AlgebraicGeometry–Schemes1amyekut/teaching/2018-19/schemes-1/n… · CourseNotes | AmnonYekutieli | 1Feb2019 CourseNotes: AlgebraicGeometry–Schemes1 Fall2018-191 AmnonYekutieli

Course Notes | Amnon Yekutieli | 1 Feb 2019

Course Notes:

Algebraic Geometry – Schemes 1

Fall 2018-19 1

Amnon Yekutieli

Contents

1. Basics 12. Sheaves of Functions on Topological Spaces 23. Sheaves on Topological Spaces 64. Stalks 75. Morphisms of Sheaves 96. Sheafification 117. Gluing Sheaves and Morphisms between Them 178. Vector Bundles 279. Ringed Spaces and Sheaves of Modules 3410. Vector Bundles and Locally Free Sheaves 4511. Nonabelian First Cohomology 4612. Operations on Sheaves 5313. Locally Ringed Spaces 5714. Schemes 59References 63

1. Basics

Lecture 1, 17 Oct 2018

Before starting with the actual material, lets us go quickly over some basic ideas that wewill need. I hope all these are familiar to all students; if not, then we will have to see howto close the gaps.

The first few weeks will be on geometry in general, but from the point of view of locallyringed spaces.

Everybody needs to know a sufficient amount of elementary topology. Some algebraictopology will be required (homology, cohomology and fundamental groups).

Categories, functors and natural transformations will be used a lot. I am assuming thatall students have already been exposed to these notions. For instance, all should understandthis statement:

• Let Top∗ and Grp be the categories of pointed topological spaces and of groups,respectively. The fundamental group is a functor

π1 : Top∗ → Grp .

1The whole course. Removed empty pages between lectures.

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If not, then we will have to see how to close this gap. (Maybe go over material from [Ye3].)Differential geometry will serve as an introductory model for locally ringed spaces. (A

preparation for the more complicated schemes.) Everybody should have some knowledgeon this topic (C∞ manifolds and maps between them, tangent bundles, etc.) Knowledge ofcomplex analytic geometry will be very useful.

2. Sheaves of Functions on Topological Spaces

Consider a topological space X . We do not make any conditions on X , especially wedon’t assume X is Hausdorff. But at first you can pretend, to help intuition, that X is atopological subspace of n (with its usual topology).

Given an open subset U ⊆ X , consider the continous functions

f : U → .

Let us denote this set of functions by Γ(U,OX ).

We know that Γ(U,OX ) is a commutative -ring.Let V ⊆ U be a smaller open set. We get a continous function

f |V : V → .

The opration f 7→ f |V is a ring homomorphism

restV/U : Γ(U,OX ) → Γ(V ,OX ).

If W ⊆ V is another smaller open set, then of course

( f |V )|W = f |W .



We see that the restriction homomorphisms satisfy

restW/V restV/U = restW/U : Γ(U,OX ) → Γ(W ,OX ).

This means that OX is a presheaf of -rings on X .Here is a categorical interpretation of this statement. Let Rngc/ be the category of

commutative -rings.Let Open(X) be the category of open sets of X , where the morphisms are inclusions.

Thus if V ⊆ U then there is one arrow V → U; and if V * U then there are no arrowsV → U. The presheaf OX is a functor

Γ(−,OX ) : Open(X)op → Rngc/.

But in fact much more is true.Suppose U ⊆ X is an open set, and we are given an open covering

U =⋃i∈I

Vi .

Let f , g ∈ Γ(U,OX ), i.e.

f , g : U → ,

and assume that

f |Vi = g |Vi

for all i.



Then of course f = g.Now assume that we are given

fi ∈ Γ(Vi ,OX )

such thatfi |Vi∩Vj = fj |Vi∩Vj

for all i, j.

Because the various fi agree on double intersections, there is a function

f : U →

such thatf |Ui = fi .

Of course this function f is unique (by the previous discussion). But also f is continous.This is because continuity is a local property, and on each of the open sets Ui we know thatf is continous.

Thusf ∈ Γ(U,OX ).

Let us summarize these two further properties of OX :(a) Let U ⊆ X be an open set, let U =

⋃i∈I Vi an open covering, and let

f , g ∈ Γ(U,OX )



be such that f |Vi = g |Vi for all i. Then f = g.(b) Let U ⊆ X be an open set, let U =

⋃i∈I Vi be an open covering, and let

fi ∈ Γ(Vi ,OX )

be such thatfi |Vi∩Vj = fj |Vi∩Vj

for all i, j. Then there exists

f ∈ Γ(U,OX )

such that f |Vi = fi for all i.These are the sheaf axioms. They tell us that OX is a sheaf of rings on X .Because rings have underlying abelian groups, axioms (a) and (b) can be stated in terms

of exact sequences.(∗) For every open set U ⊆ X and every open covering U =

⋃i∈I Vi the sequence of

abelian groups

0→ Γ(U,OX )ρ−→

∏i∈I

Γ(Vi ,OX )δ0−δ1

−−−−−→∏j,k∈I

Γ(Vj ∩ Vk ,OX )

is exact.Here ρ is the product on all i ∈ I of the restriction homomorphisms

restVi/U : Γ(U,OX ) → Γ(Vi ,OX ).

The homomorphism δ1 is the product on all i = j ∈ I of the product on all k ∈ I of

restVj∩Vk /Vj: Γ(Vj ,OX ) → Γ(Vj ∩ Vk ,OX ).

And the homomorphism δ0 is the product on all i = k ∈ I of the product on all j ∈ I of of

restVj∩Vk /Vk: Γ(Vk ,OX ) → Γ(Vj ∩ Vk ,OX ).

Exercise 2.1. Prove that condition (∗) is equivalent to condition((a) and (b)

).

The next exercise gives a variation of what we did above.

Exercise 2.2. Let X be a differentiable manifold (of type C∞). For every open set U ⊆ Xlet Γ(U,OX ) be the set of differentiable functions f : U → .

Prove that the assignmentU 7→ Γ(U,OX )

is a sheaf of -rings on X . The sheaf OX is called the sheaf of differentiable functions onX .

Exercise 2.3. If you know about real or complex analytic manifolds, state and prove thecorresponding analogue of Exercise 2.2.

Exercise 2.4. This exercise is for those who know the algebraic geometry of varieties. Let be an algebraically closed field, and let X be an algebraic variety over . For every(Zariski) open set U ⊆ X let Γ(U,OX ) be ring of algebraic functions on U. Prove that theassignment

U 7→ Γ(U,OX )

is a sheaf of -rings on X . The sheaf OX is called the sheaf of algebraic functions on X .



3. Sheaves on Topological Spaces

Until now we only saw ring valued sheaves. Here are some variations.

Definition 3.1. Let X be a topological space. A presheaf of groups on X is a functor

G : Open(X)op → Grp,

where Grp is the category of groups.

Concretely, the presheaf G is the data of a group Γ(U, G) for every open set U ⊆ X ,called the group of sections of G over U, and a group homomorphism

restV/U : Γ(U, G) → Γ(V , G)

for every inclusion V ⊆ U, such that

restW/U = restW/V restV/Ufor every double inclusion W ⊆ V ⊆ U. And of course

restU/U = idΓ(U,G)for every U.

We often use the abbreviation

(3.2) g |V := restV/U (g) ∈ Γ(V , G)

for a presheaf G, an inclusion of open sets V ⊆ U, and a section g ∈ Γ(U, G).

Definition 3.3. Let X be a topological space. A sheaf of groups on X is a presheaf ofgroups G on X that satisfies the two sheaf axioms:

(a) Let U ⊆ X be an open set, let U =⋃

i∈I Vi be an open covering, and let g, h ∈Γ(U, G) be sections such that g |Vi = h|Vi for all i. Then g = h.

(b) LetU ⊆ X be an open set, letU =⋃

i∈I Vi be an open covering, and let gi ∈ Γ(Vi , G)be sections such that

gi |Vi∩Vj = gj |Vi∩Vj

for all i, j. Then there exists a section g ∈ Γ(U,OX ) such that

g |Vi = gi

for all i.

Recall that a topological group is a topological space G, that is also a group, such thatthe operations of multiplication and inversion are continous. Namely

mult : G × G→ G

andinv : G→ G

are continous functions.

Example 3.4. Let X be a topological space and G a topological group. For every open setU ⊆ X define

Γ(U, G) := continous functions g : U → G.

I claim that G is a sheaf of groups on X .That G is a presheaf is obvious. Sheaf axiom (a) is also clear, because for every point

x ∈ U we can find some i such that x ∈ Vi , and hence we have

g(x) = g |Vi (x) = g′ |Vi (x) = g′(x).

Thus g = g′.



Axiom (b) is also easy to verify. The values gi(x) at a point x ∈ U are equal, for all i ∈ Isuch that x ∈ Vi . So there is a function g : U → G. Because continuity is a local property,and g |Vi = gi , we see that g is continous. Thus g ∈ Γ(U, G).

Exercise 3.5. Let X be a topological space. Let G := GLn() for some positive integer n,with the usual topology (by the embedding GLn() ⊆

n2 ). So G is a topological group.Let G be the sheaf of groups on X from Example 3.4 for this choice of G. And let OX

be the sheaf of continous real valued function on X . Prove that for every open set U ⊆ Xthere is a group isomorphism

Γ(U, G) GLn(Γ(U,OX )),

that respects the restriction homomorphisms.

Definition 3.6. Let X be a topological space.(1) A presheaf of abelian groups on X is a functor

G : Open(X)op → Ab,

where Ab is the category of abelian groups.(2) A sheaf of abelian groups on X is a presheaf of abelian groups G that satisfies the

sheaf axioms (a) and (b) from Definition 3.3.

It is not hard to see that a sheaf of abelian groups G is the same as a sheaf of groups Gsuch that each Γ(U, G) is abelian.

Definition 3.7. Let X be a topological space.(1) A presheaf of commutative rings on X is a functor

A : Open(X)op → Rngc,

where Rngc is the category of commutative rings.(2) A sheaf of commutative rings on X is a presheaf of commutative rings A that

satisfies the sheaf axioms (a) and (b) from Definition 3.3.

If A is a sheaf of commutative rings, and we forget the multiplication of A, then weobtain a sheaf of abelian groups.

Example 3.8. Let X be a topological space and let A be a commutative ring. The constantsheaf of rings on X with values in A is the sheaf AX defined as follows. Put on A the discretetopology. Then for every U ⊆ X open we let

Γ(U, AX ) := continous functions g : U → A.

Exercise 3.9. Take a nonzero commutative ring A, say A := . Calculate the ring Γ(X , AX )

for these choices of X :(1) X := with the classical topology.(2) X := with the discrete topology.

