11. Coherent Sheaves; Invertible Sheaves.

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Coherent Sheaves. Let V = SpecmA be an affine variety over k, and let M be a

finitely generated A-module. There is a unique sheaf of OV -modules M on V such

that, for all f ∈ A,

Γ(D(f),M) = Mf (= Af ⊗A M).

The sheafMis said to be coherent. A homomorphism M → N of A-modules defines a

homomorphismM→N of OV -modules, and M _→Mis a fully faithful functor from

the category of finitely generated A-modules to the category of coherent OV -modules,

with quasi-inverse M _→Γ(V,M).

Now consider a variety V. An OV -module M is said to be coherent if, for every

open affine subset U of V , M|U is coherent. It suffices to check this condition for the

sets in an open affine covering of V .

For example, On

V is a coherent OV -module. An OV -module M is said to be locally

free of rank n if it is locally isomorphic to On

V , i.e., if every point P ∈ V has an open

neighbourhood such that M|U ≈ On

V . A locally free OV -module is coherent.

Let v ∈ V , and letMbe a coherent OV -module. We define a κ(v)-moduleM(v) as

follows:a fter replacing V with an open neighbourhood of v, we can assume that it is

affine; hence we may suppose that V = Specm(A), that v corresponds to a maximal

ideal m in A (so that κ(v) = A/m), andMcorresponds to the A-module M; we then

define

M(v) = M ⊗A κ(v) = M/mM.

It is a finitely generated vector space over κ(v). Don’t confuse M(v) with the stalk

Mv of M which, with the above notations, is Mm = M ⊗A Am. Thus M(v) =

Mv/mMv = κ(v) ⊗Am

M

m. Nakayama’s Lemma shows that

M(v) = 0⇒Mv = 0.

The support of a coherent sheaf M is

Supp(M) = {v ∈ V | M(v) _= 0} = {v ∈ V | Mv _= 0}.

Suppose V is affine, and that M corresponds to the A-module M. Let a be the

annihilator of M:

a = {f ∈ A | fM = 0}.

Then M/mM _= 0 ⇐⇒ m ⊃ a (for otherwise A/mA contains a nonzero element

annihilating M/mM), and so

Supp(M) = V (a).

Thus the support of a coherent module is a closed subset of V .

Note that if M is locally free of rank n, then M(v) is a vector space of dimension

n for all v. There is a converse of this.

Proposition 11.1. If M is a coherent OV -module such that M(v) has constant

dimension n for all v ∈ V , then M is a locally free of rank n.

144 Algebraic Geometry: 11. Coherent Sheaves; Invertible Sheaves

Proof. We may assume that V is affine, and that M corresponds to the finitely

generated A-module M. Fix a maximal ideal m of A, and let x1, . . . , xnbe elements

of M whose images in M/mM form a basis for it over κ(v). Consider the map

γ : An → M, (a1, . . . , an) _→

_

aixi.

The cokernel is a finitely generated A-module whose support does not contain

v. Therefore there is an element f ∈ A, f /∈ m, such that γ defines a surjection

Anf

→ Mf. After replacing A with Af we may assume that γ itself is surjective.

For every maximal ideal n of A, the map (A/n)n → M/nM is surjective, and hence

(because of the condition on the dimension of M(v)) bijective. Therefore, the kernel

of γ is contained in nn (meaning n × · · · ×n) for all maximal ideals n in A, and the

next lemma shows that this implies that the kernel is zero.

Lemma 11.2. Let A be an affine k-algebra. Then

∩m = 0 (intersection of all maximal ideals in A).

Proof. Suppose first that k is algebraically closed. Recall (1.9) that if a is a radical

ideal in k[X1, . . . ,Xn], then IV (a) = a. When we use the one-to-one correspondence

between points of V (a) and the maximal ideals of k[X1, . . . ,Xn] containing a, we see

that this says that a function that is in every maximal ideal containing a is, in fact,

in a. On applying this statement to the ring A = k[X1, . . . ,Xn]/a, we obtain the

lemma.

