 0 Mor (Y → X,Y 0 → X) = Mor (Y,Y 0) = {g : Y → Y 0 | f 0 ◦ g = f}. FEt´ X X Example 4.1. (1) Let X = Spec(k) where k is a field. Then ´ opp FEtX = {finite separable k-algebras} . (2) Let X = Spec(A). Then ´ FEtX = {A → B which are finite locally free opp and Q : B × B → A, (b1, b2) 7→ TrB/A(b1b2) is nondegenerate = {separable A-algebras}opp. ´ FEtX will be a Galois category only when X is connected. In general, ´ Lemma 4.2. FEtX has all finite limits and finite colimits and for a morphism of schemes 0 ´ ´ 0 0 X → X, then base change functor FEtX → FEtX0 , Y/X 7→ X ×X Y/X is an exact functor. 9 Proof. • Limits: Check there exists final object and fibre products. – Final object: id : X → X. – Fibre products: FEt´ 3 Y Y Y × Y / X X 1 3 1 Y2 3 ?

~ finite étaleas finite étale b.c. of Y1 → X $ Y2 Y3

× X • Colimits: enough to check ﬁnite coproducts and coequalizers – Coproducts: disjoint unions a / ´ – Coequalizers: Y1 / Y2 in FEtX . b Note Y = Spec (f O ). i X i,∗ Yi

a# o o f1,∗OY1 o f2,∗OY2 A b#

where A is a quasicoherent sheaf of OX -algebras, and the last map is the equalizer of a# and b#. Claim. Spec (A) → X is in FEt´ and is coequalizer of a and b. X X Proof of claim (please find your own). (*): Etale´ locally on X you can write Y = ` X. i j=1,··· ,mi (**): Y = ` X, then any morphism Y → Y 0 over X Zariski locally on i j=1,··· ,mi 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) étalesuch 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 étalebase change. (****): Explicitly compute what happens when Y and Y 0 are map are as in (**). 4.1. Fibre functor. Definition 4.3. A geometric point x of a scheme is a morphism Spec(K) →x X whre k is an lagebraically closed field. 10 Abusive notation: k = κ(x), x = Spec(κ(x)), x = Im(x). Base change gives ´ ´ ∼ Fx :FEtX → FEtx = category of finite sets → Sets

Y/X 7→ Fx(Y ) = Yx where Yx is the set of y such that ? Y y

x X commutes. ´ Theorem 4.4. If X is a connected scheme, then (FEtX ,Fx) is a Galois category! Warning. Not true if X is not connected. Deﬁnition 4.5 (Grothendieck’s Fundamental Group). For X connected,

π1(X, x) := Aut(Fx). Proof. Already know • There exist finite (co)limits. • Fx exact. • Decomposition into connected components – Fact: A finite morphism is closed (hence proper and integral). – Fact: An étalemorphism 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 FEtX are open immersions.) g 0 ´ • Lastly: Fx reflects isomorphisms. Suppose Y → Y in FEtX 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. 0 0 – degree of g is 1 by looking at the fibre over some y ∈ Fx(Y ). – degree 1 implies g isomorphism. Lemma 4.6. Let f : X0 → X be a morphism of connected schemes. Let x0 be a geometric point of X0. Set x = f(x0). Then we get a canonical continuous homomorphism of profinite groups 0 0 f∗ : π1(X , x ) → π1(X, x) such that ´ base change ´ FEtX0 o FEtX

∼ ∼ Fx0 = = Fx 0 0 f∗ Finite-π1(X , x )-Setso Finite-π1(X, x)-Sets 11 2-commutes. ∼ 0 Proof. Fx = Fx0 ◦ (base change) because x = f ◦ x .

Lemma 4.7 (Change of base point). If x1, x2 are geometric points of connected X, then ∼ π1(X, x1) = π1(X, x2) well-deﬁned up to inner automorphism. ∼ Proof. Choose isomorphism Fx1 = Fx2 . Example 4.8. Fix k ⊂ ksep ⊂ k. Then sep π1(Spec(k), Spec(k)) = Gal(k /k). Proof. The functor ´ sep FEtX → Finite- Gal(k /k)-Sets sep Y 7→ MorSpec(k)(Spec(k),Y ) = MorSpec(k)(Spec(k ),Y ) sep (the right hand side has a left action of Gal(k /k)) is an equivalence.

Example 4.9. π1(Spec(C)) = {1}.

Example 4.10. π1(Spec(R)) = Z/2Z. ˆ Example 4.11. π1(Spec(C((t)))) = Z. Example 4.12. π ( 1 ) = {1}. 1 PC ´ 1 1 Proof. Every object in FEt 1 is isomorphic to a disjoint union of copies of P . Let f : Y → P PC C C be ﬁnite ´etale,with Y connected. Have to show Y ∼ 1 . = PC By Hurwitz, 2gY − 2 = (2g 1 − 2) deg(f) + deg(R) = −2 deg(f). PC Since gY ≥ 0, this implies deg(f) = 1. 5. Lecture 5 (September 22, 2015) 5.1. Complex Varieties.

Notation. Given a scheme X that is lft (locally of ﬁnite type) over C we denote Xan 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) ⊂ n , AC n X(C) = {(a1, ··· , an) ∈ C | fi(a1, ··· , an) = 0 for i = 1, ··· , t}. Properties: • If f : X → Y is a morphism of schemes lft over C, then f an : Xan → Y an is continuous. • If f is an open/closed immersion, then f an is an open/closed immersion. • If X = n , then Xan = n with Euclidean topology. AC C Question/Remark: The properties of topology on C we need to do this construction are: 12 (1) Multivariate polynomials give continuous maps. (This is implied by the fact that 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 : Xan → Y an is proper (⇔ f closed and fibres are compact). Proof idea. • Reduce to projective case by Chow’s lemma: given X → Y proper, there exists a 0 0 projective surjection X X such that the composition X → Y is projective. n • Reduce to PY → Y by second axiom. n an n • (P ) = P (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 ⊂ Xan and f(x) ∈ open V ⊂ Y an such that open an f |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 = n ; to do this pick AC X / W affine

´etale Y / n AC

such that X = Y × n W . AC Lemma 5.3. Let A be a ring, I ⊂ A be an ideal and A/I → B ´etale. Then there exists A → B ´etalesuch that B ∼= B/IB as A/I-algebras.

∂fi Proof. Write B = A/I[x1, ··· , xn]/(f1, ··· , fn) with ∆ = det( ) invertible. Set ∂xj −1 ∂fi B = A[x1, ··· , xn]/(f1, ··· , fn) det . ∂xj Then W → n is smooth of relative dimension 0 so we can write AC W = Spec C[x1, ··· , xn, z1, ··· , zm]/(F1(z, x), ··· ,Fm(z, x))