4. Stalks

A directed set is a partially ordered set I such that for every i, j ∈ I there exists somek ∈ I with i, j ≤ k. We can view the directed set I as a category, with a single arrowri, j : i → j if i ≤ j, and no arrows otherwise.

A direct system in a category C, indexed by a directed set I, is a functor

C : I → C, i 7→ C(i) = Ci , ri, j 7→ C(ri, j).

We usually denote such a direct system by Cii∈I , leaving the ri, j implicit.



A direct limit of a direct system Cii∈I is an object C∞ ∈ C, together with a collectionof morphisms fi : Ci → C∞, such that the diagram

Ci

fi

C(ri, j )

Cj

fj// C∞

is commutative for every i → j, and such that(C∞, fii∈I

)is universal for this property. We write

limi→

Ci := C∞.

The categories Grp, Ab and Rngc have direct limits. Here is the construction:

limi→

Ci =(∐

i∈ICi

)/ ∼ ,

where ∼ is the relation ci ∼ cj for ci ∈ Ci and cj ∈ Cj whenever there are arrows i, j → ksuch that

C(ri,k)(ci) = C(rj,k)(cj) ∈ Ck .

Let X be a topological space. For a point x ∈ X let Open(X , x) be the set of openneighborhoods of x, made into a category by inclusions. Then Open(X , x)op is a directedset.

Definition 4.1. LetM be a presheaf of abelian groups on a topological space X . Let x ∈ Xbe a point. The stalk of M at x is the abelian group

Mx := limU→

Γ(U,M),

where the direct limit is on U ∈ Open(X , x)op.Likewise for a presheaf of groups and for a sheaf of commutative rings.

Exercise 4.2. With the assumptions of Exercise 3.9(1, 2), calculate the stalks (AX )x for apoint x ∈ X .




5. Morphisms of Sheaves

We will mostly work with sheaves of abelian groups; but things are the same for sheavesin Rngc, Grp and Set.

First we talk about morphisms of presheaves.

Definition 5.1. Let M and N be presheaves of abelian groups on a topological space X .A morphism of presheaves of abelian groups

φ : M→ N

is a collectionφ =

Γ(U, φ)

U∈Open(X)

of homomorphisms of abelian groups

Γ(U, φ) : Γ(U,M) → Γ(U,N ),

such that the diagramsΓ(U,M) //

restV /U

Γ(U,N )

restV /U

Γ(V ,M)Γ(V,φ)

// Γ(V ,N )are commutative for all inclusions V ⊆ U.

The category of presheaves of abelian groups on X us denoted by PAb X

In other words, a morphism of presheaves φ : M → N is a morphism of functors(natural transformation)

(Open X)op → Ab .Given a morphism φ : M→ N and a point x ∈ X , there is a group homomorphism

φx : Mx → Nx

in the stalks.We say that φ is injective (respect. surjective) if for every open setU the homomorphism

Γ(U, φ) is injective (respect. surjective).Let M be a presheaf. A subpresheaf of M is a presheaf M′ such that

Γ(U,M′) ⊆ Γ(U,M)

for every U, and they have the same restriction homomorphisms. The inclusionM′→Mis an injective morphism of presheaves.

Recall that the sheaves on X form a subset of the presheaves on X – these are presheavesthat satisfy the sheaf axioms.

Definition 5.2. Let M and N be sheaves of abelian groups on a topological space X . Amorphism of sheaves of abelian groups φ : M→ N is just a morphism of presheaves.

Thus the categoryAb X of sheaves of abelian groups on X is a full subcategory of PAb X .

Likewise for groups, rings and sets: there are full embeddings

Grp X ⊆ PGrp X ,

Rng X ⊆ PRng X



andSet X ⊆ PSet X

of the categories of sheaves in the corresponding categories of presheaves.It will be convenient to be a bit ambiguous sometimes - we shall talk about a morphism

of presheaves or sheaves, meaning any of the four kinds (Ab, Grp, Rng or Set). For this weintroduce the symbolic notation

Sh X ⊆ PSh X .(Unless this turns out to be too confusing – then we will abolish it.)

Let φ : M → N be a morphism of sheaves on X . Given a point x ∈ X there is aninduced morphism

(5.3) φx : Mx → Nx .

Definition 5.4. Let φ : M→ N be a morphism of sheaves on X .(1) We call φ an injective sheaf morphism if every point x the morphism φx is injective.(2) We call φ a surjective sheafmorphism if every point x themorphism φx is surjective.

Proposition 5.5. Let φ : M → N be a morphism of sheaves. The following conditionsare equivalent.

(i) φ is an injective sheaf morphism.(ii) φ is an injective presheaf morphism.

Exercise 5.6. Prove the last proposition.

A subsheaf of a sheafM is a subpresheafM′ ⊆Mwhich is itself a sheaf. The inclusionM′→M is an injective morphism of sheaves.

A presheaf M is called a separated presheaf if it satisfies sheaf axiom (a).

Exercise 5.7. Suppose M is a sheaf and M′ ⊆ M is a subpresheaf. Show that M′ is aseparated presheaf.

Proposition 5.5 is false for surjections! See Exercise 5.10 below. Instead we have:

Proposition 5.8. Let φ : M → N be a morphism of sheaves. The following conditionsare equivalent.

(i) φ is a surjective sheaf morphism.(ii) For evey open set U ⊆ X and every section n ∈ Γ(U,N ) there is an open covering

U =⋃

i∈I Vi and sections mi ∈ Γ(Vi ,M) such that

Γ(Vi , φ)(mi) = n|Vi

in Γ(Vi ,N ).

Exercise 5.9. Prove this proposition.

Exercise 5.10. Find an example of a sheaf homomorphism φ : M→ N on a space X withthis property: φ is a surjection of sheaves, but it is not a surjection of presheaves. (Thiscould be hard; we will see examples later.)

Update. In class today such an example was proposed by Guy. The topological spacewas X := − 0, the sheaf M was the sheaf of holomorphic (i.e. analytic) -valuedfunctions on X , the sheaf N was the subsheaf of nonzero functions, and φ : M→ N wasf 7→ exp( f ). We can view these a morphism in Ab X . Try to understand why

Γ(X , φ) : Γ(X ,M) → Γ(X ,N )

is not surjective; so φ is not surjective as a morphism of presheaves. But φ : M → Nis surjective as a morphism of sheaves. (Hint: a logarithm is defined on each contractibleopen set in X .)

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Exercise 5.11. Let M be a sheaf on X . What is Γ(,M) ?

Proposition 5.12. Let φ : M → N be a morphism of sheaves on X . The following areequivalent:

(i) φ is an isomorphism of sheaves, i.e. an isomorphism in the category of sheaves.(ii) For every point x ∈ X the morphism on stalks

φx : Mx → Nx

is bĳective.(iii) For every open set U ⊆ X the morphism

Γ(U, φ) : Γ(U,M) → Γ(U,N )

is bĳective.

Note that condition (ii) above says that φ is both injective and surjective, see Definition5.4.


6. Sheafification

[comment: (181104) made this into a new section; small changes below]Recall that by a (pre)sheaf, and a morphism of (pre)sheaves, we mean of the four kinds:

abelian groups, groups, rings or sets. (Later we will also talk about sheave of A-modules,where A is a sheaf of rings.) We shall use the generic notation C(X) ⊆ PC(X), whereC = Set,Grp,Ab, Rng. So when C = Ab this stands for Ab(X) ⊆ PAb(X), etc.

Theorem 6.1 (Sheafification). LetM be a presheaf with values in C on a topological spaceX . There is a sheaf Sh(M) on X , with a morphism of presheaves

τM : M→ Sh(M),

having this universal property:(S) For every pair (N , φ), consisting of a sheaf N and a morphism of presheaves

φ : M→ N ,

there is a unique morphism of sheaves

φ′ : Sh(M) → N

such that the diagram

MτM //

φ

""

Sh(M)

φ′

N

in PC(X) is commutative.The pair

(Sh(M), τM

)is called the sheafification of M.

Let us note, before proving the theorem, that:

Proposition 6.2. The sheafification(Sh(M), τM

)of M is unique, up to a unique isomor-

phism.


Corollary 6.4. In M is a sheaf then (Sh(M), τM) = (M, id); i.e. uniquely isomorphic.

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Exercise 6.5. Prove this corollary.

We need an auxilliary construction. Given a presheaf M, let us define the presheafGSh(M) as follows: for every open set U we take

Γ(U,GSh(M)) :=∏x∈U

Mx ,

the product on all stalks. For an open subset V ⊆ U we define

restV/U : Γ(U,GSh(M)) → Γ(V ,GSh(M))

to be the projection

(6.6)

Γ(U,GSh(M)) =∏x∈U

Mx =( ∏x∈V

Mx

)×

( ∏x∈U−V

Mx

)pr−−→

∏x∈V

Mx = Γ(V ,GSh(M)).

Note that a section m ∈ Γ(U,GSh(M)) looks like this:

(6.7) m = mxx∈U , mx ∈Mx .

Lemma 6.8. Let M be a presheaf.(1) The presheaf GSh(M) is a sheaf.(2) There is a presheaf morphism

γM : M→ GSh(M).

(3) If the presheaf M is separated, then the morphism γM is injective.(4) For every inclusion V ⊆ U of open sets, the morphism

Γ(U,GSh(M)) → Γ(V ,GSh(M))

is surjective.

A sheaf satisfying (4) above is called a flasque sheaf.

Proof. (1) Let U =⋃

i∈I Vi be an open covering.Let’s verify axiom (a) of Definition 3.3 for this covering. Let m, n ∈ Γ(U,GSh(M)) be

sections such that m|Vi = n|Vi in

Γ(Vi ,GSh(M)) =∏x∈Vi

Mx

for all i. This means that the stalks satisfy

mx = nx ∈Mx

for all x ∈ Vi . But for every x ∈ U there is some i such that x ∈ Vi . We see that

mx = nx ∈Mx

for all x ∈ U. By formula (6.7) we conclude that m = n.Now we shall verify axiom (b) of Definition 3.3 for this covering. So we are given a

collection mii∈I of sectionsmi ∈ Γ(Vi ,GSh(M))

satisfying

(6.9) mi |Vi∩Vj = mj |Vi∩Vj

for all i, j ∈ I. Let’s write

mi = mi,xx∈Vi , mi,x ∈Mx .

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From (6.9) we see that mi,x = mj,x for all x ∈ Vi ∩Vj . Hence for every x ∈ U we can define

mx := mi,x ∈Mx

where i is some index such that x ∈ Vi , and this does not depend on the choice of i. Weobtain a section

m := mxx∈U ∈ Γ(U,GSh(M))which satisfies

m|Vi = mi

for all i.

(2) For every open set U ⊆ X , a section m ∈ Γ(U,M) and a point x ∈ U let mx ∈Mx bethe image of m under the canonical homomorphism

Γ(U,M) →Mx .

