Quasi-coherent modulesNotes 5—Sheaves of modules MAT4215 — Vår 2015 To pin down the quasi-coherent sheaves, one ﬁrst establishes a collection of model-sheaves on aﬃne schemes

Quasi-coherent modules

Warning: Very, very preliminary version. Version prone to errors.Version 0.1— last update: 5/26/15 11:12:42 AMWhen you study commutative algebra may be you are primarily interested in the rings,but probably you start turning your interest towards the modules pretty quickly; theyare an important part of the world of rings, and to get the results one wants, one canhardly do without them. The category Mod

A

of A-modules is a fundamental invariantof a ring A; and in fact, is the principal object of study in commutative algebra. So alsowith schemes; for which the so called quasi-coherent O

x

-modules form an importantattribute of the scheme, if not a decisive part of the structure. They form a categoryQCoh

X

with many properties parallelling those of the category Mod

A

—an in case theschemeX is affine, i.e., X = SpecA, they are equivalent. Imposing finiteness conditionson the O

X

-modules one arrives at the catergory Coh

X

of so called coherent OX

-modulesthat in the noetherian case parallel the finitely generated A-modules.

We start out by describing the much broader concept of an OX

-module. It is donefor schemes, but the concept is meaningfull for any ringed spaced.

In the literature one finds different approaches to the quasi-coherent sheaves. Wefollow Hartshorn and introduce the quasi-coherent module first for affine schemes. IfX = SpecA one defines an O

X

-module fM associated with an A-module M , and in

the general case these modules serve as the local models for the quasi-coherent ones.There is a notion of quasi-coherence for O

X

-modules on a general locally ringed space.In some other branches of mathematics they are important, e.g., analytic geometry,but we concentrate our efforts on schemes.

Sheaves of modules

A module over a ring is just an additive abelian group equipped with a multiplicativeaction of A. Loosely speaking we can multiply members of the module by elementsfrom the ring, and of course, the well known series of axioms must be satisfied. In a

1

Notes 5—Sheaves of modules MAT4215 — Vår 2015

similar way, if X is a scheme, an OX

-module is an abelian sheaf F whose sections canbe multiplied by sections of O

X

; the multiplicator and the multiplicand of course beingsections over the same open subset.

Formally, an OX

-module structure on the abelian sheaf F is defined as a family ofmultiplication maps �(U, F ) ⇥ �(U,O

X

) ! �(U, F )—one for each open subset U ofX—making the space of sections �(U, F ) into a �(U,O

X

)-module in a way compatiblewith all restrictions. That is, for every pair of open subsets V ✓U , the natural dia-gram below—where vertical arrows are made up of appropriate restrictions, and thehorizontal arrows are multiplications—commutes

�(U, F )⇥ �(U,OX

)

✏✏

// �(U, F )

✏✏

�(V, F )⇥ �(V,OX

) // �(V, F ).

Maps , or homomorphism, of OX

-modules are just maps ↵ : F ! G between OX

-modules considered as abelian sheaves respecting the multiplication by sections of O

X

.That is, for any open U the map ↵

U

: �(U, F ) ! �(U,G) is a �(U,OX

)-homomorphism.We have now explained what O

X

-modules are and told what their homomorphismsshould be, and this organizes the O

X

-modules into a category that we denote by Mod

X

.The category Mod

X

is an additive category : The sum of two OX

-homomorphismsas maps of abelian sheaves is again an O

X

-homomorphism. So for all F and G the setHomO

X

(F,G) is an abelian groups, and the compositions maps are bilinear. Moreover,the direct sum of two O

X

-modules as abelian sheaves has an obvious OX

-structure—multiplication being defined componentwise—and is as well the direct sum in the ca-tegory Mod

X

. Infact, this argument works for arbitary direct sums (or coproducts asthey also are called). For any family {F

i

} of OX

-modules,L

i2I Fi

is an OX

-module(seeexercise �.� below).

The notions of kernels, cokernels and images of OX

-module homomorphisms nowappear naturally. All the three corresponding abelian constructs are invariant undermultiplication by sections of O

X

, and therefore they have OX

-module structures. Therespective defining universal properties (in the category of O

X

-modules) come for free,and one easily checks that this makes Mod

X

an abelian category; i.e., the kernels ofthe cokernels equal the cokernels of the kernels.

There is tensor product of OX

-modules denoted by F ⌦OX

G. As in many othercases, the tensor product F ⌦O

X

G is defined by first describing a presheaf that sub-sequently is sheafified. The sections of the presheaf, temporarily denoted by F ⌦0

OX

G,are defined in the natural way by

�(U, F ⌦0O

X

G) = �(U, F )⌦�(U,OX

) �(U,G).