∂Fj and det( )j,k=1,··· ,m invertible on W . ∂zk Implicit function theorem says an n (W , w) → (C , f(w)) is a local isomorphism. Second Proof. Use 13 Theorem 5.4 (Structure Theorem). Let A → B be a ring map étaleat 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 dP where P,Q ∈ A[z], P is monic and dz is invertible in (A[z]/(P ))Q. n n−1 Suppose P (z) = z + a1z + ··· + an ∈ C[z] and α is a simple root. Then there exists > 0 such that for |bi| < , i = 1, ··· , n, the polynomial n n−1 Pb1,··· ,bn (z) = z + (a1 + b1)z + ··· + (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 étalemap. Then we can pick Py(z) ∈ OY (Y )[z]. Corollary 5.5. Given X lft over C, there is a functor ´ an FEtX → {finite covering spaces of X } f f an (Y → X) 7→ (Y an → Xan). 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 Xan is connected, and for x ∈ X(C), there is an isomorphism top an ∧ =∼ π1 (X , 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 1 an an × exp : (AC) = C → Gm,C = C is not the analytification of any f : 1 → . AC Gm,C Sketch of proof for X smooth and projective. Claim. For any X/C smooth there exists a unique structure of a complex manifold on Xan such that • If f : X → Y is a morphism with X,Y smooth, then f an : Xan → Y an is holomorphic. • If f is étale,then f an is a local isomorphism of complex manifolds. • If X = n , then get usual complex structure on n. AC C Characterization of smooth morphism: Let f : X → S be a morphism of schemes. TFAE: (1) f is smooth. (2) For all x ∈ X, there exists x ∈ U such that X o ? _U

f étale o n S AS commutes. 14 Let X/C be a projective and smooth variety. To show ´ =∼ an FEtX → finite topological covers of X , let π : M → Xan 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 Xan coming from X,→ n , so Xan ,→ n( ) and O(r) PC P C 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 H0(M,L⊗k) gives M,→ Pn(C). By Chow’s theorem, M,→ Pn(C) is cut out by polynomials equations, which gives us Y .) The problem left over: Y an = M →π Xan is equal to f an for some f : Y → X. We can an either do Chow’s theorem for the graph Γ ,→ M × X , or use GAGA. Remark. More generally, if X and Y are projective smooth over C, any holomorphic map Xan → Y an is f an for some f : X → Y algebraic. 5.2. Applications. Recall the profinite completion ˆ Y = lim /n = `. Z n≥1 Z Z Z ` prime • π ( n ) = {1}. 1 PC • π ( n ) = {1}. 1 AC (ˆ n Z if n = 1, n 1 2n−1 • π1(A \{0}) = (A \{0} is homeomorphic to S and S in these two C 0 if n > 1. C cases respectively.) • π ( 1 \{0, 1, ∞}) = profinite completion of free group on 2 generators. This is some- 1 PC thing we don’t know how to prove without using topology!

6. Lecture 6 (September 24, 2015) 6.1. Two more examples. • If X is a smooth projective genus g curve over C, then ∼ π1(X) = proﬁnite completion of the free group on α1, ··· , αg, β1, ··· , βg

subject to [α1, β1][α2, β2] ··· [αg, βg] = 1.

• If A/C is an abelian variety of dimension g, then ∼ ˆ ⊕2g π1(A) = Z . 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 sep Gal(K /K) = π1(Spec(K), Spec(K)) → π1(X, Spec(K))

is surjective and this identifies π1(X, Spec(K)) with the quotient Gal(Ksep/K) → Gal(M/K), 16 where M is the compositum of all finite K ⊂ L ⊂ Ksep such that X is unramified in L, and M/K is Galois. The key ingredient to the proof is: Lemma 6.8. If Y → X is finite étaleand 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. π ( 1 ) = {1}. We saw this last time using topology ( is a contractible 1 AC C 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.

1 1 1 Example 6.11. π1( ) 6= {1}. A map → given by x 7→ f(x) is unramiﬁed if AFp AFp AFp f 0(x) 6= 0 for all x. An example is f(x) = xp − x. Look up Artin–Schreier coverings. In fact, 1 ∼ 1 p Hom π1( ), /p = Γ( , O)/(subgroup of f − f). AFp Z Z AFp 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(ksep/k). The middle term is sometimes called the arithmetic fundamental group. Example 6.13. X = 1 \{0, 1, ∞}. Then PQ ˆ 1 → F2 → π1(X) → Gal(Q/Q) → 1,

where F2 is the free group on 2 generators. Here we have used: ∼ Fact. π ( 1 \{0, 1, ∞}) →= π ( \{0, 1, ∞}). 1 PC 1 PQ The exact sequence gives a continuous group homomorphism Aut(Fˆ ) Gal( / ) ,→ Out(Fˆ ) = 2 . Q Q 2 ˆ Inn(F2) The method is to prove corresponding results for the functors FEt´ ← FEt´ ← FEt´ . Xk X Spec(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 X

Spec(k) / 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

0 C →H C0 →H C00, which give rise to the 2-commutative diagram

0 C H / C0 H / C00

F =∼ F 0 =∼ F 00 =∼ 0 Finite-G-Sets h / Finite-G0-Sets h / Finite-G00-Sets We want to study when the sequence of proﬁnite groups

0 G ←h G0 ←h G00 is exact. Properties of these two diagrams are related as follows: 0 H H0 G ←h G0 ←h G00 C → C0 → C00 (A) H fully faithful ⇔ X ∈ C connected and if there h surjective exists ∗C0 → H(X), then X = ∗C. 00 00 0 (B) For all X ∈ C connected, there exists h injective epi H0(X0) mono← Y 00 → X00. 0 0 ∼ ` h ◦ h trivial (C) H (H(X)) = j=1,··· ,m ∗C00 for all X ∈ C. 0 0 (D) For all X ∈ C connected, if there exists Im(h ) normal 0 0 0 0 ∼ ` ∗C00 → H (X ) then H (X ) = j=1,··· ,m ∗C00 . h surjective + kernel h is smallest normal (E) H fully faithful + essential image of H is exactly 0 0 0 ∼ ` closed subgroup those X such that H (X ) = j=1,··· ,m ∗C00 . containing Im(h0). We can prove that: 0 0 ∼ ` 0 Fact. (E) ⇔ (A) + (C) + (H (X ) = ∗C00 ⇒ there exists X ∈ C and an epi H(X) X ). Now apply this to 0 FEt´ →H FEt´ →H FEt´ Spec(k) X Xk with X/k quasicompact, quasiseparated and geometrically connected (for example, a geometrically connected k-variety) and k perfect. 18 0 0 (A) Have to show: for k /k ﬁnite separable ﬁeld extension we have Spec(k ) ×Spec(k) X = Xk0 is connected. This is clear because Xk Xk0 and Xk is connected by assumption. (B) First we have the following

0 Claim. For every Z → Xk finite étalethere exists a k/k /k finite and Y → Xk0 finite ∼ 0 étalesuch that Z = Y ×Xk0 Xk = Y ⊗k k.

If the claim is true, then the composition Y → Xk0 → X is ﬁnite ´etaleand

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

0 The multiplication map k ⊗k k → k gives a connected component 0 Spec(k) ,→ Spec(k ) ×Spec(k) Spec(k), and Z is the ﬁbre product

open+ closed Z / Y ×Spec(k) Spec(k)

open+ closed 0 Spec(k) / Spec(k ) ×Spec(k) Spec(k)

i.e., Z = Y ×Spec(k0) Spec(k). 0 About the claim: k = colimk/k0/k k , so Spec(k) = lim Spec(k0) 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 ´ ´ FEtX = colim FEtXi , 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 A -algebras = colim i . 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 X

Spec(k) / Spec(k)

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

[ σ T = s (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 = U. ∼ ` 0 (E) Suppose U → X is finite étaleand Uk = i=1,··· ,n Xk. Then there exists k/k /k (with k0/k finite) such that ∼ a Uk0 = Xk0 . i=1,··· ,n Then ! ∼ a a 0 U ← Uk0 = Xk0 = X ×Spec(k) Spec k i=1,··· ,n i=1,··· ,n ´ where the last coproduct is in FEtk. Remark. We’ve proved the exact sequence

sep 1 → π1(Xksep ) → π1(X) → Gal(k /k) → 1 for X/k as before and k not necessarily perfect.