We get a sectionmxx∈U ∈ Γ(U,GSh(M)).

It is easy to see that this construction respects restrictions, so it is a morphism of presheaves.

(3) Exercise (see below).

(4) This is clear from fromula (6.6).

Exercise 6.10. Prove item (3) above.

Definition 6.11. We call GSh(M) the Godement sheaf associated to M

Exercise 6.12. Show thatGSh : PC(X) → C(X)

is a functor, andγ : Id→ GSh

is a morphism of functors from C(X) to itself.

Definition 6.13. Let M be a presheaf, and let U ⊆ X be an open set. A section

m ∈ Γ(U,GSh(M))

is called a geometric section if there is an open covering U =⋃

i∈I Vi and sections mi ∈

Γ(Vi , (M)), such that for every x ∈ Vi the morphism

Γ(Vi , (M)) →Mx

sends mi 7→ mx .

See picture in Figure 1.We refer to the data

(Vii∈I , mii∈I

)as evidence for the geometricity of m.

Lemma 6.14. Let M be a presheaf, let U ⊆ X be an open set, and let

m = myy∈U ∈ Γ(U,GSh(M)).

The following conditions are equivalent:(i) m is a geometric section.(ii) For every point x ∈ X there is an open set V s.t. x ∈ V ⊆ U, and a section

m′ ∈ Γ(V ,M), s.t. m′ 7→ my for every y ∈ V .

Exercise 6.15. Prove this lemma.


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Figure 1.

Lemma 6.16. Let M be a presheaf on X . The assignment

Sh(M) : U 7→geometric sections of Γ(U,GSh(M))

is a subsheaf of GSh(M).

Proof. Step 1. Here we prove that Sh(M) is a subpresheaf of GSh(M). Namely that foran inclusion V ⊆ U, the morphism

restV/U : Γ(U,GSh(M)) → Γ(V ,GSh(M))

sends geometric sections to geometric sections.So let m ∈ Γ(U,GSh(M)) be a geometric section, and let m|V ∈ Γ(V ,GSh(M)) be its

restriction to V . Talk a point x ∈ V . By Lemma 6.14 there is evidence for m at x : an openset W such that x ∈ W ⊆ U, and a section m′ ∈ Γ(W ,M) such that m′ 7→ my for all y ∈ W .Then the pair (W ∩ V ,m′ |W∩V ) is evidence for m|V at x. We see that m|V is a geometricsection.

Step 2. Because Sh(M) is a subpresheaf of the sheaf GSh(M), it is automatically separated(axiom (a) holds); see Exercise 5.7.

Now for axiom (b). Let U =⋃

i∈I Vi be an open covering of an open set, and letmi ∈ Γ(Vi , Sh(M)) be a collection of sections that agree on double intersections. Letm ∈ Γ(U,GSh(M)) be the unique section such that m|Vi = mi . Like in step 1, we see thatm is a geometric section, namely m ∈ Γ(U, Sh(M)).

Remark 6.17. Here is a useful heuristic for the inclusion of sheaves

Sh(M)) ⊆ GSh(M)).

We can pretend that the elements of Γ(U,GSh(M)) are "arbitrary functions" on U, and theelements of Γ(U, Sh(M)) are the "continous functions".

Lemma 6.18. If M is a presheaf of abelian groups, then

Sh(M) ⊆ GSh(M)

is a subsheaf of abelian groups. Likewise for a presheaf of groups or rings.

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Lemma 6.20. The assignment M 7→ Sh(M) is a functor PC(X) → C(X).


[comment: (date 181104) new lemma next – was part of proof of thm]

Lemma 6.22. There is a morphism

τ : Id→ Sh

of functors from PC(X) to itself, such that for every presheaf M the diagram such that thediagram

MτM //

γM

##

Sh(M)

⊆

GSh(M)in PC(X) is commutative.

Proof. Take an open set U. For each section m ∈ Γ(U,M), the section

γM(m) ∈ Γ(U,GSh(M))

is geometric section – the pair (U,m) is a tautological evidence. We define

τM(m) := γM(m) ∈ Γ(U, Sh(M)) ⊆ Γ(U,GSh(M)).

[comment: (date 181104) new lemma next – was Exer 5.36 in prev version]

Lemma 6.23. Let M be a presheaf on X and let x ∈ X be a point. Then the function onstalks

(τM)x : Mx → Sh(M)xinduced by τM is bĳective.

Proof. Injectivity: For every open set U containing x there is a canonical morphism

Γ(U,GSh(M)) =∏y∈U

My →Mx .

So there are canonical morphisms

Γ(U,M)Γ(U,τM)−−−−−−−→ Γ(U, Sh(M)) → Γ(U,GSh(M)) →Mx .

Passing to the direct limit over all U 3 x we get a commutative diagram

Mx(τM)x

//

id

Sh(M)x // GSh(M)x //Mx

Hence (τM)x in injective.Surjectivity: Take a germ mx ∈ Sh(M)x . It is represented by some section m ∈

Γ(U, Sh(M)). This means that m ∈ Γ(U,GSh(M)) is a geometric section. So thereis evidence for m at x : there is an open set V and a section m′ ∈ Γ(V ,M) such thatx ∈ V ⊆ U and m′ = m|U . But then the germ m′x ∈Mx satisfies (τM)x(m′x) = mx .

[comment: (date 181104) new lemma next ]

15 | file: notes-190201-d2


Lemma 6.24. Let M be a presheaf on X . The morphism of sheaves

GSh(τM) : GSh(M) → GSh(Sh(M))

is an isomorphism.

Proof. This is immediate from Lemma 6.23.

[comment: (date 181104) many changes in proof below ][comment: (date 181107) new, improved (?) notation ]

A change of notation: for a "legitimate category" C, i.e. C = Set,Grp,Ab, Rng orMod Afor a ring A, we write CX for the category of sheaves with values in C, and Cpre

X for thecategory of presheaves with values in it.

Proof of Theorem 6.1. We will prove that the pair(Sh(M), τM

)defined in Lemmas 6.16

and 6.22 has the required properties.Let φ : M→ N be a morphism to a sheaf N . We get the solid commutative diagram

(6.25) Mφ

//

τM

γM

N

τN

γN

Sh(M)Sh(φ)

//

⊆

φ′

99

Sh(N )

⊆

GSh(M)GSh(φ)

// GSh(N )

in CpreX . Because τN is an isomorphism (see Corollary 6.4), there is a unique morphism

φ′ : Sh(M) → N

that makes the diagram commutative, namely

(6.26) φ′ := τ−1N Sh(φ).

It remains to verify the uniqueness of φ′. So let φ′′ : Sh(M) → N be any morphism inCpreX s.t. φ′′ τM = φ. We need to prove that φ′′ = φ′. Define

ψ ′′ := τN φ′′ : Sh(M) → Sh(N ).

In view of (6.26), it suffices to prove that ψ ′′ = Sh(φ).We have this commutative diagram

(6.27) Mφ

//

τM

N

τN

Sh(M)ψ′′

// Sh(N )

in CpreX . Passing to stalks at each point x ∈ X we get a commutative diagram

(6.28) Mxφx

//

(τM)x

Nx

(τN )x

Sh(M)xψ′′x // Sh(N )x

16 | file: notes-190201-d2


in C. The right vertical arrow is an isomorphism by Lemma 6.23. The commutativity ofdiagram (6.28), together with Lemma 6.24, say that

GSh(φ) = GSh(ψ ′′) : GSh(M) → GSh(N ).

[comment: (181107) small change below ] We end up with this commutative diagram

(6.29) Mφ

//

τM

γM

N

τN

γN

Sh(M)ψ′′

//

⊆

Sh(N )

⊆

GSh(M)GSh(φ)=GSh(ψ′′)

// GSh(N )

inCpreX . Comparing the bottom square in this diagram to the bottom square in diagram (6.25),

and noting that Sh(N ) GSh(N ) is a monomorphism, we conclude that ψ ′′ = Sh(φ), asrequired.

Exercise 6.30. Let X be a topological space and M an abelian group (or a ring, etc.).Define M to be the constant presheaf with values in M , namely

Γ(U,M) := M

for every open set U. Prove that the sheafification of M is

Sh(M) = MX ,

the constant sheaf with values in M .

Exercise 6.31. Consider X := with its classical topology, let M := X , the constantsheaf with values in .

(1) Let U ⊆ X be a connected open set (i.e. a nonempty open interval). CalculateΓ(U,M) and Γ(U,GSh(M)). Conclude that

Γ(U,M) Γ(U,GSh(M)).

(2) Conclude that for every point x ∈ X ,

Mx GSh(M)x .

Exercise 6.32. Consider X := p with its p-adic topology. This is a totally disconnectedcompact Hausdorff topological space. Let A := X , the constant sheaf with values in aring . Calculate Γ(X ,A).

7. Gluing Sheaves and Morphisms between Them

As a prelude to this abstract theory, today in class we saw two "geometric" versions.Let X be a topological space (the base), and let π : M → X be a map of spaces (a

continous function). We call (M , π) and X-space. A morphism of X-spaces f : M → N isa map f such that πN f = πM . See Figures below.

17 | file: notes-190201-d2


Given an open set U ⊆ X , a section of M over U is a map

σ : U → M

such thatπ σ = idU .

I.e. σ is a map of X-spaces. We denote by Γ(U, M) the set of sections over of M over U.See Figure:

The assignmentM : U 7→ Γ(U, M)

is a sheaf of sets on X . We call M the sheaf of sections of M .A map f : M → N of X-spaces induces a morphism

φ : M→ N

on the sheaves of sections.The first geometric tale was on gluing maps of X-spaces. We are given X-spaces

πM : M → X and πN : N → X , an open covering X = U =⋃

i∈I Ui , and for every i a

18 | file: notes-190201-d2


maps of X-spaces

fi : π−1M (Ui) → π−1

N (Ui).

The condition is that

fi |π−1M (Ui∩Uj )

= fj |π−1M (Ui∩Uj )

.

Then there is a unique map of X-spaces

f : M → N

such that

f |π−1M (Ui )

= fi

for all i. The reason: basic topology. See Figure below.

The second geometric tale today was on gluing X-spaces. We are given an open coveringX =

⋃i∈I Ui , and for every i a Ui-space

πi : Mi → Ui ,

for every i, j an isomorphism

fi, j : π−1i (Ui ∩Uj)

'−→ π−1

j (Ui ∩Uj)

of X-spaces. The condition is that

fj,k |π−1M (Ui∩Uj∩Uk )

fi, j |π−1M (Ui∩Uj∩Uk )

= fi,k |π−1M (Ui∩Uj∩Uk )

for all i, j, k. Then there is an X-space π : M → X , with isomorphisms of X-spaces

fi : π−1(Ui)'−→ Mi

such that

fi, j fi |π−1(Ui∩Uj )= fj |π−1(Ui∩Uj )

.

