Chapter 6 MODULES
6.1 The Structure Sheaf
6.2 Modules
6.3 Defining a Module on a Basis
6.4 Homomorphisms
6.5 Direct Images
6.6 Twisting
6.7 Proofs
fnfld
6.1
The Structure Sheaf.
Regular functions on an open subset of a variety were defined in Section 3.5. As we saw there, the regular
functions on an affine variety X = Spec A are the elements of its algebra A, and the variety X is completely
determined by that algebra. This isn’t true for varieties that aren’t affine, because they don’t have enough
regular functions. One has to look at the regular functions on affine open subsets. The structure sheaf of a
variety X organizes things by keeping track, implicitly, of the regular functions on every open subset.
defstrsheaf
defstrsheaftwo
6.1.1. Definition.
(6.1.2)
The structure sheaf OX on a variety X is the map
O
X
(algebras)
(opens)◦ −→
from open sets to algebras that sends a nonempty open set U to the algebra, denoted by OX (U ), of rational
functions that are regular on U . So OX (U ) is a subalgebra of the function field K of X.
Elements of OX (U ) are called sections of ∫ OX on U , and elements of OX (X), functions that are regular
everywhere, are global sections.
We make the set (opens) of open subsets of X into a category, defining morphisms between open sets to be
inclusion maps. So if V ⊂ U there is a unique morphism V → U , and if V 6⊂ U there is no such morphism.
The structure sheaf OX is a sheaf because it has the following two properties, both obvious:
strsheafprop
(6.1.3)
• OX is a contravariant functor: If V ⊂ U are nonempty open sets, then regular function on U is also regular
on V : OX (V ) ⊃ OX (U ).
(The superscript ◦ in (6.1.2) indicates that arrows, inclusions here, are reversed by OX , i.e., that the functor is
contravariant.)
S
• OX is a sheaf: If V1 , ..., Vk are nonempty open subsets that cover another open set V , i.e., if V = Vi , a
T
rational function is regular on V if and only if it is regular on each Vi : OX (V ) = OX (Vi ).
This sheaf property will be discussed in more detail below.
pnminuspoint
6.1.4. Example. Let Z be the locus of zeros of a homogeneous polynomial f of degree r on Pn . Corollary
3.4.6 describes the sections of the structure sheaf on the complement U of Z. The sections on U are fractions
h/f k , where h is homogeneous, of degree rk.
1
Obviously, one doesn’t want to compute the regular functions on every open set U , and one never does. As
we will see, the structure sheaf is determined by the regular functions on a collection of affine open sets that
cover X, such as on the standard affine covering of Pn . Moreover, one rarely needs to determine the functions
explicitly, even on this covering.
When studying the structure sheaf, one always works with affine open sets.
module
modulehoms
•
Information about an open set that isn’t affine is determined by the sheaf property.
6.2
Modules
(6.2.1) module homomorphisms
Because we will be working with modules over various rings, we define homomorphisms of modules
whose rings of scalars may be different. Let M and M ′ be modules over rings R and R′ , respectively. A
ϕ
ϕ
homomorphism M −→ M ′ consists of a homomorphism of additive groups M −→ M ′ and a ring homomorρ
phism R −→ R′ , the two maps being compatible, meaning that, for m in M and a in R,
homwithrho
ϕ(am) = ρ(a)ϕ(m)
(6.2.2)
This condition is expressed by the diagram below, in which the horizontal arrows represent scalar multiplication:
R×M −−−−→


(ρ,ϕ)y
M

ϕ
y
R×M ′ −−−−→ M ′
For example, if M is a module over a domain R and s is a nonzero element of R, the localization Ms is an
Rs -module, and there is a homomorphism M → Ms that sends m to m, the associated map R → Rs being
the inclusion.
As in this example, the ring homomorphism involved in a module homomorphism is often obvious. If so,
we may not mention it explicitly.
ϕ
The more usual concept of a homomorphism M −→ M ′ of modules over the same ring R is the case
that ρ is the identity map: ϕ(am) = aϕ(m). For clarity, we may refer to such map as a homomorphism of
R-modules.
ϕ
ρ
The case that M ′ −→ M ′ is the identity map , but R −→ R′ is arbitrary: am′ = ρ(a)m′ is called the
′
′
′
restriction of scalars from R to R in an R -module M .
The term “restriction of scalars” is adapted from the special case that R is a subring of R′ .
ϕ
ρ
A module homomorphism M −→ M ′ associated to a ring homomorphism R −→ R′ becomes a homomorphism of R-modules when M ′ is made into an R-module by restriction of scalars.
omod
(6.2.3) O-modules
As above, the structure sheaf O = OX of a variety X is the functor (opens)◦ → (algebras) that associates
to each nonempty open subset U of X the algebra O(U ) of regular functions on U , and to each inclusion
j
V −→ U of open sets the injective map O(U ) → O(V ).
An O-module M is a functor
functmod
M
(opens)◦ −→ (modules)
(6.2.4)
j
that associates to each open set U an O(U )-module M(U ), and to each inclusion of open sets V −→ U
j
∗
a homomorphism of modules M(U ) −→ M(V ) in the opposite direction, the associated homomorphism
2
of algebras being the inclusion O(U ) ⊂ O(V ). The homomorphism j ∗ is called restriction. It needn’t be
injective.
Elements of M(U ) are called sections of M on U , and elements of M(X) are global sections. If V → U
is an inclusion of open sets and m is a section on U , its image j ∗ m in M(V ) is the restriction of m to V .
The restriction of a section m may be denoted, as above, by j ∗ m. However, because the operation of restriction occurs very often, we usually abbreviate, using the same symbol m for a section and for its restriction.
Also, if an open set V is contained in two other open sets U 1 and U 2 , and if mi is a section of M on U i , we
say that m1 and m2 are equal on V if their restrictions to V are equal.
A homomorphism M → N of O-modules consists of a homomorphism of O(U )-modules M(U ) →
N (U ) for each open set U . These homomorphisms are required to be compatible with restriction when V ⊂ U ,
as is indicated by the diagram
M(U ) −−−−→ N (U )



