University of Cambridge

Mathematics Tripos

Part III Algebraic Geometry

Michaelmas, 2019

Lectures by M. Gross

Notes by Qiangru Kuang Contents

Contents

0 Introduction 2 0.1 Variety vs Scheme ...... 2 0.2 Categorical philosophy ...... 2

1 Sheaves 4 1.1 Sheafification ...... 7

2 Schemes 10 2.1 Projective schemes ...... 16 2.2 Open and closed subschemes ...... 19 2.3 Fibre products ...... 20

3 Sheaves of OX -modules 25 3.1 Morphisms into projective space ...... 27 3.2 Divisors and the Picard groups ...... 30 3.3 Cartier divisor ...... 34

4 Cohomology of sheaves 40 4.1 Čech cohomology ...... 41

Index 53

1 0 Introduction

0 Introduction

0.1 Variety vs Scheme In classical algebraic geometry, we study varieties which are points where poly- nomials vanish. Why do we need schemes? Why not varieties? 1. With varieties, we always work with algebraically closed fields. Otherwise, the ideals are not really classical geometric objects. For example, consider I = (x2 + y2 + 1) ⊆ R[x, y]. V (I) = ∅, I(V (I)) = R[x, y]. 2. Suppose one want to work on number theory. One is usually interested in n Diophantine equations, for example if I ⊆ Z[x1, . . . , xn] then V (I) ⊆ Z . 2 2 3. Consider X1 = V (x − y ) ⊆ A ,X2 = V (x). Then

2 X1 ∩ X2 = V (x, x − y ).

Consider I = (x, x − y2) = (x, y2) ⊆ k[x, y]. V (I) contains exactly one point, namely the origin, but the ideal I is not radical, reflecting the fact that X2 is the tangent to X1. Might it be reasonable to consider 2 k[x, y]/(x, y ) as the coordinate ring of X1 ∩ X2 rather than k[x, y]/(x, y)? Note y ∈ k[x, y]/(x, y2) is non-zero but y2 = 0.

0.2 Categorical philosophy What is a point? Let Set be the category of sets. Let ∗ be the set with one element. Then if X is a set there is an obvious bijection X → Hom(∗,X). Put in another way, you know X if you know Hom(Y,X) for all Y . In the category of affine varieties over k, a point is a point with coordinate ring k. Given I ⊆ k[x1, . . . , xn], giving a morphism from a point to V (I) = X is the same as giving a k-algebra homomorphism ϕ : k[x1, . . . , xn]/I → k given by xi 7→ ai ∈ k such that for all f ∈ I, f(a1, . . . , an) = 0, i.e. (a1, . . . , an) ∈ V (I). ϕ is surjective so k[X]/ ker ϕ =∼ k, so ker ϕ is a maximal ideal of k[X]. Conversely, given a maximal ideal m ⊆ k[X], k[X]/m is a finite field extension of k by Hilbert’s Nullstellensatz. So if k = k then k[X]/m =∼ k and the map ∼ ϕ : k[X] → k[X]/m = k gives a point (a1, . . . , an) = (ϕ(x1), . . . , ϕ(xn)) ∈ X. Why not also consider field extensions k ⊆ k0 and consider k-algebra homo- morphisms ϕ : k[X] → k0? Same as before we get a set of solutions in (k0)n. Note k[X]/ ker ϕ ⊆ k0 and hence k[X]/ ker ϕ is an integral domain. Thus ker ϕ is prime. For example take k[X] to be R[x, y]/(x2 + y2 + 1) and R-algebra map k[X] → R. There does not exist such map, but there is certainly a map k[X] → C, for example x 7→ 0, y 7→ i. We have ker ϕ = (x, y2 + 1) is a maximal ideal. Note that this ideal corresponds to two points (0, ±i), a Gal(C/R)-orbit of points with C-coordinates. In fact this also follows from Hilbert’s Nullstel- lensata: if A is a finitely-generated k-algebra, m ⊆ A a maximal ideal then A/m is a finite field extension of k. Let K be the field of fractions of A. Then ϕ : A,→ K defines a point in K, i.e. (x, y) satisfying x2 + y2 + 1 = 0 in K. Note ker ϕ = 0 so this point corresponds not to a maximal ideal but to a prime ideal. This may prompt to us to consider all prime ideals instead of just maximal ideals.

2 0 Introduction

Consider another example A = Z[x1, . . . , xn]/I. We may want to consider so- lutions to the system of equations I with coordinates in any ring R. Such is given by a ring homomorphism ϕ : A → R, i.e. (r1, . . . , rn) satisfying f(r1, . . . , rn) = 0 for f ∈ I. We may then consider different choices of R. We may take R = Z, a number ring, Q, or Fp. What we want: give a ring A (all rings in this course are commutative with 1), we want a geometric object X = “ Spec A00. For a ring R, the set of “R-valued points” of X should be

X(R) = HomRing(A, R).

A morphism X = Spec A → Y = Spec B should be determined by a ring homomorphism ϕ : B → A.

Definition. The category of affine scheme is the opposite category to the category of rings.

This is a purely formal definition and there is no geometry. Instead, we wantto study more general objects.

Definition. A scheme is an object which is locally an affine scheme.

This time, we do need geometry to talk about local properties!

3 1 Sheaves

1 Sheaves

Definition (spectrum). Let A be a ring. Then the spectrum of A is

Spec A = {p ⊆ A : p a prime ideal}.

If I ⊆ A is an ideal, define

V (I) = {p ⊆ Spec A : p ⊇ I}.

Proposition 1.1 (Zariski topology). The set V (I) form the closed sets of a topology on Spec A, called the Zariski topology.

Proof. V (0) = Spec A, V (A) = ∅. If {Ij}j∈J is a collection of ideals then X \ V ( Ij) = V (Ij) j∈J j∈J and finally V (I1I2) = V (I1) ∪ V (I2).

One direction is trivial. For the other, if p ⊇ I1 ∩ I2 then p ⊇ I1 or p ⊇ I2.

Example. A = k[x1, . . . , xn] where k = k, I ⊆ A, then the maximal ideals in V (I) is in one-to-one correspondence with points of the variety V (I) in affine n-spece.

So far we only have a topology. To talk about geometry, we need a good notion of functions. This is where we need sheaves. From now on fix a topologial space X.

Definition (presheaf). A presheaf of abelian groups on X F consists of data 1. for every open set U ⊆ X, an abelian group F(U),

2. whenever V ⊆ U, a group homomorphism ρUV : F(U) → F(V ) such that ρUU = id, ρVW ◦ ρUV = ρUW when W ⊆ V ⊆ U.

The maps ρUV are called the restriction maps and for V ⊆ U, s ∈ F(U), we often write s|V for ρUV (s).

Remark. This is just a contravariant functor F : Open(X) → Ab where Open(X) is the category of open sets with inclusions.

Definition (morphism of presheaves). A morphism of presheaves f : F → G on X is a collection of homomorphisms fU : F(U) → G(U) with, for V ⊆ U,

4 1 Sheaves

the commutative diagram

f F(U) U G(U)

ρUV ρUV f F(V ) V G(V )

Definition (sheaf). A presheaf F is a sheaf if it satisfies

1. if U ⊆ X has a cover {Ui} and s ∈ F(U) such that s|Ui = 0 for all i then s = 0.

2. if U ⊆ X has a cover {Ui} and si ∈ F(Ui) given with si|Ui∩Uj =

sj|Ui∩Uj for all i, j then exists s ∈ F(U) with s|Ui = si for all i.

Remark. 1. If F is a sheaf then ∅ has an empty cover and hence F(∅) = 0. 2. The two axioms S1, S2 together can be described as saying for each open cover {Ui} of U,

β α Y 1 Y 0 F(U) F(Ui) F(Ui ∩ Uj) β i 2 i,j

is exact, where

α(s) = (s|Ui )

β1((si)) = (si|Ui∩Uj )

β2((si)) = (sj|Ui∩Uj )

Exactness means that α is injective (S1), and β1 ◦ α = β2 ◦ α, and for Q any (si) ∈ i F(Ui) with β1((si)) = β2((si)) then exists s ∈ F(U) with α(s) = (si), i.e. α is the equaliser of β1, β2. The sheaf axioms make sense when Ab is replaced by any category with equalisers, for example Set. Example. 1. If X is any topological space, set

F(U) = {f : U → R : f continuous} ρUV (f) = f|V Then F is a sheaf.

2. Let X = C with the Euclidean topology. Set F(U) = {f : U → C : f bounded holomorphic function}.

Then F satisfies S1 but not S2. For example take Ui = {z ∈ C : |z| < i}, fi : Ui → C the identity map. Take i ∈ N, then the glued function is not bounded.

5 1 Sheaves

3. Take any group G and set F(U) = G for all U, ρUV = id. F is the constant presheaf. It is a not a sheaf if G 6= 0 as F(∅) = G. To get a sheaf, give G the discrete topology and define

G(U) = {f : U → G : f continuous}.

Note if U is connected then G(U) = G. If {Ui}i∈I are disjoint then a Y G( Ui) = G(Ui).

4. Let X be an algebraic variety, U ⊆ X a Zariski open set. Define

OX (U) = {f : U → k : f a regular function}.

Then OX is a sheaf, the structure sheaf of X.

Definition (stalk). Let F be a presheaf on X, p ∈ X. Then the stalk of F at p is

Fp = {(U, s): U open neighbourhood of p, s ∈ F(U)}/ ∼

where (U, s) ∼ (V, s0) if there exists W ⊆ U ∩ V with p ∈ W such that 0 s|W = s |W . It can be described as the direct limit

F = lim F(U). p −→ p∈U

Note a morphism f : F → G induces a homomorphism

fp : Fp → Gp

(U, s) 7→ (U, fU (s))

We call (U, s) the germ of s at p.

Proposition 1.2. Let f : F → G be a morphism of sheaves. Then f is an isomorphism if and only if fp is an isomorphism for all p ∈ X.

Proof. Only if is obvious. For if, assume fp is an isomorphism for all p ∈ X. Need to show each fU : F(U) → G(U) is an isomorphism as can then construct −1 −1 −1 f via (f )U = (fU ) .

• fU injective: suppose s ∈ F(U) with fU (s) = 0. Then fp(U, s) ∈ Gp for any p ∈ U, and fp(U, s) = (U, fU (s)) = 0. Thus (U, s) = 0 ∈ Fp, which

means that exists p ∈ Vp ⊆ U such that s|Vp = 0. Now {Vp : p ∈ U} form an open cover of U so by S1, s = 0.

• fU surjective: let t ∈ G(U). Then tp = (U, t) ∈ Gp. By surjectivity of fp, let sp ∈ Fp with fp(sp) = tp. In other words, exists a neighbourhood Vp of

p and a germ (Vp, sp) such that (Vp, fVp (sp)) = (U, t) ∈ Gp. By shrinking

Vp if necessary, we can assume fVp (sp) = t|Vp . {Vp} is an open cover of U. On Vp ∩ Vq,

fVp∩Vq (sp|Vp∩Vq − sq|Vp∩Vq ) = t|Vp∩Vq − t|Vp∩Vq = 0.

6 1 Sheaves

By injectivity already proved, fVp∩Vq is injective so sp|Vp∩Vq = sq|Vp∩Vq

for all p, q ∈ U. By S2, there exists s ∈ F(U) with s|Vp = sp for all p so

fU (s)|Vp = fVp (s|Vp ) = fVp (sp) = t|Vp

so by S1, fU (s) = t.

Note the asymmetry: injectivity comes directly from injectivity on stalks, while surjectivity follows from both surjectivity on stalks and injectivity on sections. This will appear often in the future.

1.1 Sheafification Given a presheaf F, there is a sheaf F + an a morphism θ : F → F + satisfying the following universal property: for any sheaf G and morphism ϕ : F → G, there exists a unique morphism ϕ+ : F + → G with ϕ = ϕ+ ◦ θ. + + F is called the sheaf associated to F and θp : Fp → Fp is an isomorphism for all p. Define  s : U → ` F s(p) ∈ F   p∈U p: p,  F +(U) = for each p ∈ U, exists p ∈ V ⊆ U, t ∈ F(V )  such that s(q) = (U, t) ∈ Fq for all q ∈ V 

We define θU (s) to be p 7→ (U, s) ∈ Fp for all p ∈ U. The verification of the universal property and other properties is leftas exercises on example sheet 1.

Definition. Let f : F → G be a morphism of presheaves. • The presheaf kernel of f is the presheaf

(ker f)(U) = ker fU .

• The presheaf cokernel of f is the presheaf

(coker f)(U) = coker fU .

• The presheaf image of f is the presheaf

(im f)(U) = im fU .

Remark. If f : F → G is a morphism of sheaves then ker f is also a sheaf: any sub-presheaf of a sheaf satisfies S1, and given si ∈ (ker f)(Ui) with si|Ui∩Uj = sj|Ui∩Uj , we can glue to get s ∈ F(U). But then

fU (s)|Ui = fUi (s|Ui ) = fUi (si) = 0 so by S1 fU (s) = 0 so s ∈ (ker f)(U).

7 1 Sheaves

1 Example. Let X be the variety Pk. Let P,Q ∈ X be distinct points. Let G be the sheaf of regular functions on X. Let F be the sheaf of regular functions on X vanishing at P and Q. Have an obvious inclusion i : F → G and F(U) = G(U) if U ∩ {P,Q} = ∅. Then

1 1 G(P ) k (coker i)(P ) = = = k F(P1) 0 1 1 G(P \{P }) k[x] (coker i)(P \{P }) = =∼ = k F(P1 \{P }) (x) Similarly 1 (coker i)(P \{Q}) = k and 1 (coker i)(P \{P,Q}) = 0. Note U = P1 \{P },V = P1 \{Q} cover P1 so if coker i were a sheaf, any 1 sU ∈ (coker i)(U), sV ∈ (coker i)(V ) would give an element of (coker i)(P ). In particular 1 (coker i)(P ) = k ⊕ k 6= k. Absurd. This is not a bug but a feature of the theory. At the end of the course we will find ourselves secretly calculating the sheaf cohomology in this example.

Definition (sheaf kernel/image/cokernel). Let f : F → G be a morphism of sheaves. Then the sheaf kernel is the presheaf kernel. The sheaf image is the sheaf associated to the presheaf image and the sheaf cokernel is the sheaf associated to the presheaf cokernel.

These are the categorical notions of kernel/image/cokernel. Exercise. The sheaf image im f is a subsheaf of G.

