Notes on Math 511 (Algebraic Geometry)

Li Li

April 27, 2009 2

(Week 1, two classes.)

0.1 Goal of the lecture.

Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is considered to be one of the greatest mathematicians of the 20th century. He is the chief designer of modern algebraic geometry. Now he is almost 81, and it is my great honor to present here an introduction to modern algebraic geometry which has been greatly influence by his work. Some textbook on introduction of algebraic geometry (or any field of mathematics) tries to minimize or erase the trace of history. This is some sense is reasonable: a science should be coherent and be independent of who have discovered it. Calculus is a typical example. We barely remember who have contributed to the development of it. Calculus has been born for more than 300 years if we count from Newton and Leibniz, and it becomes so mature and standard that it is easily accessed by a high school student. But algebraic geometry, on the other hand, is not old enough to forget about how it was created by force, and in my point of view, is not yet mature. (One evidence is that we are still stuck in curve, surface and dimension 3 manifold and barely know anything in higher dimensional; the other evidence is that the new objects kept born: manifolds, varieties, schemes, algebraic spaces, stacks, etc; third evidence, I did not successfully explained to my wife what is algebraic geometry, and I believe that algebraic geometry is far from accessible to general audience.) So in this class we follow instead the order of history, and that is roughly what Hartshorne did in his book. Start from varieties, and when we are familiar with their properties, we proceed to schemes. My suggestion for the class: be skeptical to the material, always ask me or yourself, if it is necessary, is there a better approach? On the other hand, you should be open minded, do not refuse to accept a concept simply because it is complicated or ugly. You might appreciate it when you are used to it :-) This class is not complex algebraic geometry, hence almost no transcendental method will be introduced, and I believe you have learned or will learn something like Hodge theory in other classes. Exercise are essential to understand the material. Some exercises you should do it at least once in your lifetime instead of simply believe it. Asking the following: affine varieties and projective varieties, Zariski topology, rational maps, birational equivalence, blow-up, valuation rings, Hilbert polynomials, sheaves, schemes, coherent sheaves, divisors, invertible sheaves, ample line bundles(invertible sheaves), linear system, derived functors, cohomology of a sheaf, Cech cohomology, Ext functor and Serre duality, higher direct images. Chapter 1

Varieties

1.1 Affine varieties

It is simply the set of points that is defined by one or several polynomial equations. For example, let k be your favorite field, consider a parabola in k2 defined by y − x2 = 0. The polynomial y − x2 is in k[x, y]. We should expected that to study the zero locus of a polynomial is closely related to the study of the polynomial in the ring k[x, y]. Indeed, to understand the interplay between the geometry and algebra is the central goal of algebraic geometry. We will see today such a relation.

n Let k be an algebraic closed field, the most common choice is C. We call it the base field. Ak (or n n n A ) is the affine n-space, or k if you like. The polynomials on A form a ring, say A = k[x1, . . . , xn], n it is called the coordinate ring of A , xi are called coordinates. Given any subset T of A, we define the zero set of T to be the common zeros of all elements in T , i.e. Z(T ) = {P ∈ An|f(P ) = 0, ∀f ∈ T } It is obvious that if a is the ideal generated by T , then Z(a) = Z(T ). Example 1.1.1. (1) x2 + y2 = 0. (2) x and x2. (3) the defining equations for (t3, t4, t5).

Definition 1.1.2. A subset Y of An is an algebraic set if Y = Z(T ) for some subset T ⊆ A.

Remark 1.1.3. T = ∅, T = (1), etc.

Proposition 1.1.4. The union of finite many algebraic set is an algebraic set. The intersection of any family of algebraic set is algebraic set.

Example 1.1.5. The union of a point (0, 0) and the line x = 1.

Now notice each algebraic set is determined by not necessarily unique subset T . For example, (x, y) and (x, y2) and (x2, xy, y2) in k[x, y]. Can one specify a canonical subset? The answer is yes by the following.

Theorem 1.1.6. There is a one-one correspondence between algebraic sets in An and radical ideals in A, given by Y 7→ I(Y ) := {f ∈ A|f(P ) = 0, ∀P ∈ Y } and a 7→ Z(a).

3 4 CHAPTER 1. VARIETIES

√ Proof. Enough to show that (1) for any ideal a, I(Z(a) = a and (2) for any algebraic set Y , Z(I(Y )) = Y . For (2), if not equal, then ∃f ∈ I(Y ), z ∈ Z(I(Y )), s.t. f(z) 6= 0, this is impossible. (1) is the famous Hilbert’s Nullstellensatz.

Theorem 1.1.7 (Hilbert’s Nullstellensatz (German: “theorem of zeros”)). Let k be an algebraically closed field, a be an ideal in A = k[x1, . . . , xn], then √ I(Z(a)) = a.

A special case: when a is maximal, hence proper, Z(a) is not empty, suppose (a1, . . . , an) ∈ Z(a), then (x1 − a1, . . . , xn − an) ⊇ a, by the maximality of a, equality holds.

Theorem 1.1.8 (Weak Nullstellensatz). Let k be an algebraically closed field, the maximal ideal in A = k[x1, . . . , xn] are the ideals (x1 − a1, . . . , xn − an).

Remark 1.1.9. Note it is not true when k is not algebraically closed, say R, the nullstellensatz is not true. Eg. (x2 + 1) is maximal in R[x].

On the other hand, the√ weak Nullstellensatz implies the strong one, by the following proof. It is easy to see I(Z(a)) ⊇ a, so we need to show the inclusion in the other direction. If we take a finite set of generators f1, . . . , fm of a, then we need to show that, if g ∈ k[x1, . . . , xn] vanishes at ` P the zero locus f1 = ··· = fm = 0, then g = hifi.

Here is the lifting trick of Zariski: consider k[x1, . . . , xn+1]. Define ideal

b = ak[x1, . . . , xn+1] + (1 − gxn+1).

We first show that b must be (1). Otherwise it is proper, by weak Nullstellensatz it is in a maximal ideal m = (x1 − a1, . . . , xn+1 − an+1). Then 1 − gxn+1 is in the maximal ideal m, so are f1, . . . , fm. But this is impossible since f1 = ··· = fm = 0 at (a1, . . . , an) implies g(a1, . . . , an) = 0.

Now since 1 ∈ b, 1 = h1f1 +···+hmfm +hm+1(1−gxn+1). Plug in xn+1 = 1/g, then eliminate ` P 0 the denominator, we will obtain g = hifi. Now we need to prove Theorem 1.1.8, we need Noether’s Normalization Lemma: let k be any field. R be an integral domain finitely generated over k. There are algebraically independent elements x1, . . . , xn in R, such that R is an integral extension of k[x1, . . . , xn]. (I am not suppose to say it now, but geometrically, the Lemma says that any variety is a branched covering of an affine space.)

Proof of Theorem 1.1.8. Let m be a maximal ideal. Then R = k[x1, . . . , xn]/m is a field. Let d be its transcendental degree over k. Then S = k[y1, . . . , yd] ⊂ R such that R is integral over S and is a field. This easily implies S itself is a field (check it!), which is only possible if d = 0.

Then R is integral over k, but k is algebraically closed (!), so R = k, hence k+m = k[x1, . . . , xn], in particular ai + mi = xi for each i = 1, . . . , n. So m ⊇ (m1, . . . , mn) = (x1 − a1, . . . , xn − an), but the right hand side is already maximal, so equality holds. 1.1. AFFINE VARIETIES 5

Now we introduce Zariski topology and irreducibility, and define variety.

Definition 1.1.10. The Zariski topology on An is defined as follows: a closed set in An is to be a algebraic set. The Zariski topology on an algebraic set in An is induced from the Zariski topology of An.

(Check it is a topology.)

Definition 1.1.11. A nonempty subset Y of a topological space X is irreducible if it cannot be expressed as the union Y = Y1 ∪ Y2 where Y1 and Y2 are closed proper subsets of Y . The empty set is not considered to be irreducible.

Example 1.1.12. Open subset of an irreducible space. Closure of a irreducible subset.

Definition 1.1.13. An affine (algebraic) variety is an irreducible closed subset of An. An open subset of an affine variety is a quasi-affine variety.

Proposition 1.1.14. An algebraic set Y on An is a variety iff I(Y ) is prime.

Now is more about topology:

Definition 1.1.15. A topological space X is noetherian if it satisfies descending chain condition for closed subsets.

(Note that it is dual to the definition of noetherian rings or modules.) Homework: any algebraic set is noetherian.

Proposition 1.1.16. In a noetherian topological space X, any nonempty closed subset Y can be uniquely written as a finite union Y = Y1 ∪ · · · ∪ Yr of maximal irreducible closed subsets. They are called the irreducible components of Y .

We will be very sketch about dimension:

Definition 1.1.17. Let X be a topological space, the dimension of X is the supremum of all integer n such that there is a chain Z0 ⊂ · · · ⊂ Zn of distinct irreducible closed subsets of X.

Algebraically, we need to wait until we define the function field k(X) of X. Then it can be shown that dim X = tr.degk(X). For affine variety, k(X) is the fraction field of the coordinate ring of X. Homework: 1.1, 1.2*, 1.3, 1.6, 1.7, 1.9*. 6 CHAPTER 1. VARIETIES

(Week 2, two classes.)

I noticed that problem 1.9 assigned last week is a consequence of problem 1.8, which is proved in 7.1 (affine dimension theorem) first paragraph. The proof needs Theorem 1.11A, Krull’s Haup- tidealsatz (principal ideal theorem): every minimal prime ideal containing f has height 1. This is too involved and we will not prove it in class.

1.2 Projective varieties

In early 19th century it was realized that it was inadequate and misleading to consider only affine algebraic sets. It turns out to be much simpler to consider projective algebraic sets in many situa- tions. We start from projective spaces, consider the corresponding rings and an affine covering, then define projective algebraic sets and the corresponding ideals, finally the projective Nullstellensatz.

Recall that the projective n-space is the set of equivalent classes of (n+1)-tuples (x0, x1, . . . , xn) 6= (0,..., 0) in kn modulo the equivalence relation

∗ (x0, . . . , xn) ∼ (λx0, . . . , λxn), λ ∈ k . Note that it is uninteresting to consider functions on Pn, so instead we consider homogeneous polynomials on Cn+1. Why homogeneous? because we want the polynomial to have a well-defined zero set in Pn.

n P has an affine covering as follows: Ui = {(x0 : x1 : ··· : xn)|xi 6= 0} for i = 0, . . . , n. Each Ui is an affine space by the isomorphism:

x0 x0 xˆi xn (x0 : ··· : xn) 7→ ( , ,..., ,..., ). xi xi xi xi

Let S = k[x0, . . . , xn], regarded as a graded ring, graded by the degrees of homogeneous polynomials. Some facts: S = ⊕d≥0Sd, Sd · Se ⊆ Sd+e. We only consider the homogeneous ideals, i.e. a = ⊕(a ∩ Sd). The sum, intersection, radical of a homogeneous ideal is again homogeneous. A homogeneous ideal is prime if fg ∈ a implies f ∈ a or g ∈ a. Let T be an set of homogeneous elements in S. The zero set of T is

Z(T ) = {P ∈ Pn|f(P ) = 0, ∀f ∈ T }.

Definition 1.2.1. A subset Y in Pn is called an algebraic set if Y = Z(T ) for a subset T of homogeneous elements in S. Proposition 1.2.2. The union of finite many algebraic sets is an algebraic set; the intersection of a family of algebraic sets is an algebraic set.

Zariski topology is defined in such a way that the algebraic set are closed subset. The algebraic set has induced Zariski topology. Theorem 1.2.3 (Homogeneous Nullstellensatz). There is a bijection between (algebraic sets in n √ P ) and (homogeneous ideals a ⊂ k[x0, . . . , xn] that a = a and a 6= (x0, . . . , xn)). 1.3. MORPHISMS 7

The irreducible algebraic sets corresponds to homogeneous prime ideals. Homework: 2.2, 2.4, 2.12(d-Uple embedding), 2.14* (Segre imbedding), 2.15* (Quadric sur- face), 2.16 (intersection of two varieties).

1.3 Morphisms

To define the category of varieties, we need to define morphisms. Example 1.3.1. The isomorphism between the affine line A1 and the parabola y = x2. More n+1 n generally, the graph of a function f ∈ k[x1, . . . , xn] in A is isomorphic to A .

We use the definition in Mumford, which is equivalent to Hartshorne’s:

Definition 1.3.2. A map f : X → Y between two affine varieties X ⊆ Am and Y ⊆ An is called a morphism if there exists n polynomials f1, . . . , fn ∈ k[x1, . . . , xm] such that

f(x) = (f1(x), . . . , fn(x)), ∀x = (x1, . . . , xm) ∈ X.

One shortcoming: For projective varieties, we cannot define morphisms in exactly the same way. Example: a smooth conic xz = y2 in P2 maps isomorphically to P1 by [x : y : z] → [x : y] = [y : z], but there is no surjective morphism P2 → P1. (explain using intersection theory.) The other shortcoming: the definition seems to depend on the embedding we choose. Recall how we handle morphisms in the category of differential manifolds: once we have built up differential structures on two manifolds X and Y , we difine a map f : X → Y to be differentiable if the pull back of any differential function on Y is a differentiable function on X. Grothendieck’s philosophy is: to study a space is ‘equivalent’ to study the set of functions on the space; to study a morphism from X to Y is ‘equivalent’ to study the map from the set of functions on Y to the set of functions on X. Let me explain in more details.

Definition 1.3.3. (1) If Y is a quasi-affine variety in An. A function f : Y → k is regular at a point y ∈ Y if there is an open neighborhood U with y ∈ U ⊆ Y , and polynomials g, h ∈ k[x1, . . . , xn] such that h is nowhere zero on U and f = g/h on U. (2) If Y is a quasi-projective variety in Pn. A function f : Y → k is regular at a point y ∈ Y if there is an open neighborhood U with y ∈ U ⊆ Y , and homogeneous polynomials g, h ∈ k[x0, . . . , xn] of the same degree, such that h is nowhere zero on U and f = g/h on U. We say f is regular on Y if it is regular at every point on Y . (i.e. regularity is a local property)

Now we may define morphisms between two varieties.

Definition 1.3.4. For two (quasi-affine or quasi-projective) varieties X and Y , a morphism φ : X → Y is a continuous map such that for every open set V ⊆ Y and every regular function h on V , the function h ◦ φ : φ−1V → k is regular. An isomorphism is a morphism which admits an inverse morphism. 8 CHAPTER 1. VARIETIES

Homework: Let C be defined by x2 − y3 = 0 and define f : C → A1 by (x, y) → x/y. Then f is not a morphism, though it is bijective and bicontinuous. Now we introduce some rings of functions on associated with any variety.