There is also a sheaf of OX

-homomorphisms between F and G. The definitiongoes along lines slightly different from the definition of the tensor product. Recallthe sheaf Hom (F,G) of homomorphisms between the abelian sheaves F and G whose

— 2 —


sections over an open set U is the group Hom(F |U

, G|U

) of homomorphisms betweenthe restrictions F |

U

and G|U

. Inside this group one has the subgroup of the maps beingO

X

-homomorphisms, and these subgroups, for different open sets U , are respected bythe restriction map. So they form the sections of a presheaf, that turns out to be asheaf, and that is the sheaf HomO

X

(F,G) of OX

-homomorphisms from F to G.

Problem �.�. Assume that F and G are OX

-modules and that ↵ : F ! G is a mapbetween them. Show that the kernel, cokernel and image of ↵ as a map of abeliansheaves indeed are O

X

-modules, and that they respectively are the kernel, cokerneland image of ↵ in the category of O

X

-modules as well. Show that a complex of OX

-modules is exact if and only it is exact as a complex of abelian sheaves. X

Problem �.�. Let F and G be two OX

-modules on the scheme X. Show that thestalk (F ⌦O

X

G)x

at the point x 2 X is naturally isomorphic to the tensor productF

x

⌦OX,x

G

x

of the stalks F

x

and G

x

. Show that the tensor product is right exact inthe category of O

X

-modules. X

Problem �.�. Show that the sheaf-hom HomOX

(F,G) of two OX

-modules as definedabove is a sheaf. Show that the stalk at a point x 2 X of the sheaf-hom HomO

X

(F,G)equals HomO

X,x

(Fx

, G

x

). Show that HomOX

(F,G) is right exact in the second variableand left exact in the first. X

Problem �.�. Show that the category Mod

X

has arbitrary products and coproducts,by showing that the products and coproducts in the category of abelian sheaves Sh

X

are OX

-modules and are the products, respectively the coproducts, in the categoryMod

X

. X

Example �.�. — A bunch of wild examples. The OX

-modules (or at least someof them) play a leading leading role in the theory of schemes, and shortly we shallsee a long series of examples. These will all be so called quasi-coherent sheaves. Theexamples we now describe are a bunch of wild examples, intended to show that O

X

-modules without any restrictive hypothesis are very general and often unmanageableobjects.

Recall the Godement construction from Notes1. Given any collection of abeliangroups {A

x

}x2X indexed by the points x of X. We defined a sheaf A whose sections

over an open subset U wasQ

x2U A

x

, and whose restriction maps to smaller open subsetswere just the projections onto the corresponding smaller products. Requiring that eachA

x

be a module over the stalk OX,x

makes A into an OX

-module; indeed, the space ofsections �(U,A) =

Qx2U A

x

is automatically an �(U,OX

)-module, the multiplicationbeing defined componentwise with the help of the stalk maps �(U,O

X

) ! OX,x

. Clearlythis module structures is compatible with the projections, and thus makes A into anO

X

-module. e

— 3 —


Example �.�. — Modules on spectra of DVR’s. Modules on the prime spectrumof a discrete valuation ring R are particularly easy to describe. Let K denote the fractionfield of R. The scheme X = SpecR has only two non-empty open sets, the whole spaceX it self and the singleton {⌘} where ⌘ denotes the generic point. The singleton {⌘} isunderlying set of the open subscheme SpecK. The data of an O

X

-module F consist ofan R module M , that is the space �(X,F ) of global sections of F , and a vector spaceN over K, being the space of sections �({⌘}, F ). The restriction map can be just anyR-module homomorphism M ! N .

This homomorphism can very well be zero, and in that case M and N can be com-pletely arbitrary modules. Again this illustrates the versatility of general O

X

-modules.e

Problem �.�. Assume that p1, . . . , pr is a set of primes, and let Z(pi

) as usual denote thelocalization at prime ideal (p

i

). Let X be the scheme obtained by glueing the schemesX

i

= SpecZ(pi

) together along their common open subschemes SpecQ. Describe theO

X

-modules on X. X

Problem �.�. Let A =Q

1in

K

i

be the product of finitely many fields. Describe thecategory Mod

X

. X

Quasi-coherent sheaves

In The Oxford English Dictionary there several nuances of the word coherent are given,but the one at top is:That sticks or clings firmly together; esp. united by the force of cohesion. Said of asubstance, material, or mass, as well as of separate parts, atoms, etc.