Now drop the assumption that k is algebraically closed, and consider a maximal

ideal m of A ⊗k k al . Then

A/m ∩ A 8→ A ⊗k k al = k al .

Therefore A/m ∩ A is an integral domain. Since it is finite-dimensional over k, it

is a field, and so m ∩ A is a maximal ideal in A. Thus if f ∈ A is in all maximal

ideals of A, then its image in A⊗k al is in all maximal ideals of A, then its image in

A ⊗ k al is in all maximal ideals of A ⊗ k al , and so is zero.

For two coherent OV -modules M and N, there is a unique coherent OV -module

M⊗OV

N such that

Γ(U,M⊗OV

N) = Γ(U,M) ⊗Γ(U,OV ) Γ(U,N)

for all open affines U ⊂ V . The reader should be careful not to assume that this

formula holds for nonaffine open subsets U (see example 11.4 below). For a such a

U, one writes U = ∪Ui with the Ui open affines, and defines Γ(U,M⊗OV

N) to be

the kernel of

 

i

Γ(Ui,M⊗OV

N) ⇒

 

i,j

Γ(Uij ,M⊗OV

N).

Define Hom(M,N) to be the sheaf on V such that

Γ(U,Hom(M,N)) = HomOU (M,N)

(homomorphisms of OU-modules) for all open U in V . It is easy to see that this is a

sheaf. If the restrictions ofMand N to some open affine U correspond to A-modules

Algebraic Geometry: 11. Coherent Sheaves; Invertible Sheaves 145

M and N, then

Γ(U,Hom(M,N)) = HomA(M,N),

and so Hom(M,N) is again a coherent OV -module.

Invertible sheaves. An invertible sheaf on V is a locally free OV -module L of rank

1. The tensor product of two invertible sheaves is again an invertible sheaf. In this

way, we get a product structure on the set of isomorphism classes of invertible sheaves:

[L] · [L_

] = [L⊗L_

].

The product structure is associative and commutative (because tensor products are

associative and commutative, up to a canonical isomorphism), and [OV ] is an identity

element. Define

L∨ = Hom(L,OV ).

Clearly, L∨ is free of rank 1 over any open set where L is free of rank 1, and so L∨ is

again an invertible sheaf. Moreover, the canonical map

L∨ ⊗L→OV , (f, x) _→ f(x)

is an isomorphism (because it is obviously an isomorphism over any open subset where

L is free). Thus

[L∨

][L] = [OV ].

For this reason, we often write L−1 for L∨.

From these remarks, we see that the set of isomorphism classes of invertible sheaves

on V is a group — it is called the Picard group, Pic(V ), of V .

We say that an invertible sheaf L is trivial if it is isomorphic to OV — then L

represents the zero element in Pic(V ).

Proposition 11.3. An invertible sheaf L on a complete variety V is trivial if and

only if both it and its dual have nonzero global sections, i.e.,

Γ(V,L) _= 0 _= Γ(V,L∨).

Proof. We may assume that V is irreducible. Note first that, for any OV -module

M on any variety V , the map

Hom(OV ,M) → Γ(V,M), α_→ α(1)

is an isomorphism.

Next recall that the only regular functions on a complete variety are the constant

functions (see 5.28 in the case that k is algebraically closed), i.e., Γ(V,OV) = k_

where k_ is the algebraic closure of k in k(V ). Hence Hom(OV ,OV) = k_, and so a

homomorphism OV →OV is either 0 or an isomorphism.

We now prove the proposition. The sections define nonzero homomorphisms

s1 : OV → L, s2 : OV →L∨

.

We can take the dual of the second homomorphism, and so obtain nonzero homomorphisms

OV

→s1 L s∨

→2 OV .