Definition (exact sequence of sheaves). We say f : F → G is injective if ker f = 0, surjective if im f = G. We say a sequence of sheaves

f i f i+1 ··· F i−1 F i F i+1 ···

is exact if ker f i = im f i−1 for all i.

f is injective if and only if fU is injective for all U but the analogous state- ment is false for surjectivity. If F 0 ⊆ F is a subsheaf, we write F/F 0 for the sheaf associated to the presheaf U 7→ F(U)/F 0(U). This is coker(F 0 ,→ F).

Lemma 1.3. Let f : F → G be a morphism of sheaves. Then

(ker f)p = ker fp

(im f)p = im fp

8 1 Sheaves

for all p ∈ X.

Proof. We have a natural map (ker f)p → ker fp: if (U, s) ∈ (ker f)p where s ∈ (ker f)(U) then

fp(U, s) = (U, fU (s)) = (U, 0) = 0 so (U, s) ∈ ker fp. For injectivity, if (U, s) = 0 ∈ ker fp then (U, s) = 0 ∈ Fp. Shrinking U if necessary, we can assume s = 0. Then (U, s) = 0 ∈ (ker f)p. For surjectivity, if (U, s) ∈ ker fp then (U, fU (s)) = 0 ∈ Gp. Shrinking U if necessary, we can assume fU (s) = 0 and hence (U, s) ∈ (ker f)p since s ∈ (ker f)(U). Now we prove the statement about image sheaf. Let im0 f be the presheaf 0 image so im f is the associated sheaf. We have an isomorphism (im f)p = 0 (im f)p, so enough to show the natural map (im f)p → im fp is an isomorphism. For injectivity, if (U, s) = 0 ∈ Gp then shrinking U we can assume s = 0. 0 Then (U, s) = 0 ∈ (im f)p. For surjectivity, if (U, s) ∈ im fp then exists (V, t) ∈ Fp such that fp(V, t) = (U, s) so exists a neighbourhood W ⊆ U ∩ V of p such 0 that fV (t)|W = fW (t) = s|W . Thus (W, s|W ) ∈ (im f)p.

Proposition 1.4. Let f : F → G be a morphism of sheaves. Then f is injective (surjective respectively) if and only if fp : Fp → Gp is injective (surjective respectively) for all p.

Proof. fp is injective for all p if and only if ker fp = 0 for all p, if and only if (ker f)p = 0 for all p, if and only if ker f = 0, since 0 → ker f is an isomorphism on stalks, if and only if f is injective. Analogous for im f by noting the inclusion im f → G.

It is strongly recommended at this point to do all problems in II.1 of Hartshorne.

Definition. Let f : X → Y be a continuous map of topological spaces. Let F be a sheaf on X, G a sheaf on Y . We define f∗F to be the sheaf with

−1 (f∗F)(U) = F(f (U))

and obvious restriction maps. We define f −1G to be the sheaf associated to the presheaf

U 7→ lim G(V ) = {(V, s): V ⊇ f(U) , s ∈ G(V )}/ ∼ −→ open V ⊇f(U)

0 0 0 0 where (V, s) ∼ (V , s ) if exists W ⊆ V ∩V with f(U) ⊆ W and s|W = s |W .

−1 Example. If f : {p} → X and G is a sheaf on X then f G = Gp. More −1 generally if i : Z,→ X and F is a sheaf in X we write F|Z = i F. In particular if Z is open then F|Z is easy to describe: F|Z (U) = F(U).

Notation. If s ∈ F(U), we say s is a section of F over U. We often write Γ(U, F) for F(U), thinking of Γ(U, ·) as a covariant functor PreShX → Ab.

9 2 Schemes

2 Schemes

The first goal is to for a ring A construct a sheaf O = OSpec A on Spec A, the sheaf of “regular functions” on Spec A. We recall some facts about localisation. Let S ⊆ A be a multiplicatively closed subset, i.e. 1 ∈ S and if s1, s2 ∈ S then s1, s2 ∈ S. Define S−1A = {(a, s): a ∈ A, s ∈ S}/ ∼ 0 0 00 00 0 0 a where (a, s) ∼ (a , s ) if exists s ∈ S such that s (as − a s) = 0. We write s for the equivalence class of (a, s). This is the localisation of A at S. Example. 2 −1 1. Let S = {1, f, f ,... } for some f ∈ A. We write Af = S A. −1 2. Let p ⊆ A a prime ideal, S = A \ p. We write Ap = S A, called A localised at p. There is a canonical homomorphism

A → Ap a a 7→ 1

and the image of p generates the unique maximal ideal of Ap, which we write as pAp. A ring A is local if it has a unqiue maximal ideal, so Ap is local.

Now we construct O = OSpec A. In particular we want Op = Ap. Define  s : U → ` A s(p) ∈ A   p∈U p: p,  O(U) = for each p ∈ U, exists p ∈ V ⊆ U and a, f ∈ A  a  such that f∈ / q, s(q) = f ∈ Aq for all q ∈ V and the restriction maps are restrictions of functions. This is defined locally so O is a sheaf (of rings).

Proposition 2.1. For any p ∈ Spec A, Op = Ap. Proof. We have a well-defined map

φ : Op → Ap (U, s) 7→ s(p) a Surjectivity: any element of Ap can be written as f for some a ∈ A, f ∈ A \ p. Let D(f) = Spec A \ V ((f)) = {q ∈ Spec A : f∈ / q} a which is an open neighbourhood of q. Now f defines a section of O over D(f) via a s(q) = ∈ A f q for all q ∈ D(f). In particular φ is surjective. Injectivity: let p ∈ U ⊆ Spec A and s ∈ O(U) with s(p) = 0 ∈ Ap so a (U, s) ∈ ker φ. By shrinking we can assume s(q) = f for some a ∈ A, f∈ / q for all q ∈ U. There exists f 0 ∈/ p such that f 0a = 0. Let V = D(f 0)∩U which is an 0 a open neighbourhood of p. Then (V, s|V ) = 0 since f a = 0 implies f = 0 ∈ Aq for all q ∈ V . Thus (U, s) = 0 ∈ Op so φ is injective.

10 2 Schemes

∼ Proposition 2.2. For any f ∈ A, O(D(f)) = Af . In particular O(Spec A) = A. Proof. Define

ψ : Af → O(D(f)) a a 7→ (p 7→ ∈ A ) f n f n p

a a We first show ψ is injective: if ψ( f n ) = 0 then f n = 0 ∈ Ap for all p ∈ D(f). Thus for each p, exists h∈ / p such that ah = 0 in A. Let

I = {g ∈ A : ga = 0} so h ∈ I, h∈ / p so I * p. This is true for all p ∈ D(f), so V (I) ∩ D(f) = ∅. Thus \ √ f ∈ p = I, p∈V (I) n n a the radical of I. Thus f ∈ I for some I. Thus f a = 0 so f = 0 ∈ Af . To show surjectivity, let s ∈ O(D(f)). Cover D(f) with open sets Vi on ai which s is represented by gi where ai ∈ A, gi ∈/ q for all q ∈ Vi. By example sheet 1 question 1, the basic open sets form a basis for the topology of Spec A, so we can asume Vi = D(hi) for some hi ∈ A. Since D(hi) ⊆ D(gi) and

D(hi) ⊆ D(gi) ⇐⇒ V ((hi)) ⊇ V ((gi)) p p ⇐⇒ (hi) ⊆ (gi) n ⇐⇒ hi ∈ (gi) for some n

n ai ciai n so hi = cigi for some ci ∈ A. Since = n , we can assume gi = hi . But also gi hi n n D(hi) = D(hi ), we can replace hi by hi and assume that D(f) is covered by ai sets D(hi) on which s is represented by . In fact, we claim that D(f) can be hi covered by a finite number of the D(hi): [ \ X D(f) ⊆ D(hi) ⇐⇒ V ((f)) ⊇ V ((hi)) = V ( (hi)) i∈I i∈I sX ⇐⇒ f ∈ (hi) i∈I m X ⇐⇒ f ∈ (hi) i∈I

m P for some n. This means that f = bihi where all but a finite number of bi’s are zero. So we can just take these hi with bi 6= 0 and then {D(hi)} still cover D(f). Thus we can now assume we have a finite covering {D(hi)} of D(f) with s ai ai aj represented by on D(hi). On D(hi) ∩ D(hj) = D(hihj), note , both hi hi hj represent s so ai aj = ∈ Ahihj hi hj

11 2 Schemes

which follows from injectivity of ψ on the open set D(hihj). Therefore there exists n such that n (hihj) (hjai − hiaj) = 0. n+1 n Choose n large enough to work for all i, j. Rewrite the equation as hj (hi aj)− n+1 n n+1 n hi (hj ai) = 0. Replace each hi by hi and ai by hi ai, we can assume m P hjai − hiaj = 0 for all i, j. We also have f = bihi as in the proof of the P claim. Let a = biai. Then for any j,

X X m hja = hjbiai = hibiaj = f aj. i i

a aj a Thus n = on D(hj) ∩ D(f), i.e. in the ring Afh . Thus n ∈ Af defines a f hj j f a section of O(D(f)) whose restriction to D(hj)∩D(f) agrees with s so ψ( f n ) = s and ψ is surjective.

Definition (ringed space). A ringed space is a pair (X, OX ) with X a topological space and OX a sheaf of rings on X. A morphism between ringed spaces f :(X, OX ) → (Y, OY ) is data 1. f : X → Y continuous,

# 2. f : OY → f∗OX a morphism of sheaves of rings, i.e. for all U ⊆ X, # −1 fU : OY (U) → OX (f (U)). Example.

1. Let X be a topological space and OX (U) = {ϕ : U → R continuous}. Note f : X → Y induces f :(X, OX ) → (Y, OY ) by

# −1 fU (ϕ) = ϕ ◦ f : f (U) → R. In other words, there is no additional information in the ringed space structure and all data is encoded in the topological spaces themselves.

2. Let X be a variety and OX the sheaf of regular functions on X. Given # f : X → Y a morphism of varieties, we again get fU which acts by pullback.

Both examples have the feature that OX,p is a local ring with maximal ideal {(U, f) ∈ OX,p : f(p) = 0}: if f(p) 6= 0 then by shrinking U we can assume f is −1 nowhere zero so (U, f ) ∈ OX,p. This inspires us to define

Definition (locally ringed space). A locally ringed space (X, OX ) is a ringed space such that OX,p is a local ring for all p ∈ X. A morphism between locally ringed spaces f :(X, OX ) → (Y, OY ) is # a morphism of ringed spaces such that fp : OY,f(p) → OX,p is a local homomorphism for all p ∈ X.

Here

# fp : OY,f(p) → OX,p −1 # (U, s) 7→ (f (U), fU (s))

12 2 Schemes

and a homomorphism ϕ :(A, mA) → (B, mB) between local rings is local if −1 −1 ϕ (mB) = mA. Note that we always have ϕ (mB) ⊆ mA. The local homomorphism condition thus says that, for example in the two examples above, if a function vanishes at f(p) then its pullback vanishes at p.

Example. (Spec A, OSpec A) is a locally ringed space.

Definition (affine scheme). The category of affine schemes is the category whose objects are (Spec A, OSpec A) for all rings A and morphisms are mor- phisms as locally ringed spaces.

Theorem 2.3. The category of affine schemes is equivalent to Ringop.

Proof. Need to show

1. if ϕ : A → B is a homomorphism then we get a morphism of locally ringed # spaces (f, f ) : (Spec B, OSpec B) → (Spec A, OSpec A).

# 2. any morphism (f, f ) : (Spec B, OSpec B) → (Spec A, OSpec A) is con- tructed as in 1 from some ring homomorphism ϕ : A → B.

Let’s do 1 first. Given ϕ : A → B, define

f : Spec B → Spec A p 7→ ϕ−1(p) f is continuous as for I ⊆ A,

f −1(V (I)) = {p ∈ Spec B : ϕ−1(p) ⊇ I} = {p ∈ Spec B : p ⊇ ϕ(I)} = V (ϕ(I))

For p ∈ Spec B, we obtain

ϕp : Aϕ−1(p) → Bp a ϕ(a) 7→ s ϕ(s) which makes sense as if s∈ / ϕ−1(p) then ϕ(s) ∈/ p. It is also well-defined. Note ϕp is a local homomorphism: indeed the maximal ideal of Bp is generated by the image of p, which we often write as pBp and similarly the maximal ideal of −1 −1 −1 Aϕ−1(p) is ϕ (p)Aϕ−1(p) and clearly ϕp (pBp) = ϕ (p)Aϕ−1(p). Thus given V ⊆ Spec A, we get

# −1 fV : OSpec A(V ) → (f∗OSpec B)(V ) = OSpec B(f (V )) (p 7→ s(p)) 7→ (q 7→ ϕq(s(f(q))))

−1 a # where p ∈ V, q ∈ f (V ). If locally s is represented on W by f then fV (s) is −1 ϕ(a) # locally represented on f (W ) by ϕ(f) . This defined f : OSpec A → f∗OSpec B

13 2 Schemes

# and since f induces ϕq on stalks, this defines a morphism of locally ringed spaces. # Conversely, suppose given (f, f ) : (Spec B, OSpec B) → (Spec A, OSpec A), we get # ϕ = fSpec A : Γ(Spec A, OSpec A) = A → Γ(Spec B, OSpec B) = B and we need to show ϕ gives rise to (f, f #) obtained from the construction in the first part. We have a commutative diagram

ϕ A B

# fp Af(p) Bp where ϕ is induced by the map on sections, the vertical maps are passing to stalks, which are easily seen to be localisations, and the bottom map is the induced map on stalks so is a local homomorphism. Since pullback of the maximal ideal in the localisation gives the prime ideal, we have ϕ−1(p) = f(p). Note that this depends crucially on the fact that we have a local homomorphism. # # Thus we conclude that f is induced by ϕ and fp = ϕp. Thus f agrees with the map constructed from ϕ in the first part. Now we make a slight modification to the definition of affine scheme:

Definition (affine scheme). An affine scheme is a locally ringed space iso- morphic in the category of locally ringed spaces to (Spec A, OSpec A) for some A.

Definition (scheme). A scheme is a locally ringed space (X, OX ) with an

open cover {(Ui, OX |Ui )} with each (Ui, OX |Ui ) an affine scheme. A morphism of schemes is a morphism as locally ringed spaces.