Again, the proof is just basic topology, with complicated bookkeeping. A partial figure is:

19 | file: notes-190201-d2


Exercise 7.1. Draw a full picture of this gluing procedure, with these 3 open sets.

By the first tale the X-space M that we get here is unique up to a unique isomorphism.The theorems that we want are abstract versions of the concrete geometric constructions

above.

Definition 7.2. Let M be a sheaf on a space X and let U ⊆ X be an open set. Therestriction of M to U is the sheaf M|U on U such that

Γ(M|U ,V) := Γ(M,V)

for every open set V ⊆ U, andrestM |U

W/V:= restMW/V

for every W ⊆ V ⊆ U open.

Theorem 7.3 (Gluing Sheaf Morphisms). LetM andN be sheaves on a topological spaceX , let X =

⋃i∈I Ui be an open covering, and let

φi : M|Ui → N |Ui

be morphisms of sheaves satisfying the condition

φi |Ui∩Uj = φ j |Ui∩Uj : M|Ui∩Uj → N |Ui∩Uj .

Then there is a unique morphism of sheaves

φ : M→ N

such thatφ|Ui = φi : M|Ui → N |Ui .

Theorem 7.4 (Gluing Sheaves). Let X be a topological space, let X =⋃

i∈I Ui be an opencovering, for every i let Mi be a sheaf on Ui , and for every i, j let

φi, j : Mi |Ui∩Uj

'−→Mj |Ui∩Uj

be an isomorphism of sheaves on Ui ∩Uj . The condition is that

φ j,k |Ui∩Uj∩Uk φi, j |Ui∩Uj∩Uk

= φi,k |Ui∩Uj∩Uk

as isomorphismsMi |Ui∩Uj∩Uk

'−→Mk |Ui∩Uj∩Uk

,

for all i, j, k.Then there is a sheaf M on X , together with isomorphisms

φi : M|Ui

'−→Mi ,

20 | file: notes-190201-d2


such thatφi, j φi |Ui∩Uj = φ j |Ui∩Uj : M|Ui∩Uj

'−→Mj |Ui∩Uj .

Moreover, that sheafM, with the collection of isomorphisms φi, is unique up to a uniqueisomorphism.

We will give a full proof next week.

Lecture 4, 7 Nov 2018

I owe you a nice example of the topological – or geometric – gluing.

Example 7.5. The base space is X = S1, the circle. We take the covering X =⋃

i∈I Ui

with I = 0, 1, 2 shown below.

Let Z := [−1, 1] ⊆ , the closed line segment. So X × Z is the ordinary, untwisted,band. Let φ : Z → Z be the homeomorphism (or better yet, diffeomorphism) ψ(z) := −z.(If you don’t know about manifolds with boundary and their diffeomorphisms, then taketake Z to be the open line segment.)[comment: (21Nov2018) In the book [Lee] there is a good discussion of manifolds withboundary.]

For i ∈ I we define the space (or differentiable manifold)

Mi := Ui × Z ,

with the obvious mapπi : Mi → Ui .

The gluing data (what will soon be called the 1-cochain...) φi, j is

φi, j := id× id : (Ui ∩Uj) × Z → (Ui ∩Uj) × Z

for (i, j) ∈ (0, 1), (1, 2), and

φ0,2 := id×ψ : (U0 ∩U2) × Z → (U0 ∩U2) × Z .

These are extended (i.e. for j ≤ i) by φi, j := φ−1j,i and φi,i := id.

Because the triple intersections are empty, the condition


= φi,k |Ui∩Uj∩Uk

is satisfied automatically.The resulting X-space (or manifold over X) M is the Mobius band of course.What invariant tells us that M is not homeomorphic (or diffeomorphic) to X × Z ?The only one I know is orientability. It is easier to explain in the differentiable case (but

still not easy). Here is a sketchy explanation...

21 | file: notes-190201-d2


In the differentiable version, the manifold M has its tangent bundle TM . This is a rank 2(real differentiable) vector bundle, that is glued by very similar formulas (the differentialsof the φi, j). Indeed, for every i the tangent bundle of Mi is trivial:

TMi Mi ×2,

and in the fiber direction 2 we glue by (1,±1).For the ordinary band the tangent bundle is trivial:

T(X × Z) X × Z ×2.

But not so for M . Still, why?Here is what we do.First, in general, for a rank d vector bundle p : E → M on M we have its frame bundle

FE , that is a sheaf of sets on M . Over every open set V ⊆ M we define Γ(V ,FE ) to be theset of vector bundle isomorphisms

(7.6) σ : V ×d '−→ p−1(V).

This is a sheaf, and the set Γ(V ,FE ) is either empty (if E is not trivial above V), or it isisomorphic as a set to

Γ(V ,V × GLd()) HomMfld(V ,GLd()) Γ(V ,GLd(OM )).

Here V ×GLd() is the bundle over V , and we look at sections of it; these are the same asmorphisms V → GLd() in the category Mfld of differentiable real manifolds; and also asthe sections on V of the sheaf of groups GLd(OM ), whereOM is the sheaf of differentiablemanifolds. If there is one frame σ, then we get all other frames by the action ofV ×GLd()

on V ×d .[comment: (21Nov2018) some changes below]

Now the topological group GLd() has two connected components (according to thedeterminant). Hence for a small connected open setV (small enough so that Γ(V ,FE ) , )the space p−1(V) has two connected components – see (7.6). Let conn(FE ) be the sheaf ofsets on M associated to the presheaf

V 7→ π0(Γ(V ,FE )),

the set of connected components. This is a locally constant sheaf of sets: it is locallyisomorphic to the constant sheaf of sets 1,−1. (On small open sets V ⊆ M this is theconstant sheaf.)

There are two options: either the sheaf conn(FE ) is the constant sheaf, or it is not. Thisis detected by the monodromy representation.

Suppose S is a locally constant sheaf of sets on a path connected space Y , that’s locallyisomorphic to the constant sheaf 1,−1. Then there is a representation

ρS : π1(M) → G,

where G is the 2-element group, seen as permutations of 1,−1. The monodromy ρS iseither trivial; and then S is the constant sheaf, and

Γ(Y , S) = 1,−1;

or ρS is not trivial, and thenΓ(Y , S) = .

Getting back to Mobius, the explicit gluing that we made shows that ρS , for Y := M ,E := TM and S := conn(FE ), is not trivial!

Geometrically this says that the manifold M is not orientable – an orientation of M is bydefinition a global section of conn(FE ). So either there are two or none. An orientation iswhat we need to integrate on a manifold (to get a consistent sign for the Jacobian matrix).

22 | file: notes-190201-d2


By “legitimate category”, or “very concrete category” we mean a category C that admitsinfinite products, infinite direct limits, finite fiber products, and a faithful functor

F : C→ Set

that respects the previous constructions. As we know, the categories Set, Grp, Ab, Rng andMod A for a ring A, all have these good properties. (Warning: F might not respect initialobjects and epimorphisms.)

We write CX for the category of sheaves with values in C, and CpreX for the category of

presheaves with values in it.Another general fact on sheaves, related to Definition 5.4.

[comment: (21Nov2018) There was a mistake earlier in item (1) below. It is now correct.The proof is given here: Proposition 7.18.]

Proposition 7.7. Let φ : M→ N be a morphism in CX .(1) Assume C = Ab. The morphism φ is surjective iff it is a categorical epimorphism

in CX .(2) φ is injective iff it is a categorical monomorphism in CX .


Before proceeding, I see that there is something we talked about in class that was nottyped in the notes. This is the equalizer diagram formulation of the sheaf axioms.

Recall that an equalizer sequence (also called a cartesian sequence) in a category C is adiagram

C0ε

−−−→ C1

δ0

−−−→−−−→δ1

C2

such that when we write it like this:

C0ε //

ε

C1

δ0

C1δ1

// C2

is a cartesian diagram, or synonymously a pullback diagram, or equivalently that

C0 C1 ×C2 C1,

the fibered product. The pair (C0, ε) is somtimes called the kernel of C1

δ0

−−−→−−−→δ1

C2.

In Set we know that

ε : C0'−→

c ∈ C1 | δ

0(c) = δ1(c).

Hence it is the same when C is a very concrete category (the forgetful functor F repectsfiber products).

Exercise 7.9. Prove that the kernel ε is a monomorphism in C.

We have seen that:

Proposition 7.10. A presheaf M ∈ CpreX is a sheaf iff for every open set U ⊆ X and every

open covering U =⋃

i∈I Vi the diagram

Γ(U,M)ε

−−−→∏i∈I

Γ(Vi ,M)δ0

−−−→−−−→δ1

∏j,k∈I

Γ(Vj ∩ Vk ,M)

is an equalizer sequence in C.

23 | file: notes-190201-d2


Here ε is the product on all i ∈ I of the restriction morphisms

restVi/U : Γ(U,M) → Γ(Vi ,M).

The morphism δ1 is the product on all i = j ∈ I of the product on all k ∈ I of

restVj∩Vk /Vj: Γ(Vj ,M) → Γ(Vj ∩ Vk ,M).

And the morphism δ0 is the product on all i = k ∈ I of the product on all j ∈ I of

restVj∩Vk /Vk: Γ(Vk ,M) → Γ(Vj ∩ Vk ,M).

We now provide proofs of the gluing theorems.

Proof of Theorem 7.3: gluing sheaf morphisms. Let V ⊆ X be an open set. Defining Vi :=V ∩Ui , we get an open covering V =

⋃i∈I Vi . Consider the following solid diagram

(7.11) Γ(V ,M) ε //

Γ(V,φ)

∏i∈I Γ(Vi ,M)

δ0//

δ1//

Γ(Vi,φi )

∏j,k∈I Γ(Vj ∩ Vk ,M)

Γ(Vj∩Vk,φ j ) = Γ(Vj∩Vk,φk )

Γ(V ,N ) ε //∏

i∈I Γ(Vi ,N )δ0

//

δ1//

∏j,k∈I Γ(Vj ∩ Vk ,N )

in the category C. This is commutative, by the compatibilty condition

φ j |Uj∩Uk= φk |Uj∩Uk

.

Therefore there is a unique morphism Γ(V , φ) on the dashed vertical arrow.As the open set V varies, we obtain a morphism of sheaves

φ : M→ N .

If V ⊆ Ui for some index i, then Vi = V , and therefore by the commutativity of the leftsquare in (7.11) – and neglecting all indices other than i – we see that

Γ(V , φ) = Γ(Vi , φi).

This means thatφ|Ui = φi .

The uniqueness of this φ is also because it is the only morphism that makes (7.11)commutative.

