LINE BUNDLES on PROJECTIVE SPACE We Wish to Show That Any Line Bundle Over Pn K Is Isomorphic to O(M)

DANIEL LITT

n We wish to show that any line bundle over Pk is isomorphic to O(m) for some m; we give two proofs below, one following Hartshorne, and the other assuming some knowledge of sheaf cohomology. Throughout, let X be an integral scheme.

∗ Definition 1 (Cartier Divisors). We define OX to be the sheaf of groups which is the subsheaf of OX consisting of units, and KX be the sheaf associated to the presheaf KX : U 7→ Frac(Γ(U, OX )), which as we will see is actually just this presheaf in the integral case. (This can be generalized to non-integral schemes ∗ by taking the total ring of fractions instead of the fraction field). Let KX be the sheaf of groups consisting of the units of KX . Then there is a short exact sequence

∗ ∗ ∗ ∗ 0 → OX → KX → KX /OX → 0;

∗ ∗ ∗ ∗ ∗ a Cartier Divisor is a global section of KX /OX . Elements of the image of the map Γ(X, KX ) → Γ(X, KX /OX ) are principal divisors.

Remark 1 (Explicitly writing down a Cartier Divisor). By our explicit construction of the quotient sheaf, a ∗ Cartier Divisor is an equivalence class [{Ui, fi}] where the Ui are an open cover of X, fi ∈ Γ(Ui, KX ), and for each i, j, we have that fi/fj (where it is deﬁned) is in the image of the map

∗ ∗ Γ(Ui ∩ Uj, OX ) → Γ(Ui ∩ Uj, KX ).

The equivalence relation here is that {Ui, fi} ∼ {Vi, gi} if the two collections have a common reﬁnement, or,

equivalently, if {Ui ∩ Vj, fi|Ui∩Vj } ∪ {Ui ∩ Vj, gj|Ui∩Vj } is a Cartier divisor.

In this set-up, we may write the product of two Cartier divisors {Ui, fi}, {Vi, gi} as {Ui ∩ Vj, fi · gj}.

We will soon motivate this deﬁnition. Before continuing, we wish to note quickly that

Lemma 1. The group of Cartier divisors modulo principal divisors, that is

∗ ∗ ∗ coker(Γ(X, KX ) → Γ(X, KX /OX )),

1 ∗ is isomorphic to H (X, OX ).

1 ∗ Proof. By the long exact sequence in sheaf cohomology, it suffices to show that H (X, KX ) = 0. ∗ ∗ We claim in fact that the sheaf KX is simply the constant sheaf S defined by Γ(U, S) = K , where K is the field of rational functions on X (the residue field of the generic point of X). This is a sheaf as X is irreducible. Indeed, let Spec(A), Spec(B) ⊂ X be any two affine opens of X. As X is irreducible, both Spec(A), Spec(B) ∗ contain the generic point η of X; thus Frac(A) ' OX,η ' Frac(B). So KX agrees with S on basic opens, and thus agrees with S. ∗ But then KX is a flasque sheaf, and thus has vanishing cohomology. 1 We may now motivate Cartier divisors as a tool in our study of line bundles.

Deﬁnition 2. We have noted before that isomorphism classes of line bundles over a scheme for a group, with the ring of regular functions as the identity, tensor product as the operation, and dualizing as the inverse. We call this group the Picard group of a scheme X, or Pic(X).

Lemma 2 (Line Bundles are Cartier Divisors). There is a natural isomorphism

∗ ∗ ∼ L : Γ(X, KX /OX ) −→ {Invertible subsheaves of KX }

Proof. We first define the desired map L. Given a Cartier divisor D = {Ui, fi}, let L(D) be the subsheaf −1 −1 −1 of KX generated by fi on Ui. Note that on intersections Ui ∩ Uj, fi/fj is invertible, so fi , fj generate the same sub-module. The map is trivially well-defined across equivalence classes. Furthermore, note that given an invertible subsheaf of KX , say L ,→ KX , we may recover a divisor by letting Ui be a cover of X on which L is trivial, and letting fi be the inverse of a local generator on Ui, giving a Cartier divisor {Ui, fi}. These maps are mutually inverse by construction.

We must now check that the map is a homomorphism, e.g. that given two Cartier divisors D1,D2, we −1 −1 have that L(D1 · D2 ) ' L(D1) ⊗ L(D2) , with the natural embedding in KX . But indeed, if D1 = {Ui, fi} −1 and D2 = {Ui, gi}, then both sides are locally generated by fi gi. Furthermore, it is clear that {X, 1} maps to OX ,→ KX , completing the proof.

Lemma 3 (The Picard Group is the Cartier Class Group). We claim that the map above induces an isomorphism 1 ∗ ∼ H (X, OX ) −→ Pic(X).