Definition 1.3.5. Let Y be a variety. (1) Define O(Y ) to be the ring of all regular functions on Y .

(2) Define Oy,Y to be the ring of germs of regular functions on Y near y. (i.e. two regular function near y are equivalent if they are identical after restriction to a neighborhood of y.) (3) The function field K(Y ) is defined as follows: an element is an equivlent class of pairs hU, fi where U is a nonempty subset and f is a regular function on U, and we identify hU, fi with hV, gi if f = g on U ∩ V . (You should check K(Y ) is indeed a field.) The elements in K(Y ) are called rational functions on Y .

For affine varieties, the above rings are easy to describe. Let us look at the example Y = A1 first. What is O(Y ), Oy,Y ,K(Y )? In general, we have the following

Theorem 1.3.6. Let Y ⊆ An be an affine variety with coordinate ring A(Y ). Then (a) O(Y ) = A(Y ). (b) There is 1-1 correspondence between points of Y and maximal ideals of A(Y ), given by y → my, the ideal of functions vanishing at y. ∼ (c) Oy,Y = A(Y )my , dim Oy,Y = dim Y . (d) K(Y ) is isomorphic to the quotient field of A(Y ). (hence dim Y = tr.degK(Y ).)

Proof. (b) follows from weak Nullstellensatz. For (a), there is obviously an injective map A(Y ) → O(Y ), difficulty is surjectivity. So we prove (c) first, in this local situation, the injective map

A(Y )my → Oy,Y is surjective by definition. Since dim Oy,Y = dim A(Y )my = height(my), but height(my) = dim Y − dim(A(Y )/my) = dim Y . (d) follows from (c) and some algebraic fact, read Hartshorne for details. Now how do we finish the proof of surjectivity in (a)? Note \ \

A(Y ) ⊆ O(Y ) ⊆ Oy,Y = A(Y )my .

But the leftmost equals the rightmost (why?). Thus A(Y ) = O(Y ).

There is a similar statement on projective varieties, I suggest you to read Theorem 3.4 in Hartshorne. Homework: 3.2*, 3.6, 3.15, 3.17. 1.4. RATIONAL MAPS 9

(Week 3: 2 classes)

Now we will show an important fact, which guarantees that we do not lose any information to consider the coordinate rings instead of affine varieties.

Proposition 1.3.7. The contravariant functor

α : (affine varieties)→ (f.g. integral domains over k) by X 7→ A(X) is an equivalence of categories.

Proof. Given f ∈ Hom(X,Y ), it induces f ∗ ∈ Hom(A(Y ),A(X)) by pullback g ∈ A(Y ) to g ◦ f ∈ A(X). Conversely, given h : A(Y ) → A(X), how can we find f such that f ∗ = h? Suppose A(Y ) = k[y1, . . . , yn]/IY and A(X) = k[x1, . . . , xm]/IX . Then let f¯i = h(¯yi), and take any preimage fi of f¯i. We have a morphism m n f˜ = (f1, . . . , fn): A → A

Now we show f˜ maps X into Y , that is to show, ∀g ∈ IY , ∀P ∈ X, we have g(f˜(P )) = 0.

g(f˜(P )) = g(f1(P ), . . . , fn(P )) = g(h(¯y1)(P ), . . . , h(¯yn)(P )) = h(g(¯y1,..., y¯n))(P ) = 0 because h maps IY to IX . The third equality we use the fact that h is a homomorphism of algebras. Then α is a fully faithful functor, to show it gives an equivalence of categories, we need to show any f.g. integral domains over k actually occurs as the coordinate ring of an affine variety, i.e. k[x1, . . . , xm]/I for a prime ideal I, but this is obvious.

1.4 Rational maps

One main difference between algebraic geomtry and differential geometry(or topology) is that it is more rigid:

Lemma 1.4.1. Let X and Y be two varieties, suppose two morphisms ϕ, ψ : X → Y coincide on a open subset U ⊂ X, then ϕ = ψ.

Proof. W.l.o.g, we can assume Y = Pn. Consider the map

ϕ × ψ : X → Pn × Pn

When restrict to U, the image is in the diagonal ∆ ⊂ Pn × Pn. Since ∆ is a closed subset (defined −1 by {xiyj − xjyi|0 ≤ i < j ≤ n}) and (ϕ × ψ) is continuous, (ϕ × ψ) (∆) is a closed subset of X containing U, hence is X. This implies ϕ = ψ. (note in the proof, “any nonempty open subset is dense” is not true in differential geometry.) 10 CHAPTER 1. VARIETIES

Definition 1.4.2. Let X,Y be varieties. A rational map ϕ : X 99K Y is an equivalence of pairs hU, ϕU i where U is a nonempty open subset of X, ϕU is a morphism from U to Y , and where hU, ϕU i and hV, ϕV i are equivalent if ϕU = ϕV on U ∩ V . The rational map ϕ : X 99K Y is dominant if for some pair hU, V i the image is dense in Y .

The lemma implies that the relation is indeed an equivalence relation: if ϕU = ϕV on U ∩ V and ϕV = ϕW on V ∩ W , then ϕU = ϕW on U ∩ V ∩ W , therefore by the lemma the equality still holds on U ∩ W .

Definition 1.4.3. A birational map ϕ : X 99K Y between varieties is a rational map which admits an inverse rational map: ψ : Y 99K X such that ψ ◦ ϕ = idX and ϕ ◦ ψ = idY as rational maps. In this case, we say X and Y are birational.

Now observe that a rational dominant map ϕ : X 99K Y induces a field morphism ϕ∗ : K(Y ) → K(X). Indeed, an element g in K(Y ) is a regular function on an open subset V of Y . Suppose ϕ|U is a morphism, then substituting U by U ∩ ϕ−1V , we can assume ϕ : U → V is a morphism. Then ϕ∗(g) is a regular function on U, hence is a rational function on X. The main theorem of this section:

Theorem 1.4.4. The contravariant functor X 7→ K(X), ϕ 7→ ϕ∗ gives an equivalence between following two categories: A. Category of varieties and dominant rational maps. B. Category of finitely generated field extensions of the base field k.

Proof. We construct an inverse functor from B to A. For each f.g. field extension K = k(y1, . . . , yn) where y1, . . . , yn are not necessarily algebraically independent, consider k[y1, . . . , yn]. This is a quotient ring of the polynomial ring k[z1, . . . , zn], hence is the coordinate ring of some vareity Y ⊆ An. For a field morphism θ : K(Y ) → K(X), we need to define a dominant rational map ϕ : X 99K n Y . we may assume Y is affine in A , whose coordinates are y1, . . . , yn. If fi = θ(yi) are regular on X for all 1 ≤ i ≤ n, then ϕ = (f1, . . . , fn) defines the desired map. If fi are not all regular, replacing X by a nonempty open affine subset U on which all fi are regular. Then we have a ring homomorphism A(Y ) → A(U), by Proposition 1.3.7 it induces a morphism U → Y , which extends to a rational map X 99K Y .

In the above proof we need a very useful fact to reduce a problem on variety to a problem on affine variety:

Proposition 1.4.5. On any variety Y , there is a base for the topology consisting of affine open subsets, i.e., ∀P ∈ Y and a open subset U 3 P , there is an affine open subset V s.t. P ∈ V ⊆ U.

Proof. Reduce to the case Y = U ⊆ An. If Y is affine there is nothing to prove. Otherwise define Z = Y¯ − Y which is a closed subset of An, hence is defined by an ideal I. There is f ∈ I s.t. f(P ) 6= 0. Then we use the following fact: 1.4. RATIONAL MAPS 11

Fact: affine variety−Z(f) =affine variety. Actually, suppose X ⊆ An is defined by the ideal I, then X − Z(f) can be embedded into An+1 by 1 (x1, . . . , xn) 7→ (x1, . . . , xn, ) f(x1, . . . , xn) which is defined by the ideal I + (1 − fxn+1). Remark 1.4.6. We are slow in this week since the presentation on homeworks took longer than I expect. For next week, only two problems instead of three will be presented.

Extra material. I want to digress a little bit from the main theme and talk about Gr¨obnerbases. The reason is because some students are concerning about how to compute a set of generators for the kernel of a ring homomorphism. First we briefly review Gr¨obnerbasis. I highly suggest reading Hal Schenck’s book “Compu- tational algebraic geometry”, and also learn to use Macaulay 2.

1. Fix a polynomial ring k[x1, . . . , xn], and fix an order on the monomials. Given a polynomial f, in(f) is the lagest monomial appears in f, called the initial monomial of f. P P a1 an b1 bn PExampleP1.4.7. Using graded reverse lexicographic: x1 . . . xn > x1 . . . xn if ai > bi, or if ai = bi and the right most nonzero entry of (a1 − b1, . . . , an − bn) is negative.

3 3 4 3 5 2 4 5 3 5 2 4 5 3 5 2 x1x2 > x1x2, x1x2x3 > x1x2x3. Then in(x1x2x3 − x1x2x3) = x1x2x3. 2. Definition of Gr¨obnerbasis.

Definition 1.4.8. A subset {f1, . . . , fn} of an ideal I called a Gr¨obnerbasis for I if the ideal < in(f)|f ∈ I >=< in(f1), . . . , in(fn) >.

3. S-pairs. For f, g monic polynomials, define LCM(in(f), in(g)) LCM(in(f), in(g)) S(f, g) = f − g in(f) in(g)

4. The algorithm to find a Gr¨obnerbases is not hard to describe–Buchberger algorithm.

Start from G = {f1, . . . , fk}. Repeat adding into G the remainder S of S(f, g) modulo G for each pair of elements in G, until no more nonzero element can be added. 5. To find a set of generators for the kernel of

(f1, . . . , fn): k[x1, . . . , xm] → k[y1, . . . , yn], consider the ideal (y1 − f1, . . . , yn − fn) in the larger ring k[x1, . . . , xm, y1, . . . , yn]. Fix any order that xi < yj, ∀i, j. Find its Gr¨obnerbasis. The elements in Gr¨obnerbasis which contains only x variables form a Gr¨obnerbasis for the kernel. Example 1.4.9. Find a set of generators for kerk[x, y, z] → k[t] given by (x, y, z) 7→ (t3, t4, t5). Fix Lex order t > z > y > x. Start from G = {t5 − z, t4 − y, t3 − x}. 12 CHAPTER 1. VARIETIES

S(t5 − z, t4 − y) = ty − z, and G = {t5 − z, t4 − y, t3 − x, ty − z}; S(t5 − z, t3 − x) = t2x − z, and G = {t5 − z, t4 − y, t3 − x, t2x − z, ty − z}; S(t5 − z, t2x − z) = t3z − zx = z(t3 − x) ≡ 0 mod(G), G unchanged; S(t5 − z, ty − z) = t4z − zy = z(t4 − y) ≡ 0 mod(G), G unchanged; S(t4 − y, t3 − x) = tx − y, G = {t5 − z, t4 − y, t3 − x, t2x − z, ty − z, tx − y}; S(t5 − z, tx − y) = t4y − zx = (ty − z)t3 + z(t3 − x) ≡ 0, G unchanged; S(t4 − y, t2x − z) = t2z − yx, G = {t5 − z, t4 − y, t3 − x, t2z − yx, t2x − z, ty − z, tx − y}; S(t5 − z, t2z − yx) = t3yx − z2 = z2 − yx2, G = {t5 − z, t4 − y, t3 − x, t2z − yx, t2x − z, ty − z, tx − y, z2 − yx2}; S(t5 − z, z2 − yx2) = t5yx2 − z3 ≡ 0; S(t4 − y, t2z − yx) = t2yx − zy = y(t2x − z) ≡ 0; S(t4 − y, ty − z) = t3z − y2 ≡ zx − y2, G = {t5 − z, t4 − y, t3 − x, t2z − yx, t2x − z, ty − z, tx − y, z2 − yx2, zx − y2}; S(t5 − z, zx − y2) = t5y2 − z2x ≡ 0; S(t4 − y, tx − y) = t3y − yx ≡ 0; S(t4 − y, z2 − yx2) = t4yx2 − z2y ≡ 0; S(t4 − y, zx − y2) = t4y2 − zyx ≡ 0; S(t3 − x, t2z − yx) = tyx − zx ≡ 0; S(t3 − x, ty − z) = t2z − yx ≡ 0; S(t3 − x, tx − y) = t2y − x2 ≡ tz − x2, G = {t5 − z, t4 − y, t3 − x, t2z − yx, t2x − z, tz − x2, ty − z, tx − y, z2 − yx2, zx − y2}; S(t3 − x, t2x − z) = tz − x2 ≡ 0; S(t3 − x, z2 − yx2) = t3yx2 − z2x ≡ 0; S(t3 − x, zx − y2) = t3y2 − zx2 ≡ 0; S(t2x − z, ty − z) = tx − y ≡ 0; ... Finally we get G = {t5 − z, t4 − y, t3 − x, t2z − yx, t2x − z, tz − x2, ty − z, tx − y, z2 − yx2, zy − x3, zx − y2}. So the last three form Gr¨obnerbasis for the kernel.

As you noticed, this computation is not suitable for human. (In this example we can use much simpler discussion by hand.) That is why I encourage you to run the following code in Macaulay 2.

S=QQ[t] R=QQ[x,y,z] 1.4. RATIONAL MAPS 13

F=map(S,R,{t^3,t^4,t^5}) ker(F)

--the following is the output-- 2 2 2 3 o48 = ideal (y - x*z, x y - z , x - y*z) 14 CHAPTER 1. VARIETIES

(Week 4, 2 classes)

As a consequence of Theorem 1.4.4, we have

Corollary 1.4.10. Let X, Y be two varieties, TFAE: (i) X and Y are birational, (ii) there are isomorphic open subsets U ⊆ X and V ⊆ Y , (iii) K(X) =∼ K(Y ) as k-algebras.

Proof. (ii)⇒(iii) by definition, (iii)⇒(i) by above theorem. For (i)⇒(ii), I found Hartshorne a bit confusing. It should be like this: denote ϕU : U → Y and ψV : V → X be regular. It is easy −1 −1 −1 −1 −1 −1 to check ϕU ψV (U) and ψV ϕU (V ) are isomorphic. Indeed, ϕU ψV (U) = {x ∈ U|ϕU (x) ∈ V } −1 −1 −1 −1 and ψV ϕU (V ) = {y ∈ V |ψV (y) ∈ U}. So for a point x ∈ ϕU ψV (U), its image ϕ(x) lies inside −1 −1 −1 −1 −1 −1 ψV ϕU (V ). Therefore ϕ induces a morphism ϕU ψV (U) → ψV ϕU (V ) and ψ induces an inverse morphism.