Proof of Theorem 7.4: gluing sheaves. Recall that we are given an open covering X =⋃i∈I Ui , a sheaf Mi on Ui , and an isomorphism

φi, j : Mi |Ui∩Uj

'−→Mj |Ui∩Uj

for every i, j. The condition is that


= φi,k |Ui∩Uj∩Uk.

Take a point x ∈ X . Let us denote by Mi,x the stalk of Mi at x. There is an objectMx ∈ C, together with an isomorphism

φi,x : Mi,x'−→Mx

for every i, such thatφ j,k,x φi, j,x = φi,k,x .

Moreover, the object Mx , with its collection of isomorphisms φi,x, is unique (up to aunique isomorphism).

24 | file: notes-190201-d2


Let us define the sheaf M on X as follows:

Γ(V , M) :=∏x∈V

Mx .

(This will eventually be the Godement sheaf of M.) On every Ui there is a morphism ofsheaves

φi : Mi → M|Ui ,

and it gives rise to an isomorphism of sheaves

(7.12) GSh(Mi)'−→ M|Ui .

A sectionmxx∈V ∈ Γ(V , M)

will be called geometric relative to the collection Mi if for every x ∈ V there is an opensetW s.t. x ∈ W ⊆ V ∩Ui for some i, and a section m ∈ Γ(W ,Mi), s.t. φi,y(m) = my ∈My

for all y ∈ W .Now let M be the subpresheaf of M defined as follows:

Γ(V ,M) ⊆ Γ(V , M)

is the subset of all geometric sections, in the relative sense as above. As we already knowfrom previous calculations, M is a subsheaf of M; and M GSh(M).

For every index i, the isomorphism (7.12) identifies Mi with M|Ui , as the subsheavesof geometric sections of M|Ui . This is the isomorphism

φi : M|Ui

'−→Mi

that we want. By construction these satisfy

φi, j φi |Ui∩Uj = φ j |Ui∩Uj .

The uniqueness (up to unique isomorphism) of M is a consequence of Theorem 7.3,and is left as an exercise.

Exercise 7.13. Finish the proof above.


[comment: (21Nov2018) (1) No lecture 14 Nov. (2) There are corrections above.]I did not mean to introduce the next definition now, but it is needed for the proof of

Proposition 7.18 (the correction of Prop 7.7(1)).Note that for M,N ∈ Abpre

X and an open set U ⊆ X the set of morphisms M|U → N |Uin Abpre

U is itself an abelian group. We denote it by

HomAbpreU(M|U ,N |U ).

Also recall that AbU (sheaves) is a full subcategory of AbpreU (presehaves).

Definition 7.14. Let X be a topological space and let φ : M→ N be a homomorphism inAbpre

X . The cokernel of φ is the presheaf

Cokerpre(φ) ∈ AbpreX

defined byCokerpre(φ)(U) := Coker

(Γ(U, φ) : Γ(U,M) → Γ(U,N )

).

25 | file: notes-190201-d2


Definition 7.15. Let X be a topological space and let φ : M→ N be a homomorphism inAbX . The cokernel of φ is the sheaf

Coker(φ) := Sh(Cokerpre(φ)) ∈ AbX .

There is a canonical homomorphism

π : N → Coker(φ)

in AbX that’s induced from the homomorphism

N → Cokerpre(φ)

in AbpreX .

Proposition 7.16. Let φ : M→ N and be a homomorphism in AbX .(1) The canonical homomorphism π : N → Coker(φ) in AbX is surjective.(2) The composition π φ is the zero homomorphism.(3) Let ψ : N → P be a homomorphism in AbX such that ψ φ = 0. Then ψ factors

uniquely through Coker(φ). Namely there is a unique morphism

ψ : Coker(φ) → P

in AbX such thatψ = ψ π.

(4) For every point x ∈ X the sequence

Mxφx−−→ Nx

πx−−→ Coker(φ)x → 0

in Ab is exact. In other words, πx induces an isomorphism

Coker(φx)'−→ Coker(φ)x .

Exercise 7.17. Prove Proposition 7.16.

Recall that the trivial abelian group is denoted by 0. It is both the initial and terminalobject of the category Ab. Given a space X , let 0X ∈ AbX be the constant sheaf with valuesin the group 0. This sheaf is both the initial and terminal object of the category AbX .

Proposition 7.18. Let φ : M→ N be a homomorphism in AbX . The following conditionsare equivalent:

(i) φ is surjective (Definition 5.4).(ii) φ is a categorical epimorphism in AbX .(iii) The sheaf Coker(φ) is the constant sheaf 0X .

Proof.(i)⇒ (ii): This is easy, and was done in class by Yotam.

(ii)⇒ (iii): Let P := Coker(φ). We consider these two homomorphisms in AbX :

ψ0, ψ1 : N → P ,

ψ0 := 0 and ψ1 := π. These both satisfy

ψi φ = 0.

But φ is a categorical epimorphism, so we must have ψ0 = ψ1. This means that π = 0.According to Proposition 7.16(1) the homomorphism π is surjective. Hence P = 0X .

(iii)⇒ (i): We are given that Coker(φ) = 0X , so for every point x ∈ X the stalk is

Coker(φ)x = 0 ∈ Ab .

26 | file: notes-190201-d2


Using 7.16(4) we conclude thatφx : Mx −→ Nx

is surjective. Thus φ is a surjection of sheaves.

8. Vector Bundles

We are going to discuss real vector bundles on spaces and manifolds. This will take usa step closer to a modern understanding of geometry. Later we will use similar ideas forschemes.

Convention 8.1. We shall work in one of the following categories:• The category Top of topological spaces and continous maps between them. Here = , the field of real numbers.• The categoryMfld of real differentiable manifolds and differentiable maps betweenthem, where by differentiable we mean of class C∞. Here = .• The categoryVar of quasi-projective algebraic varieties over an algebraically closedfield .

We shall denote by Sp any of these categories, and refer to an object of Sp as a space.

The category Sp has finite products. These products respect the forgetful functor

Sp→ Set .

Warning: the forgetful functorVar→ Top

does not respect products. On the other hand, the forgetful functor

Mfld→ Top

does respect products.In Top and Var we have fiber products: given f : Y → X and g : Z → X the fiber

product is the closed subset

(8.2) Y ×X Z = (y, z) | f (y) = g(z) ⊆ Y × Z

with the induced structure of a closed subspace.Fiber products in Mfld are more delicate, because the closed subset in (8.2) is not a

submanifold in general.Recall that a map f : Y → X in Mfld is called a submersion if for every point y ∈ Y the

linear map on tangent spaces

d( f )y : Ty Y → T f (y) X

is surjective. (See [Lee].)

Example 8.3. The inclusion f : U → X of an open subset is a submersion in Mfld.

Exercise 8.4. Prove that if f : U → X is the inclusion of an open set in Sp, then for everyg : Z → X in Sp the fiber product exists, and it is

U ×X Z g−1(U) ⊆ Z .

Example 8.5. A submersion f : Y → X in Mfld of relative dimension 0 (i.e. dim(Y ) =dim(X)) is a local diffeomorphism.

Lemma 8.6. Given maps f : Y → X and g : Z → X in Mfld such that either f or g is asubmersion, the closed subset

(y, z) | f (y) = g(z) ⊆ Y × Z

is a submanifold. It is the categorical fiber product Y ×X Z in Mfld.

27 | file: notes-190201-d2


Exercise 8.7. Prove the lemma. (Hint: use the Implicit Function Theorem.)

Remark 8.8. From a contemporary point of view, Mfld is the wrong category. It is a relicfrom decades ago, and it is “deficient”.

There actually is a theory of “differentiable spaces”, very recent, due to D. Joyce. It isnew and not many people are aware of it. The manifolds are the nonsingular objects in thecategory of differentiable spaces.

Remark 8.9. What are the “manifolds” in Var ? These are the nonsingular varieties. Theyhave tangent spaces, and one can talk about submersions in Var. But these have anothername: smooth maps of varieties. Lemma 8.6 holds in Var. See [Har, Prop III.10.4].

A smooth map f of relative dimension 0 is called an étale map. I is usually not a localisomorphism!

In algebraic geometry we also talk about smooth maps f : Y → X between singularvarieties.

If we are lucky (it is a matter of time) we will be able to talk about smooth maps betweenschemes.

Next is a nonstandard definition.

Definition 8.10. A map Y → X in Sp is called Sp-fibered, and Y is called Sp-fibered overX , if for every Z → X in Sp the fibered product Y ×X Z exists in Sp.

Thus when Sp = Top or Sp = Var this is an empty condition (all maps are Sp-fibered),and when Sp = Mfld all submersions a fibered (by Lemma 8.6).

For every n ≥ 0 we have the affine n-dimensional space An(). As a set it is n. It isviewed as an object of Sp, either as a topological space with the usual metric topology, oras a differentiable manifold with the usual differentiable structure, or as an algebraic varietywith the Zariski topology with the usual algebro-geometric structure, as the case may be.

For n = 0, A0() is a single point, that we denote by 0. Note that A0() is the terminalobject of Sp.

For n = 1, A1() is a commutative ring object in Sp. There are maps

(8.11)

add : A1() × A1() → A1(), add(a, b) := a + b,

mult : A1() × A1() → A1(), mult(a, b) := a ·b,

0 : A0() → A1(), 0(0) := 0 ∈ ,

1 : A0() → A1(), 1(0) := 1 ∈

in Sp that satisfy the axioms of a commutative ring. After applying the forgetful functor toSet we recover the familiar operations of the ring .

For other values of n ∈ the space An() is an A1()-module in Sp, i.e. literally avector space. This just means that there are maps

(8.12)

add : An() × An() → An(), add(v, w) := v + w ∈ n,

mult : A1() × An() → An(), mult(a, v) := a ·v ∈ n,

0 : A0() → An(), 0(0) := 0n ∈ n

in Sp that satisfy the axioms of a module (for the ring structure (8.11). After applying theforgetful functor to Set we recover the familiar operations of the -module n.

The idea of a vector bundle is meant to provide a relative version of (8.12): a collectionof vector spaces that move – continously or smoothly or algebraically – over a base X .

Fixing a space X ∈ Sp (the base), we can talk about the category Sp/X of X-spaces,or spaces over X . An object of Sp/X is a pair (Y , πY ) where πY : Y → X is a map in Sp,

28 | file: notes-190201-d2


called the structure map. A morphism

g : (Y , πY ) → (Z , πZ )

in Sp/X is a map g : Y → Z satisfying

πZ g = πY .