146 Algebraic Geometry: 11. Coherent Sheaves; Invertible Sheaves

The composite is nonzero, and hence an isomorphism, which shows that s∨

2 is

surjective, and this implies that it is an isomorphism (for any ring A, a surjective

homomorphism of A-modules A → A is bijective because 1 must map to a unit).

Invertible sheaves and divisors. Now assume that V is nonsingular. For a divisor

D on V , the vector space L(D) is defined to be

L(D) = {f ∈ k(V )

× | div(f) + D ≥ 0}.

We make this definition local:de fine L(D) to be the sheaf on V such that, for any

open set U,

Γ(U,L(D)) = {f ∈ k(V )

× | div(f) + D ≥ 0 on U} ∪ {0}.

The condition “div(f)+D ≥ 0 on U” means that, if D =

_

nZZ, then ordZ(f)+nZ ≥

0 for all Z with Z ∩ U _= ∅. Thus, Γ(U,L(D)) is a Γ(U,OV )-module, and if U ⊂ U_,

then Γ(U_ ,L(D)) ⊂ Γ(U,L(D)). We define the restriction map to be this inclusion.

In this way, L(D) becomes a sheaf of OV -modules.

Suppose D is principal on an open subset U, say D|U = div(g), g ∈ k(V )×. Then

Γ(U,L(D)) = {f ∈ k(V )

× | div(fg) ≥ 0 on U} ∪ {0}.

Therefore,

Γ(U,L(D)) → Γ(U,OV ), f_→ fg,

is an isomorphism. These isomorphisms clearly commute with the restriction maps

for U_ ⊂ U, and so we obtain an isomorphism L(D)|U → OU. Since every D is

locally principal, this shows that L(D) is locally isomorphic to OV , i.e., that it is an

invertible sheaf. If D itself is principal, then L(D) is trivial.

Next we note that the canonical map

L(D) ⊗L(D_) →L(D + D_), f⊗ g _→ fg

is an isomorphism on any open set where D and D_are principal, and hence it is an

isomorphism globally. Therefore, we have a homomorphism

Div(V ) → Pic(V ), D_→ [L(D)],

which is zero on the principal divisors.

Example 11.4. Let V be an elliptic curve, and let P be the point at infinity. Let

D be the divisor D = P. Then Γ(V,L(D)) = k, the ring of constant functions, but

Γ(V,L(2D)) contains a nonconstant function x. Therefore,

Γ(V,L(2D)) _= Γ(V,L(D)) ⊗ Γ(V,L(D)),

— in other words, Γ(V,L(D) ⊗L(D)) _= Γ(U,L(D) ⊗L(D)).

Proposition 11.5. For an irreducible nonsingular variety, the map D _→ [L(D)]

defines an isomorphism

Div(V )/PrinDiv(V ) → Pic(V ).

Proof. (Injectivity). If s is an isomorphism OV → L(D), then g = s(1) is an

element of k(V )× such that

(a) div(g) + D ≥ 0 (on the whole of V );

Algebraic Geometry: 11. Coherent Sheaves; Invertible Sheaves 147

(b) if div(f) + D ≥ 0 on U, that is, if f ∈ Γ(U,L(D)), then f = h(g|U) for some

h ∈ Γ(U,OV ).

Statement (a) says that D ≥ div(−g) (on the whole of V ). Suppose U is such that

D|U admits a local equation f = 0. When we apply (b) to −f , then we see that

div(−f) ≤ div(g) on U, so that D|U + div(g) ≥ 0. Since the U’s cover V , together

with (a) this implies that D = div(−g).

(Surjectivity). Define

Γ(U,K) =

_

k(V )× if U is open an nonempty

0 if U is empty.

Because V is irreducible, K becomes a sheaf with the obvious restriction maps. On

any open subset U where L|U ≈ OU, we have L|U ⊗K ≈ K. Since these open sets

form a covering of V , V is irreducible, and the restriction maps are all the identity

map, this implies that L⊗K ≈ K on the whole of V . Choose such an isomorphism,

and identify L with a subsheaf of K. On any U where L ≈ OU, L|U = gOU as a

subsheaf of K, where g is the image of 1 ∈ Γ(U,OV ). Define D to be the divisor such

that, on a U, g−1 is a local equation for D.