So the coherence of a sheaf should mean that there are some strong relations betweenthe sections over different open sets, at least over sets from some sufficiently largecollection of open sets. In our context the open affine subsets stand out as obviouscandidates to form such a collection, and indeed, a quasi-coherent sheaf on the schemeX will turn out1 to have the following coherence property: If U ✓V are two affine opensets in X there is a canonical map

�(U, F )⌦�(U,OX

) �(V,OV

) ! �(V, F ) (1)

sending s⌦ f to ⇢UV

(s) ·f , where ⇢UV

as usual indicates the restriction maps in OX

. Thesalient point is that for a quasi-coherent sheaf F this map is an isomorphism. In factthe converse holds true as well, so the map in (1) being an isomorphism for all pairsV ✓U of open affine sets is equivalent to F being quasi-coherent.

1One might have used this coherence property as the definition, but because of obscure reasons wechoose another definition.

— 4 —


To pin down the quasi-coherent sheaves, one first establishes a collection of model-sheaves on affine schemes. To each A-module M one associates an O

X

-module fM on

X = SpecA, and on a general scheme X, a quasi-coherent module F is going to beone that locally is of the form f

M .The coherent modules are the quasi-coherent modules that satisfy a certain finite-

ness condition that in the noetherian case boils down to the OX

-module being finitelygenerated. These coherent O

X

-modules are the OX

-modules most frequently encounte-red in algebraic geometry and hence they have gotten the shortest name, even thoughit is being quasi-coherent that means satisfying a coherence property.Coherent modules over a ring Let A be a ring and let M be an A-module.The module M is of finite presentation if for some integers n and m there is an exactsequence

A

n //A

m //M

// 0 .

One says that M is coherent if the following two requirements are fulfilled

⇤ M is finitely generated.

⇤ The kernel of every surjection A

n ! M is finitely generated.

Being coherent is the strongest of the the three properties, next comes being of finitepresentation and being finitely generated is the weakest. However, in the case A is anoetherian ring—which frequently is the case in algebraic geometry—a module M beingcoherent is equivalent to M being finitely generated. The notion of coherence comesfrom the theory analytic functions where coherent non-noetherian rings are frequent.

As an example of what coherence entails, we mention that the ring A itself is acoherent A-module if and only if all finitely generated ideals are of finite presentation.

Problem �.�. Show that last statement about A being coherent over itself X

Problem �.�. Show that if A is a noetherian ring, then the three conditions of cohe-rence, finitely generation and being of finite presentation on an A-module M coincide.

X

Quasi-coherent sheaves on affine schemes

For each A-module M we shall exhibit an OX

-module fM ; the construction of which

completely parallels what we did when constructing the structure sheaf OX

on X =SpecA. One defines a presheaf on X, temporarily denoted by M

0, simply by lettingthe sections over an open U be

�(U,M 0) = M ⌦A

�(U ;OX

). (2)

The restriction maps of M 0 are induced by the restriction maps of the structure sheaf

— 5 —


OX

. That is, if U ✓V are two open subsets and ⇢UV

: �(U,OX

) ! �(V,OX

) the restric-tion map of the structure sheaf, the restrictions of M 0 are simply be the maps id

M

⌦ ⇢

U

V

.This clearly yields an O

X

-module, multiplication being performed in the second factorof the tensor product. A priori the O

X

-module M

0 is only a presheaf, so we let fM be

the sheafification.For any A-module homomorphism ↵ : M ! N there is an obvious way of obtai-

ning an OX

-module homomorphism ↵ : fM ! eN ; indeed, the maps ↵⌦ id�(U ;O

X

) are�(U,O

X

)-homomorphisms from �(U,M 0) to �(U,N 0) compatibel with the restrictions,and thus induce a map between M

0 and N

0. The map ↵ is the associated map betweenthe sheafifications. Clearly one has ]

� � = e� � e

, and the “tilde-operation” is thereforea functor Mod

A

! ModOX

.The three main properties of f

M are listed below. They are completly analogous tothe statements in proposition ?? on page ?? in Notes2 about the structure sheaf O

X

;and as well, the proofs are mutatis mutandis the same.

⇤ Stalks: Let x 2 SpecA be a point whose corresponding prime ideal is p. The stalkfM

x

of M at x 2 X is fM

x

= Mp = M ⌦A

Ap.

⇤ Sections over distinguished open sets: If f 2 A one has �(D(f), fM) = M

f

=

M ⌦A

A

f

. In particular it holds true that �(X,

fM) = M .