Example. 1. Let k be a field. Then Spec k consists of a singleton {0} and a sheaf over the point, which is just the field k. We write Spec k = ({0}, k). What does a morphism f : Spec k → X for X a scheme mean? First it # picks a point f(0) = x ∈ X. Second we get a local homomorphism fx : # # −1 OX,x → OSpec k,0 = k, meaning that ker fx = (fx ) (0) = mx ⊆ OX,x, where mx is the maximal ideal of OX,x. In particular we get a factorisation

OX,x → OX,x/mx ,→ k

We call k(x) = OX,x/mx the residue field of x in X. Conversely, if given an inclusion k(x) ,→ k, we get a morphism f : Spec k → X with f(0) = x and

# fU : OX (U) → (f∗k)(U) ( (U, s (mod m )) x ∈ U s 7→ x 0 x∈ / U

14 2 Schemes

We can ask the opposite question: what does it mean to give a mor- # phism f : X → Spec k? f is constant but we need f : k → f∗OX , i.e. # fSpec k : k → OX (X) so Γ(X, OX ) has the structure of a k-algebra and via composition with restriction

k → Γ(X, OX ) → Γ(U, OX )

so OX becomes a sheaf of k-algebras. We say X is a scheme over Spec k. A morphism of schemes over Spec k is a commutative diagram

f X Y

Spec k

# i.e. f : OY → f∗OX is a morphsim of sheaves of k-algebras. More generally we can replace Spec k by any scheme Z and talk about schemes over Z and their morphisms. √ 2. Affine variety: let k be a field and A = k[x1, . . . , xn]/I with I = I. In other words, A is a finitely generated k-algebra with no nilpotents. The homomorphism k → A gives a map Spec A → Spec k. We say Spec A is an affine variety over k. If k0 is a field extension of k, a diagram

Spec k0 Spec A

Spec k

is the same as giving a k-algebra homomorphism A → k0, i.e. this is the 0 n same thing as giving (a1, . . . , an) ∈ (k ) with f(a1, . . . , an) = 0 for all f ∈ I. With X = Spec A, we write X(k0) for the set of all such diagrams. More generally, we usually fix a base scheme S and consider the category of schemes over S. Given X → S, T → S, we write X(T ) to be the set of morphisms T → X, the set of T -valued points of X. 3. An example of a scheme that is not a variety: D = Spec k[t]/(t2) = ({(t)}, k[t]/(t2)). t is a “function” on a single point which squares to 0. What information is contained in D? Let X be any scheme over Spec k and consider a morphism f : D → X over Spec k. This specifies a point x = f((t)) ∈ X and a local homomorphism of k-algebras

# 2 fx : OX,x → OD,(t) = k[t]/(t )

mx → (t) 2 mx → 0 2 ∼ so we obtain a map mx/mx → (t) = k where the isomorphism is as a k-vector space. We also have a corresponding k-algebra homomorphism 2 ∼ OX,x → k[t]/(t ) → k[t]/(t) = k

15 2 Schemes

with kernel mx. This gives an inclusion k(x) = OX,x/mx ,→ k of k- ∼ 2 algebras, i.e. k(x) = k. Note mx/mx is a OX,x/mx-vector space and 2 ∼ thus mx/mx → k = k(x) is a map of k-vector spaces, i.e. an element 2 ∗ of (mx/mx) . This is the Zariski tangent space to X at x. Thus giving D → X is the same as giving a point x ∈ X with k(x) = k plus a Zariski tangent vector at x. glued scheme (special case of example sheet 1 Q14): given schemes X1,X2 and U1 ⊆ X1,U2 ⊆ X2 open subsets with induced scheme structure (Ui, OXi |Ui ) and an isomorphism f : U1 → U2, we can then glue X1,X2 along f to get a 0 0 0 ∼ 0 0 ∼ scheme X with open subsets X1,X2 with Xi = Xi,X1 ∩ X2 = Ui. n 1 Example. Define Ak = Spec k[x1, . . . , xn]. In particular Ak = Spec k[x]. Glue

1 1 U1 = Ak \{0} ⊆ Ak = X1 1 1 U2 = Ak \{0} ⊆ Ak = X2 where 0 denotes the maximal ideal at 0 via id : U1 → U2, then we obtain the affine line with double origin. This is pathological in some sense asinthe classical topology this is non-Hausdorff. Later we will be able to pin down precisely what the pathology is. −1 Alternatively, glue U1 to U2 via x 7→ x : since

−1 Ui = D(x) = Spec k[x]x = Spec k[x, x ] we specify this map via the ring homomorphism k[x, x−1] → k[x, x−1], x 7→ x−1. 1 This gives a glued scheme Pk, the projective line.

2.1 Projective schemes The gluing contruction brings about a whole class of schemes that are not affine, called projective schemes. They are geometric objects associated to graded rings. L Let S = Sd be a graded ring, meaning that Sd · Se ⊆ Sd+e. For d≥0 L example S = k[x1, . . . , xn] = d≥0 Sd where Sd is the space of homogeneous polynomials of degree d. If f ∈ Sd, we say f is homogeneous of degree d and L write deg f = d. Define S+ = d>0 Sd, the irrelevant ideal. An ideal I ⊆ S is homogeneous if it is generated by homogeneous elements. Define

Proj S = {p ⊆ S homogeneous prime ideals not containing S+}.

1 Example. Suppose k is algebraically closed and define Pk = Proj k[x0, x1]. The maximal ideals of k[x0, x1] are of the form (x0 − a0, x1 − a1), which is homogeneous if and only if a0 = a1 = 0. Hence no maximal ideal of k[x0, x1] lies in Proj k[x0, x1]. By results in commutative algebra, k[x0, x1] has dimension 2 and all other primes of k[x0, x1] are either 0 or principal. Consider p = (f) with f homogeneous. Since k = k, f can be factored into linear factors. So if p is prime, in fact p = (a1x0 − a0x1) for some a0, a1 ∈ k not both 0. Note that the generators of the ideal are written in this way to show that (a0, a1) is only defined by up to scaling by an element of k×, so these points are in one-to-one correspondence with points of (k2 \{0})/k∗.

16 2 Schemes

Back to the construction of projective schemes. We start to define a topology and a structure sheaf on Proj S. For I ⊆ S homogeneous, define

V (I) = {p ⊆ Proj S : p ⊇ I}.

Check these are the closed sets for a topology in Proj S. Fix p ∈ Proj S. Let

T = {f ∈ S \ p : f homogeneous} ⊆ S

−1 which is a multiplicatively closed set, and let S(p) ⊆ T S be the subring of a elements of degree 0, where we define deg b = deg a − deg b. We require the numerator and the denominator to be the same degree so that it defines a well- defined function on Proj S. Similarly if f ∈ S is homogeneous write S(f) ⊆ Sf for the set of elements of degree 0 in Sf . We now define the structure sheaf OProj S. For U ⊆ S, define  `  s : U → S(p): s(p) ∈ S(p),  p∈U   for each p ∈ U, exists p ∈ V ⊆ U  OProj S(U) =  and a, f ∈ S homogeneous of the same degree   a  such that f∈ / q, s(q) = f ∈ S(q) for all q ∈ V

This defines OProj S with OProj S,q = S(q), making (Proj S, OProj S) a locally ringed space.

Proposition 2.4. For f ∈ S+ homogeneous, define

D+(f) = {p ∈ Proj S : f∈ / p},

an open subset of Proj S. Then D+(f)’s cover Proj S and ∼ (D+(f), OProj S|D+(f)) = Spec S(f).

In particular (Proj S, OProj S) is a scheme.

Proof. Example sheet 2.

n Definition. If A is a ring we define PA = Proj A[x0, . . . , xn] with deg xi = 1 and deg a = 0 for all a ∈ A.

n If A = k is an algebraically closed field then the points of Pk corresponding to the largest ideals in Proj S are in one-to-one correspondence with points in (kn+1 \{0})/k∗.

Definition (closed point). A point x ∈ X a scheme is a closed point if {x} is closed.

In Spec A, the closed points are precisely the maximal ideals as for any p ∈ Spec A, {p} = V (p). In this language, the set of closed points in Proj S are in one-to-one correspondence with (kn+1 \{0})/k∗.

17 2 Schemes

Example (weighted projective space). Let S = k[x0, . . . , xn] but with deg xi = wi for some positive integers w0, . . . , wn. Then Proj S is the weighted projective n space W P (w0, . . . , wn). Consider W P2(1, 1, 2). Consider

Spec S(xi) = D+(xi) = {p ∈ Proj S : xi ∈/ p}.

For S(x2), there are several ways to make the numerator having the same degree as x2 and x2 x x x2 k[U, V, W ] S = k[ 0 , 0 1 , 1 ] ∼ . (x2) = 2 x2 x2 x2 (UW − V ) k[U,V,W ] 2 3 Note Spec (UW −V 2) is the quadratic cone UW = V in Ak which has a singular point. n n Example. Let M = Z and MR = M ⊗Z R = R . Let ∆ ⊆ MR be a compact convex lattice polytope, i.e. there exists a finite set V ⊆ M such that ∆ is the convex hull of V in MR. Set

C(∆) = {(m, r): m ∈ r∆, r ∈ R, r ≥ 0} ⊆ MR ⊕ R. This is a rational polyhedral convex cone. Consider the monoid ring

M p S = k[C(∆) ∩ (M ⊕ Z)] = k · z , p∈C(∆)∩(M⊕Z)

0 0 with multiplication zp · zp = zp+p . S is graded via deg z(m,r) = r and check that (m1,r1) (m2,r2) (m1+m2,r1+r2) deg z · z = deg z = r1 + r2.

We define P∆ = Proj S. This is a projective toric variety. Example.

1. Take ∆ to be the convex hull of {0, e1, . . . , en} where e1, . . . , en is the standard basis for M. Then there is an isomorphism of rings

k[x0, . . . , xn] → k[C(∆) ∩ (M ⊕ Z)] ( z(0,1) i = 0 xi 7→ z(ei,1) i 6= 0

In other words C(∆)∩(M⊕Z) is generated freely by (0, 1), (e1, 1),..., (en, 1). n Thus P∆ = P . 2. Take ∆ to be the convex hull of (0, 0), (0, 1), (1, 0), (1, 1). One can check that C(∆) ∩ (M ⊕ Z) is generated as a monoid by (0, 0, 1), (1, 0, 1), (1, 1, 1) and (0, 1, 1) with only one relation (0, 0, 1) + (1, 1, 1) = (1, 0, 1) + (0, 1, 1)

so k[U, V, W, Z] S =∼ (UW − VZ) so Proj S is a quadratic surface.

18 2 Schemes

2.2 Open and closed subschemes

Definition (open subscheme). An open subscheme of a scheme X is (U, OX |U ) for U ⊆ X an open subset.

Check U can still be covered by affine schemes so is a scheme.

Definition (open immersion). An open immersion f : X → Y is a mor- phism which induces an isomorphism of X with an open subscheme of Y .

Note that this convention is different from that in differential geometry, where an immersion is a map inducing injective maps on tangent spaces and can itself be non-injective.

Definition (closed immersion). A closed immersion f : X → Y is a mor- phism which is a homeomorphism of X onto a closed subset of Y , and the # induced morphism f : OY → f∗OX is surjective.

Definition (closed subscheme). A closed subscheme of Y is an equivalence class of closed immersions where f : X → Y, f 0 : X0 → Y are equivalent if there exists an isomorphism i : X → X0 with f = f 0 ◦ i.

Example. 1. Let Y = Spec A, I ⊆ A an ideal and X = Spec A/I. Then have the quo- tient map ϕ : A → A/I inducing f : X → Y . This is a homeomorphism of X with V (I) ⊆ Y . Also OY → f∗OX is surjective since it is surjective on stalks: OY,p = Ap and ( 0 p ∈/ V (I) (f∗OX )p = (A/I)p p ∈ V (I)

The point is that the space Spec A/I contains less information than the k[x,y] 2 scheme structure. For example consider Spec (x2,xy) . We have V (x , xy) = V (x) so the closed points can be identified with the y-axis. However, x is nilpotent in the quotient ring so the origin behaves differently from the origin in the affine line. The origin in this case is an example ofan embedded point.

1 2 2. Non-example: consider the morphism Ak → Ak induced by k[x, y] → k[t] x 7→ t2 y 7→ t3

Then the map is not surjective so this is not an immersion, although the 2 3 2 image V (y − x ) is closed in Ak and the map is a homeomorphism onto its image.

19 2 Schemes

2.3 Fibre products Let C be a category and consider a diagram in the category

Y g f X Z

Then the fibre product, if exists, is an object W equipped with morphisms p : W → X, q : W → Y with f ◦ p = g ◦ q such that for any object W 0 and morphisms ρ0 : W 0 → X, q0 : W 0 → X such that f ◦ p0 = g ◦ q0 then there exists a unique map h : W 0 → W such that p0 = p ◦ h, q0 = q ◦ h. In other words, the following diagram commutes:

0 W 0 q h

q p0 W Y p g f X Z

If it exists then it is unique up to a unique isomorphism. If W exists we write it as X ×Z Y .

Example. Let C be Set, the category of sets. Then

X ×Z Y = {(x, y) ∈ X × Y : f(x) = g(y)}.

Digression on Yoneda lemma and general approach to construct universal objects.

Definition. Let C be a category. Write hX for the contravariant functor hX : C → Set,Y 7→ Hom(Y,X) and

hX (f : Y → Z) : Hom(Z,X) → Hom(Y,X) ϕ 7→ ϕ ◦ f

Recall that a natural transformation T : F → G of (contravariant) functors F,G : C → D is data T (X): F (X) → G(X) for each object X in C and for each morphism f : X → Y in C a commutative diagram

F (f) F (Y ) F (X)

T (Y ) T (X) G(f) G(Y ) G(X)

Theorem 2.5 (Yoneda lemma). The set of natural transformations between hX : C → Set and (contravariant) G : C → Set is G(X).

20 2 Schemes

Sketch proof. Given η ∈ G(X), we define for each object Y a map

hX (Y ) → G(Y ) f 7→ G(f)(η)

Conversely given T : hX → G, set η to be T (X)(idX ).

Corollary 2.6. NatTrans(hX , hY ) = hY (X) = Hom(X,Y ).

Remark. When X is a scheme, we write X(T ) for the set of T -valued points of X, which is precisely hX (T ).

Definition (representable functor). A contravariant functor F : C → Set ∼ is representable if F = hX for some object X in C.