j ∗
j∗y
y
M(V ) −−−−→ N (V )
The simplest examples of O-modules are the free O-module On , whose sections on an open set U form
the free O(U )-module O(U )n .
Multiplication by a global section α of O, a regular function on X, defines a homomorphism O → O: If U
is an open set, α will be a regular function on U , so multiplication by α is an O(U )-linear map O(U ) → O(U ).
Thus a homomorphism Om → On of free modules will be given by an m×n-matrix of global sections of O.
defqcoh
6.2.5. Definition.
An O-module M is quasicoherent if it satisfies the following two conditions:
the coherence property (compatibility with localization)
Let U be an open subset of X, and let R = O(U ) be the algebra of regular functions on U . Let s be a nonzero
element of R, and let Us be the open subset of U points at which s 6= 0. If M is an O-module and M(U ) is
the R-module M , then M(Us ) is its localization Ms .
If this property is true for a particular open set U , we may say that the coherence property holds for U .
the sheaf property (sections are determined locally)
Let U be an open subset of X and let {U i } be open subsets that cover U . Let mi ∈ M(U i ) be sections of M
on U i . If mi = mj on U i ∩ U j for all i and j, there is a unique section m of M on U such that m = mi on
U i for all i.
This property is also called the sheaf axiom. A functor that has this property is a sheaf.
For the structure sheaf O, the coherence property asserts that, with notation as above, O(Us ) = Rs . This
is true when U is an affine open set, and Theorem 6.3.1 below asserts that it is true for arbitrary open sets.
The sheaf property is trivial for the structure sheaf: If {U i } cover U and if a rational function α is regular
on each U i then α is regular on U .
coverpone
6.2.6. Example. The projective line P1 is covered by the two standard affine open sets U0 and U1 , and with
our usual notation, O(U0 ) = C[t], O(U1 ) = C[u], and O(U0 ∩ U1 ) = C[t, u], where u = t−1 . The sheaf
property asserts that a global section of O is be determined by elements a(t) ∈ C[t] and b(u) ∈ C[u] such that
a(t) = b(u) in C[t, u]. Since u = t−1 , a and b must be equal constants. The only rational functions that are
regular everywhere on P1 are the constants. I think we knew this.
The coherence property is more fundamental than the sheaf property. As Theorem 6.3.6 below shows, the
sheaf property follows from the coherence property on an affine variety X. The sheaf property is needed only
to determine the sections of an O-module on the subsets that aren’t affine.
When working with O-modules, one should always work with affine open sets. Though we will be interested in the global sections of a module, the non-affine opens are just along for the ride most of the time.
We make a few more preliminary remarks here.
3
As the example above shows, a module may have few sections on a non-affine open set. The global sections
of OP1 are the constants, and they tell us very little about the structure sheaf. On the other hand, the regular
functions on the affine line A1 form the polynomial ring C[t], and that ring determines the structure of A1
completely.
In the coherence property, saying M(Us ) “is” the localization Ms is inaccurate. It would be better to say
that Ms and M(Us ) are canonically isomorphic. When M is a functor, the inclusion of Us ⊂ U gives us a
module homomorphism M(U ) = M → M(Us ), and since s is invertible on Us , this map extends to a homomorphism Ms → M(Us ). The coherence condition requires that this homomorphism be an isomorphism. To
avoid cluttering up our notation, we often identify canonically isomorphic objects.
emptyset
6.2.7. Proposition. The only section of an OX -module M on the empty set is the zero section: M(∅) = 0.
In particular, OX (∅) = 0.
proof. This follows from the sheaf property. The empty set ∅ is covered by the empty covering, the family
indexed by the empty set. The only set of sections on the empty covering is the empty set. The sheaf property
asserts that this empty set of sections determines a unique section on ∅. So M(∅) contains a unique element.
That element is zero.
sheafonpoint
somemodules
6.2.8. Example. The affine variety Spec C is a point; let’s call it p. It has two open sets: the whole space p
and the empty set, and since O(p) = C, the global sections M(p) of an O-module M form a complex vector
space, while M(∅) = 0. The sheaf and the coherence properties are trivial, so to define M, one can assign
the vector space M(p) arbitrarily. Speaking informally, we may say that a module on the point is a complex
vector space.
(6.2.9)
the twisting modules
The twisting modules are among the most important modules on projective space.
Let X be the projective space with coordinates x0 , ..., xn . If g and h are homogeneous polynomials in x of
the same degree, the fraction g/h defines a rational function on X. The nonzero elements of the function field
of X are the equivalence classes of such fractions (3.4.5). To define the twisting modules O(d), we consider
homogeneous fractions, fractions of homogeneous polynomials that aren’t necessarily of the same degree. The
degree of a nonzero homogeneous fraction g/h is the difference deg g − deg h. It can be any integer. The
set of homogeneous fractions of degree d, together with the zero element, will be denoted by Kd . So K0 is the
function field of X.
An element of Kd is regular at a point p if it can be represented as a fraction g/h whose denominator h
isn’t zero at p, and an element is regular on an open set U if it is regular at every point of U . When we write
a fraction in the form g/h where g and h are homogeneous and relatively prime, it will be regular at p if and
only if h(p) 6= 0. This agrees with the definition given before for rational functions. For example, the only
homogeneous polynomials that don’t vanish at any point of the standard affine U0 are the powers of x0 . So the
homogeneous fractions that are regular on U0 have the form g/xk0 .
The twisting module O(d) on Pn is defined as follows:
The sections of O(d) on an open set U are the homogeneous fractions of degree d
that are regular on U .
Odismodule
6.2.10. Lemma. (i) The twisting module O(d) on projective space X is a quasicoherent O-module.
(ii) The sections of O(d) on the standard affine open set U0 form a free module of rank one over the polynomial
ring O(U0 ), with basis xd0 .
Thus O(d) seems quite similar to the structure sheaf. However, O(d) is only locally free. The sections of
O(d) on the standard open set U1 form a free O(U1 )-module with basis xd1 . The bases on the two open sets
are related by the factor (x0 /x1 )d .
(A property P is said to be true “locally” on a topological space X if there is an open covering {U i } of X
such that P is true on every U i .)
4
proof of Lemma 6.2.10 (i) It is easy to show that O(d) is a functor and that the sheaf property holds, We verify
the coherence property. Let R be the coordinate ring of an open subset U of X, and let s be a nonzero element
of R. We must show that the module of sections of O(d) on Us is obtained by localizing the module of sections
on U . The essential point to show is that, if u is a section of O(d) on Us , then for sufficiently large k, sk u is a
section on U .
Say that we write s = q/r and u = g/h as homogeneous fractions, in which r doesn’t vanish on U and
h doesn’t vanish on Us . Then sk u is regular on U if q k /h is regular on U . So it is enough to show that, if a
homogeneous polynomial h is regular on Us , then for large k, q k h is regular on U .
We may assume that h is an irreducible polynomial. Let Y be the closed complement of U in X, and
let Z = VX (q) be the zero locus of q. The union Y ∪ Z is the complement of Us in X, and it contains the
irreducible closed set C = VX (h). Therefore C is contained in Y or in Z. If C ⊂ Y , then 1/h is regular on
Us . If VX (h) ⊂ Z = VX (q), then by the Strong Nullstellensatz, h divides a power q k of q, and g k /h is regular
on U .
sectionsofOn
6.2.11. Proposition. When d ≥ 0, the global sections of the twisting module O(d) on Pn , n > 0, are the
homogeneous polynomials of degree d. When d < 0, the only global section of O(d) is zero.
proof. A global section u of O(d) will restrict to a section on the standard affine open set U0 , which means
that u is a homogeneous fraction whose denominator is a power of x0 . Similarly, restriction to U1 shows that
u is a homogeneous fraction whose denominator is a power of x1 . So u can have no denominator.
As we see here, it is often convenient to describe a quasicoherent O-module M by its sections on an
affine open covering {U i }, such as on the standard covering of projective space. When M(U i ) is known,
the coherence property determines the sections on localizations Usi , and the sections on other open sets are
determined by the sheaf property. This is the standard procedure to describe a module.
The product uv of homogeneous fractions of degrees r and s is a homogeneous fraction of degree r + s,
and if u and v are regular on an open set U , so is uv. Therefore multiplication defines a homomorphism of
O-modules
oros
O(r)×O(s) → O(r + s)
(6.2.12)
Multiplication by a homogeneous polynomial f of degree d defines an injective homomorphism
multbyfone
f
O(k) −→ O(k + d).
(6.2.13)
f
When we set k = −d, we obtain a homomorphism O(−d) −→ O. Or, setting k = 0 gives us a homomorphism
f
O −→ O(d).
idealsinstrsheaf
(6.2.14)
ideals
An ideal I in the structure sheaf of a variety X is a quasicoherent submodule of O = OX . For example
the structure sheaf has a maximal ideal at a point p. We’ll denote this ideal by mp , using the same notation
as for maximal ideals of algebras. The sections of mp on an open set U are as follows: If U contains p, then
mp (U ) is the ideal of O(U ) of regular functions that vanish at p, while if U doesn’t contain p, mp (U ) is the
unit ideal O(U ).
The main example is the ideal of functions that vanish on a closed subset Y of X. Its sections on an open
set U are the regular functions on U that are zero at every point of Y ∩ U . Let I denote this ideal. The sheaf
property for I is fairly obvious. Let’s assume that the coherence property has been shown for the structure
sheaf O (see Theorem 6.3.1 below), and deduce it for I.
Let s be a nonzero element of R = O(U ). The coherence property for O asserts that O(Us ) = Rs . Let
I = I(U ) be the ideal of R of regular functions that vanish on Y ∩ U . We must show that I(Us ) is the
localization Is . Because I ⊂ I(Us ) and s is invertible in Rs , Is ⊂ I(Us ). Let α be an element of I(Us ). So
α is an element of Rs that is zero on Y ∩ Us . Since s is invertible in Rs , we may clear the denominator in α
to reduce to the case that α is in R. Let Z be the zero locus of s in U . Since α is zero on Y ∩ Us and s is zero
on Z, αs is zero on Y ∩ U . So αs is in I, and α is in Is .
5
We’ll describe the ideals in the structure sheaf of the projective plane X = P2 that correspond to plane
curves more completely. We restrict to the plane only to focus attention. An analogous discussion can be made
for the ideals of hypersurfaces in projective spaces of arbitrary dimension.
curveideal
6.2.15. Lemma. Let C be the plane curve defined by an irreducible homogeneous polynomial f of degree d,
and let I be the ideal of C.
(i) The sections of I on an open set U are the rational functions that can be written as fractions g/h of
homogeneous polynomials, such that f divides g and h doesn’t vanish at any point of U .
f
(ii) Let O(−d) be the twisting module. The image of the multiplication map O(−d) −→ O is the ideal I.
Thus, as O-module, I is isomorphic to the twisting sheaf O(−d). The ideals of all plane curves of degree d
are isomorphic O-modules.
proof. We represent a rational function u as a fraction g/h of relatively prime homogeneous polynomials.
To be a section of I on U , u must satisfy two conditions: First, u must be a regular function on U . So the
denominator h cannot vanish at any point of U . Suppose that this is so. Then there are two cases. If the
intersection C ′ = C ∩ U isn’t empty, it will be a dense open subset of the irreducible closed set C. The
numerator g must vanish at every point of C ′ , and since C ′ is dense, it will vanish on C. This is true if and
only if f divides g. If C ′ is empty, any regular function u on U will be a section of I. In that case f won’t
vanish at any point of U , and therefore g/h can also be written as the fraction f g/f h whose numerator is
divisible by f . This proves (i), and (ii) follows easily.
basicmodprop
(6.2.16)
the sheaf property as an exact sequence
Let U be an open subset of a variety X and U i be open subsets that cover U . The sheaf property for an
OX -module and for this covering can be expressed as an exact sequence
sheafzero
(6.2.17)
α
0 → M(U ) −→
Y
β
M(U i ) −→
i
Y
M(U i ∩ U j )
i,j
Let
in which the maps α and β are explained as follows. To simplify notation,
let’s suppose that there are finitely
Q
1
k
i
many sets forming the covering,
say
U
,
...,
U
.
Then
elements
of
M(U
) can be viewed as k-dimensional
Q
i
j
vectors, and elements of M(U ∩ U ) can be viewed as k × k matrices.
definition of α: Let s be a section of M(U ). The i th coordinate of the vector α(s) is the restriction of s to U i .
Q
definition of β: Let m = (m1 , . . . , mk ) be an element of M(U i ). The i, j entry zij of the matrix β(m)
is the difference of the restrictions of mj and mi to U i ∩ U j . Using our abbreviated notation for restricted
sections, zij = mj − mi on M(U i ∩ U j ).
The sheaf property asserts that the map α is injective, and that its image is the kernel of β. Thus sections of
M over U correspond bijectively to vectors (m1 , ..., mk ) with mi ∈ M(U i ), such that mi = mj on U i ∩ U j .
ilessthanj
6.2.18. Note. A vector (m1 , ..., mk ) of sections of M is in the kernel of β if the property mi = mj on
U i ∩ U j is true for the pairs of indices i, j with i < j. The index pairs with i ≥ j are redundant because the
subsets U j ∩ U i and U i ∩ U j are the same, and it is trivial that mi = mi on U i ∩ U i .
twoopens
6.2.19. Corollary. Let M be a quasicoherent O-module and let U be an open set that is covered by two open
sets U 0 and U 1 . A pair of sections m0 , m1 on the two open sets determines a section on U if m0 = m1 on
U 0 ∩ U 1 , or if the sequence
−,+
0 −→ M(U ) −→ M(U 0 )⊕M(U 1 ) −→ M(U 0 ∩ U 1 )
is exact.
6
moduleonpone
6.2.20. Example. We go back to the covering of the projective line P1 by the standard affine open sets
U0 = Spec C[t] and U1 = Spec C[t−1 ]. Their intersection U01 is the spectrum of the Laurent polynomial
ring C[t, t−1 ], whose elements are combinations of powers of t, negative powers included. Let M be a
quasicoherent O-module, and let Mi = M(Ui ). The coherence property of M tells us that M01 can be
obtained by localization from M0 and also from M1 :
M0 [t−1 ] ≈ M01 ≈ M1 [t].
A global section of M is determined by a pair of elements m0 ∈ M0 and m1 ∈ M1 that become equal in this
common localization.
basis
6.3
Defining a Module on a Basis
The theorems in this section show that, in order to determine a quasicoherent O-module M, one doesn’t
need to know the sections on every open set. It is enough to know the sections on a particular affine covering.
The proofs are in Section 6.7.
onecoverenough
ocoherent
6.3.1. Theorem. Let {U i } be an affine open covering of a variety X, and let M be an O-module. Suppose
that M has the sheaf property, and that the coherence property is true for the open sets U i . Then the coherence
property is true for every open subset, and therefore M is quasicoherent.
6.3.2. Corollary. The structure sheaf O has the coherence property.
The corollary follows from the fact that the coherence property for O is true for affine open sets.
A collection B of open subsets of a topological space X is a basis if every open subset of X can be covered
by subsets that are in B. We may consider a basis as a subcategory of the category of open sets, morphisms
being inclusions.
defgb
6.3.3. Definition. A set B of open subsets of a variety X is a good basis if it has these properties:
• There is a covering of X by affine open sets that are in B.
• If U and V are in B, then U ∩ V is in B.
• If an affine open set U is in B, then every localization Us of U is in B.
The localizations Xs of an affine variety X form a good basis. If {U i } is a covering of an arbitrary variety X
by affine open sets, the affine open sets that are contained in one of the sets U i form a good basis (see 3.5.11).
In particular, the set of all affine open subsets of X is a good basis. These are the most important examples.
Let
absheafbasis
(6.3.4)
M
B ◦ −→ (modules)
be a functor such that M(U ) is an O(U )-module for every U in B. We say that M is a quasicoherent Bmodule if the sheaf property (6.7.3) is true for every covering of an open set U by B-open sets {U i }, and if the
coherence property holds for open sets in B.
sheafonbasis
6.3.5. Theorem. Let B be a good basis for the topology on a variety X, and let M be a quasicoherent Bmodule. There is a quasicoherent O-module, unique up to unique isomorphism, whose restriction to B is M.
The analogue of this theorem is true for sheaves of abelian groups on any topological space and for any basis,
but never mind.
The theorem is almost obvious. Suppose that M is given on a good basis, and let U be any open set. We
cover U by opens U i that are in B, and then the intersections U i ∩ U j will be in B too. The sheaf property for
the O-module that we hope to define requires that the sequence (6.2.17)
Y
Y
0 → M(U ) →
M(U i ) →
M(U ij
7
be exact. This identifies M(U ) as the kernel of a map between modules that are given. But we must verify
that this defines M(U ) up to unique isomorphism, that it gives a functor, and that this functor has the sheaf
property. These points are explained in the last section of the chapter. Then the coherence property follows
from Theorem 6.3.1.
The next theorem and its corollary show that the sheaf property for an affine variety is a consequence of
the coherence property.
sheafforaffine
6.3.6. Theorem. Let s1 , ..., sk be elements of a ring A that generate the unit ideal. Let M be an A-module,
and let Mi and Mij denote the localizations Msi and Msi sj , respectively. The sequence
Y
β Y
α
0 → M −→
Mi −→
Mij
analogous to (6.2.17) is exact.
The relation of this theorem with O-modules is as follows: Say that U = Spec A and let U i = Usi . Let
M be an O-module that has the coherence property for U , and let M = M(U ). Then M(U i ) = Mi and
M(U i ∩ U j ) = Mij . The theorem asserts that the sequence (6.2.17) is exact, as is required by the sheaf
property.
The next corollary tells us that the complication caused by working with O-modules is justified only for
varieties that aren’t affine, because a quasicoherent O-module on an affine variety is determined completely
by its global sections.
moduleonaffines
6.3.7. Corollary. On an affine variety X = Spec A, quasicoherent O-modules and A-modules are equivalent
concepts.
Thus, to define a quasicoherent O-module M, it suffices to give its sections on the affine open subsets,
and to verify the coherence property for those subsets. Since the affine open sets form a good basis, Theorem
6.3.5 tells us that the module is determined. This is the standard procedure for defining an O-module. In fact,
if {U i } is an affine open covering of X, the modules Mi = M(U i ) determine M. However, one can’t assign
those modules arbitrarily because the open subsets U i intersect. If V is an affine open set that is a localization
of U i and of U j , say V = Usi and V = Usj′ , the localized modules (Mi )s and (Mj )s′ are isomorphic.
moduleonponeagain
6.3.8. Example. (modules on the projective line) We go back to the notation of Example 6.2.20. The two
standard affine open sets U0 , U1 and their intersection U01 are spectra of the rings A0 = C[t], A1 = C[t−1 ],
and A01 = C[t, t−1 ]. Let M be a quasicoherent O-module on P1 , and let M0 and M1 and M01 be the modules
of sections on these open sets. The coherence property gives us isomorphisms
M0 [t−1 ] ≈ M01 ≈ M1 [t]
The open subsets of P1 that are contained in U0 or in U1 are affine, and they form a good basis B. If M0 and
M1 are modules over A0 and A1 , respectively, and if an isomorphism M0 [t−1 ] ≈ M1 [t] is given, one can use
the coherence property to define an O-module M.
For instance, suppose that M0 and M1 are free modules of rank r over A0 and A1 . Then M01 will be a free
A01 -module of rank r. A basis B0 of M0 is also a basis of the A01 -module M01 , and a basis B1 of M1 is also
a basis of M01 . As bases of M01 , B0 and B1 will be related by an invertible A01 -matrix P , and this matrix
determines the module up to isomorphism.
When r = 1, P will be an invertible 1 × 1 matrix in the Laurent polynomial ring A01 = C[t, t−1 ], a unit
of that ring. The units are scalar multiples of powers of t, and the scalar can be absorbed into one of the bases.
Therefore quasicoherent O-modules of rank 1 are determined, up to isomorphism, by powers of t. They are
the twisting modules on P1 .
modulemeanscoherentsheaf
6.3.9. Terminology. There are O-modules that aren’t quasicoherent, but they are much less important, and
we won’t use them. To simplify terminology, and because the adjective “quasicoherent” is ugly, we will, from
now on, refer to a quasicoherent O-module simply as an O-module. A finite O-module M is one such that
M(U ) is a finite O(U )-module for every affine open subset U of X. Elsewhere, finite O-modules are called
coherent sheaves.
8
finiteoncover
6.3.10. Proposition. Let {U i } be an affine open covering of a variety X, let M be an O-module, and let
Ai = O(U i ). If Mi = M(U i ) is a finite Ai -module for every i, then M is a finite O-module.
The proof is in Section 6.7.
exactseq
6.4
Homomorphisms
ϕ
As above, a homomorphism M −→ N of O-modules consists of homomorphisms of O(U )-modules
ϕ(U )
M(U ) −→ N (U )
j
for every open set U , that are compatible with restriction: If V −→ U is an inclusion of open sets, the diagram
below commutes:
ϕ(U )
hommod
(6.4.1)
M(U ) −−−−→ N (U )