⇤ Sections over arbitraty open sets. For any open subset U of SpecA covered bythe distinguished sets {D(f

i

)}i2I , there is an exact sequence

0 // �(U, fM) ↵ //Q

i

M

f

i

⇢

//Q

i,j

M

f

i

f

j

. (3)

Only the second of the three properties needs a separate proof, the two othersfollow formally. The first follows since both the stalks M

x

and the localizations Mp

are direct limits of the same modules over the same inductive system (indexed by thedistinguished open subsets D(f) containing x), and the third is just the general exactsequence expressing the space of sections of a sheaf over an open set in terms of thespace of sections over members of an open covering.

Lemma �.� For any two A-modules M and N , the association � ! � is an isomor-phism Hom

A

(M,N) ' HomOX

(fM,

eN).

In the lingo of category theory the lemma is expressed by saying that the tilde-operation is a fully faithful functor (trofast full funktor), fully meaning the map inthe lemma is surjective and faithful that it is injective. Loosely speaking, the “tilde-operation” gives an isomorphism of Mod

A

with a subcategory of ModOX

. It is a strictsubcategory; most of the O

X

-modules are not of the form fM , but those that are, form

the category of quasi-coherent OX

-modules, that we shortly return to.

— 6 —


Example �.�. The example of an discrete valuation ring is always useful to consider,and we continue exploring example �.� above. The O

X

module given by the dataM ! N is of the form f

M if and only if N = M ⌦R

K and the restriction map is thecanonical map M ! M ⌦

R

K sending m to m⌦ 1. e

Assume that an OX

-module F is given on X = SpecA, and let M denote theglobal sections of F ; that is, M is the A-module M = �(X,F ). There is a naturalmap f

M ! F of OX

-modules. By an appropriate glueing lemma—the one for mapsbetween sheaves—it suffices to tell what the map does to sections over members of abasis for the topology, e.g., over open distinguished sets. The sheaf F being an O

X

-module, multiplication by f

�1 in the space of sections �(D(f), F ) has a meaning since�(D(f),O

X

) = A

f

. Hence we may send the section mf

�n 2 M

f

of fM to the section of

F over D(f) obtained by multiplying the restriction of m to D(f) by f

�n; i.e., we sendm to f

�n · m|D(f) . For later reference we state this observation as a lemma, leaving

the task of checking the details to the zealous student:

Lemma �.� Given a quasi-coherent sheaf F on the affine scheme X = SpecA. Thenthere is a unique O

X

-module homomorphism ✓

F

: �(X,F ) ! F inducing the identityon the spaces of global sections.

Moreover, it is natural in the sense that if ↵ : F ! G is a map of OX

-moduleinducing the map a : �(X,F ) ! �(X,G) on global sections, one has ✓

G

� a = ↵ � ✓F

.

Problem �.�. Check that this is a well defined map (there is a choice involved in thedefinition). X

Lemma �.� In the canonical identification of the distinguished open subset D(f) withSpecA

f

, the OX

-module fM restricts to f

M

f

.

Proof: As �(D(f), fM) = M

f

, by the comment preceding the lemma, there is a mapfM

f

! fM |

D(f) that on distinguished open subsets D(g)✓D(f) induces an isomorphismbetween the two spaces of sections, both being equal to the localization M

g

. o

The “tilde-functor” is really a no-nonsense functor having almost all properties onecan desire. It is fully faithful, as we have seen, but it also is an additive as well asan exact functor. Moreover, it takes the tensor product M ⌦

A

N of two A-modules tothe tensor product f

M ⌦OX

eN of the corresponding O

X

-modules, and if M is of finitepresentations, the A-module of homomorphisms Hom

A

(M,N) to the sheaf of homo-morphisms HomO

X

(fM,

eN). However, this is nor true if M is not of finite presentation,

the only lacking desirable property of the functor g(�).To verify the first of these claims, assume given an exact sequence of A-modules:

0 //M

0 //M

//M

00 // 0.

— 7 —


That the induced sequence of OX

-modules

0 // fM

0 // fM

//gM

00 // 0

is exact is a direct consequence of the three following facts. The stalk of a tilde-modulefM at the point x with corresponding prime ideal p is Mp, localization is an exactfunctor, and finally, a sequence of abelian sheaves is exact if and only if the sequenceof stalks at every point is exact.

For the tensor product, observe that by the definition of the tensor product oftwo O

X

-modules, there is a canonical map M ⌦A

N ! �(X,

fM ⌦O

X

eN), and by the

comment preceding lemma �.� above it induces a map (M ⌦a

N)e! fM ⌦O

X

eN . This

turns out to be an isomorphisms since stalk by stalk it is an isomorphisms (the stalk ofthe tensor product being the tensor product of the stalks, you did exercise �.� didn’tyou?).