Fact. π1(Xk) = π1(Xksep ). 1 Example 7.2. Let X = Gm,k = Ak\{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 FEt´ . This implies Xk ∼ ˆ π1(X ) = lim µ(k) = (1). k n Z 20 So we have / / / / 0 π1(Xk) π1(X) π1(Spec(k)) 1

=∼ =∼ =∼ 0 / Zˆ / ? / Gal(k/k) / 1 and get × Gal(k/k) → Out(Z) = Zˆ . 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 ∼ ˆ ⊕2 π1(Xk) = Z . Again multiplying by n gives a coﬁnal system in FEt´ . The exact sequence is Xk

OX t ˆ ⊕2 0 / Z / π1(X) / Gal(k/k) / 1. Then ˆ ⊕2 Y Gal(k/k) → Aut(Z ) = GL2(Z`). `

8. Lecture 8 (October 1, 2015) Today we will talk about ramification theory and topological invariance of the étaletopol- ogy. 8.1. Ramification theory. A will be a discrete valuation ring with fraction field K = f.f.(A). Let L/K be finite separable, and B the integral closure of A in L. By a 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]. n Y Fact. [L : K] = eifi. 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(Ksep/K) by passing to the limit. (Write Ksep = S L where L runs over Ksep/L/K Galois, then for each L we get PL ⊂ IL ⊂ DL = Gal(L/K) and then sep P = lim PL → I = lim IL → Gal(K /K) = lim Gal(L/K) = lim DL.) Both P and I are normal in G and ∼ sep (1) G/I = Gal(κA /κA). ∼ Q (2) It = I/P = `6=char(κ ) Z`. (*) A (3) The short exact sequence sep 1 → It → G/P → Gal(κA /κA) → 1 sep gives an action of Gal(κA /κA) on It which is via the cyclotomic character. (4) P = {1} if char(κA) = 0. The isomorphism (*) is noncanonical. Canonically, sep It = lim µe(κ ). e A ∼ We will soon explain why IL/PL = µeL (κL) at the finite level. 1/e Example 8.3. Let A = κ[[t]] with κ = κ of characteristic 0. The extensions are Le = K(t ) 1/e 1/e 1/e (e ≥ 1) with Gal(Le/K) = µe(κ) via (t 7→ ζt ) ↔ ζ. (Note A = κ[[t]] ⊂ Be = κ[[t ]].) 1/e 1/e Given σ ∈ Gal(Le/K), the corresponding root of unity is the ζσ such that σ(t ) = ζσt . 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 2 σ(π) = θg(σ) · π mod π Bm

where π ∈ Bm is a uniformizer and θg(σ) ∈ Bm is any lift of θ(σ) ∈ κ(m). The reason this 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}. Apply with A = O (completion) or A = Oh (henselization). We get i bX,xi i X,xi

ai : Spec(f.f.(Ai)) → X. Hence ˆ ∼ π1(ai) Z = π1(Spec(f.f.(Ai))) → π1(X), well-defined up to inner conjugation on the right hand side, measures the ramification of finite étalecover 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 ˆ ⊕m ab ab Z → π1(X) → π1(X) → 0. 1 Example 8.7. Let X = Ak and X = Gm,k, so m = 1 and x1 = 0. Suppose you knew the 1 lemma and π(Ak) = {1}. Then π1(X) is topologically generated by 1 conjugacy class. This is rather weak. ∼ ˆ If we use that π1(X) = Z (see earlier), then =∼ =∼ inertia at 0 → π1(X) ← inertial at ∞. The isomorphism between the two inertia is via inverse. This is compatible with the 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 étaletopology. Theorem 8.8. Let f : X → Y be a universal homeomorphism of schemes. Then the base change functor schemes étaleover Y → schemes étaleover X ´ ´ =∼ is an equivalence. Same for FEtY → FEtX and if X and Y are connected then π1(X) → π1(Y ). 23 0 0 0 Definition 8.9. f : X → Y is a universal homeomorphism if for all Y → Y , X ×Y Y → Y is a homeomorphism. Fact. f is a universal homeomorphism if and only if f is integral, surjective and universally injective. Example 8.10. (1) Thickenings: i : X → X0 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: • I2 = 0 (square zero ideals), or • In = 0 for some n (nilpotent ideal), or • for all x ∈ I, there exists n = n(x) such that xn = 0 (locally nilpotent ideal). Etale´ cohomology doesn’t see multiplicities. (2) Purely inseparable field extensions. (3) Suppose f : Spec(A) → Spec(B) where A, B are Fp-algebras, and there exists a g : Spec(B) → Spec(A) such that f ◦ g = Spec(B → B, b 7→ bq), g ◦ f = Spec(A → A, a 7→ aq), where q = pn. Here you get the theorem because base change by g will be the quasi-inverse functor.

39 Example 8.12. Let A = Fp[x1, ··· , xn]/I and B = Fp[x1, ··· , xn]/I . Clearly there pm is a surjection B → A. To get a map A → B, we send xi 7→ xi 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 ´ ´ FEtX → FEtX0

Y 7→ Y0 = X0 ×X Y is an equivalence.

=∼ Remark. Even Xét → X0,ét (small étalesites). We often call this the topological equivalence of étaletopology. For example, we see that the étalesite 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 ´ ´ FEtX → FEtX0 24 Y 7→ Y0 = X0 ×X Y is an equivalence.

This is an amazing theorem! I will spend some time discussing what this theorem means, how we will use it, and ﬁnally sketch the proof. If we want to prove this theorem 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 étaleis given and we try to construct Y → X finite étalewith 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

n+1 Xn = Spec(A/m ) ×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 ´etalescheme 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 _ _ _ ··· o ? Z2 o ? Z1 o ? Z0

h2 h1 h0 _ _ _ ··· o ? X2 o ? X1 o ? 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 F coherent O -modules and isomorphisms F ⊗ n Xn n+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 étaleby:

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, −1 resp. étalefor all n 0, then f is smooth, resp. étaleat all points of f (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 h A = colim B = colim Bq (A→B,q) (A→B,q) where the colimit is over the category of A → B étaleand 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/In 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 N ∧ for j = 1, ··· , m, and ai ≡ bi mod I 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 = Ch where C is a Noetherian local ring essentially of finite type over Z (more generally if approximation holds for A → A∧). 0 ∧ Proof. By deformation and algebraization, we have Y → 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 ´etalewhose 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 Y 0

Spec(Ai) o Yi Y O 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 0 of Y , so equal to our given Y0. 9.2.5. Limits. Any A henselian is a filtered colimit of A’s as above. 27 10. Lecture 10 (October 8, 2015)

10.1. Purity of branch locus. Suppose X is a curve over Zp, with special fibre X0 (which is a curve in characteristic p), generic fibre Xη over Qp, and geometric fibre Xη over Qp. We want to analyze how close π1(Xη) is to π1(X0). This will have to do with the purity 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). 0 0 (3) x is closed in Xy and no x x (specialization) in Xy except x = x. (4) For some (any) affine opens Spec(A) / X

f Spec(B) / Y and x ∈ Spec(A) corresponding to p ⊂ A, the ring map B → A is quasifinite at p. Definition 10.2. • f : X → Y is locally quasifinite (lqf) if f is quasifinite at all x. • f : X → Y is quasifinite if f is locally quasifinite and quasicompact. Let me state the first version of purity of branch locus, which is really about the 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, 0 0 (5) for all x x with dim(OX,x0 ) = 1 we have f unramified at x . Then f is étaleat x. The flatness assumption is what makes the proof easy. Now we will translate this into algebra, which requires lots of technical garbage: • Etale´ locally a quasifinite morphism becomes finite. • Zariski’s main theorem. Lemma 10.4 (Algebra problem). Let A → B be a finite flat ring map, with A a Noetherian local ring. If • dim(A) ≥ 2, • Spec(B) → Spec(A) is étaleover 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 ∼ ⊕n Proof. Flatness implies B = A for some n, so we can define the trace pairing QB/A : B × B → A by (b1, b2) 7→ TrB/A(b1b2) (trace of the matrix of multiplication by b1b2 on B). Let

discB/A = det(the matrix of QB/A with respect to some basis). We know that the branch locus := {p ∈ Spec(A) | Spec(B) → Spec(A) not ´etaleat 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, 0 0 (6) for all x x with dim(OX,x) = 1, f is unramified at x . Then f is étaleat 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 étaleover 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 = mAB. • Case II: dim(A) ≥ 3. To prove this Grothendieck uses a local Lefschetz theorem. 2 Pick f ∈ mA\mA. Contemplate _ Spec(A) = XXo ? 0 = Spec(A/fA) O O