j∗y
j∗y
ϕ(V )
M(V ) −−−−→ N (V )
m
For example, if m is a global section of an O-module M, the map O −→ M that sends a regular function
f
α to αm is an example of a homomorphism. In particular, the map O −→ O(d) of multiplication by a
homogeneous polynomial f of degree d is a homomorphism.
A sequence of homomorphisms
exseqsheaves
M→N →P
(6.4.2)
of O-modules on a variety X is exact if the sequence of sections
exactsections
(6.4.3)
M(U ) → N (U ) → P(U )
is exact for every affine open subset U of X. This sequence isn’t required to be exact for open subsets that
aren’t affine. However, the next theorem shows that the section functor M
M(U ) is always left exact. The
proof will be given in Section 6.7.
leftexactsection
6.4.4. Theorem. Let X be a variety, and let
0→M→N →P
be an exact sequence of O-modules. For every open subset U of X, the sequence of sections
0 → M(U ) → N (U ) → P(U )
is exact.
When
0→M→N →P→0
is a short exact sequence of O-modules, the sequence of global sections
0 → M(X) → N (X) → P(X)
is often not exact when a zero is added on the right. Cohomology, the tool for analyzing the failure of exactness,
will be discussed in Chapter 7.
9
cokerandimage
6.4.5. Example. Let X be the projective line with coordinates x0 , x1 , and consider the sequence of homomorphisms
u
v
0 → O(−2) −→ O(−1) ⊕ O(−1) −→ O → 0
where u sends a section s of O(−2) to x0 s ⊕ x1 s and v sends a section a ⊕ b of O(−1) ⊕ O(−1) to
x1 a − x0 b . In matrix notation, u = (x0 , x1 )t and v = (x1 , −x0 ). You will be able to verify that this sequence
is exact. Hence O can be viewed as the cokernel of u, and also as the image of v. Since O(−1) has no nonzero
global section, while O(X) = C, v does not define a surjective map of global sections.
ϕ
The kernel K, the cokernel C, and the image I of a homomorphism M −→ N are defined by the standard
procedure: looking on affine open sets. If U is an affine open set, then K(U ), C(U ), and I(U ) are the kernel,
cokernel, and image of the homomorphism M(U ) → N (U ). The coherence property carries over because
localization is an exact operation, and because we have Theorem 6.3.1.
kernelsheaf
6.4.6. Corollary. Let K be the kernel of a homomorphism of O-modules M → N . For every open set U ,
affine or not, K(U ) is the kernel of the map M(U ) → N (U ).
The analogous statements for the cokernel and image are false.
ϕ
A homomorphism of O-modules M −→ N is injective if its kernel is zero, which means that the map
ϕ(U )
M(U ) −→ N (U ) is injective for every open set U . On the other hand, ϕ is surjective if its cokernel is zero,
which means that ϕ(U ) is surjective for every affine open set U .
ϕ
If {U i } is an affine open cover of a variety X, a homomorphism M −→ N of OX -modules is surjective
if the maps M(U i ) → N (U i ) are surjective for every i.
quotmod
quotient modules
(6.4.7)
When M is a submodule of an O-module N , the inclusion M ⊂ N is a homomorphism. The quotient
module N /M is the cokernel of the inclusion M ⊂ N . For example, if mp is the maximal ideal of a point p
of a variety X, then O/mp is the residue field at p, which is made into an O-module κp in an obvious way: If
an open set U contains p, then κp (U ) is the residue field k(p), and if U doesn’t contain p, then κp (U ) = 0.
directimage
6.5
Direct Images
f
Let Y −→ X be a morphism of varieties. The direct image f∗ N of an OY -module N is the OX -module
defined as follows: If U is an open subset of X, then
def
[f∗ N ](U ) = N (f −1 U ).
The OX -module structure on f∗ N is obtained by restricting scalars. We spell this out:
Let V = f −1 U , and let A = OX (U ), B = OY (V ), and NB = N (V ). (The subscript B is a reminder
ρ
that N is a B-module.) The restriction of f to a morphism V → U gives us a homomorphism A −→ B, and
we make NB into an A-module NA by restricting scalars from B to A. Then [f∗ N ](U ) = NA . As abelian
groups, NB and NA are the same. So if α is in A and n is in NA , then
def
αn = ρ(α)n
fstarcoh
f
6.5.1. Lemma. Let Y −→ X be a morphism of varieties. The direct image f∗ N of an OY -module N is an
OX -module.
Thus the direct image is a functor
fstarfunctor
(6.5.2)
f∗
(OY −modules) −→ (OX −modules) .
proof of Lemma 6.5.1. It is easy to show that f∗ is a functor. We must check the sheaf and coherence properties.
10
i
−1 i
the sheaf property: Let {U i } be a covering
of an open
Q
Q set Ui . Thej inverse images V = f U cover V =
−1
i
f U . The sequence 0 → N (V ) → N (V ) −→ Q
N (V ∩ V ) described
Q in (6.7.3) is exact, and it is the
same sequence of abelian groups as 0 → [f∗ N ](U ) → [f∗ N ](U i ) −→ [f∗ N ](U i ∩ U j ).
the coherence property: With notation as above, let s be a nonzero element of A. The coherence property
asserts that [f∗ N ](Us ) is the localization (NA )s of the A-module NA . Let s′ be the image of s in B. There
are two cases: If s′ 6= 0, then Vs′ = f −1 Us . Since N is quasicoherent, N (Vs′ ) is the localization (NB )s′ , and
when we restrict scalars in, we obtain the localization (NA )s . Thus [f∗ N ](Us ) = N (Vs ) = (NB )s′ = (NA )s ,
as required. If s′ = 0, Vs′ is the empty set, and NB,s′ = N (Vs′ ) = 0. In this case, s annihilates NA , so
NA,s = 0 too.
extbyzero
(6.5.3)
extension by zero
i
Let Y −→ X be the inclusion of a closed subvariety into a variety X. The direct image i∗ N of an OY module N is called the extension by zero of N . If if U is an open subset of X, then since i is an inclusion
map, i−1 U = U ∩ Y . Therefore
[i∗ N ](U ) = N (U ∩ Y )
The term “extension by zero” refers to the fact that, if an open set U of X doesn’t meet Y , then U ∩ Y is the
empty set, and the module of sections of i∗ N on U is zero. I think of the extension by zero as an operation
that is essentially the identity. The space is the only thing that changes.
cokernotsheaf
6.5.4. Example. (the residue field) (i) Let p be a point of a variety X, with residue field k(p) (= C), and
let i denote the inclusion of p into X. A module on the one-point space p = Spec k(p) is simply a vector
space, and the residue field is such a module. We denote it by k(p) too. The extension by zero i∗ k(p) from the
closed set p is the O-module that was denoted by κp above (6.4.7). If an open subset U of X contains p, then
κp (U ) = k(p), and if U doesn’t contain p, κp (U ) = 0.
(ii) Let X be the projective line P1x . The maximal ideal mp fits into an exact sequence of O-modules
idealsheafatp
πp
0 → mp → O −→ κp → 0
(6.5.5)
The spaces of global sections of these O-modules happen to form a short exact sequence: mp (X) = 0, and
O(X) = κp (X) = C.
(iii) Let p and p′ be distinct points of X, and let κp and κp′ denote the extensions by zero of the residue fields
at these points. The intersection mp ∩ mp′ of the maximal ideals fits into an analogous exact sequence of
O-modules
idealsheafatpq
π
0 → mp ∩ mp′ → O −→ κp ⊕ κp′ → 0.
(6.5.6)
To see that π is surjective, we cover X by the open sets U = X − p′ and U ′ = X − p, both affine lines.
It suffices to show that the restriction of π to these affine open sets is surjective (??), and this is true. On U ,
π = πp , and on U ′ , π = πp′ . The sequence of global sections of (6.5.6) is left exact:
0→0→C→C⊕C
It doesn’t remain exact with a zero on the right.
izero
i
6.5.7. Proposition Let Y −→ X be the inclusion of a closed subvariety Y into a variety X, and let I ⊂ OX
be the ideal of Y . An OX -module M is isomorphic to the direct image i∗ N of an OY -module if and only if I
annih8ilates M.
proof. It suffices to prove this when X is an affine variety, say X = Spec A. Let I = I(X), so that
Y = Spec B, where B = A/I, and let M = M(X). The assertion becomes this: An A-module M has the
structure of a B-module if and only if IM = 0.
11
includeopen
(6.5.8) inclusion of an open subvariety
f
When Y −→ X is the inclusion of an open subvariety into X, there is a simple operation that goes in the
opposite direction from the direct image. With notation as above, let M be an OX -module. An open subset V
of Y is also open in X, so M(V ) is defined. An OY -module f ∗ M called the inverse image is defined by
[f ∗ M](V ) = M(V ).
Thus the OY -module f ∗ M is obtained from M by looking only at the open sets that are contained in Y . The
sheaf and coherence properties are obvious.
The inverse image operation seems to be trivial, but the next example shows that it is interesting.
fstarforloc
6.5.9. Example. Let X be the affine variety Spec A, let Y be a localization Y = Xs = Spec As , and let
f
Y −→ X be the inclusion map. The OX -module M is determined by its global sections, the A-module
M = M(X), and an OY -module N is determined by its global sections, an As -module N = N (Y ).
Here [f∗ N ](X) = NA , is the module obtained from the As -module N by restricting scalars to A, while
[f ∗ M](Y ) = [f∗ N ](Xs ) = M(Xs ) is the localization Ms of M . Thus [f∗ f ∗ ]M(X) is the A-module
obtained from the localization Ms by restricting scalars to A.
One can describe f∗ f ∗ M as the OX -module obtained from M by allowing coefficients that have poles on the
complement of Y .
maptofstaffstar
f
6.5.10. Proposition. Let Y −→ X be the inclusion of an open subvariety into a variety X, let M be an
OX -module, and let N be an OY -module.
(i) There is a canonical homomorphism of OX -modules M → f∗ f ∗ M.
(ii) The OY -modules f ∗ f∗ N ≈ N and N are canonically isomorphic.
(iii) The functors f∗ and f ∗ are adjoint: Homomorphisms of OY -modules f ∗ M → N correspond bijectively
to homomorphisms of OX -modules M → f∗ N .
proof. (i) We must give homomorphisms of OX (U )-modules M(U ) → [f∗ f ∗ M](U ) for every open subset
U of X, compatibly with restrictions to smaller open sets. Let V = U ∩ Y (= f −1 U ). Then V ⊂ U , so we
have a restriction map M(U ) → M(V ). The required map is the composition
maptofstarfstar
def
def
M(U )−→M(V ) = [f ∗ M](V ) = [f∗ f ∗ M](U )
(6.5.11)
(ii) Let V be an open subset of Y . Then
def
def
[f ∗ f∗ N ](V ) = [f∗ N ](V ) = N (V ∩ Y ) = N (V )
ϕ
(iii) Suppose a homomorhism M −→ f∗ N is given. If U is an open subset of X, then [f∗ N ](U ) = N (U ∩Y ),
so ϕ defines a map M(U ) → N (U ∩ Y ). Then if V is open in Y , ϕ give us a map [f ∗ M](V ) = M(V ) →
N (V ∩ Y ) = N (V ). This is the homomorphism ψ on V . Conversely, suppose that a homomorphism
ψ
f ∗ M −→ N is given. If U is an open subset of X, we need to find a map M(U ) → f∗ N (U ). Let
V = U ∩ Y . So V is a subset of U . The required map is the composition
ψ
M(U ) → M(V ) = f ∗ M(V ) −→ N (V ) = [f∗ N ](U )
exampledirectimage
6.5.12. Example. Let X = Pn , let f denote the inclusion U0 ⊂ X of the standard affine open subset into X.
The direct image f∗ OU0 of the structure sheaf on Y is an OX -module whose sections are rational functions
that are allowed to have poles along the hyperplane H at infinity. For example, the sections of f∗ OU0 on the
standard affine U1 : {x1 6= 0} are the rational functions that are regular on U0 ∩U1 . They are the homogeneous
fractions of degree zero of the form g(x)/xi0 xj1 .
12
Note. An inverse image functor f ∗ that is adjoint to f∗ can be defined for any morphism of varieties, but its
description involves a limit and a tensor product. We’ll use the inverse image only for the trivial case of an
inclusion of an open subvariety.
directlimits
(6.5.13)
direct limits
A directed set M• is a a sequence of maps
dirset
M0 → M1 → M2 → · · ·
(6.5.14)
It could be a sequence of maps of sets, but we’ll be interested in directed sets of rings or modules.
S
A directed set M• has a direct limit lim
M• , which is a set of equivalence classes on the union Mk .
−
→
Elements m ∈ Mr and m′ ∈ Ms are called equivalent if their images in Mn are equal for sufficiently large
n. So an element of −
lim
→ M• will be represented by an element of Mk for some k, and elements m ∈ Mr and
m′ ∈ Ms represent the same element of the limit if their images in Mn are equal for large n.
examplelimit
6.5.15. Example. An increasing family M0 ⊂ M1 ⊂ · · · of submodules of an R-module S is a directed
S
set whose limit is the union Mk . The quotients S n = S/MnSof an increasing family of submodules form a
directed set whose maps are surjective, and whose limit is S/( Mk ).
limitmodule
6.5.16. Lemma Let (6.5.14) be a directed of rings or of modules over a ring A. With the structure inherited
from the modules Mn , the direct limit lim
−→ M• is a ring or a module.
ϕ
ϕn
A homomorphism M• −→ M•′ of directed sets of R-modules consists of homomorphisms Mn −→ Mn′
for each n that make a (commutative) diagram
· · · −−−−→ Mn −−−−→ Mn+1 −−−−→ · · ·