We can redefine fibre product as follow: the fibre product in a categoryis an object which represents the functor

T 7→ Hom(T,X) ×Hom(T,Z) Hom(T,Y )

f g (with X −→ Z ←− Y given). The advantage of phrasing universal property in terms of representable func- tors is that it enables us to prove identities easily. For example, in Set we have ∼ (A ×B C) ×C D = A ×B D which is simply ((a, c), d) 7→ (a, d). Then we get this identity for all fibre products (if exist) for free: the two functors

T 7→ (hA(T ) ×hB (T ) hC (T )) ×hC (T ) hD(T )

T 7→ hA(T ) ×hB (T ) hD(T ) are naturally isomorphic by the map on sets just written down, so by Yoneda they are represented by isomorphic objects.

Theorem 2.7. Fibre products exist in the category of schemes.

Proof. This is an prolonged construction and we proceed in several steps. Step 1: suppose X = Spec A, Y = Spec B,S = Spec R. Assume given f g A ←− R −→ B inducing X → S ← Y . Tensor product is pushout in the category of rings. C

A ⊗R B B g f A R

Thus Spec A⊗R B = Spec A×Spec R Spec B in the category of affine schemes. In general, fibre products in a category are not the same as those in a subcategory,

21 2 Schemes since we have different test objects. But if T is an arbitrary scheme, giving a map T → Spec A is the same as A → Γ(T, OT ) so we can replace Spec C with an arbitrary scheme T . Thus X ×S Y = Spec A ⊗R B. Step 2: We will construct the general fibre proudct via gluing schemes (ex- ample sheet 1 question 14). We will also want to glue morphisms: given an open cover {Ui} of a scheme X and morphisms fi : Ui → Y with fi|Ui∩Uj = fj|Ui∩Uj then exists a unique morphism f : X → Y such that f|Ui = fi. This is an exercise. Step 3: Given X,Y → S such that X ×S Y exists with projections p1 : X ×S Y → X, p2 : X ×S Y → Y , suppose U ⊆ X is an open subscheme. −1 Then p1 (U) ⊆ X ×S Y is U ×S Y : given maps f1 : T → U, f2 : T → Y , by composition we get a map T → X so by universal property of fibre product there is a map h : T → ×SY . As p2 ◦ h = i ◦ f1, must have h factors through −1 p1 (U).

f T 2

h

−1 p2 f1 p1 (U) X ×S Y Y

p1 U i X S

Step 4: Let {Xi} be an open cover of X and suppose Xi ×S Y exists for each i. Then claim X ×S Y exists. −1 Proof. Let Xij = Xi ∩ Xj,Uij = p1 (Xij) = Xi ×S Y . Then Uij = Xij ×S Y by step 3 and hence by uniqueness of fibre product up to a unique isomorphism, we get an isomorphism ϕij : Uij → Uji. As an exercise, check these isomorphism satisfy the hypotheses of example sheet 1 question 14. We can then glue the schemes Xi ×S Y along the Uij’s to get a scheme X ×S Y . 0 0 Need to show this is the fibre product. Suppose given p1 : T → X, p2 : 0 −1 T → Y . Set T1 = (p1) (Xi) so get a map θi : Ti → Xi ×S Y by universal property. We have an open immersion Xi ×S Y,→ X ×S Y so denote also by θi the composition of θi with the inclusion. On Ti ∩ Tj, θi and θj agree since they factor through Xij ×S Y ⊆ Xi ×S Y and Xji ×S Y ⊆ Xj ×S Y respectively. Thus using step 2, we can glue the θi’s to get θ : T → X ×S Y .

Step 5: Using step 4 we can construct X ×S Y when S and Y are affine by taking {Xi} to be an affine cover. Similarly we obtain X ×S Y for X,Y arbitrary by reversing the roles of X and Y . Step 6: For X,Y,S arbitrary, let {Si} be an affine cover of S. Take Xi,Yi to be the preimages of Si in X,Y . Then Xi ×Si Yi exists by step 5 and Xi ×Si Yi = Xi ×S Yi by the universal property. Glue as before. scheme-theoretic fibres of morphisms Give a set-theoretic map f : X → Y , one way to describe the fibre of y ∈ Y is to form the fibre product ∼ −1 {y} ×Y X = {(y, x): y = f(x)} = f (X).

This is completely analogous for schemes. Given a morphism of schemes f : X → Y and y ∈ Y , we have k(y) = OY,y/my and a morphism Spec k(y) → Y

22 2 Schemes with image y. We define

Xy = Spec k(y) ×Y X to be the scheme-theoretic fibre of f at y. Example. Let X = Spec k[x],Y = Spec k[t] and a morphism f : X → Y induced by k[t] → k[x], t 7→ x2. Suppose for the moment k = k. Let y = (t − a) for some a ∈ K. Then k(y) = k[t]/(t − a) so Spec k(y) → Spec k[t] is induced k[t] by the quotient map k[t] → (t−a) . Then k[t] k[x] X = Spec ⊗ k[x] = Spec y (t − a) k[t] (x2 − a) by noting that (A/I) ⊗A M = M/IM. If a 6= 0 and char k 6= 2 then by Chinese remainder theorem k[x] k[x] k[x] = √ × √ (x2 − a) (x − a) (x + a) so the fibre are just two points. If a = 0 or char k = 2 we get a ring with k[x] √ nilpotents ((x+ a)2) . In some ways this indicates ramification at 0, and this confirms our previous remark that the scheme captures more information than the underlying set. If we instead take y = (0) ∈ Y , then k(y) = k[t](0)/m(0) = k(t) so k(t)[x] √ X = Spec k(t) ⊗ k[x] = Spec = Spec k( t). y k[t] (x2 − t)

Definition (integral scheme). A scheme is integral if for every open set U ⊆ X, OX (U) is an integral domain.

Definition (morphism (locally) of finite type). Let f : X → Y be a mor- phism of schemes. Then f is locally of finite type if there is a covering of −1 Y by open affine subsets Vi = Spec Bi such that for each i, f (Vi) can be covered by open affines Uij = Spec Aij with each Aij a finitely-generated Bi-algebra. −1 We say f is of finite type if the cover {Uij} of f (Vi) can always be taken to be finite. 1 Exercise. Let Ui = Pk for i ∈ Z and identify 0 ∈ Ui with ∞ ∈ Ui+1. Let ` X = Ui/ ∼. Show the morphism X → Spec k is locally of finite type but i∈Z not of finite type. We can now redefine a variety to be

Definition (variety over k). Let k be a field. A variety over k is a scheme X over Spec k such that X is integral and X → Spec k is finite type.

Note. This still allows the affine line with double origin. More exercises on properties of schemes and morphisms can be found on example sheet 2.

23 2 Schemes quick remarks on separated and proper morphisms The line with dou- ble origin is non-Hausdorff in the Euclidean topology, but as Zariski topology is so coarse, Hausdorffness does not suit the algebraic geometry world. Instead, we use the characterisation that X is Hausdorff if and only if ∆X ⊆ X × X is closed.

Definition (separated morphism). Let f : X → Y be a morphism of schemes and ∆ : X → X×Y X the diagonal morphism induced by f : X → Y and id : X → X. We say f is separated if ∆ is a closed immersion.

Exercise. The affine line with two origins in not separated.

Proposition 2.8 (valuative criterion for separatedness). Let f : X → Y be a morphism with X Noetherian. Then f is separated if and only if the following condition holds: for any field K and any valuation ring R ≤ K (R is an integral domain with field of fractions K and for all x ∈ K∗ either x ∈ R or x−1 ∈ R), let T = Spec R,U = Spec K and U → T the morphism induced by R,→ K. Given any commutative diagram

U X h f T Y

there is at most one h making the diagram commute.

One may ask the dual question when such h exists.

Definition (proper morphism). A morphism f : X → Y is proper if it is separated, of finite type and universally closed, i.e. for all Y 0 → Y , the 0 0 morphism X ×Y Y → Y is a closed map.

Proposition 2.9 (valuative criterion for properness). Let f : X → Y be a morphism of finite type with X Noetherian. Then f is proper if and only if in the set up of the criterion for separatedness, there exists a unique morphism h making the diagram commute.

Example. If X is a projective variety over Spec k (i.e. a closed subscheme of n Pk ) then X → Spec k is proper.

24 3 Sheaves of OX -modules

3 Sheaves of OX-modules

Definition (sheaf of modules). Let (X, OX ) be a ringed space. A sheaf of OX -modules is a sheaf F of abelian groups such that F(U) has the structure of OX (U)-module compatible with restriction, i.e. if V ⊆ U, s ∈ OX (U), m ∈ F(U) then (s · m)|V = s|V · m|V . A morphism of sheave of OX -modules f : F → G is a morphism of sheaves such that fU : F(U) → G(U) is an OX (U)-module homomorphism.

Kernels, cokernels and images of such morphisms are sheaves of OX -modules. We have

HomOX (F, G) = {f : F → G : f morphism of OX -module} an OX (X)-module. We define the Hom sheaf to be

HomOX (F, G)(U) = HomOX |U (F|U , G|U ),

an OX (U)-module. Then HomOX (F, G) is a sheaf of OX -modules.

We define F⊗OX G to be the sheaf associated to the presheaf U 7→ F(U)⊗OX (U) G(U). Given f : X → Y a morphism of ringed spaces, F a sheaf of OX -module, G a sheaf of OY -modules, then

1. f∗F is naturally a sheaf of OY -modules as f∗F is a sheaf of f∗OX -modules, # and the morphismf : OY → f∗OX gives f∗OX an OY -algebra structure.

−1 −1 2. f G is an f OY -module, but not necessarily an OX -module. However −1 ∗ # by adjunction between f and f on example sheet 1, f : OY → f∗OX # −1 gives a map f : f OY → OX . Then we define

∗ −1 −1 f G = f G ⊗f OY OX

which is an OX -module.

Definition ((locally) free sheaf, line bundle, invertible sheaf). A sheaf of L OX is free if it is isomophic to i∈I OX for some index set I. If #I = r < ∞ we say the sheaf is free of rank r. A sheaf F is locally free of rank r if exists {Ui} open cover of X with

F|Ui free of rank r for all i. Finally we say F is a line bundle or invertible sheaf if it is locally free of rank 1.

Example. Let X = Spec A and M be an A-module. We obtain a sheaf Mf as follow.  s : U → ` M s(p) ∈ M   p∈U p: p,  Mf(U) = for each p ∈ U, exists p ∈ V ⊆ U and m ∈ M, a ∈ A  m  such that a∈ / q, s(q) = a ∈ Mq for all q ∈ V

25 3 Sheaves of OX -modules

Proposition 3.1.

1. (Mf)p = Mp.

2. Mf(D(f)) = Mf . In particular Γ(X, Mf) = M.

Proof. Exercise.

Definition ((quasi-)coherent sheaf). Let X be a scheme and F a sheaf of OX -modules. We say F is quasi-coherent if exists a cover {Ui = Spec Ai}

of X and Ai-modules Mi such that F |Ui = Mfi. If further Mi is a finitely genertated Ai-module for each i we say F is coherent.

Remark. The category of quasi-coherent (coherent respectively) sheaves on X = Spec A is equiavlent to the category of A-modules (finitely generated A- modules respectively). This is not obvious and the key point is to know that if F is a quasi-coherent sheaf on Spec A then F = Γ(^X, F).

Example. 1. A locally free sheaf is quasi-coherent, and coherent if finite rank: if F

is locally free then there exists an open cover {Ui} such that F|Ui = L ∼ L^ i∈I OUi . If Ui = Spec A then F|Ui = i∈I A. 2. Kernels, cokernels and images of morphisms of quasi-coherent sheaves are quasi-coherent. It suffices to show M 7→ Mf is exact so, for example, ker(Mf → Ne) =∼ ker(^M → N). If X is Noetherian then the same holds for coherent sheaves (as sub- modules of finitely generated modules over a Noetherian ring is finitely generated).

Example. Note that if L is a line bundle with a trivialising cover {Ui} then on Ui ∩ Uj we have

∼= ∼= OUi |Ui∩Uj L|Ui∩Uj OUj |Ui∩Uj where the isomorphisms are given by trivialisations on Ui and Uj respectively.

Thus for each i 6= j, we get an automorphism ϕij : OUi∩Uj → OUi∩Uj of OUi∩Uj - ∗ modules. ϕij must be given by ϕij(s) = gijs for some gij ∈ OX (Ui ∩ Uj) where

∗ OX (U) = {f ∈ OX (U): f is invertible}.

We call the gij’s transition functions. Furthermore On Ui ∩ Uj ∩ Uk, we have ϕik = ϕjk ◦ ϕij. Given f : X → Y and L a line bundle on Y , we can explicitly describe ∗ f L using transition functions. Let {Ui} be a trivialising cover for L and let −1 Xi = f (Ui) and fi : Xi → Ui the restriction of f. Then

∗ ∗ −1 f (L|U ) = f (OU ) = f OU ⊗ −1 OX = OX i i i i i i f OUi i i

26 3 Sheaves of OX -modules

∗ ∼ so f (L)|Xi = OXi . Thus the pullback of a line bundle is a line bundle, and # ∗ the transition functions are f (gij) ∈ OX (Xi ∩ Xj). More generally the pullback of a locally free sheaf is a locally free sheaf.

Example. Let L1, L2 be line bundles with a common trivialising cover {Ui} and let gij, hij be the transition functions for L1 and L2. Then

1. the transition functions for L1 ⊗OX L2 are gij · hij: if M1,M2 are rank 1 free A-modules, α : M1 → M1, m 7→ gm, β : M2 → M2, m 7→ hm then α ⊗ β : M1 ⊗ M2 → M1 ⊗ M2 is given by

m1 ⊗ m2 7→ gm1 ⊗ hm2 = (g · h)(m1 ⊗ m2).

∨ −1 ∨ 2. set L1 = L1 = HomOX (L1, OX ). Then the transition maps of L1 are −1 gij : if M is a rank 1 free A-module and α : M → M, m 7→ gm then one might guess that M ∨ → M ∨ is given by ϕ 7→ gϕ. However, now we are going in the wrong way because of the dualising, so we invert g. ∨ ∼ Remark. As a result L ⊗OX L = OX for any line bundle L.

Definition (Picard group). Let X be a scheme. Denote by Pic(X), the Picard group, to be the set of isomorphism classes of line bundles on X.

This is a group with multiplication (L1, L2) 7→ L1 ⊗OX L2 and inverse L 7→ L∨. This is the most important invariant associated to a scheme.