Proof. We first check that the map is well-defined and injective. This is exactly the claim that the kernel of the map ∗ ∗ ∼ Γ(X, KX /OX ) −→ {Invertible subsheaves of KX } → Pic(X) ∗ ∗ ∗ is exactly the image of Γ(X, KX ) → Γ(X, KX /OX ), where the last map sends an invertible subsheaf of KX to its isomorphism class. Equivalently, we wish to show that a divisor D is principal if and only if L(D) ' OX . ∗ −1 −1 If D is principal and given by some f ∈ Γ(X, K ), L(D) is simply f OX , so sending 1 to f gives an isomorphism OX ' L(D). Similarly, given such an isomorphism, the inverse of the image of the element 1 gives the desired principal divisor. To see surjectivity, it suffices to show that the map

{Invertible subsheaves of KX } → Pic(X) is surjective—that is, that every line bundle on X admits an embedding into KX . Let L be a line bundle.

Note that by our proof of Lemma 1, the sheaf KX is the constant sheaf S. Furthermore, for U with L trivial, we have Γ(U, L⊗KX ) ' Γ(U, KX ), so L⊗KX is the constant sheaf S as well, giving a map L → L⊗KX ' KX , where the ﬁrst map is injective by integrality. This completes the proof.

n n So to characterize line bundles on projective space Pk , it would suffice to compute the group Pic(Pk ), and to find which element corresponds to, say, O(1). However, we need more computational tools before we can do this. From here on we assume that X is separated, noetherian and normal. Furthermore, for convenience we work in slightly less generality than Hartshorne; rather than assuming that the scheme is locally factorial, n we assume it admits an open cover by the spectra of UFDs (all of these conditions are satisfied by Pk ). 2 Definition 3 (Weil Divisors). Prime divisors of X are closed integral subschemes of codimension one. The group of Weil divisors, denoted Div(X), is the free Abelian group generated by such primes.

∗ ∗ Let K = Γ(X, KX ). If p is a prime divisor of X with generic point η, the local ring Oη,X is a discrete valuation ring (as e.g. it is an integrally closed noetherian local ring of dimension one) with valuation ∗ vp : K → Z. ∗ ∗ P Then there is a natural map K → Div(X) given by sending f ∈ K to vp(f)p. We must show that this sum is finite. Choose an open affine U = Spec(A) on which f is regular (e.g. localize an open affine at the denominator of f); its complement is closed and thus by Noetherianness contains only finitely many prime divisors of X. By our choice of U, vp(f) ≥ 0 for p ∈ U; those p for which it is nonzero are precisely those containing Af, of which there are finitely many. It is clear that this map is a homomorphism. Divisors in the image of the map K∗ → Div(X) are called principal divisors. The cokernel of this map is the class group, denoted Cl(X). Before proceeding, we need a lemma.

∗ Lemma 4. Let A be a UFD with ﬁeld of fractions K, and f ∈ K an element such that vp(f) = 0 for all p. Then f is a unit in A.

Q ni 0 Q Proof. It suﬃces to show that f is in A, as vp(1/f) = −vp(f). Let f = u fi /u gj with fi, gj irreducible 0 and fi 6= gj for all i, j, and u, u units. Up to units, there is a unique way to do this, as A is a UFD. But then the product in the denominator must be empty, as otherwise v(gj )(f) 6= 0 for any gj. But then f ∈ A, as desired. It is as yet unclear why Weil divisors should be a useful tool; the next lemma should clear this up.

Lemma 5 (Cartier Divisors are the same as Weil Divisors). There is a natural isomorphism

∼ ∗ ∗ Div(X) −→ Γ(X, KX /OX ) which induces an isomorphism ∼ 1 ∗ Cl(X) −→ H (X, OX ).

Proof. Let D be a Weil divisor, and let Spec(Ai) be an open affine cover of X where the Ai are UFDs. P Restricting D to each Ai and writing it as ni,jpi,j, we may pick generators fi,j of each prime pi,j of Ai Q ni,j appearing in D, and send D to the Cartier divisor {Spec(Ai), j fi,j }. This is a Cartier divisor as the ratios on intersections Spec(Ai) ∩ Spec(Aj) are units, by lemma 4 above. P To map in the other direction, let {Ui, fi} be a Cartier divisor; we send it to vp(fi), where we choose an fi defined on an open Ui containing p. The value is independent of this choice by the definition of a

Cartier divisor. To see that the sum is finite, note that each fi has finitely many non-zero valuations by a previous argument—we may assume there are only finitely many fi by Noetherianness. It is clear that these maps are mutually inverse. Furthermore, principal divisors map to principal divisors, giving the second isomorphism. We can now directly compute the class group of projective space. P Lemma 6 (Computing Class Group using Weil Divisors). Let the degree of a Weil divisor D = nipi on n P Pk be given by deg(D) = ni deg(pi), where deg(pi) is the degree of the homogeneous prime of k[x0, ..., xn] 3 corresponding to pi. It is clear that deg is a homomorphism Div(X) → Z. The degree function induces an isomorphism n Cl(Pk ) → Z.