Example 11.6. Suppose V is affine, say V = SpecmA. We know that coherent

OV -modules correspond to finitely generated A-modules, but what do the locally free

sheaves of rank n correspond to? They correspond to finitely generated projective Amodules

(Bourbaki, Commutative Algebra, II.5.2). The invertible sheaves correspond

to finitely generated projective A-modules of rank 1. Suppose for example that V is

a curve, so that A is a Dedekind domain. This gives a new interpretation of the ideal

class group:i t is the group of isomorphism classes of finitely generated projective

A-modules of rank one (i.e., such that M ⊗A K is a vector space of dimension one).

This can be proved directly. First show that every (fractional) ideal is a projective

A-module — it is obviously finitely generated of rank one; then show that two ideals

are isomorphic as A-modules if and only if they differ by a principal divisor; finally,

show that every finitely generated projective A-module of rank 1 is isomorphic to

a fractional ideal (by assumption M ⊗A K ≈ K; when we choose an identification

M ⊗A K = K, then M ⊂ M ⊗A K becomes identified with a fractional ideal).

[Exercise:P rove the statements in this last paragraph.]

Remark 11.7. Quite a lot is known about Pic(V ), the group of divisors modulo

linear equivalence, or of invertible sheaves up to isomorphism. For example, for any

complete nonsingular variety V , there is an abelian variety P canonically attached to

V , called the Picard variety of V , and an exact sequence

0 → P(k) → Pic(V ) → NS(V ) → 0

where NS(V ) is a finitely generated group called the N´eron-Severi group.

Much less is known about algebraic cycles of codimension > 1, and about locally

free sheaves of rank > 1 (and the two don’t correspond exactly, although the Chern

classes of locally free sheaves are algebraic cycles).

Direct images and inverse images of coherent sheaves. Consider a homomomorphism

A → B of rings. From an A-module M, we get an B-module B ⊗A M,

which is finitely generated if M is finitely generated. Conversely, an B-module M

148 Algebraic Geometry: 11. Coherent Sheaves; Invertible Sheaves

can also be considered an A-module, but it usually won’t be finitely generated (unless

B is finitely generated as an A-module). Both these operations extend to maps of

varieties.

Consider a regular map α: W → V , and let F be a coherent sheaf of OV -modules.

There is a unique coherent sheaf of OW-modules α∗F with the following property:

for any open affine subsets U_ and U of W and V respectively such that α(U_) ⊂

U, α∗F|U_ is the sheaf corresponding to the Γ(U_ ,OW)-module Γ(U_,OW) ⊗Γ(U,OV )

Γ(U,F).

Let F be a sheaf of OV -modules. For any open subset U of V , we define

Γ(U, α∗F) = Γ(α−1U,F), regarded as a Γ(U,OV )-module via the map Γ(U,OV ) →

Γ(α−1U,OW). Then U _→ Γ(U, α∗F) is a sheaf of OV -modules. In general, α∗F will

not be coherent, even when F is.

Lemma 11.8. (a) For any regular maps U →α V

β→

W and coherent OW-module

F on W, there is a canonical isomorphism

(βα)

∗F ≈→

α

∗

(β

∗F).

(b) For any regular map α: V → W, α∗ maps locally free sheaves of rank n to locally

free sheaves of rank n (hence also invertible sheaves to invertible sheaves). It

preserves tensor products, and, for an invertible sheaf L, α∗(L−1)

∼=(α

∗

L)−1.

Proof. (a) This follows from the fact that, given homomorphisms of rings A →

B → T , T ⊗B (B ⊗A M) = T ⊗A M.

(b) This again follows from well-known facts about tensor products of rings.

Algebraic Geometry: 12. Differentials 149

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