3.1 Morphisms into projective space n Fix a base scheme Spec k and Pk = Proj k[x0, . . . , xn]. Let Sch/k be the cate- gory of schemes over k. Let F : Sch/k → Set be the functor

 O⊕(n+1) L  Ob(Sch/k) 3 T 7→ surjections T  / ∼ for L a line bundle on T (f : T → T ) 7→ (O⊕(n+1) L) 7→ (f ∗O⊕(n+1) = O⊕(n+1) f ∗L) 1 2 T2  T2 T1  with an isomorphism being a commutative diagram

⊕(n+1) OT L1

∼=

L2

Surjectivity is by right exactness of tensor product.

n ∼ Theorem 3.2. The contravariant functor F is represented by Pk , i.e. F = h n Pk .

n n 1 n ∈ h n ( ) Proof. If this holds there is a universal object on Pk , i.e. Pk Pk Pk cor- n ⊕(n+1) responds to some element in F ( ), i.e. a surjection O n L. Further, Pk P 

27 3 Sheaves of OX -modules

n following the proof of Yoneda, given T : hP → F the natural transformation n giving the equivalence and f : X → P in hPn (X), we get a commutative diagram

n n T (P ) n n hP (P ) F (P )

⊕(n+1) 1 n (O n L) P P 

h n (f) F (f) P

⊕(n+1) ∗ f (OX  f L)

T (X) hn(X) F (X) P

⊕(n+1) ∗ n so the element of F (X) corresponding to f ∈ hP (X) is just OX  f L. n ⊕(n+1) Representability means that comes with a universal object O n L, P P  ⊕(n+1) 0 such that for any X and OX  L , there exists a unique morphism f : X → Pn such that ⊕(n+1) ∗ ∼ ⊕(n+1) 0 (OX  f L) = (OX  L ).

n ⊕(n+1) n We will write the universal object in as O n O n (1). has P P  P P an open cover {D+(xi) : 0 ≤ i ≤ n} and D+(xi) = Spec S(xi) where S = x0 xˆi xn k[x0, . . . , xn],S = k[ ,..., ,..., ]. We take this to be the trivialising (xi) xi xi xi cover with transition functions

xi ∗ g = ∈ O n (D (x ) ∩ D (x )) = S ij P + i + j (xixj ) xj

xi as D+(xi) ∩ D+(xj) = D+(xixj). We will later see why we choose and not xj xj ⊕(n+1) . Have a sheaf morphism O n → O n (1) defined on D+(xi) by xi P P

O⊕(n+1) → O D+(xi) D+(xi) xk ek 7→ xi where ek is 1 on kth coordinate and 0 elsewhere. This is well-defined on overlaps:

xi − · : OD+(xi)|D+(xixj ) → OD+(xj )|D+(xixj ) xj x x k 7→ k xi xj

xi Further this map is surjective on D+(xi) as ei 7→ = 1, so for any ϕ ∈ xi

OD+(xi)(D+(xi)), ϕ · ei 7→ ϕ. This implies surjectivity on stalks. ⊕(n+1) Now suppose given T a scheme over k and ϕ : OT  L, we want to construct a morphism f : T → Pn such that ∗ ⊕(n+1) ∼ ⊕(n+1) (f (O n O n (1))) = (O L) P  P T 

28 3 Sheaves of OX -modules

⊕(n+1) Let ϕ(ei) = si ∈ Γ(T, L) where ei = (0,..., 1,..., 0) ∈ Γ(T, OT ). Define

Zi = {t ∈ T :(si)t ∈ mtLt} which makes sense as Lt is a OT,t-module. Claim that Zi is a closed set: being closed can be checked on an open cover {Uj}, i.e. by checking that Uj ∩ Zi is closed in Uj for all j, so can take affine cover which also trivialises L. Consider si open subset Spec A and L = Ae = OSpec A and si ∈ A induces 1 ∈ Ap for all p ∈ si Spec A. thus p ∈ Ui if and only if 1 ∈ mpAp, if and only if si ∈ A ∩ mpAp = p. Thus (Spec A) ∩ Zi = V ((si)) T is a closed subset. Note that Zi = ∅ by surjectivity of ϕ. Let Ui = T \ Zi open. Then

OUi → L|Ui

1 7→ si is an isomorphism because it is isomorphism on stalks and we denote its inverse s by s 7→ . Now define fi : Ui → D+(xi) = Spec S by giving a k-algebra si (xi) homomorphism x0 xn k[ ,..., ] → Γ(Ui, OUi ) xi xi x s k 7→ k xi si

On Ui ∩ Uj, want to show fi|Ui∩Uj = fj|Ui∩Uj : # (f ) : S = Γ(D (x ) ∩ D (x ), O n ) → Γ(U ∩ U , O ) i Ui∩Uj (xixj ) + i + j P i j T x s k 7→ k xi si x x x s s s k = k i 7→ k i = k xj xi xj si sj sj and (f )# : S → Γ(U ∩ U , O ) j Ui∩Uj (xixj ) i j T x s k 7→ k xj sj x x x s s s k = k j 7→ k j = k xi xj xi sj si si n so the ring maps agree and we may glue the fi’s to get a morphism f : T → P . The intuition is as follow: sections si of a line bundle are analogous to homogeneous coordinates xi of projective space. In particular they are not functions although their quotients are. In other words, t 7→ (s0(t), . . . , sn(t)) is defined only up to scaling. What we’re really doing is todefine f : T → Pn, t 7→ (s0(t): ··· : sn(t)) piecewise on affine charts. ∗ ∼ # xi si Note that f O n (1) = L since f ( ) = on Ui ∩ Uj, but the transition P xj sj maps for L on Ui ∩ Uj are

OUi |Ui∩Uj → L|Ui∩Uj → OUj |Ui∩Uj si 1 7→ si 7→ sj

29 3 Sheaves of OX -modules

x so they agree. Note that had we used transition functions j , we would get a xi n different line bundle (the tautological bundle OP (−1)) whose pullback is not isomorphic to L. ⊕(n+1) n For uniqueness, suppose given ϕ : OT  L and a morphism g : T → P such that ∗ ⊕(n+1) ∼ g (O n O n (1)) = ϕ. P  P n ⊕(n+1) n Let si = ϕ(ei) ∈ Γ(T, L) and xi ∈ Γ(P , OP (1)) the image of ei ∈ Γ(Pn, O n ). ∗ ∗ −1 P n n Then the given isomorphism gives si = g xi (recall g OP (1) = g OP (1)⊗g−1O n ∗ P OT so g xi equals to the image of xi ⊗ 1 under sheafification). Now note that si vanishes at a point t ∈ T , i.e. si ∈ mtLt, if and only if xi vanishes at the point −1 g(t). Thus if Ui = {t ∈ T : si ∈/ mtLt} then necessarily g (D+(xi)) = Ui. It # xj sj # xj sj is thus enough to show fi = g|U . But g ( ) = ∈ OT (Ui) and f ( ) = i xi si i xi si by definition.

⊕(n+1) Let s0, . . . , sn ∈ Γ(X, L) be images of ei under OX . If this morphism is surjective (as in the case above), we say s0, . . . , sn generate L globally, and say L is generated by global sections. By the proposition we get a morphism X → Pn, and understanding morphisms into projective space is important to understand varieties.

3.2 Divisors and the Picard groups The aim of this section is to define two divisors 1. Weil divisors: codimension 1 subvariety. 2. Cartier divisor: a subvariety locally defined by a single equation. We begin by defining dimension and codimension.

Definition (dimension of a topological space). The dimension of a topo- logical space X is the length d of the longest chain

Z0 ( Z1 ( Z1 ··· ( Zd ⊆ X

of irreducible closed subsets of X.

Definition (Krull dimension). The Krull dimension of a ring A is

dim A = dim Spec A

which equals to the length d of the longest chain of prime ideals

p0 ( p1 ( ··· ( pd

of A.

Definition (codimension). If Z ⊆ X is an irreducible closed subset then the codimension of Z in X, codim(Z,X), is the length d of the longest chain

Z = Z0 ( Z1 ( ··· ( Zd ⊆ X

30 3 Sheaves of OX -modules

of irreducible closed subsets. More generally, if Z ⊆ X is closed then

codim(Z,X) = inf codim(Z0,X). Z0⊆Z irreducible

Definition (height). If p ∈ Spec A we define the height of p to be

ht p = codim(V (p), Spec A)

which is the length d of the longest chain of primes

p0 ( ··· ( pd = p. Remark. Codimension and dimension don’t always behave as expected even for Noetherian affine schemes. But if B is a finitely generated k-algebra and integral domain, then for any p ∈ Spec B, ht p + dim B/p = dim B or equivalently, codim(V (p), Spec B) + dim V (p) = dim Spec B. Thus for a variety X (i.e. an integral scheme of finite type over k), if Z ⊆ X is an irreducible closed subset then codim(Z,X) + dim Z = dim X. Moreover if η ∈ Z is the generic point of Z, i.e. the unique η ∈ Z such that {η} = Z (see example sheet 3) then dim OX,η = codim(Z,X).

Proposition 3.3. If X is a Noetherian topological space (i.e. satisfies the descending chain condition for closed subsets) then there exists a decomposi- tion X = X1 ∪ · · · ∪ Xn into irreducible closed subsets. Moreover if Xi * Xj for i 6= j then the decomposition is unique up to reordering. Proof. Exercise.

Definition (regular local ring). A local ring (A, m) is a regular local ring if 2 dimA/m m/m = dim A.

For dealing with divisors we assume X is a Noetherian integral scheme over Spec k which is regular in codimension one, i.e. for any x ∈ X with dim OX,x = 1, OX,x is a regular local ring. Remark. It is a standard commutative algebra fact (Atiyah, MacDonald chap- ter 9) that if (A, m) is a regular Noetherian local ring which is a domain and dim A = 1 then A is a discrete valuation ring, i.e. if K is the field of frac- tions of A then there exists a group homomorphism v : K∗ → Z such that v(mk \ mk+1) = k. In particular A = {x ∈ K∗ : v(x) ≥ 0} ∪ {0} m = {x ∈ K∗ : v(x) ≥ 1} ∪ {0}

31 3 Sheaves of OX -modules

Example.

1. k[x](x−a) is a DVR with valuation

∗ v : k(x) → Z f (x − a)k 7→ k g

if f, g ∈ k[x] and (x − a) - f, g. ∗ k a 2. The number theoretic analogue is Z(p) with v : Q → Z, p b 7→ k. We emphasise again the assumptions put on scheme X.

Definition (prime divisor). A prime divisor of X is a closed subvariety, i.e. an integral closed subscheme of X, of codimension 1.

Let Div X be the free abelian group generated by prime divisors on X. An element of Div X is call a divisor. Let K(X) be the function field of X (see example sheet 2). Note that K(X) is the field of fractions of A whenever Spec A ⊆ X is an affine open subset. It is also OX,η, the stalk at the generic point. For Y a prime divisor, let η ∈ Y ⊆ X be its generic point. Then

dim OX,η = codim(Y,X) = 1.

∗ Thus OX,η is a DVR with valuation vY : K(X) → Z that measures the order of zero/pole of a function along Y . In particular

∗ OX,η = {f ∈ K(X) : vY (f) ≥ 0} ∪ {0}.

Define the divisor of zeros and poles of f ∈ K(X)∗ to be X (f) = vY (f) · Y. Y prime divisor

A priori (f) may not be in Div X as the sum might be infinite. But we have

Lemma 3.4. With X satisfying the assumptions, if f ∈ K(X)∗ then vY (f) = 0 for all but a finite number of prime divisors.

s Proof. First pass to an open affine U = Spec B ⊆ X with f = a , s ∈ B, a ∈ B \{0}. By passing to D(a) we can assume f ∈ B. Let Z = X \ U. Then Z is a Noetherian topological space so Z has a finite decomposition into irreducible components. Thus Z contains a finite number of prime divisors, so we can replace X with Spec B as any prime divisor Y intersecting U must have its generic point ηY in U, and U ∩ Y is a prime divisor of U. Now vY (f) ≥ 0 for Y ⊆ X since f ∈ B and vY (f) > 0 if and only if Y ⊆ V ((f)). But V ((f)) is a proper closed subset of X and hence has a finite decomposition into irreducible components. Thus there are only finitely many prime divisors with vY (f) > 0.

∗ Note K(X) → Div X, f 7→ (f) is a group homomorphism as vY is.

32 3 Sheaves of OX -modules

Definition (linearly equivalent divisor, principal divisor). Two divisors D,D0 ∈ Div X are linearly equivalent if exists f ∈ K(X)∗ with D−D0 = (f) and we write D ∼ D0. If D ∼ 0, i.e. D = (f) for some f we say D is a principal divisor.

Definition (class group). The class group of X, Cl X, is Div X modulo the principal divisors.

Remark. If Q ⊆ K is a finite field extension, let OK ⊆ K be the integral closure of Z in K, i.e. the ring of integers. Then from algebraic number theory we know OK is a Dedekind domain and hence Spec OK satisfies our assumptions (except for being a k-scheme). Then Cl Spec OK = Cl OK as defined in number theory.

Theorem 3.5. Let A be a Noetherian domain. Then A is a UFD if and only if X = Spec A is normal, i.e. A is integrally closed in its field of fractions, and Cl X = 0.

Proof. A UFD is integrally closed in its field of fractions. Also A is a UFD if and only if every height 1 prime is principal. Thus it is enough to show that if A is integrally closed, then Cl X = 0 if and only if every prime p ⊆ A of height 1 is principal. • ⇐= : given a prime divisor Y ⊆ X, Y corresponds to a prime p with Y = V (p). If p is generated by f then (f) = 1 · Y . Thus every prime divisor is principal so Cl X = 0. • =⇒ : suppose Cl X = 0, p ⊆ A a height 1 prime and Y = V (p) a prime divisor. Then Y is a principal divisor so exists f ∈ K(X)∗ such that (f) = 1 · Y . Since vY (f) = 1, f generates pAp = mpOX,p. 0 For any other prime divisor Y = V (q), vY 0 (f) = 0 so f ∈ Aq = OX,q and moreover f is a unit in Aq. Now apply the following proposition (Matsumura, Commutative Algebra, Thm 38, pg 124): if A is an integrally closed Noetherian domain then \ A = Ap ⊆ A(0). p∈Spec A ht p=1

Thus f ∈ A so f ∈ A ∩ pAp = p. To show f generates p, let g ∈ p. Then 0 vY (g) ≥ 1 since g ∈ pAp. Also vY 0 (g) ≥ 0 for all Y so g v 0 ( ) = v 0 (g) − v 0 (f) ≥ 0 Y f Y Y

g g Thus f ∈ Aq for all q of height 1, so f ∈ A by Matsumura. Then g g = f · f ∈ (f). Thus p = (f).