ϕn+1 y
ϕn y
′
−−−−→ · · ·
· · · −−−−→ Mn′ −−−−→ Mn+1
A sequence → M• → M•′ → M•′′ → of such homomorphisms is exact if the sequence → Mn → Mn′ →
Mn′′ → is exact for every n.
limitmodule
6.5.17. Proposition (i) A homomorphism of directed sets induces a homomorphism of limits.
(ii) If a sequence M• → M•′ → M•′′ of homomorphisms of directed sets is exact, then the limit sequence
′′
′
lim
−→ M• is exact.
−→ M• → lim
−→ M• → lim
(iii) If M• is a directed set of R-modules and N is another R-module, then lim
−→ (M• ⊗R N ) is canonically
isomorphic to (lim
−→ M• ) ⊗R N .
Let X be a variety, and let M• = {M0 → M1 → M2 → · · · } be a directed set of OX -modules. The
limit lim
−→ M• is defined by [lim
−→ M• ](U ) = lim
−→ [M• (U )], for every open set U .
limitsheafagain
6.5.18. Proposition. Let M• be a directed set of O-modules.
(i) The direct limit M• is an O-module.
(ii) Direct and inverse images are compatible with limits: lim
(f∗ M• ) ≈ f∗ (lim
M• ) and lim
(f ∗ M• ) ≈
−
→
−
→
−
→
f ∗ (lim
−→ M• ).
(iii) If N is an O-module, then (lim
−→ M• ) ⊗O N is isomorphic to lim
−→ (M• ⊗O N )
13
twistmodule
tensorproduct
6.6
(6.6.1)
Twisting
tensor products
Tensor products of modules over a ring are defined in the background chapter. We have supressed their use
so far, but now is the time to learn the definition.
Tensor products are compatible with localization: If M and N are modules over a domain A and s is a
nonzero element of A, then (M ⊗A N )s and Ms ⊗As Ns are isomorphic. This allows us to extend tensor
products to O-modules. The tensor product is defined by the standard procedure. If U is an affine open set,
then
def
[M ⊗O N ](U ) = M(U ) ⊗O(U ) N (U )
Because this tensor product is over the structure sheaf O, functions move through the tensor symbol: If α is a
function, then mα ⊗ n = m ⊗ αn.
Various canonical isomorphisms, such as the isomorphism M ⊗O O ≈ M given by m ⊗ 1
over from A-modules to O-modules.
deftensor
m, carry
6.6.2. Lemma. For any open set V there is a canonical map of O(V )-modules
ǫ
M(V ) ⊗O(V ) N (V ) −→ [M ⊗O N ](V )
proof. Let {V i } be an affine covering of V . We need some notation: Let Ri = O(U i ), Mi = M(U i ),
Ni = N (U i ), and T i = [M ⊗O N ](U i ), and let Rij , Mij , Nij and Tij be defined analogously, for sections
on the intersections U i ∩ U j . Then
M(V i ) ⊗O(V i ) N (V i ) ≈ Mi ⊗ri Ni ≈ Ti
We form a diagram
0 −−−−→
M(V ) ⊗O(V ) N (V ) −−−−→