? _ ? Spec(A)\{m} = U o ? U0 = Spec(A/fA)\{m} 29 Note A/fA is regular local of dimension 1 less, so by induction we have the result ´ ´ for A/fA. Reformulated, then result is the statement that FEtX0 → FEtU0 is an equivalence. If A is henselian, then we have ´ ∼ ´ ∼ ´ FEtX = FEtX0 = FEtSpec(κA). So we’re done if we can show ´ ´ FEtX → FEtU0 is an equivalence when A is henselian or something. Theorem 10.7. 1 Let (A, m) be a local ring and f ∈ m a nonzerodivisor. Let X = Spec(A) and U the punctured spectrum of A. Let X0 = Spec(A/fA) and U0 the punctured spectrum of A/fA. Let V be finite étaleover U. Assume (1) f is a nonzerodivisor, 1 (2) Hm(A) is a finite A-module, 2 (3) a power of f annihilates Hm(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. i depth(A) ≥ t ⇔ Hm(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, y2] ⊂ B = C[x, y]. This looks like the plane mapping to the cone via quotienting out by {±1}. Then B is not finite étaleover 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/fA)\{m} _ _

_ Spec(A) = XXo ? 0 = Spec(A/fA)

1PH: 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 ´etaleschemes 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 étalecover 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.

1 N 2 Theorem 11.1. Assume A is henselian, Hm(A) is ﬁnite, and f kills Hm(A) for some N. Then ´ ´ FEtU → FEtU0 is fully faithful, i.e., π1(U0) π1(U) if U and U0 are connected. In the situation of Theorem 11.1, if A is strictly henselian and if local purity holds for A/fA, 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 ´ ´ FEtU → FEtU0

V 7→ V0 = V ×U U0 is fully faithful. n+1 Set Vn = V ×U Un where Un = V (f ) ⊂ U.

_ _ _ _ V o ? ··· o ? V2 o ? V1 o ? V0

U o ? _··· o ? _U o ? _U o ? _U _ 2 _ 1 _ 0 _ open Spec(A) o ? _··· o ? _Spec(A/f 3A) o ? _Spec(A/f 2A) o ? _Spec(A/fA)

´ bij. Lemma 11.2. The functor is fully faithful if and only if for all V ∈ FEtU , 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) / subsets of π0(V1) / subsets of π0(V0)

bij. bij. bij. bij. set of idempotents set of idempotents ∼ set of idempotents ∼ set of idempotents / ··· / = / = / of Γ(V, OV ) of Γ(V2, OV2 ) of Γ(V1, OV1 ) 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 Hp and short exact sequences 1 p−1 p p 0 → R lim H (OVn ) → H → lim H (OVn ) → 0 and 0 p ∧ p p+1 0 → H (H (OV ) ) → H → Tf H (OV ) → 0, where P ∧ p • H (OV ) is the derived completion of H (OV ) with respect to F , and for any A- module M one has a short exact sequence, 1 n 0 ∧ 0 → R lim M[f ] → H (M ) Mc → 0 f n where Mc = lim M/f nM is the usual completion, and M ∧ = R lim(M → M) (at degrees −1 and 0 respectively) is the derived completion. • T (M) = lim M[f n] is the f-Tate module of M: f ←−n ×f · · · → M[f 2] → M[f] = {x ∈ M | fx = 0}. We will take p = 0: 0 0 • H = lim H (OVn ) by the ﬁrst exact sequence. 0 • If H (OV ) has bounded f-power torsion (e.g. when V is Noetherian), then

0 0 ∧ 0 H (H (OV ) ) = H\(OV ). We get 0 0 1 0 → H \(V, OV ) → lim H (OVn ) → Tf H (OV ) → 0. ´ ´ Corollary 11.4. To show FEtU → FEtU0 is fully faithful it suffices to show for all V → U finite étalethat: 0 (1) H (OV ) is f-adically complete. 1 (2) f-power torsion on H (OV ) is bounded. 32 1 0 Proof of (1) when A is complete. Hm(A) finite and V → U finite étaleimply that H (V, OV ) is finite over A. This uses finiteness theorems in coherent cohomology. Example 11.5. 0 • If dim A = 1, then H (U, OU ) is never finite: pick x ∈ m nonzerodivisor, and then 1 0 xn ∈ H (U, OU ) for all n ≥ 1. 0 • If A is a normal domain of dim ≥ 2, then H (U, OU ) = A is finite. We have 1 Hm(A) = 0. In general, there is an exact sequence

0 0 1 0 → Hm(A) → A → H (U, OU ) → Hm(A) → 0 and i =∼ i+1 H (U, OU ) → Hm (A) if i ≥ 1. (Look in Hartshorne for

0 0 0 1 0 → HZ (X, F) → H (X, F) → H (U, F) → HZ (X, F) → · · · .) Proof of (2). We have to show π : V → U ´etaleﬁnite implies

1 1 1 H (V, OV ) = H (U, π∗OV ) = H (U, some vector bundle on U)

has bounded f-torsion. This follows from the next lemma. 1 ∼ 2 N Lemma 11.6. If H (U, OU ) = Hm(A) is annihilated by f for some N, then for any ﬁnite locally free sheaf E on U there is an M = M(E) such that f M annihilates H1(U, E). Proof. For all u ∈ U there exists u ∈ U 0 ⊂ U such that open

=∼ ⊕r E|U 0 ← O |U 0 . ϕ U We can pick U 0 = D(g) for some g ∈ A. ⊕r N1 There exists a map α : OU → E such that α|U 0 = g ϕ. Dually, there exists a map ⊕r N2 −1 β : E → OU such that β|U 0 = g ϕ . Thus the composition

β ⊕r α E → OU → E

N1+N2 is multiplication by g (maybe after increasing N1 and N2). Now for all u ∈ U, there exists g ∈ A such that u ∈ D(g) and gN1+N2 f N ∈ Ann(H1(U, E)). The ideal generated by all these gN1+N2 is m-primary, so mN101 · f N ⊂ Ann H1(U, E) and hence N +N 1 f 101 ∈ Ann H (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 ⊂ k0 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 i 0 i 0 H (X, OX ) ⊗R R = H (XR , OXR0 ). To prove connectedness, we use 0 0 0 0 H (X, OX ) ⊗k k = H (Xk , OXk0 ). 0 Note H (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 k0. 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: 2 1 1 π1( ) 6= π1( ) × π1( ). AFp AFp AFp 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 0 morphism with geometrically connected ﬁbres. Let s s 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. 0 Proof. Use that given a specialization s s in S Noetherian, we can find 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 étalecover. 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 / YB / Y / X

# Spec(L) / Spec(B) / 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) / π1(X0)

# { G 35 0 Proof. Let G acting on Y → Xη be the corresponding finite étalecover. 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 0 by B ⊂ L (see Step 5), we may assume we have G acting on Y → Xη. Let Y → X be the normalization of X in f.f.(Y 0). Then we get Y 0 / Y

πη π Xη / 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. −1 Let y1, ··· , yn ∈ Y0 be the generic points of Y0. Then π ({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) / f.f.(Y ) O O

? ? OX,ξ / 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 Ab- hyankar’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 • e = ramification index e: πA = (unit)πB, • 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 0 generated subextension κ /`/κ is separably generated: there exists x1, ··· , xr ∈ ` such that

` ⊃ κ(x1, ··· , xr) ⊃ κ where the first extension is finite separable and the second extension is purely transcendental. Under this definition, it is true (but not obvious) that a separably generated extension is separable! Proposition 13.1. TFAE: 36 • κ0/κ is separable. • κ0 is geometrically reduced over κ. 0 • κ ⊗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. 0 • dimκ0 (Ωκ0/κ) = Tr(κ /κ).