When it comes to hom’s however, the situation is somehow more subtle. If M isnot of finite presentation, it is not true that Hom

A

(M,N) localizes. There is always acanonical and obvious map

HomA

(M,N)⌦A

A

f

! HomA

f

(Mf

, N

f

), (4)

that sends ↵ to ↵⌦ idA

f

, but without some finiteness condition (like being of finitepresentation) on M , it is not necessarily an isomorphism. Even in the simplest caseof a infinitely generated free module M =

Li2I Aei that map is not surjective. An

element in HomA

(M,N)⌦A

A

f

is of the form ↵⌦ f

�n. An element in HomA

f

(Mf

, N

f

),however, is given by its values m

i

⌦ f

�n

i on the free generators ei

, and the salient pointis that the n

i

’s may tend to infinity, and no n working for all i’s can be found.

Problem �.��. Show that the map (4) above is an isomorphism when M is of fi-nite presentation. Hint: First observe that this holds when M = A. Then use apresentation of M to reduce to that case. X

If M is of finite presentations, one has The global sections of the sheaf-hom HomOX

(fM,

eN)

equals HomA

(M,N), and there is a map

HomA

(M,N)e! HomOX

(fM,

eN).

In case M is of finite presentation, the maps in (4) are isomorphisms, and the mapabove induces isomorphisms between the spaces of sections of the two sides over anydistinguished open subset. One concludes that the map is an isomorphism, and onehas Hom

A

(M,N)e' HomOX

(fM,

eN).

In short, we have established the important proposition:

Proposition �.� Assume that A is a ring and let X = SpecA. The functor from thecategory Mod

A

of A-modules to the category ModOX

of OX

-modules given by M ! fM

enjoys the following three properties

— 8 —


⇤ It is a fully faithful additive and exact functor.

⇤ One has a canonical isomorphism (M ⌦A

N)e' fM ⌦O

X

eN .

⇤ If M is of finite presentation, one has a canonical isomorphism HomA

(M,N)e'HomO

X

(fM,

eN).

Problem �.��. Show that the tilde-functor is additive; i.e., takes direct sums of mo-dules to the direct sum and sums of maps to the corresponding sums. X

Example �.�. Assume that A is an integral domain and that K is the field of fractionsof A. Show that the O

X

-module eK is a constant sheaf in the strong sense, that is

�(U, eK) = K for any non empty open U ✓X and the restriction maps all equal theidentity id

K

. Hence eK is a flabby sheaf. e

Functotiality The setting is that of a map between the two affine schemes X andY ; say � : X ! Y . We let X = SpecA and Y = SpecB and we let �] : B ! A be themap corresponding to �.

If M is an A-module, can one describe the sheaf �⇤fM on Y ? The answere is notonly yes but the description is very simple. The A-module M can be considered a B-module via the map B ! A and as such is denoted by M

B

. Clealy this is a functorialconstruction in M . In this setting one has

Proposition �.� One has �⇤fM = gM

B

.

Proof: There is an obvious map gM

B

! �⇤(fM) as �(Y,�⇤(fM)) = �(X,

fM) = M .

The argument that follows shows it is an isomorphism, and it is as usual sufficientto verify that sections over distinguished open sets of the two sides coincide. Thecrucial observation is that ��1

D(f) = D(�]

f)—indeed, a prime ideal p✓A satisfiesf 2 �

]�1(p) if and only if �](f) 2 p. As f acts on M

B

as multiplication by �](f), oneclearly has (M

B

)f

= M

�

](f) = �(D(�](f)),M). o

Quasi-coherence sheaves on general schemes

Having established the sheaves on affine space that serve as local models for thequasi-coherent sheaves, we are now ready for the general definition. If X is a schemeand F an O

X

-module, one says that F is quasi-coherent (kvasikoherent) OX

-module,or quasi-coherent sheaf for short, if there is an open affine covering {U

i

}i2I of X,

say U

i

= SpecAi

, and modules M

i

over A

i

such that F |U

i

' fM

i

. Phrased in slightlydifferent manner, O

X

-module F is quasi-coherent if the restriction F |U

i

of F to eachU

i

is of type tilde of an A

i

-module. In particular, the modules fM on affine schemes

SpecA are all quasi-coherent.The restriction of a quasi-coherent sheaf F to any open set U ✓X is quasi-coherent.