Example 1.4.11. A2 → A2 given by f(x, y) = (xy, y) is a birational morphism. Take U = A2 \{y = 0} and take V = A2 \{y = 0}, then f : U → V is an isom. Example 1.4.12. Normalization. (example: A = k[x, y]/(y2 − x3), the normalization is A[t] where t = y/x, or t2 = x. Integral closure has the same fraction field, so normalization is birational.) Example 1.4.13. Any variety X of dimension r is birational to a hypersuface Y in Pn. (Emphasize here that k is algebraically closed.)

Proof. We need some algebraic statement to completes the proof. Consider the function field K(X), by an algebraic fact we can find a transcendence base x1, . . . , xr such that K(X) is a finite separable extension of k(x1, . . . , xr). Moreover, we can find a ‘primitive element’ y, i.e. K(X) = m m−1 k(x1, . . . , xr, y). Since y is algebraic over k(x1, . . . , xr), it is satisfies a0y + a1y + ··· + am = 0, each ai ∈ k(x1, . . . , xr), clear the denominants, we can assume ai ∈ k[x1, . . . , xn].

Example 1.4.14. blow-ups.

Definition 1.4.15. The blow-up of An at the origin O is the closure of the embedding

An \ O,→ An × Pn−1 ¡ ¢ (x1, . . . , xn) 7→ (x1, . . . , xn), [x1 : ··· : xn] i.e. the subvariety defined by equations xiyj = xjyi.

We will come back to this notion in chapter 2. Here we only mention some basic properties for the blow-up π : Afn → An: (1) π−1(P ) is a point if P 6= O. In fact, Afn \ π−1(O) =∼ An \{O}. (2) π−1(O) =∼ Pn−1. It is called the exceptional divisor of the blow-up. 1.5. NONSINGULAR VARIETIES. 15

(3) The points in π−1(O) are in 1-1 corresponds to the lines through O. (For each point ((0,..., 0), [y1 : ··· : yn]) we associate the line (λy1, . . . , λyn).) (4) Aen is irreducible. (Pure topological fact: the closure of an irreducible subset is irreducible.) (5) Aen is nonsingular. (We will show this in the next section) Homework: 4.3, 4.4*(rational variety), 4.5, 4.6*(Cremona Transformation).

1.5 Nonsingular varieties.

n Definition 1.5.1. Let Y ⊆ A be an affine variety, f1, . . . , ft a set of generators of IY . Y is nonsingular at a point y ∈ Y if the rank of the Jacobian matrix (∂fi/∂xj(P )) is n − dim Y . Y is nonsingular if it is nonsingular at every point.

More intrinsically, Definition 1.5.2. Let A be a noetherian ring with maximal ideal m and residue field k. A is a 2 regular local ring if dimk m/m = dim A. (by Nakayama, it is the same to say the minimal number of generators equals the Krull dimension.) Theorem 1.5.3. Let Y ⊂ An be an affine variety and P ∈ Y . Then Y is nonsingular at P iff OP,Y is regular.

Proof. Let R = k[x1, . . . , xn], and P is defined by the maximal ideal aP = (x1 − a1, . . . , xn − an). n (1) Consider the map R → k , f 7→ (∂f/∂x1, . . . , ∂f/∂xn)(P ). It induces an isomorphism

2 n θ : aP /aP → k

The rank of the Jacobian matrix is the dimension of image of IY under the composition

2 n IY ⊂ aP → aP /aP → k ,

2 2 which is the dimension of (IY + aP )/aP .

(2) On the other hand, let m be the maximal ideal of OP . Since OP = (A/IY )aP , we have

2 2 m/m = aP /(aP + IY ).

This is a very useful fact, it holds because

2 (aP /IY )aP 2 2 m/m = 2 = (aP /(aP + IY ))aP = aP /(aP + IY ). ((aP + IY )/IY )aP

Compare (1)(2), rank(Jacobian matrix)+dim m/m2 = n. So the rank= n−dim Y iff dim m/m2 = dim Y , i.e. iff Oy,Y is regular.

Now we can define nonsingularity for arbitrary variety. 16 CHAPTER 1. VARIETIES

Definition 1.5.4. A variety Y is nonsingular at point P if the local ring OP,Y is regular. The variety is nonsingular if it is nonsingular at every point.

Theorem 1.5.5. Let Y be a variety. The set of singular points is a proper closed subset of Y .

Proof. We need a lemma, saying that dim m/m2 ≥ dim A, i.e. the rank of Jacobian matrix is at most n − r. So the points where the Jacobian matrix has rank n − r must form a closed subset. The rest is to show Y must be nonsingular on a nonempty open subset. First we may replace the variety by a birational equivalent variety, so we only need to consider n a hypersurface in A defined by a irreducible polynomial f(x1, . . . , xn) = 0.

If Sing(Y ) = Y , then ∂f/∂xi = 0 on Y , but it has degree lower than f, so ∂f/∂xi = 0. In p p char 0, this is impossible. In char p, this implies f is a polynomial in xi , for each i. Then f = g for some g, contradicts to the fact that f is irreducible.

Remark 1.5.6. Zariski tangent.

Homework: 5.1, 5.2, 5.5, 5.10*(Zariski tangent).

At this stage we are familar with the basic notion of algebraic varieties. I skip §6 (curves) and §7 (intersection theory) now and will come back later. Now proceed to the notion of schemes. Chapter 2

Schemes

(week 5, two classes)

2.1 Sheaves

We start from (pre)sheaf of abelian groups, and you may replace ’abelian groups’ by ’sets’, ’groups’, ’rings’,etc.

Definition 2.1.1. Let X be a topological space. A presheaf F of abelian groups consists of the data (a) for every open U ⊂ X, an abelian group F(U),

(b) for every pair of open sets V ⊂ U, a ‘restriction’ map resUV : F(U) → F(V ) satisfying the following axioms: (0) F(∅) = 0, the zero abelian group,

(1) resUU : F(U) → F(U) is the identity map,

(2) If W ⊂ V ⊂ U are open sets, then resUW = resVW ◦ resUV . An element s ∈ F(U) is called a section of F on U.

Sometime it is more intuitive to write f|V instead of resUV f. The axiom (0) should be replaced by the final object in a category if you wish to consider sheaves of sets, etc.: in case of (Set) it is a point, in case of (Group) it is the trivial group (1), in case of ’rings’, it is the trivial ring, etc. Let T op(X) be the category whose objects are open subsets and morphisms are inclusion maps. Here is a fancy definition of presheaf if you like category language: a contravariant functor

F : T op(X) → Ab from the category T op(X) to the category of abelian groups. 17 18 CHAPTER 2. SCHEMES

Definition 2.1.2. A presheaf F is called a sheaf if it satisfies

Identity axiom: If {Ui → U} is an open covering of an open set U and s, t ∈ F(U) such that s|Ui = t|Ui for all i, then s = t.

Gluability axiom: If {Ui → U} is an open covering of an open set U and si ∈ F(Ui) such that si|Ui∩Uj = sj|Ui∩Uj for all i, j, then there is an element s ∈ F(U) such that s|Ui = si.

A fancy definition using category theory: the above two condition is to say the following diagram is exact (i.e. it gives an equaliser): Y Y F(U) → F(Ui) ⇒ F(Ui ∩ Uj) i i,j Example 2.1.3. Sheaf of functions, continuous functions, differential functions/forms, holomorphic functions/forms. Example 2.1.4. Sheaf of regular functions on a variety. Example 2.1.5. Constant presheaf (why it fails to be a sheaf?) and constant sheaf. For example the constant sheaf Z is to assign to each open set U the abelian group Hom(U, Z), i.e. the direct product of Z, one for each connected component of U. Example 2.1.6. Skyscraper sheaf, that is F(U) = A iff x ∈ U. Equivalently, skyscraper sheaf is a sheaf whose stalks are all zero except at point x. Example 2.1.7. presheaf assign to each U the cohomology Hi(U, Z). (fail the identity axiom: look at the Mayer–Vietoris sequence i−1 i i i H (U1 ∩ U2) → H (U1 ∪ U2) → H (U1) ⊕ H (U2) for example, cover S1 by two invervals, both of which are contractable.) Example 2.1.8. presheaf of functions with compact support. (fail the Gluability axiom.)

Definition 2.1.9. Let F be a presheaf on X and P is a point on X. The stalk FP of F at P is lim F(U), −→ U3P for all open sets U containing p.

Elements in FP are called germs of sections of F at the point P .

To define the category of (pre)sheaves, we need to define morphisms. Definition 2.1.10. Let F and G be (pre)sheaves on X. A morphism ϕ : F → G consists of morphisms of abelian groups ϕ(U): F(U) → G(U) for all open set U, such that for all pairs V ⊂ U, the following commute: ϕ(U) F(U) / G(U)

 ϕ(V )  F(V ) / G(V ) where the horizontal arrows are restriction maps in F and G. An isomorphism is a morphism which has a two-sided inverse. 2.1. SHEAVES 19

(In class, Lance pointed out that, in category language, the above definition is exactly saying that ϕ is a natural transformation from the functor F to the functor G.) Unlike presheaf, a sheaf can be determined by its local data. By taking the direct limit, a morphism between sheaves ϕ : F → G induces a morphism between stalks ϕP : FP → FG.

Proposition 2.1.11. A morphism between sheaves ϕ : F → G on a topological space X is an isomorphism iff ϕP : FP → GP are isomorphisms for all P ∈ X.

Proof. If ϕ is an isomorphism then clearly ϕP is an isomorphism. So the converse is really what we should show. Assume ϕP is an isomorphism for all P . To show ϕ has an inverse, it is sufficient to show ϕ(U): F(U) → G(U) is an isomorphism for all U.

(1) ϕ(U) is injective. If a section s ∈ F(U) maps to 0 ∈ G(U) by ϕ(U), then ϕP (sP ) = 0. but ϕP is isomorphic, so sP = 0. Now we claim that sP = 0(∀P ∈ U) implies s = 0. This is because by the definition of germ we can find a open set UP 3 P such that s|UP = 0, those UP gives a covering of U, therefore by Identity Axiom of sheaf, s = 0. −1 (2) ϕ(U) is surjective. Given any t ∈ G(U), each germ tP has a unique inverse sP := ϕP (tP ). We hope to glue those sP together using Gluability Axiom. Take an open set UP 3 P and a section s(P ) ∈ F(UP ) such that s(P ) gives the germ sP . ϕ(UP ) maps s(P ) to a section in G(UP ) which gives the germ tP , therefore by shrinking UP if necessary, we can assume ϕ(UP ) maps s(P ) to t|UP . Now we show those s(P ) can be glued together, i.e. they satisfy the Gluability Axiom: s(P ) = s(Q) after restricting to UP ∩ UQ. But ϕ(UP ∩ UQ) sends both to t|UP ∩UQ , by the injectivity we proved in (1), s(P ) = s(Q).

Definition 2.1.12. Let F be a presheaf. There is a unique sheaf F + and a morphism θ : F → F + satisfying the following ‘universal property’: any sheaf G with a morphism ϕ : F → G factors uniquely through θ, i.e. there is unique morphism ψ : F + → G such that ϕ = ψ ◦ θ. F + is the sheafification of F.

Proof. (quote from Mumford) The sheafification is the best possible sheaf you can get from a presheaf. The construction is first identify things which have the same restrictions, then add all things which can be patched together.

+ Construction: define a section in F (U) to be a map s : U → ∪P FP such that

(i) s(P ) ∈ FP (therefore Identity Axiom follows), (ii) for any P ∈ U, there is a neighborhood U 3 P and a section t ∈ F(U) such that the germ tQ = s(Q). (this guarantee the Gluability Axiom.) It is your excise to check above constructed sheaf satisfies universal property.

Now we define kernel, cokernel, image for a morphism of (pre)sheaves:

Definition 2.1.13. (1) For a morphism of presheaves ϕ : F → G, the kernel, cokernel, image of ϕ are presheaves defined by U → ker(ϕ(U)),U → coker(ϕ(U)),U → im(ϕ(U)). (2) For a morphism of sheaves, the kernel, cokernel and image are the sheafification of above. 20 CHAPTER 2. SCHEMES

(3) A morphism ϕ : F → G of sheaves is injective if ker(ϕ) = 0; ϕ is surjective if imϕ = G.

(4) A sequence Fi−1 → Fi → Fi+1 is exact if kerϕi = imϕi−1.

Note: injective/surjective/exact can be checked at stalks. Example 2.1.14. Here we give an example (in complex geometry) of an exact sequence of sheaves on C \ 0, and after taking global sections it is no longer exact.

0 → Z → O → O∗ → 0 where Z is the constant sheaf, O is the sheaf of regular functions, O∗ is the sheaf of regular functions which are nowhere zero, and O → O∗ is given by f 7→ exp(2πif). The sequence is exact, but after we take the global sections, it is not exact on the right, for example z is a global section of O∗, 1 but 2πi log z is not a well defined global section of O. Where goes wrong? We know that log z is not well defined because C \ 0 is not simply connected. So if we shrink an open set to a simply connected one, the sequence of global sections on the shrinked open set will be exact.

Next Thursday I will go to MSRI, and you can have a discussion session on the following problems: 1.2(Exactness can be checked on stalks), 1.14 (support), 1.15 (sheaf Hom), 1.21(examples of sheaves). Other fundamental problems: 1.9, 1.10, 1.12. 2.2. SCHEME 21

(week 6, one class)

Definition 2.1.15. Given f : X → Y a continuous map of topological spaces.

−1 (1) Let F be a sheaf on X. Then the direct image sheaf f∗F is defined by V → F(f V ). (2) Let G be a sheaf on Y . The inverse image sheaf f −1F is the sheafification of the presheaf defined by U → limV ⊃f(U) G(V ). If f : Z → X is an inclusion map, then f −1F is called the restriction of F to Z, denoted by F|Z .

−1 ∗ One reason to study f∗F and f F, or more generally f and the higher direct images of a sheaf, is to ’move’ the sheaf F on a different space which might be easier to study (eg. the Leray spectral sequence Hp(Y,RqfF) abuts Hp+q(X, F), and BBDG decomposition theorem), the other reason is by comparing a sheaf with the induced sheaf on another space and obtain the information of the morphism f itself (eg. a proof of Zariski’s main theorem, using the fact that if f : X → Y −1 is a projective morphism of Noetherian schemes, and f∗OX = f∗OY , then f (y) is connected for any y ∈ Y . In case when both X and Y are integral scheme and Y is normal, we can check that the condition f∗OX = OY is satisfied, so all fibers are connected.)