S 1/pn Example 13.2. Fp ⊂ n Fp(t ) is separable: the only derivation of Fp is trivial! p p p Example 13.3. Let κ = Fp(t, s, r). Then the variety X : tx + sy + rz = 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 → 3 over Spec( (t, s, r)). AFp Fp

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

o B B1 r O O O e eij ? ? $ L o A A1 r L ⊗K K1 O ei

$ K K1 where:

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

• mi1, ··· , mimi are the maximal ideals of B1 lying over mi. We have new invariants:

• ei = ramiﬁcation of A → (A1)mi ,

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

Theorem 13.4 (Abhyankar’s lemma). Assume e is prime to the characteristic of κA 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 e eij = . gcd(e, ei) This is not yet in the Stacks Project. 37 Example 13.5. x / (z, ··· , z)

5 Q5 2 3 4 A[x]/(x − t) / k[[z]] (z, ζ5z, ζ z, ζ z, ζ z) O i=1 O 5 O 5 5

A = k[[t]] / A[y]/(y5 − t) y_ Q5 where ζ5 ∈ k is a primitive 5-th root of unity. Here i=1 k[[z]] is the normalization of A[x, y]/(x5 − t, y5 − t).

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 ´etale.

Y → X → Spec(A) 3 η G Yη → Xη

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

o X XA1

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.(X ) with respect to O , we see that Y → X is unram- A1 XA1 ,ξ 1 B1

iﬁed in codimension 1. By purity, Y1 → XA1 is ´etale.This means G is a quotient of π1(X0) as desired. 38 13.3. Applications. Let us now apply this famous theorem of Grothendieck. An example of a variety in characteristic p that lifts to characterisitc 0 is complete intersection.

n 13.3.1. Complete intersections. Let X ⊂ Pk 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, ··· , βgi 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  ⊕2 if char(k) = 0, Zb Q ⊕2 π1(E) = `6=p Z` × Zp if char(k) = 0,E ordinary, Q ⊕2  `6=p Z` × {0} if char(k) = 0,E supersingular. Example 13.8. If A is an abelian variety, then Y ⊕2g f π1(A) = Z` × Zp `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 : Xan → San is a ﬁbre bundle: for all s ∈ San = S(C), there exists s ∈ U ⊂ San an −1 ∼ open such that (f ) (U) = Xs × U as homeomorphism of spaces over U. i an an Corollary 14.1. The sheaves R f∗ (Z) are locally constant on S (such a thing is often i called a local system) with stalk H (Xs, Z) at s. In particular, there is a monodromy representation i an i ρ : π1(S , s) → AutZ(H (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 an each x ∈ S\S there is a well-defined conjugacy class in π1(S , 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. Etale´ cohomology version. Let f : X → S be a smooth proper morphism of i Noetherian schemes. Let ` be a prime number invertible on S. Then we get R fét,∗(Z`) locally constant (technically we haven’t discusses étalesites yet, but fét is the étaleanalogue of the analytification for the complex topology), with i i R fét,∗(Z`)s = Hét(Xs, Z`) is a finitely generated Z`-module. There is a monodromy representation i i ρ` : π1(S, s) → AutZ` (Hét(Xs, Z`)). i Theorem 14.4 (Grothendieck). ρ` 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 ρi : Gal(Ksep/K) → Aut (Hi (X , )) XK /K,` Z` ét K Z` when restricted to an inertia subgroup has the following properties, provided ` is invertible in κA: (1) the image of wild inertia is finite: #ρi (P ) > ∞. XK /K,` (2) if τ ∈ I is an element mapping a topological generator of I , then ρi (τ) is t XK /K,` quasi-unipotent. Recall {1} ⊂ P ⊂ I ⊂ D ⊂ Gal(Ksep/K), where P is the wild inertia, I is the inertia, D ∼ Q is the decomposition, P = I/P = 0 0 is the tame inertia. I ` 6=char(κA) Z` 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 i cohomology and the local systems R 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 n AutZ` (M) = Autcont(M) = lim Aut(M/` M) is a proﬁnite group and the kernel of the continuous

AutZ` (M) → Aut(M/`M) is a pro-` group. n Proof. Omitted. (This is clear from the description lim Aut(M/` M).) i sep i Lemma 14.7. If X/K is a variety, then ρX/K,` : Gal(K /K) → AutZ` (Hét(XK , Z`)) is continuous. Proof. By definition, i i n Hét(XK , `) = lim Hét(XK , /` ) Z n Z Z sep and the action of Gal(K /K) on these is continuous. i n sep Remark. Even though Hét(XK , Z/` Z) is finite, continuity is not automatic because Gal(K /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(Ksep/K) where P is the wild inertia, I is the inertia and D is the decomposition (if K is henselian local, then D = Gal(Ksep/K)), together with representations i sep i ρX/K,` : Gal(K /K) → Aut(Hét(XK , Z`)). i i Last time we showed that ρX/K,`(P ) is finite, using ` 6= char(κ) and the continuity of ρX/K,`. i Theorem 15.1. If τ ∈ I maps to a topological generator of It = I/P , then ρX/K,`(τ) is quasi-unipotent. i We will do this by mapping Aut(Hét(XK , Z`)) into i ∼ Aut (H (X , `) ⊗ `) = GL i ( p), Q` ét K Z Z` Q BX Q 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, −1 α × • στσ and τ map to the same element of It where α ∈ Zb is α = (α2, α3, α5, α7, ··· ) with α` ≡ 1 (mod `) but α` 6= 1.

Aside: Suppose G is a profinite group, τ ∈ G, α ∈ Zb. Then τ α is defined. 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 α profinite completion, we get Zb → G by α 7→ τ . Proof of lemma. There are canonical maps D → Gal(κsep/κ) and sep θcan : I → lim µn(κ ) n prime to p where p = char(κ) (or 1 when char(κ) = 0) (and the limit on the right is noncanonically Q isomorphic to `06=p Z`0 ), such that −1 θcan(στσ ) = σ(θcan(τ)) ∼ for τ ∈ I and σ ∈ D. Note that θcan factors through It, and any σ ∈ D acts on It = sep Q Q × limn prime to p µn(κ ) ' `0 Z`0 by multiplication by some ασ ∈ `06=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, n sep where ζ` and ζ`n are primitive `- and ` -th roots of 1 in κ . 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, sep and in fact it holds for any continuous ρ : Gal(κ /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(στσ−1) (here we need the fact that ρ is defined on D, not just I). Since #ρ(P ) < ∞ and στσ−1 and τ α have the same image mod P , this implies by thinking plus group theory that ρ((στσ−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 gα with α as in the previous lemma, then g is quasi-unipotent.