Indeed, it will suffices to verify this for X an affine scheme, and by lemma �.� therestriction of a sheaf of tilde-type to a distinguished open set is of tilde-type. As any

— 9 —


open U in an affine scheme is the union of distinguished open subsets, it follows thatF |

U

is quasi-coherent.For F to be quasi-coherent, we require that F be locally of tilde-type for just one

open affine cover. However, it turns out that this will hold for any open affine cover, orequivalently, that F |

U

is of tilde-type for any open affine subset U ✓X. This is a muchstronger than the requirement in the definition, and it is somehow subtle to prove. Asa first corollary we arrive at the a priori not obvious conclusion that the modules ofthe form f

M are the only quasi-coherent OX

-modules on an affine scheme. We shall alsosee that quasi-coherent modules enjoy the coherence property (1) on page 4 that wasthe point of departure for our discussion.

The story starts with a lemma that establishes the coherence property (1) in a veryparticular case; i.e., for sections over distinguished open sets of a quasi-coherent O

X

-module on an affine scheme X = SpecA. For any distinguished open set D(f)✓X itholds that �(D(f),O

X

) = A

f

, and consequently there is for any OX

-module a canonicalmap �(X,F )⌦

A

A

f

! �(D(f), F ) sending s⌦ af

�n to af

�n · s|D(f). It turns out to be

an isomorphism whenever F is quasi-coherent:

Lemma �.� Suppose that X = SpecA is an affine scheme and that F is a quai-coherent O

X

-module. Let D(f)✓X be a distinguished open set. Then one has

⇤ �(D(f), F ) ' �(X,F )⌦A

A

f

.

⇤ Let s 2 �(X,F ) be a global section of F and assume that s|D(f) = 0, then

sufficiently big powers of f kill s, that is, for sufficiently big integers n one hasf

n

s = 0.

⇤ Let s 2 �(D(f), F ) be a section. Then for a sufficiently large n, the section f

n

s

extends to a global section of F . That is, there exists an n and a global sectiont 2 �(X,F ) such that t|

D(f) = f

n

s

Proof: The first statement is by the definition of localization equivalent to the twoothers.

The sheaf F being quasi-coherent by hypothesis, and the affine scheme X = SpecAbeing quasi-compact, there is a finite open affine covering of X by distinguished setsD(g

i

) such that F |D(g

i

) ' fM

i

for some A

g

i

-modules Mi

. The section s of F restricts tosections s

i

of F |D(g

i

) over D(gi

), that is, to elements s

i

of Mi

.Further restricting F to the intersections D(f)\D(g

i

) = D(fgi

) yields the equalityF |

D(fgi

) = (fMi

)f

, and by hypothesis, the section s restricts to zero in �(D(fgi

), F ) =(M

i

)f

. This means that the localization map sends s

i

to zero in (Mi

)f

. Hence s

i

iskilled by some power of f , and since there is only finitely many g

i

’s, there is an n withf

n

s

i

= 0 for all i; that is, (fn

s)|D(g

i

) = 0 for all i. By the locality axiom for sheaves, itfollows that f

n

s = 0.Assume now a section s 2 �(D(f), F ) is given. We are to see that f

n

s extends toa global section of F for large n. Each restriction s|

D(fgi

) 2 �(D(fgi

), F ) = (Mi

)f

is

— 10 —


of the form f

�n

s

i

with s

i

2 M

i

= �(D(gi

), F ), and by the usual finiteness argument,n can be chosen uniformly for all i. This means that s

i

= f

n

s and s

j

= f

n

s mach onthe intersection D(f) \ D(g

i

) \ D(gj

), and by the first part of the lemma applied toSpecA

g

i

g

j

, one has fN(si

� s

j

) = 0 on D(gi

)\D(gj

) for a sufficiently large integers N .Hence the different f

N

s

i

’s patch together to give the desired global section t of F . o

Theorem �.� Let X be a scheme and F an OX

-module. Then F is quasi-coherent ifand only if for all open affine subsets U ✓X = SpecA, the restriction F |

U

is isomorphicto a an O

X

-module of the form fM for an A-module M .

Proof: As quasi-coherence is conserved when restricting OX

-modules to open sets,we may surely assume that X itself is affine; say X = SpecA. Let M = �(X,F ). Wesaw in lemma �.� on page 7 that there is a natural map f

M ! F that on distinguishedopen sets sends mf

�n to f

�n

m|U

, but by the fundamental lemma �.� above, this isan isomorphism between the spaces of sections over the distinguished open sets. Hencethe two sheaves are isomorphic. o

Applying this to an affine scheme, yields the important fact that any quasi-coherentsheaf F on an affine scheme X = SpecA is of the form f

M for an A-module M .