2.2 Scheme

I found the following history of scheme interesting. Italian school had often used the somewhat foggy concept of ”generic point” when proving statements about algebraic varieties. What is true for the generic point is true for all points of the variety except a small number of special points. In the 1920s, Emmy Noether had first suggested a way to clarify the concept: start with the coordinate ring of the variety (the ring of all polynomial functions defined on the variety); the maximal ideals of this ring will correspond to ordinary points of the variety, and the non-maximal prime ideals will correspond to the various generic points. In the 1930s, Wolfgang Krull turned things around and took a radical step: start with any commutative ring, consider the set of its prime ideals, turn it into a topological space by introducing the Zariski topology, and study the algebraic geometry of these quite general objects. Others did not see the point of this generality and Krull abandoned it. (It reminds me that Walle throw off the diamond ring and kept the beautiful box.) In the 1950s, Jean-Pierre Serre, Claude Chevalley and Masayoshi Nagata, motivated largely by the Weil conjectures relating number theory and algebraic geometry, pursued similar approaches with prime ideals as points. According to Pierre Cartier, the word scheme was first used in the 1956 Chevalley Seminar, in which Chevalley was pursuing Zariski’s ideas. Alexander Grothendieck then gave the decisive definition. He defines the spectrum of a com- mutative ring as the space of prime ideals with Zariski topology, but augments it with a sheaf of rings: to every Zariski-open set he assigns a commutative ring, thought of as the ring of ”polynomial functions” defined on that set. These objects are the ”affine schemes”; a general scheme is then 22 CHAPTER 2. SCHEMES obtained by ”gluing together” several such affine schemes, in analogy to the fact that projective varieties can be obtained by gluing together affine varieties. So, what are the new stuff that belong to schemes? Generic points, nilpotent function (from which we can define infinitesimal objects), connection with arithmetic number theory. Now we are about to define the spectrum. Definition 2.2.1. Let R be a commutative ring with a unit element 1. (1) SpecR is the set of prime ideals P ( R. (2) We can augment the Zariski topology on SpecR: the closed sets are of the form

V (I) := {P ⊇ I|P ∈ SpecR}, for any ideal I of R.

To verify the closed set indeed give a topology, we notice the following fact in commutative algebra:

Lemma 2.2.2. Let I1,I2,Jα be ideals of R. Then

(1) V (I1) ∪ V (I2) = V (I1 ∩ I2) P (2) ∩αV (Jα) = V ( α Jα). √ √ (3) V (I1) ⊆ V (I2) iff I1 ⊇ I2. Example 2.2.3. Speck for a field k. Example 2.2.4. Speck[x]/(x2), is set theoretically the same as Speck. But we expect them to be different, right? Well, they will be assigned different sheaves of rings. Example 2.2.5. Let k be algebraically closed. Speck[x], Speck[x, y]. Generic points and closed points. Topology. Example 2.2.6. SpecZ, its dimension, closed points and generic point. Topology. Compare it with Speck[x].

Now we associate with X = SpecR a natural sheaf of rings: the structure sheaf OX . Let Ap be the localization of A at a prime ideal p.

Definition 2.2.7. The structure sheaf OX (or O if no confusion arises) for X = SpecR `is a sheaf of rings defined as follows: for each open set U ⊂ X, a section in O(U) is a map s : U → p∈U Rp, such that s(p) ∈ Rp and for any p ∈ U, there is an open set V 3 p, elements a, f ∈ R, such that for any q ∈ V , f∈ / q and s = a/f ∈ Rq.

Remark: check that O(U) is a ring and OX indeed defines a sheaf. Usually SpecR means the set of prime ideals with Zariski topology and the structure sheaf O. Now consider a special kind of open sets: Definition 2.2.8. For any f ∈ R,define D(f) to be the complement of the closed set V (f) in SpecR. Then p ∈ D(f) iff f∈ / p. 2.2. SCHEME 23

c Note that these D(f) form√ a base for the Zariski topology. Indeed, if p ∈ V (I) and without c loss of generality assume I = pI, to find√ a D(f) such that p ∈ D(f) ⊆ V (I) is equivalent to find a f ∈ R satisfying f∈ / p and (f) ⊆ I. But p ∈/ V (I) means p + I, we can take f to be an element in I \ p.

Lemma 2.2.9. Any affine scheme is quasi-compact.

−1 n Recall Rf is the localization S R for S = {f |n ∈ Z≥0}. Proposition 2.2.10. Consider X = SpecR and its structure sheaf O.

(1) The stalk Op of the sheaf O is isomorphic to the local ring Rp.

(2) For f ∈ R, the ring O(D(f)) is isomorphic to Rf . (3) The ring of global sections O(X) = R.

Proof. (1) For any a/f ∈ Rp, f∈ / p hence p ∈ D(f), then a/f defines a section in O(D(f)), hence determines an element in Op. It is well defined: if a/f ∼ b/g, then ∃h∈ / p, h(ag − bf) = 0, a/f = agh/fgh = bfh/fgh = b/g on D(fgh). So a/f and b/g induce the same element in the stalk Op.

Now` construct Op → Rp. By our definition of O, an element in x ∈ Op is induced by s : U → p∈U Rp such that s(p) ∈ Rp. Then send a to s(p). Easy to see it is well defined. It is an exercise to check that we have defined mutually inverse morphisms. (3) is a special case of (2).

(2) There is a natural map ϕ : Rf → O(D(f)). m m ϕ is injective: suppose ϕ(a/f ) = 0, then a/f restrict to every stalk Rp is 0, so for each p, there is h∈ / p, ah = 0, h ∈ ann(a). So ann(a) * p, i.e. p ∈/ annp(a). Since p is arbitrary, V (ann(a))∩D(f) = ∅, so V (ann(a)) ⊆ V (f), which is equivalent to f ∈ ann(a), i.e. f k ∈ ann(a), then a/f m = af k/f m+k = 0.

ϕ is surjective: take s ∈ O(D(f)). Assume s| = si = ai/hi finite many i ∈ [1, k] such O(Dhi ) m m that ∪D(hi) = X. Since si = sj on D(hihj), hi hj (aihj − ajhi) = 0. We may assume m is the same for any pair of i, j since we have only finite many i, j.

m+1 m By replacing hi by hi and hi ai by ai, we may assume hiaj = hjai. n n P P n Since f ∈ (h1, . . . , hk), f = bihi. Let a = biai. Check that a/f = ai/hi on each D(hi). n So a/f is the expected element in Rf . To complete the proof, we need to show that we do not lose anything by assuming s| = O(Dhi ) si = ai/hi. For si = ai/gi, we can consider aihi/gihi in a smaller open set D(gihi) instead. The finiteness follows from the quaisi-compactness of an affine scheme. 24 CHAPTER 2. SCHEMES

(Week 7, two classes.) To glue together the spectrum, we need functoriality of A → (SpecA, O). We want a category containing those pairs (SpecA, O) and their glues. For this purpose, we define ringed space and locally ringed space.

Definition 2.2.11. (1) A ringed space is a pair (X, OX ) consisting of a topological space X and a sheaf of rings OX (structure sheaf). ] A morphism of ringed spaces from (X, OX ) to (Y, OY ) is a pair (f, f ), where f : X → Y is a ] continuous map and f : OY → f∗OX is a morphism of sheaves of rings on Y .

(2) A locally ringed space (X, OX ) is a ringed space that for every P ∈ X, the stalk OX,P is a local ring. A morphism (f, f ]) of local ringed space is a morphism of ringed spaces such that the induced ] map fP : OY,f(P ) → OX,P is a local homomorphism, i.e. the inverse image of the maximal ideal is a maximal ideal. (3) A morphism (f, f ]) of locally ringed spaces is an isomorphism if f is a homeomorphism of the underlying topological spaces and f ] is an isomorphism of sheaves.

] Remark 2.2.12. We need to explain the induced map fP . Since ] −1 f : OY (U) → OX (f (U)).

The neighborhoods of f(P ) form a directed system, so we can take the direct limit, we obtain

−1 OY,f(P ) = lim OY (U) → lim OX (f (U)). U3f(P ) U3f(P )

The right hand side naturally maps to OX,P . The composition gives a map OY,f(P ) → OX,P . Proposition 2.2.13. (a) For a ring A, (SpecA, O) is a locally ringed space. (b) A homomorphism φ : A → B of rings induces a morphism of locally ringed spaces

(SpecB, OSpecB) → (SpecA, OSpecA).

(c) Any morphism of locally ringed spaces from SpecB to SpecA is induced by a ring homo- morphism A → B.

Proof. (a) is proved before. (b) We need to check (i) the induced map f : SpecB → SpecA by mapping a prime ideal q to φ−1(q) is contin- uous. This is because for any closed set V (I) ⊆ SpecA, the inverse image f −1(V (I)) = {q ∈ SpecB|φ−1(q) ⊇ I} = {q ∈ SpecB|q ⊇ φ(I)} = V (φ(I)).

] (ii) the nature morphism f : OSpecA → f∗OSpecB induces local homomorphisms OSpecA,f(q) → OSpecB,q for every q ∈ SpecB. What is the natural morphism? It is determined by the ring homomorphism φ : A → B. This map induces morphism of stalks Af(q) → Bq, which is sufficient 2.2. SCHEME 25 to determine f ]. (One need to check those morphisms of stalks indeed glue together.) Such f ] induces the morphism of local rings Af(q) → Bq, the inverse of the maximal idealq ¯ is the maximal ideal f(q), therefore f ] is a local homomorphism. (c) Let (f, f ]) be the morphism from SpecB to SpecA. Then f ] induces an A-module morphism φ = f ](SpecA): A → B which is also a ring morphism. We’ll show φ induces (f, f ]). (i) To check φ−1(q) = f(q), consider the following commutative diagram

φ A / B

 ]  fq Af(q) / Bq

] ] −1 −1 but fq is local, so (fq ) (¯q) = f(q), then φ (q) = f(q). ] (ii) The morphism Aφ−1q → Bq induced by φ coincides with the one induced by f . This ] follows immediately by noticing in the above diagram φ determines fq .

The building block for scheme is affine scheme, defined as follows: Definition 2.2.14. (1) An affine scheme is a local ringed space which is isomorphic to SpecA for some ring A.

(2) A scheme is a locally ringed space (X, OX ) in which every point has an open neighborhood U such that (U, OX |U ) is an affine scheme. X is called the underlying topological space, OX the structure sheaf. (3) A morphism of schemes is a morphism as local ringed space. An isomorphism is a morphism with two sided inverse.

One way to construct schemes is using Gluing Lemma (exercise 2.12). Here are two examples: 1 Example 2.2.15. Projective line Pk. Speck[x] glues with Speck[y] by Speck[x, 1/x]. Example 2.2.16. Non-separated affine line by gluing two pieces of Speck[x] along Speck[x, 1/x].

Now we define projective schemes as a ‘generalization’ of projective variety.

Definition 2.2.17. Let S be a graded ring S and S+ = ⊕d>0Sd.

(1) Define the set P rojS to be the homogeneous ideals p which do not contain S+. (2) Define the topology of P rojS as follows: the closed sets are of the form

V (I) = {p ∈ P rojS|p ⊇ I}, where I is a homogeneous ideal of S.

(3) Define the structure sheaf O on P rojS as follows: for p ∈ P rojS, denote S(p) be the degree −1 0 elements in the localization T S where T is the multiplicative` system of homogeneous elements not in p. Then a section in O(U) is a map s : U → S(p), such that s is locally a quotient of elements in S, i.e. ∀p ∈ U, there is an open neighborhood V 3 p and homogeneous elements a, f ∈ S of the same degree, such that for any q ∈ V , f∈ / q, and s(q) = a/f. 26 CHAPTER 2. SCHEMES

Above indeed gives a scheme. To show this, we need to show OP rojS,p is a local ring, and find an affine cover for P rojS. Actually, we have

Proposition 2.2.18. Let S be a graded ring.

(1) Op is isomorphic to the local ring S(p).

(2) For any f ∈ S+, denote the open set D+(f) = {p|f∈ / p}. Those open sets cover P rojS, and ∼ (D+(f), O|D+(f)) = SpecS(f).

Proof. For (1), the proof is similar to the affine case. We can easily define a map Op → S(p) and a natural map in the other direction. The rest is to check they are inverse to each other.

For (2), we need to show (i) D+(f) and SpecS(f) are homeomorphic as topological spaces, and

(ii) isomorphism of sheaves O|D+(f) and OSpecS(f) . For (i), given any ideal I of S that f∈ / I, consider the map I to

n {g/f |g ∈ I, deg g = n deg f} = ISf ∩ S(f).

Check that it gives an homeomorphism ϕ : D+(f) → SpecS(f). ] For (ii), what is the natural morphism ϕ : OSpecS(f) → ϕ∗(O|D+(f))? Notice that the stalk of ∼ the left hand side at ϕ(p) is OSpecS(f),ϕ(p) = S(p), which is naturally isomorphic to the stalk of the right hand side at p. These isomorphisms determine an isomorphism ϕ] of sheaves.

n Example 2.2.19. Fix a ring A. Define the projective space PA := P rojA[x0, . . . , xn] where each xi has degree 1.

1 PZ = P rojZ[x0, x1]. The graded ring S = Z[x0, x1], and S+ = (x0, x1). Then prime ideals 1 (2), (2, x0) are in PZ, but (2, x0, x1) is not.

Example 2.2.20. Weighted projective space P(a0, . . . , an)(ai ∈ Z+).

(n+1) ∗ Consider the quotient (Ak \ 0)/k under the equivalent relation

a0 an (z0, . . . , zn) ∼ (t z0, . . . , t zn) for t ∈ k∗. To have a well-defined zero set, the definition polynomial must be ‘weighted homoge- a an neous’, i.e. f(t 0 z0, . . . , t zn) is proportional to f(z0, . . . , zn). By assigning deg zi = ai, the degree of each monomial of f must be the same. It is nature to guess the weighted projective space is Proj of the graded ring S = k[z0, . . . , zn] with deg zi = ai. Let look at a concrete example P(1, 1, a). Proposition 2.2.18 help us to find out an affine a covering for P(1, 1, a). D+(x0) is isomorphic to the spectrum of S(x0), which is k[x1/x0, x2/x0], 2 isomorphic to Ak. Similar as D+(x1). D+(x2) is more interesting. It is the spectrum of S(x2) = a a−1 a k[x0/x2, x0 x1/x2, . . . , x1/x2] which is a singular surface. It is an affine cone over the rational normal curve of degree a in Pa, i.e. the image of P1 → Pa given by (x : y) 7→ (xa : xa−1y : ··· : ya). We know the affine cone is singular for a > 1 because the dimension of the Zariski tangent space at the origin is a + 1. 2.3. FIRST PROPERTIES OF SCHEMES 27

We state the following without proof.