33 3 Sheaves of OX -modules

Proposition 3.6. Suppose X satisfies the assumptions. Let Z ⊆ X be a proper closed subset and U = X \ Z. Then

1. there exists a surjective homomorphism

Cl X → Cl U X X niYi 7→ ni(Yi ∩ U)

with Yi ∩ U = ∅ interpreted as 0.

2. If codim(Z,X) ≥ 2 then the homomorphism is an isomorphism. 3. If Z is irreducible of codimension 1 then there is an exact sequence

Z Cl X Cl U 0

where the first map is Z → Cl X, 1 7→ [Z]. Proof. 1. If Y is a prime divisor of X then Y ∩ U is a prime divisor of V (provided it is not empty). If f ∈ K(X)∗ = K(U)∗ then the image of (f) under the map is (f)U since vY (f) = vY ∩U (f). The map is surjective as any prime divisor Y of U satisfies Y = Y ∩ U with Y a prime divisor on X. 2. Div X = Div U. 3. ker(Cl X → Cl U) consists of divisors contained only in Z. As Z is irre- ducible, there is precisely one such prime divisor.

n ∼ Proposition 3.7. Cl Pk = Z and is generated by the divisor class of H = V ((xi)).

n ∼ n n Proof. Note Pk \ H = D+(xi) = Ak . But Cl Ak = 0 since k[y1, . . . , yn] is a n UFD. Thus we have a surjection Z → Cl Pk , 1 7→ [H]. A rational function on n f Pk is a fraction g with f, g ∈ k[x0, . . . , xn] homogeneous of the same degree. Thus if d · H ∼ 0 then there is a rational function ϕ with (ϕ) = d · H, which is impossible unless d = 0.

3.3 Cartier divisor

Definition (sheaf of rational functions). Let X be a scheme. We define the sheaf of rational functions on X, KX , to be the sheaf associated to −1 the presheaf U 7→ S(U) Γ(U, OX ) where S(U) ⊆ Γ(U, OX ) is the set of elements whose stalks in OX,x are non-zero divisors for each x ∈ U.

Remark. If X is integral then this is the constant sheaf U 7→ K(X). ∗ Let KX ⊆ KX be the subsheaf of invertible elements. Then we have an ∗ ∗ inclusion OX ,→ KX (since it is an inclusion at the presheaf level).

34 3 Sheaves of OX -modules

Definition (Cartier divisor). A Cartier divisor on a scheme X is a global ∗ ∗ section of KX /OX . A Cartier divisor is principal if it is the image of the homomorphism ∗ ∗ ∗ Γ(X, KX ) → Γ(X, KX /OX ). Two divisors are linearly equivalent if their difference is principal. We write CaCl(X) to be the group of Cartier divisors modulo principal divisors. Remark. We have a short exact sequence

∗ ∗ ∗ ∗ 0 OX KX KX /OX 0 so the divisor group measures the failure of the global section functor to be exact. Note that locally, however, the last map in the short exact sequence is surjec- tive on sections so a Cartier divisor can always be represented by {(Ui, fi)}i∈I ∗ fi where {Ui}i∈I is an open cover of X, fi ∈ Γ(Ui, K ) and on Ui ∩ Uj, |U ∩U ∈ X fj i j ∗ OX (Ui ∩ Uj).

Relationship to Weil divisors We assume X satisfies the assumptions. ∗ ∗ Then there exists a homomorphism Γ(X, KX /OX ) → Div X descending to a homomorphism CaCl(X) → Cl X: given a Cartier divisor {(Ui, fi)} and Y a prime divisor on X, choose i so that Ui ∩ Y 6= ∅, so Ui ∩ Y can be thought of a prime divisor on Ui. Let nY = vUi∩Y (fi). Note this is independent of choice of ∗ i: if Uj ∩ Y 6= ∅ then on Ui ∩ Uj, fj = fi · f with f ∈ OX (Ui ∩ Uj) and then

vY (fj) = vY (fi · f) = vY (fi) + vY (f) = vY (fi). P Then set D = nY · Y (to check this is a finite sum, use quasi-compactness of X to assume {Ui} is a finite cover). Note that a principal divisor can be represented by {(X, f)} with f ∈ ∗ ∗ Γ(X, KX ) = K(X) , which is mapped to (f) ∈ Div X. Thus the map descends to a map CaCl X → Cl X.

Proposition 3.8. If X satisfies the assumptions and all local rings of X ∗ ∗ are UFDs (i.e. X is locally factorial) then the map Γ(X, KX /OX ) → Div X is an isomorphism.

Remark. If X is non-singular, i.e. all OX,x are regular then OX,x is a UFD. This is a theorem of Serre. As a counterexample, Spec k[x, y, z, w]/(xy − zw) is not locally factorial, essentially because it is singular at (x, y, z, w). Proof. Need to construct the inverse map. Let D be a Weil divisor on X. Let x ∈ X. Then D induces a Weil divisor Dx on Spec OX,x: indeed we have a natural morphism Spec OX,x → X induced by localisation of any affine neighbourhood of x and can take inverse image of prime divisors under this map. Since OX,x is a ∗ UFD, Cl Spec OX,x = 0 and hence exists fx ∈ K(X) such that (fx) = Dx. Now consider (fx) on X. This may differ from D on prime divisors not containing x.

Nevertheless there exists an open neighbourhood U of x on which (fx) = D|Ux .

35 3 Sheaves of OX -modules

We do this for all x and then represent D as the Cartier divisor {(Ux, fx)}x∈X . fx On Ux ∩ Uy, (fx) and (fy) agree as both agree with D|U ∩U . So is invertible i j fy in OX,p for all p ∈ Ux ∩ Uy with dim OX,p = 1. If we cover Ux ∩ Uy by open fx affines Spec A, this says ∈ Ap for all p ∈ Spec A of height 1. But since Aq fy is a UFD, hence integrally closed for all q ∈ Spec A, A is integrally closed and fx fx fy by Matsumura ∈ A. Thus ∈ Γ(Ui ∩ Uj, OX ) and same for . Thus fy fy fx fx ∗ ∈ Γ(Ui ∩ Uj, O ). fy X

Correspondence between Cartier divisors and line bundles

Definition. Let D be a Cartier divisor on X represented by {(Ui, fi)}. −1 Define OX (D) to be the subsheaf of OX -modules of KX generated by fi on Ui.

fi ∗ −1 −1 Note that as ∈ Γ(Ui ∩ Uj, O ), f and f generate the same OX - fj X i j module over Ui ∩Uj. Also note that OX (D) is a locally free sheaf of rank 1. For those interested in number theory, this defines a fractional ideal for a Dedekind domain.

Remark. The local trivialising map is

OX |Ui → OX (D)|Ui −1 s 7→ s · fi

fj ∗ so the transition map on Ui ∩ Uj is given by ∈ O (Ui ∩ Uj). Consequently fi X ∨ ∼ OX (D1) ⊗ OX (D2) = OX (D1 − D2), so we have a group homomorphism ∗ ∗ Γ(X, KX /OX ) → Pic X.

∼ Lemma 3.9. D1 ∼ D2 if and only if OX (D1) = OX (D2).

Proof. Suffices to show D is principal if and only if OX (D) = OX . If D is ∗ principal then D is represented by {(X, f)} for some f ∈ Γ(X, KX ). Thus

−1 OX (D) = OX · f → OX s 7→ sf ∼ is an isomorphism. Conversely if OX (D) = OX , 1 ∈ Γ(X, OX ) be identified with −1 some f ∈ Γ(X, OX (D)) ⊆ Γ(X, KX ). Thus {(X, f )} represents the Cartier −1 ∗ divisor D = {(Ui, gi)} since on Ui, f and gi differ by a factor in Γ(Ui, OX ), −1 as both f |Ui and gi generate OX (D)|Ui . Thus D is principal.

Corollary 3.10. There is an injective homomorphism

CaCl X → Pic X

[D] 7→ [OX (D)]

36 3 Sheaves of OX -modules

Proposition 3.11. If X is integral then the map is an isomorphism.

Proof. Omitted. The idea is to show that if X is integral then every line bundle on X is isomorphic to a subsheaf of KX .

n Example. Pk satisfies all hypotheses necessary to obtain

n n n Cl P =∼ CaCl P =∼ Pic P .

We have calculated Cl Pn = Z[H] for any hyperplane H. We will later see that n n n the image of [H] in Pic P is OP (1). We will call the image of d[H] OP (d). Since this is is a homomorphism, we have

( ⊗d O n (1) d > 0 n P OP (d) = ∨ n OP (−d) d < 0

Effective divisors

Definition (effective divisor). Assume X is integral (and for Weil divisor P satisfying the assumptions). A Weil divisor aiYi is effective if ai ≥ 0 for all i. A Cartier divisor is effective if it is represented by {(Ui, fi)} with fi ∈ Γ(Ui, OX ) for all i.

If s ∈ Γ(X, L) for L a line bundle, {Ui} a trivialising cover with ϕi : L|Ui →

OUi then we get a Cartier divisor (s)0 = {(Ui, ϕi(s|Ui ))} provided s 6= 0 as ∗ ∗ X integral implies that s ∈ K(X) = Γ(X, KX ). This is an effective Cartier ϕi(s|Ui ) divisor: ϕi(s|Ui ) ∈ Γ(Ui, OX ) and on Ui ∩ Uj, is the transition function ϕj (s|Uj ) ∗ for L, which lies in Γ(Ui ∩ Uj, OX ). This is called the divisor of zeros of s.

n Theorem 3.12. Let X ⊆ Pk be a closed subscheme and F a coherent sheaf of OX -modules. Then Γ(X, F) is a finite dimensional k-vector space.

Proof. Hartshorne II, Thm 5.19. The magic word here is properness.

n Theorem 3.13. Let X ⊆ Pk be an integral closed subscheme (i.e. a projec- tive variety). Then Γ(X, OX ) = k if k = k.

Proof. Hartshorne I, 3.4.

n Theorem 3.14. Let X be an integral closed subscheme of Pk where k = k. Let D0 be a Cartier divisor on X and L = OX (D0). Then

1. for every s ∈ Γ(X, L), s 6= 0, (s)0 is an effective Cartier divisor linearly equivalent to D0.

2. every effective divisor linearly equivalent to D0 is of the form (s)0 for some s ∈ Γ(X, L).

37 3 Sheaves of OX -modules

3. two sections s, s0 ∈ Γ(X, L) give the same divisor if and only if there exists λ ∈ k∗ with s = λs0. This says that we have a natural isomorphism ∗ ∼ P(Γ(X, L)) = (Γ(X, L) \{0})/k = |D0|. 0 0 where |D0| = {D Cartier effective : D ∼ D0} is a complete linear system. Proof.

1. We have seen (s)0 is effective so left to show it is linearly equivalent to D0. OX (D0) ⊆ KX so s ∈ Γ(X, L) corresponds to some f ∈ Γ(X, KX ) = K(X). If D0 is represented by {(Ui, fi)} then OX (D0) is locally generated −1 by fi , giving trivialisations

ϕi : OX (D0)|Ui → OUi

t 7→ t · fi

so f 7→ f ·fi and (s)0 is represented by {(Ui, f ·fi)}. Thus (s)0 = D0 +(f).

2. If D is effective and D = D0 + (f) then write D = {(Ui, fi · f)} with −1 fi ·f ∈ Γ(Ui, OX ). Then ϕi (fi ·f) is a section of OX (D0)|Ui and is equal to f. Thus f determines a global section s of OX (D0) with (s)0 = D.

0 0 0 f 3. If (s)0 = (s )0 then (s)0 = D0 + (f), (s )0 = D0 + (f ) and ( f 0 ) = 0 so f ∗ ∗ f 0 ∈ Γ(X, OX ) = k by the assumption that X is projective and k = k.

n n n Example. Let X = P and s ∈ Γ(P , OP (1)). Recall that the transition maps xi from Ui = D+(xi) to Uj = D+(xj) is . Then one section of O n (1) is given xj P x0 x0 by on Ui. The divisor of zeros is (s)0 = {(Ui, )}, and H0 = V (x0) is a xi xi n prime Weil divisor. It’s not hard to see that Γ(X, OP (1)) = S1, the space of linear homogeneous polynomials. We can refine the correspondence between divisor and line bundle ifwe expand our vocabulary.

Definition (support of an effective divisor). If D = {(Ui, fi)} is an effective Cartier divisor then define its support to be [ Supp D = {x ∈ Ui : fi ∈ mx ⊆ OX,x}. i

Then for a Weil divisor, X [ Supp aiYi = Yi.

ai6=0

Definition (linear system, basepoint-free). A linear subspace D ≤ |D| is called a linear system. We say D is basepoint-free if for all x ∈ X, exists D0 ∈ D such that x∈ / Supp D0.

38 3 Sheaves of OX -modules

We can summarise the result as follow. Assume X is a projective variety over an algebraically closed field k.

line bundle Cartier divisor ∼ line bundle L on X D ∈ CaCl X with L = OX (D) section s ∈ Γ(X, L) effective Cartier divisor (s)0 linearly equivalent to D P(Γ(X, L)) = (Γ(X, L) \{0})/k∗ complete linear system |D| s0, . . . , sn ∈ Γ(X, L) inducing a mor- linear system D spanned by s0, . . . , sn ⊕(n+1) phism OX → L L globally generated by sections basepoint-free D inducing a morphism ⊕(n+1) ϕ : X → n s0, . . . , sn, i.e. OX  L, inducing P after choosing a basis n a morphism X → P s0, . . . , sn of the corresponding vector space. Pullbacks of hyperplanes in Pn give all elements of D if sections of L define a closed immersion if |D| induces a closed immersion X → X → Pn, say L very ample Pn, say |D| is very ample if L⊗m is very ample for some m > 0, if mD is very ample for some m > 0, say L ample say |D| is ample

Table 1: Dictionary between line bundle and Cartier divisor

Remark. There exists a (subtle) criterion for ampleness on example sheet 3. There also exists a numerical criterion for ampleness (degrees of pullbacks to curves on varieties). It is important to control how many sections a line bundle has.