To prove this for Q`-vector spaces, pick a stable lattice, but we will not do this in detail. × Proof. Look at eigenvalues λ1, ··· , λt; these will be in Z` . Then we get: for all i there exists α αr αr αs j such that λi = λj. Then for all i, there exists 0 ≤ r < s such that λi = (λi ) , so αr αs−1 (λi ) = 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 × β We still have to prove that if λ ∈ Z` and λ = 1 with β ∈ Zb and β 6= 0, then λ is a root 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 n ∼ ⊕2g T`X = lim X[` ](K) = Z` with a continuous action ρ of Gal(Ksep/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 1   .. 0 ..   . .     1 1     1   .  ρ(τ) =  0 .. 0     1     1     ..   0 0 .  1 of size 2g × 2g, with the middle block of size 2g0 × 2g0. 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 g0, and T is a torus over κ: ∼ g−g0 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 sep / sep Gal(K /K) Gal(K0 /K0)

ρi ρi X/K,` X0/K0,` Aut(Hi (X , )) ∼ / Aut(Hi (X , )) ´et K Q` ´et 0,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→ ζnt , 2πi n S where ζn = e . 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 0 group of the fraction field of A will map onto that of A. Nérondesingularization is the same process needed for constructing Néronmodels. The references are Tag 0BJ7 of the Stacks Project and Artin’s paper on approximation. Theorem 16.2 (Nérondesingularization). 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’sprocedure of proving it is even more amazing. This is proved using Néronblowups, 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 Nérondesingularization to Fp[[t]] ⊂ κ[[t]]. We need to check the hypotheses are satisfied: indeed Ω1 = [[t]] dt shows the Fp[[t]]/Fp Fp 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.

1 1 Remark. In particular, K = f.f.(A) = A[ π ] = colimi Ai[ t ]. 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 ∈ m , t∈ / m2 (because [[t]] → A or [[t]] → A is smooth). Ai Ai Fp i Q i Important question: What is the structure of 1 π (Spec(A )\V (t)) = π Spec A ? 1 i 1 i 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 t π1 Spec Ai π1 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 t sep 0 → lim µn(κ ) → π1 → Gal(κ /κ) = π1(Spec(A)) → 1 n prime to p i=1 sep Q where limn prime to p µn(κ ) is non-canonically isomorphic to `6=p Z`, such that the action of Gal(κsep/κ) on the group on the left is as indicated.

t Remark. π1 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 √ √ [ n n f.f.(A[ t1, ··· , 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 t t sep whose image is pro-`, factors through π1 and maps elements of ker(π1 → Gal(κ /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 sep i • the action ρX/K,` is trivial mod `, i.e., the action of Gal(K /K) on Hét(XK , Z/`Z) 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 Step 2: Reduce to action on Hét(XK , Z/`Z) trivial (Lemma 16.6).

Step 3: Write (A, π) = colim(Ai, t) with Ai henselian regular local, κAi finitely generated over the prime field, and regular parameters (Remark). 1 Step 4 (limit): Get Xi → Spec(Ai[ t ]) proper smooth whose base change is X. i 1 i ρX /A [ 1 ],` : π1 Spec Ai → Aut(Hét(XK , Z`)). i i t t Step 5 (limit): After possibly increasing i we may assume ρi mod ` is trivial. By Xi/Ai,` the previous lemma, Im(ρi ) is pro-`. Xi/Ai,` 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 e e−1 π + λ1π + ··· + λe = 0. To finish, consider the diagram / RO AO

? ? W (κ0) / W (κ) where

• κ0 ⊂ κ is the perfection of a ﬁnitely ﬁeld extension of Fp, ∼ e 0 e−1 0 0 • R = W (κ0)[X]/(X + λ1X + ··· + λe) with λi 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 A is regular local henselian, t ∈ m /m2 , with residue field i Ai Ai

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 e e−1 π + λ1π + ··· λe = 0 with λi ∈ W (κ). 0 0 Lemma 17.1 (Krasner’s lemma). There exists N > 0 such that if λ1, ··· , λe ∈ W (κ) satisfy 0 N λi − λi ∈ p W (κ), then e 0 e−1 0 P (x) = x + λ1x + ··· + λe 0 0 2 has a root π ∈ A with π ≡ π (mod mA). Last lecture we considered Nérondesingularization, 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 0 0 N a finitely generated extension of Fp, and λ ∈ W (κ0) such that λ − λ ∈ p 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)) A Krasner / A O 0 O

? ? W (κ0) / W (κ) • Pick N as in Krasner’s lemma. 0 0 • Pick κ0, λ1, ··· , λe using Step 3 (repeatedly). e 0 e−1 0 • Set A0 = W (κ0)[x]/(x + λ1x + ··· + λe). • Because we started with an Eisenstein polynomial we see that A0 is a complete dvr with uniformizer the class x of x. 0 • We get A0 → A by mapping x to π .

Step 5: N´erondesingularization applies to A0 → A. Thus 0 (A, π ) = 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 π1(f(U)) O

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}. n 0 Proof. Choose a birational map ϕ : Pk 99K X, and let X be the normalization of Γϕ ⊂ n Pk × 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. 1 As a special case, let Y = Pk and X be a smooth projective surface over k = k. Then 1 π1(X) is a quotient of π1(Xt) for all t ∈ P . 1 Example 17.5. If one ﬁbre is a tree of P ’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 1 ∼ • X = E × P and π1(X) = π1(E), or • π1(X) is finite cyclic. Proof. No singular fibres ⇔ X = E × P1 ⇔ j-invariant of fibres is constant. The rest of the ∼ cases occur when there exists a bad fibre Xt. Then π1(Xt) = Zb and π1(X) is procyclic. We have the correspondence ⊗n ∼ nconnected cyclic finite étalecoverso nL invertible on X with L = OX o ↔ ⊗m ∼ with group Z/nZ but L 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 χ : Z/nZ → k×. More L precisely, the action of /n gives the decomposition (g O 0 ) = L . Z Z ∗ X χ:Z/nZ→k× χ 48 1 1 ∼ Let η ∈ Y = P be the generic point, and K = κ(η) = k(P ) = 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 complex- analytic 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→ zn, we can fill in a nodal curve. This complex version is not that hard to prove. I will formulate this in the correct gener- ality in algebraic geometry, with dvr’s. Definition 18.1. Let X be a locally Noetherian scheme. A strict normal crossings 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 étalecovering {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 + X3 + X2 = 0 in 2 is a normal crossings divisor. AR 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= ∅. −1 (b) Xs = f (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) For all nonempty J ⊂ I we have that the scheme-theoretic intersection XJ = T 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 ∼ ObX,x = B[[t1, ··· , tr]]/(t1 ··· tr − π) where B is a complete local R-algebra which is formally smooth over R. Fact. If κ(s) is perfect, then

(c1), (c2), (c3) ⇔ Xs is an s.n.c.d. in X. Example 18.7. Let R = κ[[t]] ⊂ A = 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 ´etalelocally 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 Xs n.c.d. ks X s.s./S KS KS 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 ∼ if and only if ObX,x = 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) / Spec(R)

& Mg / 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, • X0 → S0 = Spec(R0) an s.s.s. S-variety, 0 ∼ 0 0 • C ⊗K K = X ⊗R0 K . Additional properties we would like: • K0/K finite separable, 0 • Y → Spec(B) where B is the integral closure of R in K such that Y ⊗B Bm0 is an 0 s.s.s. variety over Bm0 for all maximal ideals m ⊂ 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 Ksep/K0/K finite separable such that some property holds for the 0 GalK0 -action on Pic (CKsep )tors. 0 0 0 0 Step 2: Let R = Bm0 with B ⊃ m as before. We will show there exists X → Spec(R ) 0 (relatively minimal model) which is proper flat, X regular, (Xs)red s.n.c.d. and without (−1)-curves, with X0 ⊗ K0 ∼= C ⊗ K0. 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; ν ∧ (2) Y has finitely many singular points y , ··· , y and O ν is normal. 1 n Y ,yi 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 Silver- man. Notation. Fix R a dvr with K = f.f.(R), and C/K a proper smooth geometrically connected curve. Proposition 19.4. There exists C / X

f Spec(K) / Spec(R) with f flat proper and X regular. n n Proof. Choose C,→ PK and let Y ⊂ PR 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 −1 such that f (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 1- dimensional schemes whose normalization is finite. Then there exists a sequence of blowups −1 f : X → Y such that f (Z)red is a strict normal crossings divisor (s.n.c.d.). This is a lot easier to show than Lipman’s theorem. Proposition 19.7. There exists C / X

f Spec(K) / Spec(R) 52 such that f is projective and ﬂat, X is regular, (Xs)red is a s.n.c.d.