ǫy
[M ⊗O N ](V )
Q
−−−−→
Mi ⊗Ri Ni −−−−→
Q
Ti
−−−−→
Q
Mij ⊗Rij Nij
Q
Tij
in which the bottom row is exact. The map ǫ is induced from the rest of the diagram.
By definition of the tensor product, ǫ is an isomorphism when V is affine. When V isn’t affine, the map needn’t
be injective or surjective.
tensornotisom
Example. The multiplication map O(r)×O(s) → O(r + s) of twisting modules on Pn induces a homomorϕ
phism O(r) ⊗O O(s) −→ O(r + s). On the standard affine open set U0 , the sections of O(r) ⊗O O(s) form
a free module with basis xr0 ⊗ xs0 , and this element maps to the basis x0r+s of O(r + s) on U0 . So ϕ is an
isomorphism.
Let X be the projective line. When k > 0, The monomials of degree k form a basis for the space of global
sections of O(k). So [O(1)](X) has dimension 2. Therefore [O(1)](X) ⊗O(X) [O(1)](X) has dimension 4,
whereas the space of sections of the tensor product O(1) ⊗O O(1) ≈ O(2) has dimension 3. The map ǫ isn’t
injective when V = X and r, s = 1, 1. When r, s = −1, 1, ϕ isn’t surjective.
Let M be an O-module on projective space, and let O(k) denote the twisting sheaf on P. The kth twist of
M is the tensor product
twist
(6.6.3)
M(k) = M ⊗O O(k)
If M is an O-module on a projective variety X, the twist M(n) is the OX -module obtained by twisting its
extension by zero to P. (see ??).
14
imageoff
6.6.4. Lemma. Let C be the hypersurface in projective space that is defined by an irreducible homogeneous
polynomial f , let I be the ideal of C, and let M be an OP -module.
(i) Multiplication by f defines a map
1⊗f
M(k) −→ M(k + d)
1⊗f
(ii) The image of the map M(−d) −→ M is the submodule MI of M.
See (6.2.13)) and (6.2.15).
When possible, we often abbreviate by writing f instead of a ⊗ f . For example, multiplication by xd0 =
1 ⊗ xd0 defines a homomorphism
twistmuzero
xd
0
M(k) −→
M(k + d)
(6.6.5)
This homomorphism is an isomorphism on U0 , where the sections of O(d) form a free module with basis xd0
xd
1
on U0 . As for the structure sheaf, this isomorphism is related to the homomorphism M(k) −→
M(k + d) by
d
the factor (x0 /x1 ) .
generators
glsechomom
(6.6.6)
generating a module
6.6.7. Lemma. Let M be an O-module on a variety X. Global sections of M correspond bijectively to
homomorphisms O → M. If m is a global section, the corresponding homomorphism sends a section α of O
ϕ
on an open set U to the section αm of M on U . In the other direction, if O −→ M is a homomorphism, the
corresponding global section of M is ϕ(1).
A set m = (m1 , ..., mk ) of global sections generates M if the map of O-modules
gen
m
Ok −→ M
(6.6.8)
that sends (α1 , ..., αk )
α1 m1 + · · · + αk mk is surjective. If the sections m generate M, then they generate
the O(U )-module M(U ) for every affine open set U . They may not generate when U isn’t affine, as example
(6.5.6) shows. Futhermore, there may not be enough global sections to generate. For instance, it is possible
that the only global section is zero.
Let X be the projective space Pd . The twisting modules O(n) = OX (n) on X can be assembled into a
directed set by repeated multiplication by x0 :
Oninclusions
x
x
x
0
0
0
···
O(2) −→
O(1) −→
O −→
(6.6.9)
and multiplication by 1 ⊗ x0 defines a directed set
cmnsequence
x
x
x
0
0
0
···
M(2) −→
M(1) −→
M −→
(6.6.10)
We’ll denote the limits of the sequences (6.6.9) and (6.6.10) by lim
−→ x0 M(n).
−→ x0 O(n) and lim
Onsequence
6.6.11. Proposition. Let j denote the inclusion of the standard affine open set U0 into Pd and let M be an
∗
O-module. Then lim
−→ x0 M(n) is isomorphic to j∗ j M.
proof. If V is an affine open set, then [j∗ j ∗ M](V ) = [j ∗ M](V ∩ U0 ) = M(V ∩ U0 ). The sections of
M(n) = M ⊗O O(n) are sums of tensors m ⊗ u, where m is in M(V ) and u is a homogeneous fraction of
x−n
0
∗
degree n, regular on V . Multiplication by x−n
0 defines a map M(n) −→ j∗ j M that sends m ⊗ u to m ⊗ α,
−n
0
where α = ux0 , which is a regular function on V ∩ U . This gives us a homomorpism of directed sets
x
x
=
=
x
−−−0−→ M(n) −−−0−→ M(n + 1) −−−0−→