39 4 Cohomology of sheaves

4 Cohomology of sheaves

If 0 F 0 F F 00 0 is a short exact sequence of sheaves of abelian groups, then

0 Γ(X, F 0) Γ(X, F) Γ(X, F 00) is exact, but in general the last map is not surjective. Can we extend this to a long exact sequence? There is a very general solution to this type of question: derived functors. In this case they are the right derived functor of Γ(X, ·), written as Hi(X, ·), the ith cohomology functor. We first give a list of properties of Hi(X, ·) and then give a sketch of its i construction. H (X, ·): ShX → Ab is a covariant functor with the following properties:

1. If 0 F 0 F F 00 0 is a short exact sequence then there are homomorphisms δ : Hi(X, F 00) → Hi+1(X, F 0) giving rise to a long exact sequence

0

H0(X, F 0) H0(X, F) H0(X, F 00) δ

H1(X, F 0) H1(X, F) H1(X, F 00) δ

H2(X, F 0) ···

2. Given a commutative diagrams of short exact sequences

0 F 0 F F 00 0

0 G0 G G00 0

we get commutative squares

Hi(X, F 00) δ Hi+1(X, F 0)

Hi(X, G00) δ Hi+1(X, G0)

3. Whenever F is flasque or flabby, i.e. all restriction maps are surjective, Hi(X, F) = 0 for i > 0. 4. H0(X, F) = Γ(X, F).

40 4 Cohomology of sheaves

The construction of sheaf cohomology is similar to other derived functors in homological algebra. Recall that an abelian group I is injective if given any diagram where the row is exact I g f 0 A i B exists g : B → I making the diagram commute. Examples include Q, and more generally any divisible group G, i.e. if g ∈ G, n ∈ Z, n > 0 then exists g0 ∈ G such that ng0 = g. It is a fact that every abelian group A has an inclusion 0 A I for some I injective. This gives an injective resolution of A which is a sequence

0 A  I0 I1 I2 ··· where In is injective and the sequence is exact at In for all n. We use this to get injective resolutions of sheaves. If F is a sheaf of abelian groups on X, then for all x ∈ X we get an inclusion 0 Fx Ix where Ix is injecitve. Let Ix also denote the corresponding sheaf on {x} and let ix : {x} ,→ X be the inclusion. Define Y I = (ix)∗Ix, x∈X we then get an inclusion 0 F I and one can show I is injective in the category of sheaves of abelian groups. This shows that the category of sheaves of abelian groups has enough injectives. We then obtain an injective resolution of F

0 1 2 0 F I0 d I1 d I2 d ··· and define ker di : Γ(X, Ii) → Γ(X, Ii+1) Hi(X, F) = . im di−1 : Γ(X, Ii−1) → Γ(X, Ii) It does satisfy properties of a cohomology functor and the properties listed above.

4.1 Čech cohomology

Let X be a topological space, F a sheaf of abelian groups on X. Let U = {Ui} be an open cover and choose a well-ordering on I. Write Ui0···ip = Ui0 ∩· · ·∩Uip . Set ˇp Y C (U, F) = F(Ui0···ip ).

i0<···

41 4 Cohomology of sheaves

Exercise. d2 = 0. We then define ker d : Cˇp → Cˇp+1 Hˇ p(U, F) = Hp(Cˇ•(U, F)) = im d : Cˇp−1 → Cˇp Example. 1. Let X = S1, F the constant sheaf Z, i.e. F(U) = {f : U → Z continuous}. 1 Take an open cover {U0 = X − {1},U1 = X − {0}} of S . Then ˇ0 C (U, F) = F(U0) × F(U1) = Z × Z ˇ1 C (U, F) = F(U0 ∩ U1) = Z × Z and the coboundary map is

d : Z × Z → Z × Z (a, b) 7→ (b − a, b − a)

so

0 ker d = Hˇ (U, F) = Z(1, 1) =∼ Z 1 2 coker d = Hˇ (U, F) = Z /Z(1, 1) =∼ Z and of course the higher cohomology groups vanish. It is not a coinci- dence that this gives the same result as the “ordinary cohomologies” for topological spaces, as long as the cover satisfies certain properties. ∨ 1 1 1 1 2. F = OP (−2) = (OP (−1) ⊗ OP (−1)) . Recall OP (1) has transition x0 map from U0 = D+(x0) to U1 = D+(x1) given by . Thus O 1 (−2) has x1 P 2 x1 transition map 2 . Let U = {U0,U1}. Then x0 0 Cˇ (U, F) = Γ(U0, F) × Γ(U1, F) x1 x0 = k[ ] × k[ ] using trivialisations on U0 and U1 x0 x1 ˇ1 x0 C (U, F) = F(U0 ∩ U1) = k[ ] x0 using trivialisation on U1 x1 x1 Then 2 x1 d(f, g) = g − 2 · f x0 where the contribution from g is just inclusion, and the contribution from f must be multiplied by the transition function. ker d = 0 as no cancellation is possible. coker d is one dimensional, spanned by x1 . Thus x0 Hˇ 0(U, F) = 0, Hˇ 1(U, F) = k.

xd−1 ˇ 1 1 x1 1 Similarly for d > 0, H ( , O 1 (−d)) is generated by ,..., d−1 . P P x0 x0 To remove the dependence on the cover, we need to take the direct limit of Čech cohomologies with respect to refinement of open cover. However this calculation soon becomes intractable. Fortunately we have

42 4 Cohomology of sheaves

Theorem 4.1. Let X be a Noetherian scheme with an open affine cover ˇ p U = {Ui}i∈I such that Ui0···ip are affine for i0 < ··· < ip. Then H (U, F) = Hp(X, F) for F a quasi-coherent sheaf.

Proof. Omitted. Here open affine subsets are analogous to contractible open subsets ofa manifold.

Remark. If X → S is a separated morphism with S affine then any affine open cover of X satisfies the hypothesis.

Theorem 4.2 (Grothendieck). Let X be a Noetherian topological space of dimension n, F a sheaf of abelian groups on X. Then Hi(X, F) = 0 for all i > n.

Proof. Hartshorne III Thm 2.7.

r Calculation of cohomology of projective space Fix X = Pk where r > 0. r We would like to calculate the cohomology of OX (m) = OP (m) = OX (mH) for m ∈ Z.

Definition (perfect pairing). A perfect pairing is a bilinear map h·, ·i : V × W → k of vector spaces such that the induce map V → W ∗, v 7→ hv, ·i is an isomorphism.

Theorem 4.3. Let S = k[x0, . . . , xr]. Then 1. there exists an isomorphism of graded S-modules

∼ M 0 S = H (X, OX (m)). m∈Z

i 2. H (X, OX (m)) = 0 for 0 < i < r. r ∼ 3. H (X, OX (−r − 1)) = k. 4. there is a perfect pairing

0 r r ∼ H (X, OX (m))×H (X, OX (−m−r −1)) → H (X, OX (−r −1)) = k.

Proof. Use Čech cohomology with standard covering U = {Ui = D+(xi) : 0 ≤ i ≤ r}. We will calculate all cohomologies at once by calculating cohomologies L of F = m∈ OX (m), as Čech cohomology respects direct sums. Z m xi Recall the transition maps for OX (m) from Ui to Uj are xm . For I ⊆ T Q j {0, . . . , r}, UI = i∈I Ui = D+( i∈I xi). The crucial trick is to identify Γ(UI , OX (m)) with the vector space with basis of Laurent polynomials

a0 ar X {x0 ··· xr : ai ∈ Z, ai = mi, ai ≥ 0 unless i ∈ I}.

43 4 Cohomology of sheaves

If M is such a monomial then it induces a section of OX (m) on UI using the M transition coming from Ui by m . If we instead use the trivialisation on Uj, we xi note m M xi M m · m = m . xi xj xj

Γ(U , F) SQ Γ(U , F) → Thus I can be identified with i∈I xi and restriction maps I 0 Γ(U 0 , F) I ⊆ I SQ ⊆ SQ I for are the natural inclusions i∈I xi i∈I0 xi with all these rings subrings of Sx0···xr . The Čech complex is thus

Y 0 Y 1 r−1 S d S d ··· d S xi0 xi0 xi1 x0···xr 0≤i0≤r 0≤i0

Note H0(X, F) = ker d0 and

0 d ((Mi)0≤i≤r) = (Mj − Mi)0≤i

0 Thus Mi = Mj for all i, j if and only if (Mi) ∈ ker d . Thus

r 0 \ ker d = Sxi ⊆ Sx0···xr . i=0

i0 ir Any homogeneous element of Sx0···xr can be written uniquely as x0 ··· xr f(x0, . . . , xr) with f ∈ S homogeneous and not divisible by any xi with i0, . . . , ir ∈ Z. This i0 ir T lives in Sxi if and only if ij ≥ 0 for j 6= i. Thus x0 ··· xr f ∈ Sxi if and only T 0 if ij ≥ 0 for all j. Thus Sxi = S, so H (X, F) = S. As this is an isomorphism of graded modules,

0 H (X, OX (m)) = Sm = homogeneous elments of S of degree m.

Next consider

r−1 Y d : Sx0···xˆk···xr → Sx0···xr k X k (Mk) 7→ (−1) Mk

Note Sx0···xr is a k-vector space with basis consisting of all Laurent monomials i0 ir i0 ir x0 ··· xr , i0, . . . , ir ∈ Z, while Sx0···xˆk···xr has a basis of monomials x0 ··· xr r−1 i0 ir with ik ≥ 0. Thus im d has a basis consisting of monomials x0 ··· xr with i0 ir at least one ij ≥ 0. So coker dr−1 has a basis {x0 ··· xr : ij < 0}. In degree −1 −r − 1, the only basis vector is (x0 ··· xr) . Thus

r ∼ H (X, OX (−r − 1)) = k.

0 For perfect pairing, note that H (X, OX (m)) = 0 for m < 0 since Sm = 0. r Also H (X, OX (−m − r − 1)) = 0 for m < 0. Thus the pairing is trivial if 0 m < 0. If m ≥ 0, we have a basis for H (X, OX (m)) given by

i0 ir X {x0 ··· xr : ij = m, ij ≥ 0}

44 4 Cohomology of sheaves and the pairing is given on the level of basis vectors by

i0 ir j0 jr i0+j0 ir +jr (x0 ··· xr ) · (x0 ··· xr ) = x0 ··· xr | P {z } P| {z } ik=m jk=−m−r−1 ik≥0 jk<0 where RHS should be interpreted as 0 if any ik + jk ≥ 0. As a sanity check, it −i0−1 −ir −1 has degree m − m − r − 1 = −r − 1. Note that x0 ··· xr is the dual i0 ir basis vector to x0 ··· xr . Hence the pairing is perfect. i It remains to show H (X, OX (m)) = 0 for 0 < i < r. Induction on r. The base case r = 1 is vacuously true. Assume true for r −1. If we localise Cˇ•(U, F) at xr as a graded S-module, the Čech complex now calculates the cohomology ∼ n of F|Ur using the open covering {Ui ∩ Ur : 0 ≤ i ≤ r}. But Ur = Ak and any quasi-coherent sheaf F on an affine scheme Y has Hi(Y, F) = 0 for all i ˇ• i > 0, so H (Ur, F|Ur ) = 0 for all i > 0 and thus C (U, F)xr has cohomology i vanishing in degree > 0. Since localisation is exact, we see H (X, F)xr = 0 for all i > 0. Thus for i > 0 every element of Hi(X, F) is annihilated by some r power of xr. Let H = V ((xr)) ⊆ P . Note that we can view H as the closed ∼ r−1 subscheme Proj S/(xr) = Proj k[x0, . . . , xr−1] so H = P . In particular we r r have a surjective map OP → i∗OH where i : H → P is the inclusion. The kernel is the ideal sheaf I of H in Pr, and as H is defined by a single equation, I is a xr Sxi line bundle. On Ui = Spec S , I is generated by , i.e. H ∩Ui = Spec . (xi) xi (xr /xi) Thus I is a line bundle with transition maps

OUi |Ui∩Uj → I|Ui∩Uj → OUj |Ui∩Uj x x x 1 7→ r 7→ r · j xi xi xr ∼ so I = OX (−1). We now have an exact sequence

−·xr 0 OX (−1) OX i∗OH 0

xr where − · xr means multiplication by on Ui under standard trivialisation. xi Tensoring with OX (m) we get an exact sequence

−·xr 0 OX (m − 1) OX (m) (i∗OH ) ⊗ OX (m) 0 since OX (m) is locally free. On the other hand (i∗OH ) ⊗ OX (m) = i∗OH (m) since transition maps for OX (m) restrict to transition maps for OH (m). Direct sum over all m,

−·xr 0 F(−1) F i∗FH 0 where F(−1) = F ⊗ OX (−1) is the same module as F with grading shifted by 1. We get a long exact sequence

j −·xr j j ··· H (X, F(−1)) H (X, F) H (X, i∗FH ) ···

j j The induction hypothesis says H (X, FH ) = H (H, FH ) = 0 as the Cech com- j plexes are the same using covers {Ui} and {Ui ∩ H}. Write H (X, FH ) for j H (X, i∗FH ). If 1 < j < r − 1 we get a short exact sequence

0 Hj(X, F(−1)) −·xr Hj(X, F) 0

45 4 Cohomology of sheaves

so multiplication by xr is an isomorphism. But we know every element of j j H (X, F) is annihilated by a power of xr, so H (X, F) = 0. Now we are left with two boundary cases. If j = 1 we get

S(−1) S S/(xr)

0 −·xr 0 0 0 H (X, F(−1)) H (X, F) H (H, FH )

H1(X, F(−1)) −·xr H1(X, F)

0 where S(−1) is the graded S-module with S(−1)d = Sd−1. The map H (X, F) → 0 0 1 H (X, FH ) is surjective so H (X, FH ) → H (X, F(−1)) is 0. Thus − · xr : H1(X, F(−1)) → H1(X, F) is surjective and hence H1(X, F) = 0. For j = r − 1, we have

r−1 r −·xr r r H (H, FH ) H (X, F(−1)) H (X, F) H (H, FH ) 0

0

r a0 ar By our calculations of H , the kernel of − · xr is generated by {x0 ··· xr : r−1 ai ≤ −1 for all i < r, ar = 1}. But this is precisely H (H, FH ). So − · xr : Hr−1(X, F(−1)) → Hr−1(X, F) is surjective and Hr−1(X, F) = 0. Remark.

1. In general, given an effective Cartier divisor D = {(Ui, fi)} where fi ∈ OX (Ui), we can view D as a closed subscheme defined by fi on Ui. Then ∼ ID/X = OX (−D).

2. Given a sheaf F on X and a divisor D, we usually write F(D) = F ⊗OX r OX (D). For example if X = P ,D = nH then F(n) = F ⊗OX OX (n), the Serre twisting sheaf . 3. It is a general fact that if i : X → Y is a closed immersion then Hj(X, F) = j j j H (Y, i∗F) for all j, and frequently write H (Y, F) instead of H (Y, i∗F).