Proposition 2.2.21. Let k be an algebraically closed field. There is a natural fully faithful functor from (Varieties/k) to (Schemes/k).

2.3 First properties of schemes

We list definitions of some properties. (1) A scheme is connected if its topological space is connected, i.e. it cannot be the disjoint union of two nonempty open subsets.

Example 2.3.1. SpecZ is` connected. SpecZ/(6) = SpecZ/(2) × Z/(3) is not connected. In general Spec(A1 × · · · × An) = SpecAi is not connected.

(2) A scheme is irreducible if its topological space is irreducible, i.e. it is not the union of two proper closed subsets. Example 2.3.2. In A2, y2 − x2 defines a reducible scheme when char6= 2. while y2 defines an irre- ducible scheme. In general SpecA is irreducible if the nilradical is prime. Moreover, each minimal prime ideal corresponds to a irreducible component. (recall that nilradical is the intersection of all primes.)

(3) A scheme X is reduced if OX (U) has no nilpotent elements for every open subset U. Equivalently, if OX,P has no nilpotent elements. Example 2.3.3. y2 defines a non-reduced scheme. SpecA is reduced if the nilradical is 0.

(4) A scheme X is integral if for every open set U, OX (U) is an integral domain. A fact: reduced+irreducible=integral. (read Hartshorne.) 28 CHAPTER 2. SCHEMES

(Week 8: two classes)

(5) A scheme is locally noetherian if there is an affine cover by SpecAi where each Ai is noetherian. A scheme is noetherian if there is a finite such cover. Being a noetherian scheme is stronger than being a noetherian top space. 2 Example 2.3.4. Spec k[x1,... ]/m is not a noetherian scheme, but it is of dimension 0. Another example is R = Spec (k + yk[x, y]), it is not noetherian since there is an ascending chain of ideals (xy) ⊂ (xy, x2y) ⊂ (xy, x2y, x3y) ⊂ · · · On the other hand, dim R = 2.

A fact: A scheme is locally noetherian iff every affine open subset has a noetherian coordinate ring.

(6) A morphism f : X → Y is locally of finite type if there is an affine covering {SpecBi} of Y , −1 such that f (SpecBi) is covered by open affine subsets {SpecAij}, and Aij is a finitely-generated −1 Bi-algebra. f is of finite type if f (SpecBi) can be covered by finite many {SpecAij}. Example 2.3.5. Let k be an algebraically closed field. Regard a variety as a scheme, then it is of finite type over k.

Example 2.3.6. Speck[x](x) is not of finite type over k. SpecOP is not in general of finite type over k.

(7) A morphism f : X → Y is a finite morphism if there is an open covering {SpecBi} of Y , for −1 each i, f SpecBi is isomorphic to SpecAi where Ai is a Bi-algebra which is a finitely generated Bi-module. In the definition of (locally) of finite type and finite, we can require the given properties for every open covering of Y instead of some open covering. A morphism is quasi-finite if for each point y ∈ Y , f −1(y) is a finite set. Example 2.3.7. A1 \ 0 → A1 is quasi-finite but not finite.

f : Speck[x1, ...] → Speck is not locally of finite type. Actually, (quasi-finite)+(proper)=(finite). Indeed, quasi-finite is “etale-locally” the same as finite, in the following sense: (cf. N´eron models, 2.3.Prop8, proved using strict henselization.) Proposition 2.3.8. Let f : X → Y be locally of finite type, quasi-finite at x ∈ X. Then there is an ´etaleneighborhood Y 0 → Y of y such that the morphism f 0 : X0 → Y 0 (obtained by base change) 0 0 0 0 0 induces a finite morphism f |U 0 : U → Y where U is an open neighborhood of the fiber of X → X above x. If f is separated, then U 0 is a connected component of X0.

(8) An open subscheme of a scheme X is a scheme U whose topological space is an open subset ∼ of X and whose structure sheaf OU = OX |U . An open immersion is a morphism f : X → Y which induces an isomorphism of X with an open subscheme of Y . (9) A f : X → Y is a closed immersion if f(X) is a closed subset of the underlying space of Y , ] and f : OX → f∗OY is surjective. A closed subscheme is an equivalent class of closed immersion 2.3. FIRST PROPERTIES OF SCHEMES 29 where f : Y → X and f 0 : Y 0 → X are equivalent if there is an isomorphism i : Y 0 → Y such that f 0 = f ◦ i. Example 2.3.9. Spec k → Spec k[x]/(x2) is a closed immersion, but Spec k[x]/(xn) → Spec k is not. On the other hand, Spec k[x]/(xn) → Spec k[x] is a closed immersion, which is the infinitesimal neighborhood of the origin in the line A1 defined by x = 0. In general, Spec A/I → Spec A is a closed immersion, and every closed subscheme of Spec A arises in this way. Remark 2.3.10. Note that closed subscheme is more delicate than open subscheme in the sense that it is not enough to specify the topological subset. It is convenient to specify a closed subscheme by an ideal sheaf IY .

Homework: 3.1* (cf. EGA I, Prop 6.3.2 pp144), 3.11, 3.12, 3.13. (10) Reduced induced closed subscheme structure, i.e. given a scheme X and a closed subset Y , there is a ’smallest’ subscheme structure on Y (in the sense of closed immersion), which can be constructed as follows. √ First assume X = Spec A, for a subset Y defined by an ideal I, then I = ∩{p : p ⊇ I} defines the reduced induced structure on Y .

For a general scheme X and a closed subset Y , take an affine cover Ui of X. Let Yi = Y ∩Ui and give it the reduced induced structure. Those Yi glue together to give a reduced induced structure on Y . (11) Dimension. The dimension of a scheme is defined to be the dimension of the underlying topological space. Codimension of a irreducible closed subset Z of X: the supremum of integers n such that there exists a chain

Z ⊂ Z1 ⊂ Z2 · · · ⊂ Zn of distinct closed irreducible subsets of X. For an arbitrary closed subset Y of X, define

codim(Y,X) := infZ⊆Y codim(Z,X) where Z runs through all closed irreducible subsets of Y . For affine integral scheme X of finite type over a field, any Y is a closed irreducible subset, then dimY + codim(Y,X) = dimX

Weird phenomenon (ex 3.21): Let R be a DVR containing its residue field(eg. k[[a]]). Let X = Spec R[t] and Y be defined by at − 1. Then dimY = 0, codim(Y,X) = 1 and dimX = 2. (12) Fiber product. In category theory, given a category (eg the category of schemes) and two morphisms X → S, Y → S in the category, the fiber product of X and Y over S is a space, denoted by X ×S Y together with two projection maps X ×S Y → X and X ×S Y → Y for which the following diagram 30 CHAPTER 2. SCHEMES commutes: / X ×S Y Y

  X / S and the space X ×Y is universal in the sense that for any Z making the following diagram commute

Z / Y

  X / S there is a unique morphism Z → X ×S Y make the following diagram commute. (omit the dia- gram...)

Theorem 2.3.11. The fiber product exists for the category of schemes.

Proof. If existence is proven, uniqueness follows from the universal property of fiber products.

(1) For X = Spec A, Y = Spec B,Z = Spec C, then X ×Z Y = Spec (A ⊗C B). (Show it first in the category of affine schemes, then the category of schemes. We need the fact that Hom(R, O(S)) =∼ Hom(S, Spec R)) (2) All we need to do is to patch the affine fiber products together. We cover S by affine charts Wk = Spec Ck. Then cover the inverse of Wk in X (resp. Y ) by affine charts Uki (resp. Vkj). Then need to show Φkij := Uki ×Wk Vkj can be glued together to a scheme. For this, check that we can naturally glue Φkij and Φk0i0j0 along

0 0 0 0 (Uki ∩ Uk i ) ×S1∩S2 (Vkj ∩ Vk j )

Then check the cocycle condition. c.f. EGA Ch1, pp.106-107. 2.4. SEPARATED AND PROPER MORPHISMS 31

(Week 9, two classes, next week is spring break.) (13) Example of fiber product: (a) Base change.

0 0 Let S be a scheme, X be an S-scheme and S an S-scheme. Then XS0 := X ×S S is a S0-scheme, this process is called base change by S0 → S.

n n Let S be a scheme. Define PS := PZ ×Spec Z S. (b) Fiber over a point. Let f : X → Y be a morphism of schemes, y ∈ Y be a point with residue field k(y). Let Spec k(y) → Y be the natural morphism. Then we define the fiber of f over the point y to be the scheme Xy := X ×Y Spec k(y) Example 2.3.12. Consider Spec k[x, y, t]/(ty − x2). Explain the reduced structure at t = 0.

2.4 Separated and proper morphisms

I will focus on applications instead, and will be rather sketchy on the proof. First we define separated morphism. Definition 2.4.1. Let f : X → Y be a morphism of schemes.

(1) The diagonal morphism is the unique morphism ∆ : X → X ×Y X such p1 ◦ ∆ = p2 ◦ ∆ = idX . (2) f is separated if ∆ is a closed immersion. Example 2.4.2. The line over k with double origins is not separated over k. Since the diagonal is not closed: any function vanish on the diagonal must vanish on 4 origins.

Glue two varieties along open subsets is not separated in general. (explain the geometry of valuative criterion). Example 2.4.3. A morphism of affine schemes is separated. Let X = Spec A and Y = Spec B, then X → X ×Y X corresponds to the ring morphism B ×A B → A, which is surjective, so X → X ×Y X is a closed immersion. Remark 2.4.4. Surjection of rings ⇒ Closed immersion of affine schemes.

But injection of rings ; surjection of schemes. For example Z → Q. Proposition 2.4.5. A morphism f : X → Y is separated iff the set-theoretic image of the diagonal morphism ∆ is a closed subset of X × X.

Proof. Obviously separatedness implies the ∆(X) is closed. So we need to prove that if ∆(X) is closed then (1) X → ∆(X) is a homeomorphism, (2) the induced morphism OX×Y X → ∆∗OX is surjective. 32 CHAPTER 2. SCHEMES

For (1), notice that the composition X → ∆(X) → X is identity.

For (2), it is local in X ×Y X. So we can restrict to an open affine subset U = Spec A of X, with its image f(U) in an open affine subset V = Spec B. Then it boils down to the affine case which we have proved.

Before we give the valuative criterion for separatedness and properness, let us review basic facts on valuation ring:

Definition 2.4.6. Let K be a field and G a totally ordered abelian group. A valuation of K with values in G is a map v : K \{0} → G s.t. (1) v(xy) = v(x) + v(y), ∀x, y ∈ K, x, y 6= 0; (2) v(x + y) ≥ min(v(x), v(y)), ∀x, y ∈ K, x, y, x + y 6= 0. If v is a valuation ring, R = {x ∈ K|v(x) ≥ 0} ∪ {0} is called the valuation ring of v. If R is a valuation ring with values in Z, then R is called a discrete valuation ring.

We need the following Fact:

Theorem 2.4.7. [Atiyah,Macdonald, Thm 5.21, ex 27] A local ring R in a field K is a valuation ring if it is maximal of the set of local rings contained in K w.r.t. the domination relation, i.e. we say B dominates A if A ⊆ B and mB ∩ A = mA.

Theorem 2.4.8 (Valuative criterion of separatedness, EGA II, 7.2.3). . Let Y be a scheme (resp. locally noetherian scheme), f : X → Y be a morphism (resp. a morphism which is locally of finite type). Then f is separated iff the following holds: X → X ×Y X is quasi-compact, and for any valuation ring R (resp. discrete valuation ring R) with quotient field K, in the following commute diagram Spec K / X

  Spec R / Y there is at most one lifting Spec R → X making the whole diagram commute.

Proof. One direction: suppose f is separated, i.e. ∆ : X → X ×Y X is a closed immersion. Let h1, h2 : Spec R → X be two lifts making the whole diagram commute. By definition of fiber product we have h = (h1, h2) : Spec R → X ×Y X. The restriction of h to Spec K is in the diagonal, hence the image of h(Spec R) is in the diagonal set-theoretically. But Spec R is reduced (reduced structure is minimal), so h factors through the diagonal X: Spec R → X → X ×Y X, which means that h1 = h2. The other direction: suppose the lifting condition is satisfied, it suffices to show that ∆(X) is a closed subset in X ×Y X.(note here that we don’t need to prove it is a subscheme, which save us lots of strength). For this, we use the following fact: 2.4. SEPARATED AND PROPER MORPHISMS 33

∆(X) is a closed subset in X ×Y X if it is stable under specialization, i.e. let z1 be a point in ∆(X), z0 be a point in z1, then z0 should be in ∆(X).

The idea is to construct a valuation ring R and f : Spec R → X ×Y X s.t. the generic point sends to z1 and the closed point sends to z0. Then p1 ◦ f, p2 ◦ f : Spec R → X give two lifting, and by the lifting condition we have p1 = p2. Hence f factors through ∆ as ∆ f : Spec R → X → X ×Y X then z0 is in the diagonal ∆(X).

So to finish the proof, we need to find a valuation ring that works. Let K = k(z1) the residue field of z1. Let O be the local ring of z0 on z1 and let R be a valuation ring of K dominating O. The existence of such an R is guaranteed by Theorem 2.4.7. Proposition 2.4.9. Assuming all schemes below are noetherian. (1) Open and closed immersions are separated. (2) A composition of two separated morphism is separated. (3) Separatedness is stable under base change. (4) If f : X → Y , g : X0 → Y 0 are separated morphisms over a base scheme S, then f × f 0 : 0 0 X ×S X → Y ×S Y are separated. (5) If f : X → Y and g : Y → Z are morphisms, if g ◦ f is separated then f is separated.

−1 (6) f : X → Y is separated iff Y has a covering {Vi} such that f (Vi) → Vi are separated.

Proof. Show (5) in class using valuative criterion.