x0−n−1 y
x−n
0 y
−−−−→ j∗ j ∗ M −−−−→
15
j∗ j ∗ M
=
−−−−→
which defines a map
ǫ
∗
lim
−→ x0 M(n) −→ j∗ j M
To show that ǫ is an isomorphism, it suffices to show that ǫ is bijective on sections on the standard affine open
sets Ui . This follows from Corollary ??.
see Proposition 6.3.10). On U0 , M(n) and j∗ j ∗ M are isomorphic to M, and multiplication by x0 is
a bijective map. So ǫ bijective on U0 . We look at sections on U1 , where [j∗ j ∗ M](U1 ) = M(U01 ). Let
A = O(U1 ) and M = M(U1 ). The coherence condition for M tells us that M(U01 ) is the localization
Ms where s = x0 /x1 , and Ms ≈ M ⊗A As . Recall that [O(n)](U1 ) is a free A-module with basis xn1 .
sends m ⊗ xn1
So the elements of [M(n)](U1 ) have the form m ⊗ xn1 with m in M . Multiplication by x−n
0
−n
−n
to m ⊗ s . This is the element of M ⊗A As that corresponds to ms
in Ms . It follows that the map
∗
1
−n
k
M(n)
→
j
j
M
is
surjective
on
U
.
If
m
⊗
s
=
0,
some
power
s
of s annihilates m. In that case
lim
∗
x
0
−→
the image of m ⊗ xn0 in [M(n + k)](U1 ) is m ⊗ xn1 xk0 = m ⊗ sk x1n+k = msk ⊗ x1n+k = 0. So m ⊗ xn1 is
1
also zero in lim
−→ x0 M(n). The map ǫ is bijective on U .
genposexact
6.6.12. Corollary. Let j denote the inclusion of the standard affine open set U0 into Pd .
(i) For any O-module M, j∗ j ∗ M is generated by global sections.
(ii) Let M be a finite O-module on a projective variety X. For large n, the twist M(n) is generated by global
sections.
∗
proof. (i) For any n ≥ 0, O(n) is generated by the global sections xn0 , ..., xnd . Since lim
−→ x0 O(n) = j∗ j O,
∗
0
j∗ j O is generated by global sections. Next, M = M(U ) is a module over the polynomial algebra A =
O(U0 ). We choose generators for M , obtaining a surjective map AN → M , where N may be infinite, and
we show that the induced map (j∗ j ∗ O)N → j∗ j ∗ M is surjective. Then since j∗ j ∗ O is generated by global
sections, so is j∗ j ∗ M.
Let V be an affine open subset of X. Because V and U0 are affine, so is V ∩ U0 (3.5.11). By definition,
[j∗ j ∗ O](V ) = O(V ∩ U0 ) and [j∗ j ∗ M](V ) = M(V ∩ U0 ). Because V ∩ U0 is affine, the fact that the map
AN → M is surjective implies that the restriction of the map ON → M to U0 is surjective, and therefore that
O(V ∩ U0 )N → M(V ∩ U0 ) is surjective. So [j∗ j ∗ O](V )N → [j∗ j ∗ M](V ) is surjective, as required.
(ii) We may assume that X = Pd . We look on the standard affine open set U0 , where M = M(U0 ) is a finite
module over A = O(U0 ). Let x1 , ..., xk be module generators for M . These generators are also sections of
(
lim
j∗ j ∗ M, and therefore they define a map Ok x) −→ j∗ j ∗ M. Then since j∗ j ∗ M = −
→ x0 M(n), xi will be in
(
M(n) for any sufficiently large n, and this will define a map Ok x) −→ M(n). Let Cn denote the cokernel
of this map. Because the elements xi generate the A-module M , and because M(n)(U0 ) = M(U0 ) = M ,
the support of Cn is contained in the hyperplane H 0 at infinity.
So for large n, the global sections of M(n) generate a submodule S large enough so that the cokernel
M(n)/S = C\ has support in H 0 . The same reasoning shows that, increasing n as necessary, the support of
Cn will be contained in the hyperplane H i for any i. Since the intersection of the hyperplanes H i is empty,
M(n) is generated by global sections.
6.7
Proofs
modproofs
We prove the theorems that were stated without proof earlier. One reason that we deferred some proofs to
the end of the chapter is that the notation gets complicated. It will help to introduce a symbol such as U to
represent a family {U i } of open sets. A morphism from one family V = {V ν } of open sets to another family
U = {U i } consists of a morphism (an inclusion), of each V ν into one of the subsets U i . Such a morphism will
be given by a map of index sets ν
i(ν), such that V ν is contained in U i(ν) . There may be more than one
morphism V → U, because a subset V ν may be contained in more than one subset U i . To define a morphism,
one must make a choice among those subsets.
We extend an O-module to families of open sets by defining
sectionsonfamily
(6.7.1)
M(U) =
Y
16
M(U i ).
f∗
f
Then a morphism V −→ U of families defines a map M(U) −→ M(V) in a way that is fairly obvious, though
our notation for it is clumsy. A section of M(U) can be thought of as a vector (ui ) with ui ∈ M(U i ). Its
restriction to V is (vν ), where vν is the restriction of ui(ν) to V ν , i(ν) being the map of indices as above.
Let U0 denote a family {U i } of open subsets that covers another open set U , and let U1 denote the family
of intersections {U i ∩ U j }, indexed by ordered pairs i, j of indices. The inclusions U i ∩ U j → U j and
U i ∩ U j → U i define morphisms of families U1 ⇒ U0 . We call the diagram that we obtain,
covdiagram
U ←− U0 ⇔ U1
(6.7.2)
a covering diagram.
With this notation, the exact sequence (6.2.17) that expresses the sheaf property for the covering becomes
sheafone
(6.7.3)
β
α
0 → M(U ) −→ M(U0 ) −→ M(U1 )
in which the maps α and β are as in (6.2.17).
Proof of Theorem 6.3.1 (about the coherence property)
We are given an O-module M that has the sheaf property, and that has the coherence property for the open
sets U i of an affine covering of the variety X. We must prove the coherence property for an arbitrary open set
V . Since the coherence property is true on U i , it is also true on any localization of U i , and we can cover V
by such localizations, say by the affine varieties V ν = Spec Aν . We relabel V as X and {V ν } as {U i }. The
statement to be proved becomes this: Suppose that the coherence property for M is true for the open sets of
an affine covering U0 = {U i } of X. If O(X) = R and M(X) is the R-module M , then for every nonzero
element s of R, M(Xs ) is the localization Ms . In particular, O(Xs ) = Rs . What is clear here is that, since s
doesn’t vanish on Xs , its inverse is in O(Xs ). Therefore Rs ⊂ O(Xs ).
Let U1 denote the family of intersections U i ∩ U j . Since M has the sheaf property, the sequence
covoneagain
(6.7.4)
β
α
0 → M(X) −→ M(U0 ) −→ M(U1 )
is exact. We may regard this as a sequence of R-modules by restriction of scalars. Since localization is an
exact operation, the localized sequence of Rs -modules
covtwo
(6.7.5)
0 → M(X)s → M(U0 )s → M(U1 )s
is exact. The family of localizations U0,s = {Usi } is an affine covering of Xs , and U1,s is the family of
intersections {Usi ∩ Usj }. Then
covthree
(6.7.6)
0 → M(Xs ) → M(U0,s ) → M(U1,s )
is an exact sequence that can be made into a sequence of Rs -modules by restrictcion of scalars. Since M is a
functor, (6.7.4) maps to (6.7.6), and since (6.7.6) is a sequence ofd RS -modules, the sequence (6.7.5) maps to
(6.7.6):
0 −−−−→ M(X)s −−−−→ M(U0 )s −−−−→ M(U1 )s