Normal and conormal bundles Recall from example sheet 3 if i : Z,→ X # is a closed subscheme then IZ/X = ker(i : OX → i∗OZ ) is a quasi-coherent sheaf of ideals (coherent if X is Noetherian). Define ∨ 2 NZ/X = IZ/X /IZ/X ∼ which is quasi-coherent (coherent respectively). Note this is also an OX /IZ/X = OZ -module. This is the conormal sheaf of Z in X which can be viewed as a sheaf on Z (analogue: I ⊆ A an ideal then I/I2 is an A/I-module). Fact: if X and Z are nonsingular varieties (i.e. all local rings are regular) ∨ then NZ/X is locally free of rank codim(Z,X). In this case we define the normal bundle ∨ NZ/X = HomOZ (NZ/X , OZ ).

46 4 Cohomology of sheaves

Definition (differential). Suppose f : X → Y is a separated morphism, i.e. ∆ : X → X ×Y X is a closed immersion then we define

Ω = ∆∗N ∨ X/Y ∆(X)/X×Y X

Quick recap of algebra: let B an A-algebra and M a B-module. An A- derivation d : B → M is a map such that 1. d(a) = 0 for a ∈ A, 2. d(b + b0) = d(b) + d(b0), 3. d(bb0) = bd(b0) + d(b)b0.

The module of relative differential ΩB/A is a B-module satisfying the following universal property: there exists a unique A-derivation d : B → ΩB/A such that for any A-derivation d0 : B → M, exists a B-module homomorphism 0 f :ΩB/A → M such that d = f ◦ d.

0 B d M

d f

ΩB/A

0 0 ΩB/A is constructed as follow: consider ϕ : B ⊗A B → B, b ⊗ b 7→ b · b . Let I = ker ϕ. Then I/I2 is a B-module and we can define d : B → I/I2 b 7→ 1 ⊗ b − b ⊗ 1

2 which makes (I/I , d) = (ΩB/A, d). Example. Suppose Y = Spec k, X a non-singular variety of dimension n over k, then ΩX/Y is a locally free sheaf of rank n on X. Just as in differential geometry where we define bundle of differential forms once we have the cotangent bundle, we define the canonical bundle of X to be dim X ωX = Λ ΩX/Y . Hence if ΩX/Y has a trivialisation on an open cover {Ui} with transition maps gij ∈ GLn(OX (Ui ∩ Uj)), then ωX is the line bundle with ∗ transition maps det gij ∈ OX (Ui ∩ Uj). It is also called the determinant line bundle of ΩX/Y . We usually write the divisor class of ωX as KX , called the canonical class.

Theorem 4.4 (Serre duality). Let X be a non-singular projective variety over Spec k of dimension n. Then for any locally free sheaf F on X, there is a natural isomorphism

i ∨ n−i ∨ H (X, F ⊗ ωX ) → H (X, F)

∨ where F = HomOX (F, OX ).

This is analogous to Poincaré duality for manifolds1.

1To account for the mysterious tensor product we need the full version of twisted Poincaré duality which uses local orientations and does not require orientability.

47 4 Cohomology of sheaves

r Example. For P = X, ωX = OX (−r − 1) so we can recover the perfect pairing by setting F = OX (m):

i r−i ∨ H (X, OX (−r − 1 − m)) → H (X, OX (m)) .

In general if X is a projective variety over k and F ia coherent sheaf on X then Hi(X, F) are finite dimensional k-vector space. We then define the Euler characteristic of F as

X i i χ(F) = (−1) dimk H (X, F). i

Note that if 0 F 0 F F 00 0 is exact then χ(F) = χ(F 0) + χ(F 00) which follows from if 0 V0 ··· Vn 0 is a long exact P i sequence of vector spaces then (−1) dim Vi = 0. From this we can derive the classical Riemann-Roch for curves. n Let X be a non-singular projective curve over a field k, i.e. X ⊆ Pk is a closed integral scheme of dimension 1 with all local rings regular.

1 Definition (genus). The genus of X is g = dim H (X, OX ).

P Definition (degree). If D is a divisor on X, D = niPi then the degree P of D is deg D = ni.

Theorem 4.5 (Riemann-Roch for curves). For D ∈ Div X,

0 0 dim H (X, OX (D)) − dim H (X, OX (KX − D)) = deg D + 1 − g.

Proof. We prove the theorem under the hypothesis k = k. By Serre duality

0 1 χ(OX (D)) = dim H (X, OX (D)) − dim H (X, OX (D)) 0 0 = dim H (X, OX (D)) − dim H (X, ωX ⊗ OX (−D)) = LHS

Now let P ∈ X a closed point. Have an exact sequence

0 IP/X OX OP 0

(recall the convention that OP denotes pushforward). Note IP/X = OX (−P ). Now tensor with a line bundle L,

0 L ⊗ OX (−P ) L OP ⊗ L 0

L(−P ) OP

Thus χ(L) = χ(L(−P )) + χ(OP ) = χ(L(−P )) + 1

48 4 Cohomology of sheaves

so χ(OX (D)) = χ(OX ) + deg D by repeated use of the above equation. Also

0 1 χ(OX ) = dim H (X, OX ) − dim H (X, OX ) = 1 − g so the result follows.

Remark. 1 ∼ 0 ∨ 0 1. By Serre duality, H (X, OX ) = H (X, ωX ) so g = dim H (X, ωX ), the definition of genus from IID Algebraic Geometry using differentials.

2. If k = C we can endow X with the Euclidean topology, making it a 2- dimensional manifold, written Xan. Then g is the genus of Xan and

1 an ∼ 0 1 Hsing(X , C) = H (X, ωX ) ⊕ H (X, OX ).

This is related to Hodge decomposition and will be covered in detail in III Complex Manifolds.

Remark. 1. Have

0 1 χ(ωX ) = dim H (X, ωX ) − dim H (X, ωX ) 0 0 = dim H (X, ωX ) − dim H (X, OX ) = g − 1

so g − 1 = deg KX + 1 − g so deg KX = 2g − 2.

0 0 2. If deg D < 0 then H (X, OX (D)) = 0. Indeed if D ∼ D then deg D = 0 0 deg D (as χ(OX (D)) = χ(OX (D )), this follow from Riemann-Roch). If s ∈ Γ(X, OX (D)), s 6= 0 then (s)0 ∼ D and (s)0 is effective, contradicting deg(s)0 < 0. Thus if deg D > 2g − 2 then

deg(KX − D) = 2g − 2 − deg D < 0,

0 0 so dim H (X, OX (KX −D)) = 0. Thus by Riemann-Roch dim H (OX (D)) = deg D + 1 − g. 3. A linear system on a curve on a curve is basepoint-free if

0 0 dim H (X, OX (D)) = dim H (X, OX (D − P )) + 1

for all P ∈ X, as follows from the exact sequence

0 OX (D − P ) OX (D) OP 0

OX (D)/mP OX (D)

and Γ(X, OX (D)) → Γ(X, OP ) is surjective if and only if L is generated by global sections at P . In particular if deg D > 2g − 1 then |D| is basepoint-free or equivalently, OX (D) is generated by global sections.

49 4 Cohomology of sheaves

4. The very ampleness criterion on example sheet 3 takes the following form for curves: D is very ample if 0 0 dim H (X, OX (D − P − Q)) = dim H (X, OX (D)) − 2 for all P,Q ∈ X not necessarily distinct. Thus if deg D > 2g then |D| is very ample. 5. As such, the most interesting behaviour happens when 0 ≤ deg D ≤ 2g−2. Example. 0 1. g = 0. Then if P ∈ X then |P | is very ample and dim H (X, OX (P )) = 2 so we get a closed embedding f : X,→ P1, necessarily an isomorphism.

2. g = 1. Fix P0 ∈ X. |3P0| is very ample and we get an embedding f : X,→ P2. The image is a plane cubic, given by a single equation h = 0 (as height 1 prime ideals are principal) where h is a homogeneous polynomial of degree 3. To see it has degree 3, suppose H = (s)0 for 2 ∗ 2 s ∈ Γ(P , OP (1)) is a hyperplane. Then 3P0 ∼ (f s)0. What are divisors of degree 0 on X? Claim that if D ∈ Div X with deg D = 0 then D ∼ P − P0 for some unique P ∈ X.

0 Proof. Consider D+P0. We have dim H (X, OX (D+P0)) = 1 so exists an 0 effective divisor P ∼ D+P0, say P = (s)0 for s ∈ H (X, OX (D+P0))\{0}. Note P has degree 1 so must be a point. This point is uniquely determined as 0 |P | = P(H (X, OX (D + P ))) = {pt}.

Then D ∼ P − P0.

For any curve we have an exact sequence

0 deg 0 Cl X Cl X Z 0 where Cl0 X is the equivalence class of degree 0 divisors. Thus if g = 1 have a bijection {closed points of X} ↔ Cl0 X, so we can transport the group structure on Cl0 X to the set of closed points. See for example III Elliptic Curves. More generally if X is of genus g, Cl0 X is in bijection with the closed points of a projective variety A of dimension g. We can also specify morphisms m : A × A → A, i : A → A that satisfy group laws.

Surfaces* Fix X a projective non-singular surface. Then divisors on X are linear combinations of curves. If two curves C,D intersect transversally we can count the number of intersections. Theorem 4.6. There exists a unique pairing

Div X × Div X → Z (C,D) 7→ C · D

satisfying the following properties: 1. if C,D are non-singular curves meeting transversally (i.e. C and D

50 4 Cohomology of sheaves

not tangent at any point of C ∩ D) then C · D = #(C ∩ D). 2. C · D = D · C.

3. (C1 + C2) · D = C1 · D + C2 · D.

4. if C1 ∼ C2 then C1 · D = C2 · D.

We thus get a pairing Cl X × Cl X → Z. Analogy with algebraic topology: a two dimensional projective surface over C can be thought of as a 4-dimensional (compact orientable) manifold. Then the pairing corresponds to cup product H2(X; Z)×H2(X; Z) → H4(X; Z) =∼ Z.

Theorem 4.7 (Riemann-Roch for surfaces). For X a non-singular surface,

0 1 0 dim H (X, OX (D)) − dim H (X, OX (D)) + dim H (X, OX (KX − D)) 1 = D · (D − K ) + 1 + P (X) 2 X a

where Pa(X) = χ(OX ) − 1 is the arithmetic genus of X.

Example. Let X = P2 so Cl P2 = Z. Then

Cl X × Cl X → Z aH · bH 7→ ab

n n−1 n n−1 n n Consider X ⊆ A × P and pr1 : A × P → A . Give A coordinates n n−1 n x1, . . . , xn and A × P coordinates x1, . . . , xn, y1, . . . , yn. Can think of A × n−1 P = Proj k[x1, . . . , xn, y1, . . . , yn], deg xi = 0, deg yi = 1. X is defined by the ideal generated by {xiyj − xjyi = 0, 1 ≤ i < j ≤ n}.

⊆ X An × Pn−1 pr ϕ 1 An Some facts:

1. X is a closed subvariety (in particular irreducible) of An × Pn−1. 2. ϕ : ϕ−1(An \{1}) → An \{0} is an isomorphism. 3. ϕ−1(0) = {0} × Pn−1. ϕ is called the blow-up of A2 at the origin. If Z ⊆ An is any closed subvariety and 0 ∈ Z then the blow-up of Z at 0 is ϕ−1(Z \{0}) ⊆ X. This is the also called the strict transformation of Z. This allows blowing up affine variety the points and then not too hardto define the blow-up of a projective variety at apoint.

2 2 Example. Let X be the blow-up of P at 6 general points P1,...,P6 ∈ P (here it means no 3 lie on a line and not all 6 lie on a conic). Let ϕ : X → P2 −1 ∼ 1 2 L6 be the blow-up. Let Ei = ϕ (Pi), Ei = P . Have Cl X = Cl P ⊕ i=1 ZEi. Here we use the convention that Cl P2 is identified as a subgroup of Cl X via

51 4 Cohomology of sheaves the pullback map ϕ∗ : Pic P2 → Pic X, and for H ∈ Cl P2 the generator, write ϕ∗H for the corresponding divisor in Cl X. One can check that ( ∗ ∗ 0 0 ∗ 0 i 6= j (ϕ D) · (ϕ D ) = D · D , (ϕ D) · Ei = 0,Ei · Ej = −1 i = j

∗ 6 |3ϕ H − E1 − · · · − E6| is very ample and embeds X as a cubic surface in P .

52 Index affine scheme, 13, 14 height, 31 ample, 39 Hodge decomposition, 49 arithmetic genus, 51 Hom sheaf, 25 homogeneous ideal, 16 basepoint-free, 38, 49 blow-up, 51 ideal sheaf, 45 injective module, 41 canonical bundle, 47 injective resolution, 41 canonical class, 47 invertible sheaf, 25 Cartier divisor, 35 irrelevant ideal, 16 linearly equivalent, 35 principal, 35 Krull dimension, 30 Čech cohomology, 41 class group, 33 line bundle, 25 closed immersion, 19 linear system, 38, 49 closed point, 17 local homomorphism, 13 closed subscheme, 19 localisation, 10 codimension, 30 locally ringed space, 12 complete linear system, 38 morphism conormal sheaf, 46 locally of finite type, 23 of finite type, 23 derivation, 47 proper, 24 derived functor, 40 separated, 24 determinant line bundle, 47 diagonal morphism, 24 natural transformation, 20 differential, 47 Noetherian topological space, 31 dimension, 30 normal bundle, 46 discrete valuation ring, 31 divisor, 32 open immersion, 19 linearly equivalent, 33 open subscheme, 19 principal, 33 divisor of zeros, 37 perfect pairing, 43 divisor of zeros and poles, 32 Picard group, 27 presheaf, 4 effective divisor, 37 morphism, 4 embedded point, 19 prime divisor, 32 Euler characteristic, 48 regular local ring, 31 fibre, 23 representable functor, 21 fibre product, 20, 21 residue field, 14 flabby, 40 Riemann-Roch theorem for curves, flasque, 40 48, 51 ringed space, 12 generic point, 31 genus, 48 scheme, 14 germ, 6 integral, 23 globally generated, 30, 39 over k, 15

53 Index section, 9 strict transformation, 51 Serre duality, 47 structure sheaf, 6 Serre twisting sheaf, 46 support, 38 sheaf, 5 cokernel, 8 transition functions, 26 image, 8 kernel, 8 variety, 23 sheaf of modules, 25 very ample, 39, 50 coherent, 26 free, 25 weighted projective space, 18 locally free, 25 quasi-coherent, 26 Yoneda lemma, 20 sheaf of rational functions, 34 spectrum, 4 Zariski tangent space, 16 stalk, 6 Zariski topology, 4

54