Notation. Let X → Spec(R) = S be as in the proposition, C1, ··· ,Cn the irreducible com- P ponents of Xs. Write Xs = riCi as Cartier divisors, where ri is the multiplicity of Ci in Xs. Then Xs is reduced if and only if ri = 1 for all 1 ≤ i ≤ r. 0 0 0 1/r 0 Claim. If char(κ(s)) = 0, then set r = lcm(ri), R = R[π ] with π = π , and X the 0 normalization of X ×Spec(R) Spec(R ). Then X0 → Spec(R0) = S0 is a semi-stable curve over R0. This is not the same thing as saying X0 is a semi-stable S0-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 S0-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. 2 p Example 19.9. Let κ = Fp(t) where p 6= 2. Then C = Spec κ[x, y]/(y − x − t) is a regular non-smooth curve over κ, as follows. One can check that Sing(C → Spec(κ)) = V (y, y2 − xp − t) = {(y, xp + t)}. p p Denote Q = (y, x + t) ∈ C. Then κ(Q) = κ[x]/(x + t), and OC,Q is a regular local ring as mQ = (y). But over the algebraic closure, 2 1/p p ∼ 2 p C ×Spec(κ) Spec(κ) = Spec(κ[x, y]/(y − (x + t ) )) = Spec(κ[x, y]/(y − x )) 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 étalelocal structure of X and use that normalization commutes with étalelocalization. We will not define “étalelocal structure”, but the picture is

(u1,U1) ··· ··· étale étale étale étale y % { # (x ∈ X)(u2,U2) ··· (uk,Uk)

where (uk,Uk) should be in a standard form. n Example 19.11. A smooth morphism is ´etalelocally like AS → S. 53 Let x ∈ Ci ∩ Cj (i 6= j) with (1) κ(x)/κ(s) separable, (2) ri, rj prime to char(κ(s)).

We have R → OX,x ⊃ mx. Let a, b ∈ mx be local equations for Ci,Cj, and n = ri and m = rj. Then π = u · anbm × where u ∈ OX,x. Because char(κ(s)) = char(κ(x)) does not divide n, after replacing X by n m an ´etalecover we may assume π = a b (adjoin n-th root of u to OX,x). Now κ(x)/κ(s) is separable, so

n m OX,x ← R[u, v]/(u v − π) a ← u b ←[ v [ defines an étalemorphism (X, x) → Spec(R[u, v]/(unvm − π)). n n m In characteristic 0, consider An = R[u, v]/(u − π) and An,m = R[u, v]/(u v − π). Final step: Compute the étalelocal structure of

0 0 n m 0 d norm An,m,d = (R [u, v]/(u v − (π ) )) where R0 = R[π] and (π0)d = π, n | d, m | d. 0 Step 1: If e = gcd(n, m) > 1 and ζe ∈ R , then e 0 ∼ Y 0 An,m,d = An/e,m/e,d/e. i=1

n/e m/e e 0d/e e 0 Why? (u v ) = (π ) in An,m,d. Step 2: If gcd(n, m) = 1, then unvm = (π0)d 0 in An,m,d, i.e., !m (π0)d π0d/m un = = , vm v

0 0 so there exists u ∈ An,m,d such that (π0)d/m (u0)m = u and (u0)n = . v 0 0 Similarly, there exists v ∈ An,m,d such that (π0)d/n (v0)n = v and (v0)m = . u Check that 0 0 0 0 0 0 0 d/nm An,m,d ≡ R [u , v ]/(u v − (π ) ). 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](Ksep). ` ←− If C has semi-stable reduction, then the action ρ is unipotent. 20.1. Method of Artin–Winters. Suppose we have a Cartesian diagram C / X

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

Let C1, ··· ,Cm be the irreducible components of Xs. Then X Xs = riCi as Cartier divisors. Let r = gcd(r1, ··· , rn). Warning. r need not be 1.

Lemma 20.1. Xs is geometrically connected over κ(s). 0 Proof. By the conditions on C, we have H (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 1 , as a variety over , PQ(i) 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 if i 6= j, Ci · Cj = Cj · Ci = deg(NCi/X ) if i = j. ∼ Remark. Because X is regular, Ci ⊂ X is Cartier and I = ICi/X is invertible and OX (Ci) = −1 ∼ I 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 X Lemma 20.2. riCi · Cj = 0 for all j.

P ∼ =∼ Proof. OX ( riCi) = 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 = riCi. Deﬁne a pairing Λ × Λ → Z X 0 X 0 X D = aiCi,D = biCi 7→ D · D = aibjCiCj which is a symmetric bilinear form.

Lemma 20.3. The pairing Λ × Λ → Z is semi-negative deﬁnite and if Z ∈ Λ with Z2 = 0, 1 then Z ∈ Z( r F ). P Proof. Say Z = siCi. Then !2 ! 2 X X X Z = siCi = siCi sjCj i i j ! X X si X = s C s C − r C i i j j r j j i j i j ! X X sirj = siCi sj − Cj ri i j6=i X si = (risj − rjsi)CiCj ri i6=j 2 X (risj − rjsi) = − C C . r r i j i

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∗) 0 → Z/ gcd(n, r)Z → Pic(X)[n] → Pic(C)[n] → . nΛ + ZF 1 ∼ Proof. ker(Pic(X) → Pic(C)) = Λ/ZF . The torsion in this is Z( r F )/ZF = Z/rZ. Hence ∼ (Z/rZ)[n] = Z/ gcd(r, n)Z. ∼ Suppose LC ∈ Pic(C)[n]. Pick L ∈ Pic(X) with LC = L|C . Then ⊗n ∼ X L = OC aiCi P for some aiCi ∈ Λ well-defined modulo ZF . Moreover X ⊗n aiCi · Cj = L · Cj = n(L· Cj) ∈ nZ. P −1 ∗ Hence aiCi ∈ α (nΛ ) is well-defined modulo ZF . On the other hand we can replace P P P P P L by L( biCi) for some biCi ∈ Λ. Then aiCi changes to aiCi + n( biCi). We −1 ∗ see that we get a well-defined class in α (nΛ ) which if zero means we can choose L to be nΛ+ZF n-torsion. Lemma 20.6. There is an exact sequence α−1(nΛ∗) 0 → → Λ/nΛ → Λ∗/nΛ∗ → G/nG → 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(Ksep/K) = I acts trivially on J[`m](Ksep) for all m ≥ `, then we can conclude m m 2g # Pic(Xs)[` ] ≥ (` ) /fixed constant where J is the Jacobian of C/K and g is the genus of C. This implies

dimF` Pic(Xs)[`] ≥ 2g. P We will see later this implies Xs is a tree of smooth curves with gi = g. 57 ∼ 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 ρ (G) = min r | . c such that H can be generated by r elements

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

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

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 , (mij ) ∗ Ln • G = coker(Λ → Λ ) with Λ = i=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 C2 C3 C4 r r r r

··· Rest ··· then we remove the middle edge to form T 0:

−2 −1 −1 −2 ◦ ◦ ◦ ◦ C1 C2 C3 C4 r r r r

··· Rest ··· Recall that 1 X g = 1 + k r . 2 i i 0 0 0 0 0 Since k1 = k2 = k3 = k4 = 0 in T but k2 = k3 = −1 in T , we have g < g. If T is connected, then it is an abstract type of genus g0. 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 ! mij ∗ X M = Im(Λ → Λ ) = Span mijxj, i 6= 2, 3; x1 − 2x2 + x3; x2 − 2x3 + x4 j and ! m0 0 ij ∗ X M = Im(Λ → Λ ) = Span mijxj, i 6= 2, 3; x1 − x2; −x3 + x4 . j We see that 0 M ⊂ M + Z(x2 − x3), so there is a surjection ∗ ∗ G = Λ /M Λ /M + Z(x2 − x3) = G/hx2 − x3i with cyclic kernel, and similarly a surjection 0 ∗ 0 G = Λ /M G/hx2 − x3i. Hence 0 ρc(G) ≤ ρc(G/hx2 − x3i) + 1 ≤ ρc(G ) + 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 K0 be a ﬁnite separable extension of K such that ∼ 2g Pic(CK0 )[`] = (Z/`Z) and C(K0) 6= ∅. 0 0 Replace R,K,C by R ,K ,CK0 . We will show C has semi-stable reduction. Pick C / X

Spec(K) / Spec(R) a regular ﬂat proper model without (−1)-curves (omitted: can contract (−1)-curves). We will show this model is semi-stable. The existence of rational point implies:

• gcd(ri) = 1 (there is an i with ri = 1). 1 • dimκ H (Xs, O) = g. 1 1 • dimκ H (Xs, O) ≥ dim H ((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 → Γ(Y, O∗ ) → Γ(C , O∗ ) → O∗ → Pic0,0(Y ) → Pic0(C ) → 0. Y i Ci Ci∩Cj ,x i i=1 x∈Ci∩Cj i=1 i6=j

0 Fact (Oort). dimF` Pic (Ci)[`] ≤ g(Ci) + pa(Ci), with equality if and only if Ci is nodal. Then dim (O∗ )[`] = 1 F` Ci∩Cj ,x ∼ 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 X 2g ≤ (g(Ci) + pa(Ci)) + 2β. Now we consider the same long exact sequence as before without ∗’s: n n 0 Y Y 1 M 1 0 → H (Y, OY ) → Γ(Ci, OCi ) → OCi∩Cj ,x → H (Y, OY ) → H (Ci, O) → 0.

i=1 x∈Ci∩Cj i=1 i6=j 60 This gives 1 1 X g = dimκ H (Xs, O) ≥ dimκ H (Y, OY ) = pa(Ci) + β. Therefore, X X 2 pa(Ci) + 2β ≤ (g(Ci) + pa(Ci)) + 2β so X X pa(Ci) ≤ g(Ci).

This shows there are no δ-invariants, i.e., all Ci’s are smooth and we have equality every- where. This ﬁnishes the proof. 22.3. Saito. We have been following Artin–Winters, but there is a paper by Saito 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 ´etalecohomology and vanishing cycle sheaves.

23. Lecture 23 (December 8, 2015) 23.1. Néronmodels. We will discuss the Néron–Ogg–Shafarevich criterion. First we need to talk about Néronmodels. 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 G extends 99K (*) to a morphism X → G. An alternative weaker condition is: If X is smooth over Spec(R) then any morphism X → A extends to K (†) a morphism X → G. It is clear that (*) implies (†). Remark. In both cases we have, if R is a dvr, then sep I sh sh sp sep A(K ) = A(f.f.(R )) = G(R ) → Gs(κ ), where I ⊂ Gal(Ksep/K) is the inertia with (Ksep)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 semi- abelian varieties. These models are also called Néron models.

Example 23.4. Consider Gm,K with R dvr. Pick π ∈ R uniformizer. Set [ n “G = π Gm,R” n∈Z 61 (construct by glueing). Then we have

× [ n × K = Gm,K (K) = G(R) = π R . n∈Z Example 23.5. Consider elliptic curves with multiplicative reduction and v(∆) = 1. As- sume we have a Weierstrass equation over R with discriminant ∆ = π · unit. Then closed subscheme of 2 defined singular point of G = PR \ by the Weierstrass equation the special fibre (over Spec(R)) is the Néronmodel 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 ∼ ⊕r K = Gm,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 Néronmodel 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(Ksep)[`∞] is trivial. 62 “Proof” of ⇐= . First we show that although

sep I sh sh sp sep A(K ) = A(f.f.(R )) = G(R ) → Gs(κ ) is not injective, writing f.f.(Rsh) = Ksh we have

sh n sep n A(K )[` ] ,→ Gs(κ )[` ]. Note that I acts trivially by assumption, so sh n ∼ 2g A(K )[` ] = (Z/`Z) where g = dim(A). 0 Since G → Spec(R) is of ﬁnite type, [Gs : Gs] < ∞ and hence 0 sep n 2gn−constant #Gs(κ )[` ] ≥ ` . 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, =⇒ t = u = 0. g = u + t + b

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 Néronmodel 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 ⊗ ` = lim A(K )[` ] ⊗ ` Z` Q n Z` Q is a Q`-vector space of dimension 2g endowed with a continuous action of sep 0 → I → Gal(K /K) Galκ = Zb → 0. By assumption, the action of I is unipotent. ∼ Q Let σ ∈ I be a topological generator, and recall the tame inertia It = I/P = `06=char(κ) Z`0 . sep Suppose τ ∈ Gal(K /K) maps to the arithmetic Frobenius in Galκ = Zb 3 1, i.e., τστ −1 = σq mod P. r Let r be the greatest integer such that (σ − 1) 6= 0 on V`(A). Consider the coinvariants and invariants I (V`)I (V`) O

∼ r = ? r Q = V`/ ker((σ − 1) ) / Im((σ − 1) ) =: S (σ−1)r

This map Q → S is not Galκ-invariant but it is if we twist ∼ Q ⊗Q` Q`(r) → S r as Galκ-representations. Note τ acts as q on Q`(1) and as q on Q`(r). Proof. τ(σ − 1)r = (τστ −1 − 1)rτ = (σq − 1)rτ = q(σ − 1)r by looking at 1 1 ∗ 1 q ∗ 0 ··· 0 qr . . .  ..   ..   ..   1  q  1  q r  0 0  σ =  .  , σ =  .  , (σ − 1) =  .  .  .. 1  .. q  .. 0  1 1 0

Deﬁnition 24.3. A q-Weil number of weight w is α ∈ Z such that |α0| = qw/2 for all conjugates α0 ∈ C of α. 64 −1 I Lemma 24.4. The eigenvalues of τ on (V`) are either: • q-Weil numbers of weight −2 or • q-Weil numbers of weight −1.

I 0 0 Proof. (V`) = V`(Gs), where G is the N´eronmodel and Gs is the special ﬁbre. Gs 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. −1 Lemma 24.5. The eigenvalues of τ on (V`)I are either: • q-Weil numbers of weight 0, or • q-Weil numbers of weight −1.

t 0 Proof. Let A = PicA be the dual abelian variety. Then there exists a nondegenerate Galκ- equivariant bilinear pairing t V`(A) × V`(A ) → Q`(1). Note ∼ t I ∗ (V`(A))I = ((V`(A )) ) (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 I −2 (V`)I (r) (V`) −1−2r O −1

∼ ? Q(r) = / S If r = 0, N´eron–Ogg–Shafarevich says A has good reduction. 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 1 1 dim(torus part of G0) ≥ # , s 0 1

I 0 using V` = V`(Gs) and the fact that weight −2 eigenvalues come from the torus part. The I 0 relation V` = V`(Gs) also shows 1 1 2 dim(abelian part of G0) + dim(torus part of G0) = 2g − # s s 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 i V = H (XK , Q`). There are two filtrations on V : • filtration coming from the nilpotent operator N = σ − 1, • weight filtration. Conjecture 24.6. The two filtrations agree. We just proved this for abelian varieties on H1. Scholze proved this for complete intersections using perfectoid spaces.