cy
ay
by
0 −−−−→ M(Xs ) −−−−→ M(U0,s ) −−−−→ M(U1,s )
The coherence property for the covering U0 implies that the map b is bijective. Therefore the map a is injective,
and since X was arbitrary, this is also true for every open subset V of X. Therefore the map c is injective.
Then it follows from the diagram that a is bijective, as the coherence property asserts.
Proof of Theorem 6.3.5 (extending a module given on a basis)
We are given a good basis B for the topology on the variety X, and a quasicoherent B-module. We must
extend M to an O-module on X. The proof isn’t difficult, but there are several things to check.
We call an open set that is a member of B a B-open set, and a covering of any open set by B-open sets a
B-covering.
17
f though
Let M be a B-module. In this proof, we denote the O-module that is to be constructed by M,
when we are done, we will use the same symbol for a B-module and for the O-module on X that it determines.
f that we are looking for must
Suppose given a B-covering U0 = {U i } of an open subset Y . The O-module M
f
f
f becomes the
have the property that M(U0 ) = M(U0 ) and M(U1 ) = M(U1 ). So the sheaf property for M
exactness of the sequence
defm
(6.7.7)
β
α
f ) −→ M(U0 ) −→ M(U1 )
0 → M(Y
f ) as a kernel, but there will be many B-coverings of a given open set. We show first that
This identifies M(Y
all such kernels are isomorphic.
kersisom
6.7.8. Lemma. Let U0 = {U i } and V0 = {V ν } be B-coverings of an open set Y . There is a canonical
isomorphism
β
β′
ker M(U0 ) −→ M(U1 ) ≈ ker M(V0 ) −→ M(V1 ) .
proof. Let U0 V0 denote the family of intersections {U i ∩ V ν }, indexed by pairs i, ν of indices, let U1 V0
denote the family {U i ∩ U j ∩ V ν }, indexed by triples i, j, ν of indices, and let U0 V1 be defined analogously.
We form a diagram
(The arrows with an asterisk in the diagram below are supposed to be double arrows.)
comdiag
(6.7.9)
U1 ←−−−− U1 V0