In a commutative diagram:

Yg

//

πY

Z

πZ

X

We shall often keep πY implicit.We need to know about finite products in Sp/X . These are just the fibered products in

Sp. To be precise:

(8.13) (Y , πY ) × (Z , πZ ) = (Y ×X Z , π),

whereY ×X Z ⊆ X × Y

is the fibered product in Sp (if it exists); and

π(y, z) := πY (y) = πZ (z) ∈ X

for (y, z) ∈ Y ×X Z .Thus in the cases Sp = Top and Sp = Var all finite products exist in Sp/X .In the case Sp = Mfld, in many important instances at least one of the maps πY or πZ

will be a submersion, so the finite product will exist. Here is a typical good situation in thiscase. Given any space Y0, consider the product

(8.14) Y := X × Y0

Define

(8.15) πY : Y = X × Y0 → X

to be the projection on the first coordinate. This is a submersion, so Y is Sp-fibered over X .

Lemma 8.16. Let (U, πU ) ∈ Sp/X , let Y0 ∈ Sp, and let (Y , πY ) ∈ Sp/X be as in (8.14)and (8.15).

(1) There is an isomorphism of sets

HomSp/X (U,Y ) = HomSp/X (U, X × Y0) HomSp(U,Y0).

It is functorial in U and Y0.(2) There is an isomorphism

U ×X Y U × Y0

in Sp. It is functorial in U and Y0.

Exercise 8.17. Prove the lemma. Give explicit formulas for the isomorphisms.

Before approaching the vectors bundles, let’s talk about more general (and less struc-tured) bundles.

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Definition 8.18. Let X ∈ Sp. A fiber bundle over X is an object (Y , πY ) in Sp/X with thisproperty: there is a space Z , an open covering X =

⋃i∈I Ui , and isomorphisms

φi : Ui × Z'−→ π−1

Y (Ui)

in Sp/Ui . The space Z is called the fiber, and the isomorphisms φi are called localtrivializations.

Example 8.19. TheMöbius band is a fiber bundle in Top (or, if we choose the open version,in Mfld). The base is X = S1, and the fiber is Z = [−1, 1] (or Z = (−1, 1)).

Proposition 8.20. Let (Y , πY ) be a fiber bundle over X with fiber Z .(1) If Sp = Mfld then πY is a submersion.(2) The object (Y , πY ) ∈ Sp/X is Sp-fibered.(3) For every point x ∈ X the fiber

π−1Y (x) x ×X Y

is an object of Sp, and it is isomorphic (noncanonically) to Z .

Exercise 8.21. Prove the proposition.


Recall that Sp is either Top, Mfld or Var, and correspondingly the field is either , or algebraically closed.

Last time we talked about the vector spaces An(), that are A1()-modules in Sp. Wealso talked about fiber bundles in Sp. Now we are going to combine these notions.

Definition 8.22. We fix a base X ∈ Sp. For every n ∈ consider the X-space

(8.23) An(X) := X × An()

with structure mapπAn(X)(x, v) := x,

the projection on the first coordinate.

Of course A0(X) = X .For n = 1, A1(X) is a commutative ring object in Sp/X . This structure is induced

from the commutative ring structure of A1() in formula (8.11). Here it is explicitly. Asmentioned above (in Prop 8.20(2)) the fiber product

A1(X) ×X A1(X) ⊆ A1(X) × A1(X)

exists in Sp. (In fact it is isomorphic to A2(X), but that’s not helpful here.) The points inA1(X) are pairs

((x, a), (x, b)

)with x ∈ X and a, b ∈ . The operations are:

(8.24)

add : A1(X) ×X A1(X) → A1(X), add((x, a), (x, b)

):= (x, a + b),

mult : A1(X) ×X A1(X) → A1(X), mult((x, a), (x, b)

):= (x, a ·b),

0X : A0(X) → A1(X), 0(x) := (x, 0),

1X : A0(X) → A1(X), 1(x) := (x, 1).

These are maps in Sp/X .For any value n ∈ , the space An(X) is an A1(X)-module object in Sp/X . This

structure is induced from formula (8.12). Again in explicit terms: the fiber product

An(X) ×X An(X) ⊆ An(X) × An(X)

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Figure 2. A rank n vector bundle E over X

exists in Sp. The points in it are((x, v), (x, w)

)with x ∈ X and v, w ∈ n. The operations

are:

(8.25)

add : An(X) ×X An(X) → An(X), add((x, v), (x, w)

):= (x, v + w),

mult : A1(X) ×X An(X) → An(X), mult((x, a), (x, v)

):= (x, a ·v),

0X : A0(X) → An(X), 0(x) := (x, 0).

The map 0X is called the zero section.

Definition 8.26. Let n be a natural number. The standard trivial rank n vector bundle overX is the space

An(X) = X × An()

with the operations defined in (8.25).

Definition 8.27. Let X ∈ Sp and n ∈ . A rank n vector bundle over X is an Sp-fiberedobject (E , πE ) in Sp/X , with these maps

addE : E ×X E → E ,

multE : A1(X) ×X E → E ,

0E : X → E

in Sp/X that are called the operations. The conditions are:(i) The operations make E into an A1(X)-module in Sp/X , i.e. the module axioms are

satisfied.(ii) Local triviality: there is an open covering X =

⋃i∈I Ui , and for every i an

isomorphism

φi : An(Ui)'−→ π−1

E (Ui)

of A1(Ui)-modules, i.e. an isomorphism in Sp/Ui that respects the operations.

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See Figure 2.Notice that for every point x ∈ X the fiber

(8.28) E(x) := π−1E (x) x ×X E

is isomorphic to An() as -modules.

Example 8.29. Assume Sp = Mfld. Let X be an n-dimensional manifold, an object ofMfld. The tangent bundle TX is a rank n vector bundle on X .

Exercise 8.30. If you don’t understand the example above, then read about it (in [Lee] oranother book) and write a proof.

Remark 8.31. This is an elaboration of Example 8.29. Unfortunately you won’t find thismaterial in [Lee], and probably not in any other textbook on differential geometry. It is"extra material". We might do the algebraic version next semester (for schemes).

Let X be an n-dimensional differentiable manifold (so Sp = Mfld and = ). Thetangent bundle is TX , and let TX be the sheaf of section of TX , i.e.

Γ(U, TX ) := Γ(U,TX) = HomSp/X (U,TX)

for every open set U ⊆ X . It is standard to refer to a section ∂ ∈ Γ(U,TX) as a vector fieldon U.

This is not mysterious, at least when X = An(). Let t1, . . . , tn be the coordinatefunctions. Then there are the constant vector fields

∂i :=∂

∂ti,

and every vector field ∂ can be written uniquely as

∂ =

n∑i=1

fi ·∂i

with fi ∈ Γ(X ,OX ). Moreover, ∂ is a derivation of A := Γ(X ,OX ), as defined in the nextparagraph, and

fi = ∂(ti).

Suppose A is a commutative -ring. A derivation of A (relative to ) is an -linearhomomorphism

∂ : A→ A

satisfying the Leibniz rule∂(a ·b) = ∂(a)·b + a ·∂(b).

The set of derivations of A is Der(A).This can be done for sheaves. Consider the sheaf of differentiable functions OX on X .

For an open set U ⊆ X we can look at derivations

∂ : OU → OU ,

namely -linear sheaf homomorphisms on U that satisfy the Leibniz rule on every openset U ′ ⊆ U (or equivalently, in all stalks at x ∈ U). These form an -module

Der(OU ) ⊆ HomX (OU ,OU ).

As U moves we get a sheaf Der(OX ) such that

Γ(U,Der(OX )) = Der(OU ).

Theorem. There is a canonical isomorphism of sheaves of -modules

Der(OX ) TX .

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Proposition 8.32. Let X be a space in Sp, and let E , F be vector bundles on X , of ranksm, n respectively.

(1) There is a vector bundle E ⊕ F on X , whose fibers are

(E ⊕ F)(x) = E(x) ⊕ F(x).

(2) There is a vector bundle E ⊗ F on X , whose fibers are

(E ⊗ F)(x) = E(x) ⊗ F(x).

(3) There is a vector bundle Hom(E , F) on X , whose fibers are

Hom(E , F)(x) = Hom(E(x), F(x)).

(4) There is a vector bundle∧k(E) on X , whose fibers are∧k(E)(x) =

∧k(E(x)),

the k-the exterior power, for 0 ≤ k ≤ m.(5) There is a vector bundle Symk(E) on X , whose fibers are

Symk(E)(x) = Symk(E(x)),

the k-the symmetric power, for 0 ≤ k.

Exercise 8.33. Prove this proposition. What are the ranks of the various vector bundlesthere?

As a prelude to next week’s lecture, do the following exercise.

Exercise 8.34. Let E be a vector bundle over X , let U ⊆ X be an open set, and let

σ, τ : U → E

be sections, i.e.σ, τ ∈ HomSp/X (U, E).

Then there is a unique section

σ + τ ∈ HomSp/X (U, E)

such that for every point x ∈ U we have

(σ + τ)(x) = σ(x) + τ(x) ∈ E(x).

(Hint: this is easy if you understand the definitions well.)

Lecture 7, 5 Dec 2018

Last time we defined vector bundles. Recall that Sp is either Top, Mfld or Var, andcorrespondingly the field is either , or algebraically closed.

Definition 8.35. Let X be a space in Sp, and let E and F be vector bundles on X . A mapof vector bundles

φ : E → F

is a map φ in Sp/X that respects the operations add and mult.We denote by Vec/X the category of vector bundles on X .

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We could require φ to respect the zero sections, but that is a consequence of the operationmult.

Note that the category Vec/X is deficient: given φ : E → F, there usually aren’t vectorbundles E ′ and F ′, and maps α : E ′ → E and β : F → F ′, such that in all fibers thesequence of -modules

0→ E ′(x)α(x)−−−→ E(x)

φ(x)−−−→ F(x)

β(x)−−−→ F ′(x) → 0

is exact.

Exercise 8.36. Find an example of the phenomenon above, and explain what goes wrong.(See Proposition 9.14 for another point of view.)

This weakness of Vec/X is an important reason to work with sheaves, as we now do.

9. Ringed Spaces and Sheaves of Modules

We now take a break from vector bundles.

Definition 9.1. A ringed space is a pair (X ,OX ), where X is a topological space X , andOX is a sheaf of commutative rings on X .

Every space X ∈ Sp, see Convention 8.1, is actually a ringed space.• When Sp = Top, OX is the sheaf of continous -valued functions.• When Sp = Mfld, OX is the sheaf of differentiable -valued functions.• When Sp = Var, OX is the sheaf of algebraic -valued functions.

In all these cases the formula for OX is this:

Γ(U,OX ) = HomSp(U,A1()),

and the ring structure comes from that of A1() in (8.11).

Definition 9.2. Let (X ,OX ) be a ringed space. A sheaf of OX -modules on X , or anOX -module, is a sheaf of abelian groupsM on X , together with a structure of a Γ(U,OX )-module on Γ(U,M) for every open set U ⊆ X , such that for every inclusion of open setsV ⊆ U the homomorphism

restMV/U : Γ(U,M) → Γ(V ,M)

is Γ(U,OX )-linear.