Similar to the separatedness, the properness for a morphism of schemes cannot be defined by requiring the inverse image of a quasi-compact set is quasi-compact, since there are too many quasi-compact sets. (recall that all affine schemes are quasi-compact). Amazingly, we can use closed morphism to define proper morphism. Definition 2.4.10. (1) A morphism is closed if the image of any closed subset is closed. A morphism is universally closed if the morphism is closed for every base change. (2) A morphism is proper if it is separated, or finite type, and universally closed. Example 2.4.11. A1 → pt. Although it is closed since a point is a closed subset of itself, but it is not ’closed in heart’. And the way to find it out is by base-change. Theorem 2.4.12 (valuative criterion for properness, EGA II, 7.3.8). Let Y be a scheme (resp. locally noetherian scheme), f : X → Y be quasi-compact separate (resp. of finite type). Then f is universally closed (resp. proper) iff for any valuation ring R (resp. discrete valuation ring R) with quotient field K, in the following commute diagram Spec K / X

  Spec R / Y 34 CHAPTER 2. SCHEMES there is at least (resp. exactly) one lifting Spec R → X making the whole diagram commute.

The proof is skipped. Again, we have the following properties: Corollary 2.4.13. Assuming all schemes below are noetherian. (1) A closed immersion is proper. (2) Composition of proper morphisms is proper. (3) Proper morphisms are stable under base change. (4) Products of proper morphisms is proper. (5) Given f : X → Y , g : Y → Z, if g ◦ f is proper and g is separated, then f is proper. (6) Properness is local on the base.

Why do we care about properness? Here are some good properties. (a) We use it to define a complete k-variety if it is proper over k.

q (b) For a proper f : X → Y and a coherent sheaf F on X, R f∗(F )) is also coherent. In particular, if X is a complete variety over k, then Hq(X,F ) is finite dimensional. Very often, we require a morphism f : X → S to be proper and flat to form a good family. So, what are the examples of proper morphisms? We will give a large class of proper morphisms: projective morphisms. You may think of them as families of projective schemes. Here is the n n definition. Recall that the projective space PY is defined by PZ ×Spec Z Y . Definition 2.4.14. (1) A morphism f : X → Y is projective if it factors into a closed immersion n n i : X → PY for some n, followed by the projection PY → Y . (2) A morphism is quasi-projective if it is a composition of an open immersion followed by a projective morphism.

Note: in EGA II 5.5 the projective morphism is defined more general as the composition of a closed immersion followed by the projection P(E) → Y where E is a quasi-coherent OY -sheaf of finite type. As pointed out by Hartshorne, two definition coincide when Y is quasi-projective over an affine scheme. As the first application of valuative criterion, we show the following Theorem 2.4.15. A projective morphism of (noetherian) scheme is proper.

Proof. We need noetherian condition to keep the proof simple, but the theorem is true in general, cf. EGA II, 5.5.3.

n Let f : X → Y be projective. First, since X → PY is proper and the composition of proper n morphisms is proper, so suffices to prove PY → Y is proper. Since properness is stable under base n change, it suffices to prove PZ → Spec Z is proper. 2.4. SEPARATED AND PROPER MORPHISMS 35

Given / n Spec K PZ

  Spec R / Spec Z

Let z1 be the point in Spec K, ξ1 the image of z1 in X.

(1) Existence of the lift. Recall that PZ is covered by affine charts Vi := Spec Z[x0/xi, . . . , xn/xi]. 2 (Draw the 3 lines xyz = 0 in P on blackboard and draw the ‘cloud’ ξ1). Assume that ξ is at a

‘good’ position, in the sense that all the functions xi/xj are invertible in Oξ1 . (if not, then ξ1 lies n in a coordinate hyperplane of PZ and we can use induction on n.) Pull back xi/xj to Spec K, we get functions fij ∈ K satisfying fijfjk = fik. Find fij with largest valuation, say f10. (In picture, x0 = 0 is the ‘farthest’ hyperplane to ξ1). Then all fi0 ∈ R, because if v(fi0) < 0, then v(f1i) = v(f10) − v(fi0) > v(f10) contradicts with the choice of f10. Thus we have a morphism

Z[x1/x0, . . . , xn/x0] → R which induces Spec R → X compatible with Spec K → X.

(2) For the uniqueness of the lift, note that in the above argument, Spec K → Vj extends to Spec R → Vj iff fij ∈ R for all i, and such an extension is unique. If there are a lifting Spec R → Vj and a lifting Spec R → Vj0 , then by restricting to Spec R → Vj ∩ Vj0 we can show that they will give the same map Spec R → X. 36 CHAPTER 2. SCHEMES

(Week 10, after spring break). Students asked about the relation of properness defined in analytic geometry and algebraic geometry. They are indeed the same, in the following sense: (cf. Hartshorne: App B, transcendental methods, and Serre’s GAGA.)

Let X be a scheme of finite type over C, Xh the associated complex analytic space (it can have nilpotent functions).

X Xh separated Hausdorff connected connected reduced reduced smooth smooth f : X → Y is proper f : Xh → Yh is proper The following theorem shows that a proper morphism is very close to a projective morphism. Theorem 2.4.16 (Chow’s lemma, for detail, cf. EGA II 5.6.1). Let X be proper over a noetherian scheme S. There exists g : X0 → X such that X0 → S is projective, and there is an dense open subset U of X such that g : f −1U → U is an isomorphism.

Proof. (A) Reduce to the case when X is irreducible. This is possible since X has finite many irreducible components. (Note that by definition of properness X → S is of finite type, so X is noetherian).

−1 (B) Suppose X is irreducible. Cover S by fine affine charts Sj. Cover each f (Sj) by finite affine charts Tjk. Since Tjk → Sj and Sj → S are both quasi-projective, so is Tjk → S. For convenience, rename those Tjk by Ui (1 ≤ i ≤ n). For each Ui, there is an open immersion Ui → Pi where Pk a projective space over S. Let U = ∩Ui (which is non-empty since X is irreducible) and consider the map φ : U → X ×S P1 ×S P2 ×S · · · ×S Pn 0 0 0 Let X be the closure of φ(U). Then there is a natural map X → X and X → P := P1 ×S P2 ×S · · · ×S Pn. 0 0 (C) Since each Pi is projective over S, so is P . X → P is a closed immersion. So X is projective over S. (D) X0 → X is isomorphic on U, so birational.

2.5 Sheaves of Modules

Recall the sheaf of differential k-forms Ωk on a differential manifold X. For any open set U ⊆ X, the set of sections Ωk(U) is not only an abelian group: you can multiply a smooth function f ∈ C∞(U) with a form ω. Therefore Ωk(U) is a C∞(U)-module. In algebraic geometry, sheaves of modules play an dominant role to understand schemes and the morphism among them.

Definition 2.5.1. Let (X, OX ) be a ringed space. A sheaf of OX -modules is a sheaf F on X such that 2.5. SHEAVES OF MODULES 37

(1) F(U) is a O(U)-module; (2) the restriction map is compatible with the module structure, i.e. for V ⊂ U, denote r : F(U) → F(V ), s : O(U) → OV , then r(fa) = s(f)r(a). A morphism f : F → G of sheaves of modules is a morphism of sheaves compatible with the module structure, i.e., F(U) → G(U) is a morphism of O(U)-modules.

Example 2.5.2. Note first that Z, C are not sheaves of OX modules in general. So naive examples are: OX , OX ⊕ OX ... 1 Example 2.5.3. More interesting examples: for X = A , what are the sheaves of OX -modules? Skyscraper sheaves Ca supported at point a, sheaf of polynomials O(−k) vanishes at 0 to degree m.

The global section of the sheaf should be a OX (X)-module, i.e. a k[x]-module. The sheaf O ⊕ O has global section k[x] ⊕ k[x]. The skyscraper sheaf Ca has global sections k, which is a k[x]-module by the action x · 1 = a. The sheaf O(−k) has global sections k[x] · xm. Actually, we claim that we can recover the sheaf from its global sections–this is not very surprising by recalling that the global section of OX is A, which determines the sheaf OX . Definition 2.5.4. Let A be a ring and M be an A-module, let X = Spec A. We define the sheaf M˜ associated to M as follows. Denote Mp the localization of M at p ∈ Spec A. ˜ ` For each open set U ⊆ Spec A, define M(U) to be the set of functions s : U → p Mp such that for each p ∈ U, sp is in Mp and s is locally a fraction m/f with m ∈ M and f ∈ A. Proposition 2.5.5. A,M,M˜ as above.

(a) M˜ is an OX module. ∼ (b) The stalk M˜ p = Mp, ∀p ∈ X. ∼ (c) For any f ∈ A, M˜ (D(f)) = Mf . (d) Γ(X, M˜ ) = M.

Proof. Similar to the proof of OX . Try it!

Let us familiar ourselves with some facts about this˜by look at the following facts: Fact 2.5.6. (Facts on ∼, the associated sheaf.) Let f : Y = Spec B → X = Spec A.

(a) M → M˜ gives an exact, fully faithful functor (A-modules)→(OSpec A-modules). ∼ ˜ ˜ (b) (M ⊗A N) = M ⊗OX N. ∼ (c) (⊕Mi) = ⊕M˜ i. (d) f(N˜) =∼ N˜. (regard the latter N as A-module.) ∗ ∼ ∼ (e) f (M˜ ) = (M ⊗A B) .

Proof. Omitted. 38 CHAPTER 2. SCHEMES

Example 2.5.7. When M is an ideal of A, M˜ is subsheaf of OX . In general, we say I is a sheaf of ideals on X if I is a subsheaf of OX . Then we can define closed subschemes as the subscheme defined by a sheaf of ideals I. The set is the support of OX /I and the sheaf of rings is the quotient ring OX /I.

Many of the notions on sheaves and notions of modules are inherited to sheaves of modules. Here we list important ones:

Fact 2.5.8. (Facts on sheaves of modules)

• kernel, cokernel, image, quotient, direct sum, direct product, direct limit, inverse limit are well-defined.

• For two sheaves of OX -modules F, G, the set of morphisms HomOX (F, G) is an OX (X)- module.

Moreover, for any open subset U ⊆ X, HomOU (F|U , G|U ) is an OX (U)-module. So we actu-

ally obtain a sheaf of modules by assigning to each U the OX (U)-module HomOU (F|U , G|U ).

We call it the sheaf Hom, and denoted by HomOX (F, G).

• The tensor F ⊗OX G is defined to be the sheafification of the presheaf

U → F(U) ⊗OX (U) G(U).

Then (⊗, Hom) are adjoint functors. Indeed, we have

Hom(F ⊗ G, H) =∼ Hom(F, Hom(G, H))

and the sheaf version Hom(F ⊗ G, H) =∼ Hom(F, Hom(G, H)). It can be proved easily.

• We can relate sheaves on two spaces if there is a morphism. Here are some notions. Let f : X → Y be a morphism of ringed spaces, F is a sheaf of OX -modules, G is a sheaf of OY -modules.

f∗(F) is a sheaf of f∗OX -module, by the morphism OY → f∗OX it is also a sheaf of OY - module. We call it the direct image of F. −1 −1 −1 f G is a sheaf of f OY -module. By morphism f OY → OX , we give the inverse image an OX -module structure by force:

∗ −1 −1 f G = f G ⊗f OY OX .

Proof. Omitted.

∗ Fact: (f , f∗) are adjoint functors, i.e.

∗ ∼ HomOX (f G, F) = HomOX (G, f∗F). 2.5. SHEAVES OF MODULES 39

Definition 2.5.9. (a) A sheaf of OX -modules F is quasi-coherent if there is an affine covering Ui = Spec Ai such that there is an Ai-module Mi with ∼ ˜ F|Ui = M.

Equivalently, any point has a neighborhood U s.t.

I J OX |U → OX |U → F|U → 0 is exact, where I,J are sets of indices.

(b) (Assuming X is noetherian) A sheaf is coherent if each Mi is a finitely generated Ai-module. Equivalently, any point has a nbhd U s.t.

m n OX |U → OX |U → F|U → 0 is exact, where m, n are integers.

Remark 2.5.10. When X is not noetherian, a sheaf F is defined to be coherent if

(1) F is of finite type over OX , and n (2) for any U, any n, the kernel of any morphism OX |U → F|U is of finite type.

Example 2.5.11. OX is coherent. (This is not necessarily coherent if X is not noetherian.)

Example 2.5.12. Let i : Y → X be a closed immersion, then i∗OY is coherent.

Example 2.5.13. Let j : U → X be an open subscheme, extending OU by zero gives a sheaf j!OU which is not quasi-coherent in general. 40 CHAPTER 2. SCHEMES

(Week 11) In the definition of quasi-coherent sheaves, it is not clear if on every affine open subset, the sheaf can be induced by a module. But it turns out to be true. (The similar situation appeared before, for finite type/ finite/quasi-compact morphism.)

Proposition 2.5.14. Let X be a scheme. An OX -module F is quasi-coherent iff for every open ∼ affine subset U = Spec A there is an A-module M that F|U = M˜ . If X is noetherian, then F is coherent if is quasi-coherent and M is a finitely generated A-module.

Proof. The proof is omitted, cf. [Hartshorne] II Prop 5.4.

Corollary 2.5.15. Let X = Spec A. There is an equivalence of categories

(A − modules) → (q.coh.OX − modules) M → M˜

When A is noetherians, we have equivalence of categories

(f.g.A − modules) → (coh.OX − modules) Fact 2.5.16. (Facts on (quasi)-coherent sheaves)

• The kernel, cokernel, image of morphisms of (quasi-)coherent sheaves are (quasi-)coherent. The extension of (quasi-)coherent sheaves is (quasi-)coherent. (all by 5-lemma)

∗ • Let f : X → Y . Then f : qcohY → qcohX . ∗ If X,Y are noetherian, then f : cohY → cohX .

If X is noetherian or f is quasi-compact and separated, then f∗ : qcohX → qcohY .

If X,Y are noetherian and f is proper, then f∗ : cohX → cohY . • The ideal sheaf I of a closed subscheme Y of X is quasi-coherent. If X is noetherian, I is coherent. • If X = Spec A. There is a 1-1 correspondence between the ideals of A and closed subschemes of X given by a → Spec A/a. In particular, any subscheme of an affine scheme is affine. (The proof uses the fact that qcoh on Spec A is equivalent to A-modules.)

Proof. We will only explain f∗. Note that pull-back is easy and push-forward is difficult. To show that the push-forward sending qcohX to qcohY , we can assume Y is affine. The condition (X noetherian or f is quasi-compact and separated) guarantees us to cover X by finitely many affine charts Ui, and Ui ∩ Uj can be covered by finite many affine charts Uijk. Then we have an exact sequence

0 → f∗F → ⊕f∗F|Ui → ⊕f∗F|Uijk . Then use the fact that the kernel of a morphism of quasi-coherent sheaves is quasi-coherent.

(Assign homework, ex 5.5, saying f∗ does not send cohX to cohY in general.) 2.5. SHEAVES OF MODULES 41

For the rest of this section we shall discuss quasi-coherent sheaves on a projective scheme. Definition 2.5.17. Let S be a graded ring and M a graded S-module. The sheaf M˜ associated to M on scheme P rojS is defined as follows.