Proof. First, we must show that the map is well-deﬁned—that is, that the degree of any principal divisor is n zero. But this is clear—a rational function on Pk is exactly an element of Frac(k[x0, ..., xn]) of degree zero, and thus its divisor has degree zero. Furthermore, the map is clearly surjective, as e.g. the divisor H = (x0) has degree 1. To see that the map is injective, we must check that any divisor of degree zero is principal. Indeed, let P D = nipi be a divisor of degree zero, and let fi be a generator of pi, which has degree deg(pi) (such a Q ni generator exists as pi is a prime of height one in a UFD). Then fi is a rational function inducing the divisor D.

n n So we now see that Z ' Cl(Pk ) ' Pic(Pk ); all that remains is to see that O(1) corresponds to 1 ∈ Z. But

this is easy—trivializing O(1) locally on Ux0 , we see that the divisor of O(1) is (x0), which has degree 1. So we’ve shown the following:

n Theorem 1. Every line bundle on Pk is isomorphic to O(m) for some m ∈ Z.

We give another proof of this claim, which doesn’t use Lemma 6.

Lemma 7 (Codimension 2 doesn’t matter). Let U be an open subscheme of X. Then there is a natural surjective map Cl(X) → Cl(U); if the codimension of X − U is greater than or equal to 2, this map is an isomorphism.

Proof. We have a diagram K∗ / Div(X) / / Cl(X)

K∗ / Div(U) / / Cl(U) where the middle arrow is given by quotienting Div(X) by the free abelian group generated by prime divisors contained in X − U, and the map Cl(X) → Cl(U) is given by the functoriality of the cokernel. Surjectivity of this map follows immediately from the diagram. To see that the map is an isomorphism if X − U has codimension greater than or equal to 2, note that primes of height one are unaﬀected by quotienting by a prime of greater height, so in fact the map

Div(X) → Div(U) is an isomorphism; principal divisors are the same as KX is a constant sheaf.

Lemma 8. Let A be a Noetherian domain. Then if A is a unique factorization domain, Cl(Spec(A)) = 0.

Proof. As A is a UFD, every prime of height one is principal. Let p be a prime divisor; then it is generated by some f, which clearly induces the divisor p.

n 1 n ∗ n From here on let X = Pk . We may now directly compute H (Pk , OX ) ' Pic(Pk ).

Lemma 9 (Computing Class Group using Cohomology). We claim that

1 n ∗ H (Pk , OX ) ' Z. 4 n Proof. Let Pk = Proj(k[x0, ..., xn]). Consider the open subscheme given by U = Ux0 ∪ Ux1 . We claim that X − U has co-dimension 2; indeed, it is the complement of the closed subscheme cut out by (x0, x1), which 1 ∗ 1 ∗ is a prime of height 2. So by lemma 7, it suﬃces to compute H (U, OX ) ' Cl(U) ' Cl(X) ' H (X, OX ). 1 ∗ By lemma 8, we have that H (Uxi , OX ) ' Cl(Uxi ) ' 0, so we may use {Ux0 ,Ux1 } as a Cech cover of U (as ∗ it is an acyclic cover for OX ). This gives the ordered Cech complex

∗ ∗ ∗ 0 → k × k → k × Z → 0

∗ ∗ ∗ ∗ ∗ ∗ n as Γ(Uxi , OX ) ' (k[x0, ..., xn]xi )0 ' k , and Γ(Ux0 ∩ Ux1 , OX ) ' (k[x0, ..., xn]x0,x1 )0 ' k × {(x0/x1) }' k∗ × Z. It is clear that the map k∗ × k∗ → k∗ × Z is surjective on the k∗ coordinate and does not touch the Z 1 n ∗ coordinate, giving H (Pk , OX ) ' Z as desired.

To see that O(1) corresponds to 1 ∈ Z through this argument, we must analyze the transition map

γ01 : Γ(Ux0 ∩ Ux1 , O(1)) → Γ(Ux0 ∩ Ux1 , O(1)) induced by the two trivializations on Ux0 ,Ux1 . But in the problem set on Proj, we showed that this is induced by multiplying by x0/x1 in one direction, and x1/x0 in the other—unwinding the Cech computation above, this corresponds exactly to 1 ∈ Z, as desired. So we have another proof of the claim.