Definition 9.3. Let (X ,OX ) be a ringed space, and let M and N be OX -modules. Ahomomorphism of OX -modules from M to N is a homomorphism of sheaves of abeliangroups

φ : M→ N

such that for every open set U ⊆ X the homomorphism

Γ(U, φ) : Γ(U,M) → Γ(U,N )

is Γ(U,OX )-linear.The category of OX -modules is denoted by ModOX .

There are forgetful functors

ModOX → ModX → ModX = AbX .

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Definition 9.4. A sequence of homomorphisms

· · · →Mi φi

−−→Mi+1 φi+1

−−−→Mi+1 → · · ·

in ModOX is called exact if for every point x ∈ X the sequence of homomorphisms

· · · →Mix

φix−−→Mi+1

x

φi+1x−−−→Mi+1

x → · · ·

in ModOX,x is exact.

Example 9.5. Let φ : M→ N be a homomorphism in ModOX .(1) φ is surjective iff

Mφ−→ N → 0

is exact. Here 0 is the zero sheaf.(2) φ is injective iff

0→Mφ−→ N

is exact.

Exactness can be checked also in ModX or AbX .In Definitions 7.14 and 7.15 we learned about the cokernel of a sheaf homomorphism

in AbX . It works also for homomorphisms in ModOX . Namely Coker(φ) is the sheafassociated to the presheaf

Cokerpre : U 7→ Coker(Γ(U, φ) : Γ(U,M) → Γ(U,N )

).

Definition 9.6. Given a homomorphisms φ : M → N in ModOX , its kernel is theOX -module Ker(φ) such that for every open set U

Γ(U,Ker(φ)) := Ker(Γ(U, φ) : Γ(U,M) → Γ(U,N )

).

Exercise 9.7.(1) Verify that the sheaf Coker(φ) is an OX -module.(2) Verify that Ker(φ) is a sheaf, and it is an OX -module.

Proposition 9.8. Given a homomorphism φ : M→ N in ModOX , the sequence

0→ Ker(φ) →Mφ−→ N → Coker(φ) → 0

in ModOX is exact.

Proof. By definition for every open set U we have an exact sequence of abelian groups

(EU ) 0→ Γ(U,Ker(φ)) → Γ(U,M)Γ(U,φ)−−−−−→ Γ(U,N ) → Γ(U,Cokerpre(φ)) → 0.

Choose some point x ∈ X . Passing to the direct limit on all exact sequences EU , where Uis an open neighborhood of x, we get an exact sequence

(Ex) 0→ Ker(φ)x →Mxφx−−→ Nx → Coker(φ)x → 0.

Here we are using the fact that the direct limit of a direct system of exact sequences is exact(see below), and that

Cokerpre(φ)x → Coker(φ)xis bijective.

Theorem 9.9. Let Eii∈I be a direct system of sequences of A-modules, and let

E := limi→ Ei .

If all the sequences Ei are exact, then so is E.

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Exercise 9.10. Prove this theorem. (Unless you already know it.)

Exercise 9.11. State and prove the categorical properties of Ker and Coker insideModOX .(Cf. Prop 7.16(3).)

For an open set U ⊆ X there is a functor

RestU/X : ModOX → ModOX |U , M 7→M|U .

Definition 9.12. Let (X ,OX ) be a ringed space. A sheaf of OX -modules E on X is calledlocally free of rank n, for some n ∈ , if there is an open covering U = Uii∈I of X , andfor every i there is an isomorphism

φi : O⊕nUi

'−→ E |Ui

of OUi -modules. We say that the covering U trivializes E , and that the collection φii∈Iis a trivialization of E on U .

Just like in module categories, an exact sequence

0→ Lφ−→M

ψ−→ N → 0

in ModOX is called split if there is a homomorphism τ : N →M such that ψ τ = idN .

Exercise 9.13. Suppose

(♥) 0→ Lφ−→M

ψ−→ N → 0

is a split exact sequence in ModOX . Show that

M L ⊕ N .

Proposition 9.14. Let

(E) 0→ Lφ−→M

ψ−→ N → 0

be an exact sequence in ModOX . Assume that N is locally free of finite rank. Then (†) islocally split; namely every point x ∈ X has an open neighborhoodU such that the sequenceof OU -modules

0→ L|Uφ−→M|U

ψ−→ N |U → 0

is split.


We now return to vector bundles.Let X be a space and (Y , πY ) an X-space. For an open set U ⊆ X let us write

(9.16) Γ(U,Y ) := HomSp/X (U,Y ),

the set of sections σ : U → Y in Sp/X of πY : Y → X over U.

Lemma 9.17. The assignmentU 7→ Γ(U,Y )

is a sheaf of sets on X . It is called the sheaf of sections of Y , and is denoted by ShSeX (Y ).

Exercise 9.18. Prove the lemma.

Proposition 9.19. Let X ∈ Sp. There is a canonical isomorphism of sheaves of ringsbetween OX and the sheaf ShSeX (A1(X)) of sections of the ring bundle A1(X).

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Proof. Recall thatA1(X) = X × A1().

As we saw in Lemma 8.16(1), for an open set U ⊆ X there is an isomorphism

Γ(U,A1(X)) HomSp(U,A1()).

And by definitionΓ(U,OX ) = HomSp(U,A1()).

We see that OX is the sheaf of sections of the bundle A1(X). The ring structure in bothcases comes from that of A1().

Proposition 9.20. Let X ∈ Sp, and let (E , πE ) be a rank n vector bundle on X . LetE := ShSeX (E), the sheaf of sections of E . Then E is a locally free OX -module of rank n.

Proof. The operations on E give E a structure of anOX -module. IfU = Uii∈I is an opencovering that trivializes E , and φii∈I is a trivialization, then this same data trivializes E ,so E is locally free of rank n.

Exercise 9.21. Let E be a vector bundle on X in Sp, with associated locally free sheaf E .Show that for every point x ∈ X there is a canonical isomorphism of -modules

E(x) ⊗OX,x Ex .

Here E(x) is the fiber at x, and Ex is the stalk at x, which is a module over the ring OX,x .

Definition 9.22. Let (X ,OX ) be a ringed space. The full subcategory of ModOX on thefinite rank locally free sheaves is denoted by LFModOX .

Theorem 9.23. Let X ∈ Sp. The assignment

ShSeX : Vec/X → LFModOX

is an equivalence of categories.

The proof will be given next week.For the proof we will need the next general result.

Theorem 9.24. The following are equivalent for a functor F : C→ D.(i) F is an equivalence.(ii) F is fully faithful, and essentially surjective on objects.

Exercise 9.25. Prove the theorem. (Hint: use the axiom of choice to construct a quasi-inverse of F.)


Today’s lecture turned out to be dedicated to reviewing earlier material and expandingit.

Our first topic is solving Exercise 9.7 For this we introduce the category Modpre OX ofpresheaves of OX -modules. It is defined just like in Definitions 9.2 and 9.3, except theobjects are presheaves.

Lemma 9.26. Given M ∈ Modpre OX , the sheaf of abelian groups Sh(M) has a uniquestructure of OX -modules such that τM : M→ Sh(M) is OX -linear.

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Proof. Take an open set U ⊆ X . An element f ∈ Γ(U,OX ) induces, by multiplication, ahomomorphism of presehaves

f ·(−) : M|U →M|U .

This is viewed as a morphism in AbpreU . Applying the functor Sh we get a morphism

f ·(−) : Sh(M|U ) → Sh(M|U )

in AbU . But Sh(M|U ) = Sh(M)|U , and so we obtain a homomorphism of abelian groups

f ·(−) : Γ(U, Sh(M)) → Γ(U, Sh(M)).

A little checking shows that this gives Γ(U, Sh(M)) a structure of a Γ(U,OX )-module. AsU changes we get an OX -module structure on Sh(M). And it respects τM.

The uniqueness can be checked in stalks.

Lemma 9.27. There is a functor

Sh : Modpre OX → ModOX

that respects the forgetful functor to AbX .

This is an immediate consequence of the previous lemma.Now back to our exercise. We are given a homomorphism φ : M→ N inModOX . On

every open set U we get an exact sequence

0→ Γ(U,Kerpre(φ)) → Γ(U,M)φ−→ Γ(U,N ) → Γ(U,Cokerpre(φ)) → 0.

This makes both Γ(U,Kerpre(φ)) and Γ(U,Cokerpre(φ)) into Γ(U,OX )-modules. As Uvaries we get

Kerpre(φ),Cokerpre(φ)) ∈ Modpre OX .

ButKer(φ) = Kerpre(φ),

so it is in ModOX . By Lemma 9.26 we know that

Ker(φ) ∈ ModOX .

This concludes Exercise 9.7.

– – –

Next we talked about Exercise 9.21. Here I forgot to tell you what is the -ringhomomorphism OX,x → . It comes from the next proposition.

Proposition 9.28. Let X ∈ Sp and x ∈ X .(1) There is a unique -ring homomorphism

evx : OX,x →

such that for every open neighborhood U of x and every function f ∈ Γ(U,OX ),with germ fx ∈ OX,x , there is equality

evx( fx) = f (x) ∈ .

(2) Let mx := Ker(evx). Then mx is the only maximal ideal of the ring OX,x , andhence this is a local ring.

We sometimes writek(x) := OX,x/mx ,

and call it the residue field of x. In our three geometries we always have k(x) = ; butlater, in schemes, this will usually not be the case.

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Now I can again give you:

Exercise 9.30. Let X ∈ Sp, and let E be a vector bundle on X , with associated locally freesheaf E . Show that for every point x ∈ X there is a canonical isomorphism of -modules

E(x) ⊗OX,x Ex .

Here E(x) is the fiber at x, and Ex is the stalk at x, which is a module over the ring OX,x .(I talked about this in class, but please write it in full.)

– – –

The last topic is about Exer 8.36. Here is a possible solution (perhaps the simplestone). It is like what Yotam and Hezi wrote, but without the irrelevant stuff (the two extracomponents).

Take X := A1() ∈ Sp and

E = F := A1(X) = X × A1() = A1() × A1().

A point in E is a pair (x, e), where x, e ∈ . The projection is πE : E → X , πE (x, e) = x.Likewise for F. The vector bundle map we choose is

φ : E → F, φ(x, e) := (x, x ·e).