Proposition �.� Assume that X = SpecA. The tilde-functor M 7! fM is an equi-

valence of categories Mod

A

and QCoh

X

with the global section functor as an inverse.Finitely presented modules correspond to coherent sheaves. In particular, if A is noethe-rian, finitely generated modules correspond to coherent sheaves.

When speaking about mutually inverse functors one should be very careful; in mostcases such a statement is an abuse of language. Two functors F and G are mutuallyinverses when there are natural transformations, both being an isomorphisms, betweenthe compositions F �G and G�F and the appropriate identity functors. In the presentcase one really has an equality �(X,

fM) = M , so that � � g(�) = id

Mod

A

. On the otherhand, the natural transformation �(X,F )e! F from lemma �.� on page 7 furnishesthe required isomorphism of functors.

Coherence

Let A be a ring and let M be an A-module. The module M is of finite presentationif for some integers n and m there is an exact sequence

A

n //A

m //M

// 0 .

One says that M is coherent if the following two requirements are fulfilled

⇤ M is finitely generated.

⇤ The kernel of every surjection A

n ! M is finitely generated.

— 11 —


The second statement is equivalent to every finitely generated submodule in M beingof finite presentation. In the case A is a noetherian ring—which frequently is the casein algebraic geometry—a module M being coherent is equivalent to M being finitelygenerated. The condition comes from the theory analytic functions where coherentnon-noetherian rings are frequent.

On a scheme X a quasi-coherent OX

-module is coherent if there is a covering ofX by open affine sets U

i

= SpecAi

such that F |U

i

= fM

i

with the M

i

’s being coherentA

i

-modules.

Problem �.��. Let X be a scheme and let F be a sheaf of OX

-modules. Shoe that Fis quasi-coherent if and only if every point x 2 X has an open neigbourhood U overwhich there are an exact sequences of O

X

-modules

OI

X

|U

// OJ

X

|U

//F |

U

// 0

In the sequnce the exponents I and J are not necessaily finite cardinals. Shpw that Fis coherent if and only if one may take I nd J finite in the above sequence. X

Theorem �.� Let X be a scheme and let F be an OX

-module on X. Then F is quasi-coherent of and only if for any pair V ✓U open affine subsets, the natural map

�(U, F )⌦�(U,OX

) �(V,OX

) ! �(V, F ) (5)

is an isomorphism.

Proof: We may clearly assume that X is affine, say X = SpecA. Assume that themaps (5) are isomorphisms. We may take V = D(f) (and U = X) and M = �(X,F ).Then from (5) it follows that �(D(f), F ) = M

f

which shows that the canonical mapfM ! F is an isomorphism over all distinguished open subsets, and therefore an iso-morphism.

To argue for the implication the other way, assume that U = SpecB and that F isquasi-coherent; that is, F = f

M for some A-module M after theorem �.�. The restrictionof a quasi-coherent sheaf is quasi-coherent, so M |

U

= eN for some B-module N , and

the map in (5) is just a map have map M ⌦A

B ! N . It induces an isomorphism overall local rings Ap = Bp (where p 2 U = SpecB) since f

M |U

' eN and therefore is an

isomorphism of B-modules. o

Closedness

The following lemma is an easy consequence of the global section functor beingexact on the category of quasi-coherent sheaves on an affine scheme once we have thegeneral machinery of cohomology at our disposal. It possible to give an ad hoc proof(e.g., see Hartshorn xxx) but a proof is quite subtle and may be not worth while doing

— 12 —


as the conceptual proof using cohomology is clean and straightforward. It is of coursealso possible to extract exactly what is needed from the general theory to prove thelemma,. . . ..

Lemma �.� Assume that X = SpecA is an affine scheme. Suppose we are given exactsequence

0 // fM

//F

//G

// 0

of OX

modules F , G and fM where f

M is quasi-coherent. The the corresponding sequenceof global sections

0 //M

// �(X,F ) // �(X,G) // 0

is exact.

Proof: Let � be a global section of G. By Zorn’s lemma there is a maximal open setU suct that there exists �0 of F over U mapping to �|

U

. We are to show that U = X,so assume the contrary. Let x 2 X and let V be an open distinguisjed neigbpourhoodof x such that �|

V

can be lifted to a section ⌧ of F over V . Then t� s lies in fM o

Proposition �.� Assume that ↵ : F ! G is a map of quasi-coherent sheaves on thescheme X. The kernel, cokernel and the image of ↵ are all quasi-coherent. The categoryQCoh

X

is closed under extensions; that is, if

0 //M

0 //M

//M

00 // 0 (6)

is a short exact sequence of OX

-modules with the two extreme sheaves M 0 and M

00 beingquasi-coherent, the middle sheaf M is quasi-coherent as well.