−1 (1) Define M(p) be the group of degree 0 elements in T M where T is multiplicative set of homogeneous elements in S not in p. (2) For open set U ⊆ P rojS, define sections s ∈ M˜ (U) to be a map a s : U → M(p) p such that s(p) ∈ M(p), and s is locally of form m/f where m ∈ M, f ∈ A are homogeneous and have the same degree.

Similar to the affine case, we have Proposition 2.5.18. Let X = Proj S and M a graded S-module. ˜ (a) Mp = M(p) for all p ∈ Proj S. ˜ ] + (b) For any f ∈ S+, M|D+(f) = M(f). (here D (f) = {p ∈ Proj S|f∈ / p}, and M(f) denotes the group of deg-0 elements in Mf .)

(c) M˜ is a quasi-coherent OX -module, and it is coherent If S is noetherian and M is finitely generated.

Now we introduce the building block of coherent sheaves on Proj S. Definition 2.5.19. For a graded ring S, let X = Proj S,

(a) define S(n) = S, by degree of each elements increases by n, i.e. for x ∈ S, degS(n)(x) = degS(x) − n. ] (b) define OX (n) = S(n). In particular, O(1) is called the twisted sheaf of Serre.

(c) for any sheaf F of OX -modules, define F(n) := F ⊗OX OX (n).

Definition 2.5.20. (a) A sheaf F of OX is locally free if it can be covered by open sets U such ∼ n that F|U = OX for some n ∈ Z. The integer n is called the rank of F. (b) A locally free sheaf of rank 1 is called an invertible sheaf.

In order to understand OX (n), let us state some terminology and facts (cf. Ex 5.18 for details). If you have learned differential geometry, you can define vector bundle on a scheme by yourself. n (The two ingredients are: (i) locally over some U = Spec R it is like AU := Spec R[x1, . . . , xn]; (ii) transition functions are linear.) To be more precise: Definition 2.5.21. A (geometric) vector bundle is a morphism π : E → X of schemes such that ∼ (1) there is a covering {U } of Y , s.t. φ : π−1U →= An , and i i i Ui (2) for any affine Spec A ⊂ U ∩ U , the automorphism θ = φ φ−1 of An = Spec A[x , . . . , x ] i j P j i V 1 n is linear, i.e. θ(a) = a, ∀a ∈ A and θ(xi) = aijxj. 42 CHAPTER 2. SCHEMES

We define a section of a vector bundle π : E → X to be a morphism s : X → E such that π ◦ s = idX . We need the construction of Spec F.

Definition 2.5.22. Let F be a quasi-coherent sheaf of OX algebra on a scheme X. We define the scheme Spec F as follows. Take an affine covering {Ui} of X and define Mi = F(Ui). (Then f Mi = F|Ui .) Define Vi = Spec Mi. It can be shown that these Vi can be glued together, and the resulting scheme is denoted by Spec F. Proposition 2.5.23. Following two categories are equivalent. ∼ ϕ :(Vector bundles of rank n on X) = (locally free sheaves of OX -modules of rank n) where functor ϕ : E → sheaf of sections of E. And the inverse functor ϕ−1 : E → Spec Sym(E∨).

Because of the above proposition, we treat“vector bundle” and “locally free sheaf” equivalently. Note that the latter is a subcategory of OX -modules, which is an abelian category. Operations in abelian categories, for example the kernel of a morphism, can be defined in (OX -modules) but is not always defined in (vector bundles).

Proposition 2.5.24. Let S be a graded ring generated by S1 as an S0-algebra. Let X = Proj S.

(a) OX (n) is an invertible sheaf. (b) For any graded S-module M,

M˜ (n) = M^(n). ∼ In particular, OX (n) ⊗ OX (m) = OX (m + n).

(c) Let T be another graded ring generated by T1 as an T0-algebra. Let φ : S → T be a ring morphism such that f(Sd) ⊆ Td, ∀d ∈ Z. Let Y = Proj T . Let U = {p ∈ Proj T |p + f(S+)} (i.e. where Proj T 99K Proj S is defined) and let f : U → X. Then ∗ ∼ ∼ f (OX (n)) = OY (n)|U , f∗(OY (n)|U) = (f∗OU )(n)

Similar to the fact that every quasi-coherent sheaves on an affine scheme can be recovered by a module, given a coherent sheaf F on Proj S, we can find a module M such that M˜ =∼ F.

Definition 2.5.25. Let S be a graded ring, X = Proj S, F a sheaf of OX -module. Define the graded S-module associated to F as

Γ∗(F) := ⊕n≥0Γ(X, F(n))

Example 2.5.26. (1) Let A be a ring and S = A[x0, . . . , xn]. X = Proj S. then Γ∗(OX ) = S. ] (Left=⊕Γ(X, OX (n)) = ⊕Γ(S(n)), then consider the restrictions to each D+(xi).) (2) We should point out a difference between affine and projective schemes. Let M be a graded S-module, consider M → Γ∗(M˜ ). Then this is in general not an isomorphism, but only isomorphism for large enough degrees. ∼ For example, notation as in (1), take M = ⊕d≥mSd. Then M˜ = OX : on an open affine subset k ` k+` ˜ D+(xi), given any section s/xi ∈ OX (D+(xi)) we can associate a section sxi /xi ∈ M|D+(xi) for ` + k ≥ d. In this example M is not isomorphic to Γ∗(M˜ ). 2.5. SHEAVES OF MODULES 43

Proposition 2.5.27. Let S be a graded ring and finitely generated by S1 as S0-algebra, let X = Proj S, F a quasi-coherent sheaf on X. Then there is a natural isomorphism

∼ ∼ β :Γ∗(F) = F.

Proof. (Idea of the proof). To define β, we only need to define it on each affine open subset D+(f) for f ∈ S of degree 1. Since on affine scheme U = Spec A,(∼, Γ) are adjoint, i.e. ˜ ∼ HomOX (M, F) = HomA(M, Γ(F)) it suffices to define a morphism

∼ Γ∗(F) (D+(f)) → F(D+(f)).

∼ d −d Γ∗(F) (D+(f)) = Γ∗(F)(f) consists of fractions m/f where m ∈ Γ(X, F(d)). But f gives a d −d section of OX (−d)|D(f), so we can send m/f to m ⊗ f as a section of F|D+(f). ∼ To show isomorphism, it suffices to show Γ∗(F)(f) = F(D+(f)). We will not prove this technical part, the idea is that given a section s ∈ F(D+(f)), for sufficiently large n the section n f s ∈ F(n)(D+(f)) can be extent to X. Please read Hartshorne Prop 5.15. 44 CHAPTER 2. SCHEMES

(Week 12)

Proposition 2.5.28. Let A be a ring.

n (a) A closed subscheme Y of PA can be defined by a homogeneous ideal I of the graded ring S = A[x0, . . . , xn].

(b) Y → Spec A is projective if Y = Proj S for some graded ring S, where S0 = A and S1 generates S as a S0-algebra.

Proof. (a) Let I be the ideal sheaf of Y . Then I = Γ∗(I ) defines Y .

(b) Let Y be defined by I, replace I by I≥2, let S = A[x0, . . . , xn]/I.

Example 2.5.29. Let A = C[x1, x2] and Y be the blow-up of Spec A at the origin defined by I = (x1, x2). Can we find an S such that Y = Proj S? 2 1 For Y is the subvariety of C × P = P rojA[y1, y2] defined x1y2 − x2y1 = 0. So we can take

S = A[y1, y2]/(x1y2 − x2y1).

∞ n This ring is isomorphic to the Rees algebra R = ⊕n=0I = A[It] ⊂ A[t], given by

S → R

yi 7→ xit (i = 1, 2)

And in general we define blow-up of a scheme along an ideal sheaf I to be

∞ n Proj (⊕n=0I )

n Definition 2.5.30. (a) Given any scheme Y → Spec Z, define O(1) on PY to be the pull-back of n O(1) on PZ. (b) Let X be a scheme over Y . An invertible sheaf L is called very ample relative to Y if there is an immersion (i.e. a closed immersion into an open subscheme, or equivalently, a open n ∗ ∼ immersion into a closed subscheme) i : X → PY for some n such that i O(1) = L .

Definition 2.5.31. Given a scheme X and a sheaf F of OX -modules. We say F is generated by (finitely many) global sections if there are finitely many global sections si ∈ Γ(X, F) such that each n stalk Fx is generated by the germs of {si}i=1’s. Equivalently, if there is a surjective morphism

⊕n OX → F.

Theorem 2.5.32 (Serre). Let X be a projective scheme over a noetherian ring A. Let O(1) be a very ample invertible sheaf on X. Let F be a coherent OX -module. Then there is an integer n0 that for all n ≥ n0, F(n) is generated by finite many global sections.

Proof. Omitted. 2.6. DIVISORS 45

Remark 2.5.33. Therefore, there is a surjection O⊕k → F(n) → 0, hence O(−n)⊕k → F → 0. Take the kernel of the surjection and repeat the above step, we get a (possibly infinite) locally free resolution resolution of F. It is more often to consider the algebra version: the graded free resolution of a graded ring S, in the form of

· · · → ⊕jS(−a1j) → ⊕jS(−a0j) → M → 0.

In case S = k[x0, . . . , xr] and M is a finitely generated S-module, M admits a minimal graded free resolution (minimal in the sense that the maps are given by matrices of homogeneous polynomials containing no non-zero constant entries.) The integers apj are determined by M. The sheaf M˜ is m-regular if apj ≤ p + m, ∀p, j. It is an interesting question to study the regularity bound for a sheaf M˜ . For example the resolution of M = (x2, y3) in k[x, y] is

2 3 0 o (x , y h) o Si (−2) ⊕ S(−3)o S(−5) o 0 x2 y3 3 −y  x2

Hence M is 4-regular (and is m-regular for any m ≥ 4). The regularity of a complete intersection of e hypersurfaces of degree d1, . . . , de is d1 + ··· de − e + 1. The regularity of an arbitrary ideal 2r−1 I ⊂ k[x0, . . . , xr] is bounded from above by (2d) where d is the largest degree of a minimal generator of I.

Theorem 2.5.34. Let k be a field, A a f.g. k-algebra, X projective over A, F a coherent OX - module. Then Γ(X, F) is a finitely generated A-module. In particular when A = k, then Γ(X, F) is a finite dimensional k-vector space.

Proof. Omitted. Idea is to reduce to the special case F = O(n). Let S0 = ⊕Γ(X, O(n)), show S0 is integral over S, then apply theorem of finiteness of integral closure to conclude that S0 is f.g. S-module.

Corollary 2.5.35. Let X,Y be schemes of finite type over k and f : X → Y be projective. Then f∗ maps coherent sheaves on X to coherent sheaves on Y .

2.6 Divisors

Let us first define the orders of zeros and poles. In [Hartshorne], it is restricted to the case when (*) the scheme is a noetherian integral separated scheme regular in codimension one, i.e. every local ring Ox of X of dimension 1 is regular. Definition 2.6.1. (Order) Let X satisfies (*), Y be a closed integral subscheme of codimension 1, let f ∈ K(X)∗ be a nonzero rational function. Then the local ring OY,X is a DVR with quotient field K(X) with valuation v. If v(f) > 0 (or f < 0), we say f has a zero (or pole) along Y of order v(f). Example 2.6.2. On P1 = Proj k[x, y], The rational function x2(x−y)/y3 (or x2(x−1) when restricts to D+(x)) has a zero along x = 0 of order 2, along x = 1 of order 1, and has a pole along y = 0 (the ∞ point) of order 3. 46 CHAPTER 2. SCHEMES

Remark 2.6.3. Order can be defined for any scheme: let X be a scheme and Y be a closed integral ∗ subscheme of codimension 1. For f ∈ K(X) , define ordY (f) to be the length of OY X/(f) as a OY X-module. (Recall the definition of the length of an A-module M is the length of the chain ∼ M ) M1 ) ··· ) Mr = 0, with Mi−1/Mi = A/pi, pi a prime ideal in A.)

Thus defined order satisfies ordV (rs) = ordV (r) + ordV (s). Definition 2.6.4. (Cycle group) P (a) Let X be an algebraic scheme. A d-cycle on X is a finite formal sum [V ] where V are ni i i d-dimensional integral subschemes of X and ni are integers. (b) The free abelian group generated by k-dimensional subvarieties is called the group of d- cycles, denoted by Zd(X). Definition 2.6.5. (Weil divisor) Let X be and n-dimensional scheme. A Weil divisor is a (n − 1)-cycle.

Now we need to define an equivalence relation among Weil divisors. Definition 2.6.6. (Rational equivalence/linear equivalence) (a) Let f ∈ K(X), define the divisor of f by X (f) = vY (f)[Y ]. An divisor that is equal to (f) is called a principal divisor. (b) Two divisors D and D0 are linearly equivalent if D − D0 is a principal divisor.

(c) The quotient of Zd(X) by the linear equivalence relation is called the divisor class group, denoted by ClX (or Chn−1X). Example 2.6.7. (a) Let A be a noetherian domain. Then A is a UFD iff X = Spec A is normal and n ClX = 0. In particular ClAk = 0. (b) Let A be a Dedekind domain(i.e., an integrally closed, Noetherian domain with Krull dimension one). Then ClSpec A is the ideal class group of A.

n P (c) In Pk , let H be any hyperplane, let Yi be hypersurfaces. for a divisor D = niYi, define n ∗ d = ni deg Yi. Then D ∼ dH. Moreover, for any f in the fraction field K(P ) , deg(f) = 0. Therefore there is an isomorphism deg : ClPn → Z. (d) Let Y ⊂ X be a closed subscheme and U = X − Y . We have an exact sequence

ChmY → ChmX → CHmU → 0, ∀m. In particular, when Y has codimension ≥ 2, ClU =∼ ClX. When Y has l irreducible components Yi of codimension 1, we have Zl → ClX → ClU → 0 P n by sending (n1, . . . , nl) to niYi. As an example, if Y is a degree d irreducible hypersurface in P , then Cl(Pn − Y ) = Z/dZ. (e) Cl(X) =∼ Cl(X × A1). 2.6. DIVISORS 47

(Week 13) Definition 2.6.8. (Cartier divisor) In below we assume that X is integral.

A Cartier divisor D ∈ CDiv(X) is defined by data (Ui, fi), where Ui form an open covering of X and fi are nonzero rational functions in the function field K(Ui) = K(X), such that fi/fj is a unit on Ui ∩ Uj.