∗
∗
y
y
∗
U0 ←−−−− U0 V0 ←−−−− U0 V1






y
y
y
Y ←−−−−
V0
∗
←−−−−
V1
Except for Y , all sets in this diagram are families of B-open sets. The bottom row and the left column
are covering diagrams, the middle row and middle column are families of covering diagrams, and the top
horizontal arrow and the right vertical arrow are families of coverings.
We apply the B-module to the diagram, omitting Y :
0 −−−−→ M(U1 ) −−−−→ M(U1 V0 )
x
x

β


comdtwo
(6.7.10)
0 −−−−→ M(U0 ) −−−−→ M(U0 V0 ) −−−−→ M(U0 V1 ))
x
x




M(V0 )
x


0
β′
−−−−→
M(V1 )
x


0
The sheaf axiom for the B-module M implies that all rows and all columns are exact. (Products of exact
sequences are exact.) It follows that the kernels of β and β ′ are canonically isomorphic.
f ) as the kernel of any one of the homomorphisms M(U0 ) → M(U1 ).
The lemma allows us to define M(Y
f ) next. Let a and m be sections O and M
f on Y , and let their
We define the O(Y )-module structure on M(Y
images in O(U0 ) and M(U0 ) be denoted by the same letters. Then am is an element of M(U0 ). Since β(m)
f ). This is the product section. The axioms for a
is zero, so is β(am). Therefore am is an element of M(Y
module are easy to verify.
18
Next, if Y ′ ⊂ Y are open sets, there will be B-coverings U′0 of Y ′ and U0 of Y that form a diagram
U′0 −−−−→


y
U0


y
Y ′ −−−−→ Y
This diagram will give us a module homomorphism ker M(U0 ) → M(U1 ) → ker M(U′0 ) → M(U′1 ) ,
f will be a functor.
so M
f ) for every Y , we must verify the sheaf property for a covering V0 = {V ν } of Y in
Having defined M(Y
which the open sets aren’t necessarily B-open. The coherence property will follow from Theorem 6.3.1. To
verify the sheaf property, we cover each V ν by B-open sets W ν,i , where the index set {i} depends on ν. Taken
together, the family W0 of all of these sets forms a B-covering of Y that fits into a diagram
comdiagthree
(6.7.11)
Y ←−−−−
∗
W0 ←−−−−


y
∗
W1


y
Y ←−−−− V0 ←−−−− V1
f gives us a diagram
in which the vertical arrows are families of coverings. Applying M
comdiagtwo
(6.7.12)
f ) −−−−→ M(W0 ) −−−−→ M(W1 )
0 −−−−→ M(Y
x
x




β
f ) −−−−→ M(V
f 0 ) −−−
f 1)
M(Y
−→ M(V
f ) is the kernel of
In this diagram, the top row is exact and the vertical arrows are injective. The fact that M(Y
β follows.
It is also true that homomorphisms of B-modules and of O-modules correspond bijectively, but we omit
the verification.
Proof of Theorem 6.3.6 (the sheaf property for an A-module)
We are given a module M over a ring A and nonzero elements s1 , ..., sk of A that generate the unit ideal,
and we are to prove that when we denote by Mi and Mij the localizations Msi and Msi sj , respectively, the
sequence
Y
Y
0→M →
Mi →
Mij
is exact.
In this sequence, the map α sends an element m of M to the vector (m, ..., m) of its images in M1 × · · · ×
Mk , and β sends a vector (m1 , ..., mk ) in M1 × · · · × Mk to the matrix (zij ), with zij = mj − mi in Mij .
exactness at M : Suppose that an element m of M maps to zero in every Mi . Then there exists an n such that
sni m = 0, and we can use the
n for all i. The elements sni generate the unit ideal. Writing
P
Psamenlarge exponent
P
n
ai si = 1, we have m =
ai s i m =
ai 0 = 0.
Q
exactness at Mi : Suppose given elements mi of Mi and that mi = mj in Mij for all i, j. We want to
find an element w in M such that w = mi in Mi for every i. We write mi = s−n
i xi with xi in M , using
the same integer n for all i. The equation mi = mj in Mij tells us that snj xi = sni xj is true in Mij , and
then (si sj )r snj xi = (si sj )r sni xj will be true in M if r is large. We adjust the notation. Let x
ei = sri xi , and
r+n
sei = si . Then
ei and sej x
ei = sei x
ej
xi = se−1
i x
in
PM . Since the elements
P si generate the unit ideal, so do the elements sei . There is an equation of the form
ai sei = 1. Let w =
ai x
ei . This is an element of M , and
X
X
x
ej =
ai sei x
ej =
ai sej x
ei = sej w.
i
i
19
ej = w is true in Mj . Since j is arbitrary, w is the required element of M .
Therefore mj = se−1
j x
Proof of Proposition 6.3.10 (about finite O-modules)
We are given an affine cover U i = Spec Ai of X and an O-module M such that Mi = M(U i ) is a
finite Ai -module for each i, and we are to show that M is a finite O-module. This means that for every affine
open set V = Spec A′ , M ′ = M(V ) is a finite A′ -module. Lemma 3.5.12 tells us that we can cover V
by localizations V j = Spec A′j , each of which is also a localization of one of the opens U i . The coherence
property shows that Mj′ = M(V j ) will be a localization of one of the finite modules Mi , and therefore it will
be a finite A′j -module. We relabel V as U and drop the primes. What must be proved is this:
Let U = Spec A and let M be an A-module. Let s1 , ..., sk be nonzero elements of A that generate the
unit ideal. With notation as in the previous proof, if Mi is a finite Ai -module for every i, then M is a finite
A-module.
We choose a finite set m = {m1 , ..., mr } of elements of M whose images generate the localization Mi for
P
ϕ
every i. These elements define a homomorphism Ar −→ M that sends (a1 , ..., ar ) to ai mi . Let C be the
cokernel of ϕ. Since m generates Mi and since localization is an exact operation, the localization Ci of C is
zero for every i. Theorem 6.3.6, applied to C, shows that C = 0.
Proof of Theorem 6.4.4 (left exactness of the section functor)
We are given an exact sequence of O-modules
0→M→N →P
We choose a covering diagram U ←− U0 ⇔ U1 in which U0 is a family of affine open subsets, and we inspect
the diagram
0
0
0






y
y
y
u
v
u′
v′
u′′
v ′′
0 −−−−→ M(U ) −−−−→ N (U ) −−−−→ P(U )






ay
a′′ y
a′ y
0 −−−−→ M(U0 ) −−−−→ N (U0 ) −−−−→ P(U0 )






by
b′′ y
b′ y
0 −−−−→ M(U1 ) −−−−→ N (U1 ) −−−−→ P(U1 )
The sheaf axiom asserts that the columns of this diagram are exact, and the middle row is exact because U0 is
a family of affines. We are to show that the top row is exact. The proof is a ‘diagram chase’. We present it,
though it is best to make such a verification oneself.
exactness at M(U ): Both u′ and a are injective, and u′ a = a′ u. So a′ u is injective, and this implies that u is
injective. Moreover, this statement is true for every open set U . Therefore u′′ is injective too.
exactness at N (U ): The elements to which we will refer are shown in this diagram:
u
z −−−−→


ay
u′
x


a′ y
y −−−−→ a′ (x)
Let x be an element in the kernel of v. We want to show that x = u(z) for some z in M(U ). Since x is in
the kernel of v, a′ (x) is in the kernel of v ′ . Since the middle row is exact, a′ (x) is the image of an element
y of M(U0 ): a′ (x) = u′ (y). We also know that b′ a′ = 0, so b′ u′ (y) = b′ a′ (x) = 0. Since b′ u′ = u′′ b, we
conclude that u′′ b(y) = 0. Since u′′ is injective, b(y) = 0. Since the first column is exact, y = a(z) for some
z in M(U ). Then a′ u(z) = u′ a(z) = u′ (y) = a′ (x). Since a′ is injective, u(z) = x.
20