Proof: If ↵ : F ! G is a map of quasi-coherent OX

-modules, on any open affinesubsets U = SpecA of X it may be described as ↵|

U

= ea where a : M ! N is aA-module homomorphism and M and N are A-modules with F |

U

= fM and G|

U

= eN .

Since the tilde-functor is exact, one has Ker↵|U

= (Ker a) . Moreover, by the samereasoning, it holds true that Coker↵|

U

= (Coker a) and Im↵|U

= (Im a) .Suppose now that an extension like (6) is given. The leftmost sheaf M 0 being quasi-

coherent lemma �.� entails that the induced sequence of global sections is exact; thatis, the upper horizontal sequence in the diagram below. The three vertical maps inthe diagram are the natural maps from lemma �.� on page 7. Since M

0 and M

00 bothare quasi-coherent sheaves, the two flanking vertical maps are isomorphisms, and the

— 13 —


snake lemma implies that the middle vertical map is an isomorphism as well. Hence M

is quasi-coherent.

0 // �(X,M

0) //

✏✏

�(X,M ) //

✏✏

�(X,M

00) //

✏✏

0

0 //M

0 //M

//M

00 // 0

o

The category QCoh

X

is an abelain category with tensor products and internal hom’s.

Functoriality

Theorem �.� Assume that � : X ! Y is a morphism of schemes and that F is aquasi-coherent sheaf on X. If X is separated and � is quasi-compact, then the directimage �⇤F is quasi-coherent on Y .

Example �.�. Let X =S

i2I SpecZ be the disjoint union of countably many copies ofSpecZ and let � : X ! SpecZ be the morphism that equals the identity on each ofthe copies of SpecZ that constitute X. Then �⇤OX

is not quasi-coherent. Indeed, theglobal sections of �⇤OX

satisfy �(SpecZ,�⇤OX

) = �(X,OX

) =Q

i2I Z. On the otherhand if p is any prime, one has �(D(p),�⇤OX

) = �(��1D(p),O

X

) =Q

i2I Z[p�1]. Itis not true that �(D(p),�⇤OX

) = �(SpecZ,�⇤OX

)⌦Z Z[p�1] hence �⇤OX

is not quasi-coherent. Indeed, elements in

Qi2I Z[p�1] are sequences of the form (z

i

p

�n

i)i2I where

z

i

2 Z and n

i

2 N. Such an element lies in (Q

i2I)⌦Z Z[p�1]) only if the n

i

’s form abounded sequence, which is not the case for general elements of shape (z

i

p

�n

i)i2I when

I is infinite. e

Proof: Being quasi-coherent is a local property for a sheaf, so we may assume that Y isaffine, say Y = SpecA. Then as � is assumed to be quasi-compact, X is quasi-compactand can be covered by finitely many open affine subsets U

i

, and s X is assumed to beseparated, we know that sets U

ij

= U

i

\ U

j

are affine as well. For any open V ✓U onehas the usual exact sequence

0 // �(V,�⇤F ) //Q

i

�(Ui

\ ��1V,O

X

) //Q

i,j

�(Uij

\ V,OX

). (7)

The sequence is compatible with the restriction maps induced from an inclusion V

0 ✓V ,hence gives rise to the following exact sequence of sheaves on X:

0 //�⇤OX

//Q

i

�

i⇤OU

i

//Q

i,j

�

ij⇤OU

ij

(8)

where �i

= �|U

i

and �

ij

= |U

ij

. Indeed, the sequence (7) is nothing but the sequenceobtained by taking sections over the open subset V of the sequence (8). Now, each ofthe sheaves �

i⇤OU i

and �

ij⇤OU

ij

are quasi-coherent by the affine case of the theorem

— 14 —


(proposition �.� on page 9). They are finite in number as the covering U

i

is finite.Hence

Qi

�

i⇤OU

i

andQ

i,j

�

ij⇤OU

ij

are finite products of quasi-coherent OX

-modulesand therefore they are quasi-coherent. The theorem the follows from proposition �.�on page 13. o

Problem �.��. Show that X =S

i2I SpecZ is not affine. Show that in the categorySch of schemes X is the coproduct of the infinitely many copies of SpecZ involved .Show that Spec

Qi

Z is the coproduct of the copies of SpecZ in the categiry A↵ ofaffine schemes. Show that these gadgeds are subtatially different. X

— 15 —

Documents

Quasi-coherent modulesNotes 5—Sheaves of modules MAT4215 — Vår 2015 To pin down the quasi-coherent sheaves, one ﬁrst establishes a collection of model-sheaves on aﬃne schemes