A Cartier divisor (Ui, fi) is principal if there is f ∈ K(X), such that f/fi is a unit on Ui. Two Cartier divisors are linearly equivalent if their difference is principal.

The Cartier divisors and Weil divisors are related as follows: Proposition 2.6.9. Let X be a n-dimensional integral scheme.

(a) For a subvariety V ⊂ X of codimension 1, let Ui be that Ui ∩ V 6= ∅, write ordV (D) = ordV (fi). There is a homomorphism CDiv(X) → Z(X) given by X D → ordV D[V ].

(b) The principal Cartier divisor goes to principal Weil divisor. So there is an induced map

CaCl(X) → Chn−1(X).

(c) If a separated noetherian scheme X is normal(resp. locally factorial, i.e. all local rings are UFD, regular local ring is locally factorial), then the morphisms in (a) and (b) are injec- tive(isomorphisms).

2 2 2 2 Example 2.6.10. (a) X = Pk. Let C be the irreducible curve defined by x0 + x1 + x2 = 0. Then 2 2 n 2 the Weil divisor n[C] corresponds to Cartier divisor (U0, ((x1/x0) + (x2/x0) )) ,(U1, ((x0/x1) + 2 n 2 2 n (x2/x1) )) ,(U2, ((x0/x2) + (x1/x2) )) . 3 2 (b) X ⊂ A defined by z = xy, then CaCl(X) = 0 and Ch1(X) = Z/2Z. Note the local ring is not UFD.

Now we are going to relate the Cartier divisor class to invertible sheaves. Let K be the constant sheaf of quotient field of X (a special case of total quotient ring as introduced in [Hartshorne]). we have the natural morphism.

ϕ : CaCl(X) → P icX( invertible sheaves on X) by sending D = {(Ui, fi)} to the subsheaf L (D) of K, such that the restriction of L (D) to Ui is −1 OUi · fi .

(1) It is well-defined since fi/fj is invertible hence OUi∩Uj fi = OUi∩Uj fj. For people who are familiar with the definition of vector bundles using transition functions, this is to say the line −1 1 1 bundle is defined by the transition function gij = φi ◦φj = fi/fj :(Ui ∩Uj)×A → (Ui ∩Uj)×A . 48 CHAPTER 2. SCHEMES

(2) ϕ−1 is defined as follows. Let K be the constant sheaf of function field on X. Given an ∼ ∼ invertible sheaf L , on any open set where L |U = OU , L |U ⊗OU K = K, a constant sheaf. But if a sheaf is constant on an open covering, then the sheaf is the constant sheaf K. (This is not the case if we are in complex geometry!) Then we have the natural subsheaf morphism L → L ⊗ K =∼ K. So we conclude

Proposition 2.6.11. (a) Let X be an integral scheme, the homomorphism ϕ : CaCl(X) → P icX is an isomorphism. ∼ −1 (b) L (D1 − D2) = L (D1) ⊗ L (D2) . ∼ (c) D1 ∼ D2 iff L (D1) = L (D2).

1 Example 2.6.12. Let X = PC = Proj C[x, y] covered by U0 = Spec C[u] and U1 = Spec C[v] for u = y/x, v = x/y. Let D = {(U0, u), (U1, 1)}. Then L (D)|U0 = OU0 · 1/u and L (D)|U1 = OU1 . L (D) corresponds to the line bundle with transition function g01 = u/1. It is easy to see that 1 and u = y/x span the sections of L (D), which are rational functions on X with poles at point [1 : 0] of order ≤ 1. This sheaf is isomorphic to O(1). (So Γ(L (D)) := {f is a rational function on X |D + (f) ≥ 0}.) ∼ Corollary 2.6.13. If X is noetherian, integral, separated locally factorial of dim n, then Chn−1X = P icX.

n Example 2.6.14. Every invertible sheaf of Pk is isomorphic to O(`) for some ` ∈ Z, because n ∼ n Chn−1P = Z. Chn−1P is generated by a hyperplane, which corresponds to the invertible sheaf O(1).

2.7 Projective morphisms

To describe a morphism from X to An is the same to specify n functions on X. How to describe a morphism ϕ : X → Pn in a similar fashion? There is no interesting functions on Pn, so we are “forced” to consider x0, . . . , xn, the global sections of O(1). The pullback of x1, . . . , xn will be ∗ sections s1, . . . , sn of L = ϕ O(1) on X. Will these sections determine ϕ? The answer is yes.

Theorem 2.7.1. Let A be a ring and let X be a scheme over A.

n ∗ (a) If ϕ : X → PA is an A-morphism, then ϕ O(1) is an invertible sheaf on X generated by ∗ the global sections si = ϕ (xi), i = 0, . . . , n.

(b) Conversely, if L is an invertible sheaf on X and s0, . . . , sn are global sections of L which n ∼ ∗ ∗ generate L , then there is a unique A-morphism ϕ : X → PA such that L = ϕ O(1) and si = ϕ (xi) under this isomorphism.

Proof. (a) Since xi’s generate O(1), si’s generate L .

(b) The morphism is given by x 7→ (s0(x): s1(x): ··· : sn(x)). To be strict, read [Hartshorne, Thm 7.1]. 2.7. PROJECTIVE MORPHISMS 49

n Example 2.7.2. The theorem can be used to find all automorphisms of Pk . Obviously P GL(n, k) = n GL(n, k)/Gm acts on Pk . Now we show that they are indeed all the automorphisms. By theorem, suppose ϕ : Pn → Pn, then ϕ∗O(1) = O(`). But ϕ is an automorphism, so ` = ±1. Since O(−1) ∗ P has no section, ` = 1. Then ϕ (xi) = aijxj, where [aij]i,j is invertible,and give an element in P GL(n, k). n Remark 2.7.3. When s0, ..., sn’s do not generate L , they define a rational map X 99K P . 50 CHAPTER 2. SCHEMES

(Week 14) Now we consider a special situation when X → Pn is an immersion.

Definition 2.7.4. An invertible sheaf L on a noetherian scheme X is ample if for every coherent n sheaf F on X, there is n0 > 0 such that for every n ≥ n0, F ⊗ L is generated by global sections.

We usually see another equivalent definition.

Theorem 2.7.5. Let X be a scheme of finite type over a neotherian ring A and L is invertible. Then L is ample ⇔ L m is very ample over Spec A for some m > 0.

Proof. Serre’s theorem shows ⇐. Please read [Hartshorne, Thm 7.6] for the beautiful proof of ⇒.

n Example 2.7.6. (a) Let X = Pk . Then O(d) is very ample iff d > 0. (b) Let X = P1 × P1. Invertible sheaves are of the form O(a) £ O(b). It is ample iff a, b > 0. (c) Consider a divisor on a curve of genus g. If deg D ≥ 2g + 1, then D is very ample. Hence D is ample if deg D > 0. This can be proved by Riemann-Roch.

Linear systems of divisors Before invertible sheaves were introduced, people study linear systems of divisors. It has the advantage of been ‘more’ geometric. Assume for the rest of the section that X is a non-singular over an algebraically closed field k. For an invertible sheaf L on X, a nonzero section s ∈ Γ(X, L ) determines an effective divisor (s). It carries the following property: (a) If D is a divisor and s 6= 0 ∈ Γ(X, L (D)), then the effective divisor (s) ∼ D. Indeed, (s) is obviously effective. Regard s as a rational function f ∈ K(X), Let D locally be defined as (Ui, fi). −1 The section s is locally defined by f/(fi ) = ffi. Then (s) = (ffi) = (f) + D,(s) ∼ D. (b) Every effective divisor linearly equivalent to D equals some (s). (c) Suppose X is projective. Two sections s, s0 have the same zero iff s0 = λs for some λ ∈ k∗. So we can add two sections, and quotient by the k∗ equivalence relation.

Definition 2.7.7. (a) A complete linear system on a variety is the set of all effective divisors linearly equivalent to a given divisor D, denoted by |D|. (b) A linear system is a linear subspace of a complete linear system.

Definition 2.7.8. A point P is a base point of a linear system d iff P is contained in the support of every divisor in d. A linear system is called base-point-free if there is no base point.

Simple observation: Let d corresponds to the subspace V ⊆ Γ(X, L ). P is a base point iff sP ∈ mP LP for all s ∈ V . Therefore d is base-point-free iff L is generated by global sections in V . 2.7. PROJECTIVE MORPHISMS 51

n Example 2.7.9. (a) Consider the projective space Pk . Take the divisor D = (x0 = 0). Then the n complete linear system |D| is the linear space Γ(Pk , L (D)). It is spanned by x0, x1, . . . , xn, so is n + 1 dimensional. But unfortunately we will call the linear system |D| of dimension n, which tell us the dimension of the target projective space induced by the linear system.

Since the the sections x0, . . . , xn has common zero only at x1 = ··· xn = 0 which is not a point n in Pk , so |D| is base-point-free.

(b) Consider a subspace V spanned by x1, . . . , xn. It has base point at (1 : 0 : ··· : 0). Which defines a rational map Pn 99K Pn−1 projecting from the point (1 : 0 : ··· : 0).

Proposition 2.7.10. Let f : X → Pn be a morphism determined by the linear system d. Then f is a closed immersion iff (1) d separates points, i.e., for any two closed points P,Q ∈ X there is a divisor D ∈ d, P ∈ D but Q/∈ D.

(2) d separates tangent directions, i.e., given a closed point P and a tangent vector t ∈ TP := 2 ∨ 2 ∨ (mP /mP ) , there is a D ∈ d that P ∈ D and t∈ / TP (D) = (mP,D/mP,D) . (If a morphism sends two nonzero tangent vector to the same image, then it must send a non-zero vector to 0.)

Example 2.7.11. Linear system of cubics.(Taken from [Beauville] complex algebraic surfaces, prop 4.9)

2 Suppose p1, . . . , pr(r ≤ 6) are points in Pk in general position, i.e. any 3 are not on a line and any 6 are not on a conic. Let d be the linear system of cubics passing through p1, . . . , pr. Let 2 π : Pr → P be the blow-up of p1, . . . , pr. Let d = 9 − r. Then d defines a closed embedding d d f : Pr → P . The image is a surface of degree d in P , called the del Pezzo surface of degree d. (The degree is 9 − r since each blow-up decrease the number of intersection of two divisors by 1.)

3 x Now we show for r = 6, f : P6 → P is indeed a closed embedding. Denote by Qij the conic passing through pk(k 6= i, j) and x.

(1) It separates points. Given x 6= y ∈ P6. We need to find a cubic curve passing through pi(1 ≤ i ≤ 6) and x but missing y. The union of a line with a conic is a cubic. So we can choose a line passing through two points pi, pj and missing y, and let the conic pass through pk(k 6= i, j) and x, and hope this conic will both miss y. It turns out, by changing the line if necessary, we can always make this happen.

2 (2) It separates tangents. For a point x ∈ P − {p1, . . . , p6} and a tangent v, we can choose a line through x that is in different direction than v and a conic. For a point x in the exceptional x x divisor E1, take the two conics Q23 and Q24 and they intersect at p1, p5, p6 and the intersection at p1 is at least of multiplicity 2. But the total intersection should be 2 × 2 = 4, so the intersection multiplicity at p1 is exactly 2. Then they have different tangent after blown up, so one of them is not v.

Proj (E) and blow up

Definition 2.7.12. Let F be a quasi-coherent sheaf of graded OX algebra on a scheme X and assume F0 = OX , F1 is a coherent OX -module, and F is locally generated by F1 as an OX -algebra. We define the scheme Proj F as follows. 52 CHAPTER 2. SCHEMES

Take an affine covering {Ui} of X and we have the natural morphism Vi = Proj SF (Ui) → U. It can be shown that these Vi can be glued together, and the resulting scheme is denoted by Proj F. Definition 2.7.13. (General definition of blow-ups) Let X be a scheme and I an ideal sheaf. Define the Rees algebra M∞ S = (I t)d d=0 as a sheaf of graded rings such that deg t = 1.

We call Proj S the blow-up of X along I , denoted by BlI X. Proposition 2.7.14. (Universal property of blow-up) Let X be a noetherian scheme, I a coherent −1 sheaf of ideals, π : X˜ → X the blow-up along I . If f : Z → X is a morphism such that f I ·OZ is an invertible sheaf on ideals on Z, then there exists a unique morphism g : Z → X˜. Theorem 2.7.15. Let X be a quasi-projective variety over k. If Z is a variety and f : Z → X is a birational morphism, then there exists a coherent sheaf of ideals I on X such that Z is isomorphic to the blow-up of X along I .

Here is a useful tool to compute the blow-up.

Proposition 2.7.16. Let I = (f1, . . . , fk) be a finitely generated ideal of a ring A. Let S = ∞ d ˜ ⊕d=0(It) . Let X = Spec A and X = BlI X = Proj S. ˜ (a) X has an affine covering by Ui := Spec S(fi) for 1 ≤ i ≤ k. ∼ \ ∞ (b) S(fit) = A[u1,..., ubi, . . . , un]/(f1 − u1fi,..., fi − uifi, . . . , fn − uifi)/fi -torson.

Proof. Part (a) follows from the standard fact on affine coverings of Proj S. Part (b). Without loss of generality assume i = 1. We have n ¯ o a ¯ ± S = a ∈ Id, d ∈ Z ∼ (f1t) d ¯ ≥0 f1

a ` ∞ where d ∼ 0 iff af1 = 0 for some ` ≥ 0, i.e. a is an f1 -torsion. f1 P d i1 ik Since a ∈ I , we can write a = ai1···ik f1 ··· fk , ai1···ik ∈ A and i1 + ··· + ik = d. Then X ³ ´ ³ ´ a f2 i2 fk ik d = ai1···ik ··· . fi f1 f1

Let ui = fi/f1, then the above is in A[u2, . . . , uk]/(f2 − u1f1, . . . , fk − ukf1). Finally, remember to quotient out the equivalence relation ∼.

Example 2.7.17. Blow up (x, y2) in C2, denoted by Ce2. By previous proposition, the blow-up is covered by

2 2 ∞ 2 U1 = C[x, y, u]/(y − ux)/(y ) -torsion = C[x, y, u]/(y − ux) 2.7. PROJECTIVE MORPHISMS 53 and 2 ∞ U2 = C[x, y, v]/(x − vy )/x -torsion = C[y, v]. 2 So A has a A1 singularity in chart U1. See Figure 2.1. The green curve is the exceptional divisor of Ce2 → C2. On the other hand, Ce2 can be obtained by blowing up the point (0:0:1:0) on the projective cone y2 − ux in P3, then cutting out the strict transform of the infinite curve defined by y2 − ux.

Figure 2.1: Blow up (x, y2) in A2.