If x = 0 then in the fiber the homomorphism E(x) → F(x) is 0. If x , 0 then thehomomorphism E(x) → F(x) is bijective. We see that the kernels and cokernels havedifferent ranks as x changes (either 0 or 1). If there were vector bundles E ′ and F ′, withmaps α : E ′→ E and β : F → F ′, such that the sequence

0→ E ′(x)α−→ E(x)

φ−→ F(x)

β−→ F ′(x) → 0

is exact for every x, then the ranks would be constant.Let’s now be smarter, and analyze this situation using sheaves. The sheaves of sections

of E and F are E and F respectively. The homomorphism φ : E → F in ModOX has akernel and a cokernel, say K := Ker(φ) and C := Coker(φ). They sit in this exact sequence

(9.31) 0→ Kα−→ E

φ−→ F

β−→ C → 0

in ModOX . We can calculate these sheaves.Let us introduce the coordinate function

t : A1() = X → ,

which is nothing but the identity of is disguise, i.e. t(x) = x. But it will be good to havenotation for it, as a global section of OX . Indeed, as a homomorphism of OX -modules,

φ : E = OX → F = OX

is multiplication by t.

Claim 1. The homomorphism φ is injective (so K = 0). Let’s see why. Take a point x ∈ X .To prove that

φx : Ex = OX,x → Fx = OX,x

is injective is the same as to prove that the germ tx ∈ OX,x is not a zero-divisor. Thereare two cases. First assume x , z, where z is the origin (i.e. t(z) = 0). In this case tx isinvertible in OX,x , so a fortiori it is not a zero-divisor.

Now take x = z. Here tz ∈ mz , so it is not invertible. Let fz ∈ OX,z be a germ, andassume that tz · fz = 0 in OX,z . We will show that fz = 0. Now fz has some representative

39 | file: notes-190201-d2


f ∈ Γ(U,OX ) on a neighborhood U of z, and fz ·tz = 0 means that on some smallerneighborhood V of z we have t |V · f |V = 0 in Γ(V ,OX ). So t(y)· f (y) = 0 for all y ∈ V .But t(y) , 0 for all y ∈ V − z, so f (y) = 0 for all y ∈ V − z. By continuity we getf (z) = 0 too. So f = 0 in Γ(V ,OX ), and thus fz = 0 in the stalk.

Claim 2. The sheaf C is supported on the closed set Z := z, so it is a "skyscraper sheaf".Indeed, by the calculations above for every x there is a short exact sequence

0→ Ex = OX,xtx ·(−)−−−−−→ Fx = OX,x → Cx → 0.

So Cx = 0 if x , z, andCz = OX,z/(tz) , 0.

The actual structure of the stalk Cx is clear only when Sp = Var. In this case the element tzgenerates the maximal ideal mz , so Cz = k(z) = . But in the other geometries there arerapidly decaying functions in mz , like exp(−t−2), that are not divisible by t; and hence Czis bigger.

From the two claims we see that for every x ∈ X − z there is an exact sequence

0→ Extx ·(−)−−−−−→ Fx → 0.

Passing to the fibers, i.e. tensoring with k(x) = , we get an exact sequence

0→ E(x)t(x) ·(−)−−−−−−→ F(x) → 0.

Just as expected.But for x = z something interesting happens. We have the short exact sequence of

OX,z-modules

0→ Eztz ·(−)−−−−→ Fz → Cz → 0.

Now the functor k(z)⊗OX,z (−) is right exact, so we get this exact sequence of k(z)-modules:

k(z) ⊗OX,z Ezt(z) ·(−)−−−−−−→ k(z) ⊗OX,z Ez → k(z) ⊗OX,z Cz → 0.

To close it up to an exact sequence we use the long exact sequence of the left derived functorTor, plus the fact that

TorOX,z

1 (k(z),Fz) = 0.We obtain this exact sequence of k(z)-modules:(9.32)

0→ TorOX,z

1 (k(z), Cz) → k(z) ⊗OX,z Ezt(z) ·(−)−−−−−−→ k(z) ⊗OX,z Ez → k(z) ⊗OX,z Cz → 0.

We need to figure out what are TorOX,z

1 (k(z), Cz) and k(z) ⊗OX,z Cz .But we already know that

k(z) ⊗OX,z Ez k(z)and

k(z) ⊗OX,z Fz k(z).Also t(z) = 0. We conclude that

k(z) ⊗OX,z Cz k(z)

andTorOX,z

1 (k(z), Cz) k(z).Thus (9.32) becomes this exact sequence

0→ k(z)−→ k(z)

0−→ k(z)

−→ k(z) → 0.

This agrees with what we saw in the more simplistic discussion above, but now we under-stand where the first k(z) comes from: it is the value of a derived functor.

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This lecture will start with a couple of general constructions on ringed spaces. Afterthat we’ll talk about homework.

Proposition 9.33. Let (X ,OX ) be a ringed space, and let M,N ∈ ModOX . There is anOX -module HomOX (M,N ) such that

Γ(U,HomOX (M,N )) = HomModOU(M|U ,N |U )

for every open set U ⊆ X .


It is convenient, and customary, to write

HomX (M,N ) := HomModOX(M,N )

andHomX (M,N ) := HomOX (M,N ).

Definition 9.35. Let (X ,OX ) be a ringed space. GivenM,N ∈ ModOX we letM⊗OX Nbe the OX -module associated to the presheaf

U 7→ Γ(U,M) ⊗Γ(U,OX ) Γ(U,N ).

Exercise 9.36. In the setting of the definition above:(1) Prove that for every point x ∈ X there is a canonical isomorphism ofOX,x-modules

(M ⊗OX N )x Mx ⊗OX,x Nx .

(2) Show that is M and N are locally free OX -modules then so is M ⊗OX N .

Exercise 9.37. Let (X ,OX ) be a ringed space over, and letM be an OX -module. Provethat M is a free OX -module of rank r iff

M OX ⊗X MX ,

where M is a free -module of rank r , and MX is the associated constant sheaf.

The next exercises were in an email last week.

Exercise 9.38. Find an example of a ringed space (X ,OX ), and a short exact sequence Ein ModOX consisting of locally free sheaves, that is not split.

Exercise 9.39. Prove that when (X ,OX ) is a compact manifold in Mfld, then every shortexact sequence of locally free sheaves is split. (Hint: partitions of unity.) (This also worksfor (X ,OX ) in Top, when X is a compact metric space.)

The next example is a solution of Exer 9.38.

Example 9.40. Here is an example of a short exact sequence

(E) 0→ Lφ−→M

ψ−→ N → 0

of locally free sheaves that is not globally split. This is in the setting of Sp = Var. Theexample will also take us on a tour into algebraic geometry.

The variety X is the projective line P1(). The homogeneous coordinates are t0, t1. Let

U0 := t0 , 0 = (1, λ) | λ ∈ ⊆ P1()

andU1 := t1 , 0 = (µ, 1) | µ ∈ ⊆ P1().

41 | file: notes-190201-d2


These are affine open sets, andU0 ∪U1 = P1().

Consider the rational functiont := t1/t0.

It has a zero of order 1 at the origin

x0 = (1, 0) ∈ U0

and a pole of order 1 at infinityx∞ = (0, 1) ∈ U1.

At all other points t does not have a zero or a pole.The rings of functions on the important affine open sets are

Γ(U0,OX ) = [t],

Γ(U1,OX ) = [t−1],

andΓ(U0 ∩U1,OX ) = [t, t−1].

The function field of X is the field

k(X) = (t).

For every nonempty affine open set U ⊆ X the field k(X) is the field of fractions of theintegral domain Γ(U,OX ).

Let k(X)× be the set of nonzero elements of k(X), i.e. its multiplicative group. For arational function f ∈ k(X)× and a point x ∈ X we write

(9.41) ordx( f ) := order of vanishing of the function f at the point x .

Thus ordx( f ) > 0 if f has a zero at x, and ordx( f ) < 0 if f has a pole at x.Note that

(9.42) OX,x =

f ∈ k(X)× | ordx( f ) ≥ 0∪ 0 ⊆ k(X)

and

(9.43) mx =

f ∈ k(X)× | ordx( f ) > 0∪ 0 ⊆ k(X).

We can consider k(X) as a constant sheaf of rings on X , and then

OX ⊆ k(X).

This inclusion allows the following description: every nonempty open set U ⊆ X we have

(9.44)Γ(U,OX ) =

f ∈ k(X)× | f has no poles in U

∪ 0

=⋂x∈U

OX,x .

For a nonzero rational function f and a subset U ⊆ X we define the divisor to be theelement

(9.45) divU ( f ) :=∑x∈U

ordx( f )· x ∈⊕x∈U

· x = Ffin(U,).

In the free -module Ffin(U,) we write d ≥ e if d(x) ≥ e(x) for all x ∈ U. Thus (9.44)can be rewriten as

Γ(U,OX ) =

f ∈ k(X)× | divU ( f ) ≥ 0∪ 0.

For V ⊆ U we have the restriction homomorphism:

Ffin(U,) → Ffin(V ,), e 7→ e|V .

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For every j ∈ we define the subsheaf O( j) ⊆ k(X) as follows:

(9.46) Γ(U,O( j)) :=

f ∈ k(X)× | divU ( f ) ≥ (− j · x∞)|U∪ 0.

This means that a nonzero rational function f belongs to Γ(U,O( j)) if these conditionshold:

• f has no poles in U − x∞.• If x∞ ∈ U, then f must vanish to order ≥ − j at x∞. I.e. a pole of order at most j if

j ≥ 0, and a zero of order at least − j if j < 0.This is a locally free OX -module. In fact,

(9.47) O( j)|U0 = OX |U0 ⊆ k(X)

and

(9.48) O( j)|U1 = OX |U1 ·tj ⊆ k(X).

It is clear that there are inclusions of OX -modules

O( j) ⊆ O( j + 1) ⊆ · · · k(X).

This means that multiplication by the rational function 1 ∈ k(X) belongs to the -module

HomX (O( j),O( j + 1)).

In fact:

(9.49) HomX (O( j),O( j + 1)) = ( ·1) ⊕ ( ·t).

We can finally exhibit the exact sequence E in ModOX . It is this:

(E) 0→ OX (−2)

[−1t

]·(−)

−−−−−−−→ OX (−1)⊕2 [ t 1 ] ·(−)−−−−−−−→ OX → 0.

Here we view the direct sums as column modules, and they are acted upon from the left bymatrices of morphisms, using formula (9.49).

Exercise 9.50. This exercise is an elaboration of the last example.(1) Explain how X = P1() is the gluing of the ringed spaces (U0,OU0 ) and (U1,OU1 ).(2) Prove formula (9.44).(3) Prove that (9.46) defines an OX -submodule of k(X).(4) Prove (9.47) and (9.48). Exhibit the gluing isomorphism for O( j) on U0 ∩U1.(5) Find a -basis for Γ(X ,O( j)). (Hint: you should get

rank(Γ(X ,O( j))) = j + 1

for j ≥ 0, andrank(Γ(X ,O( j))) = 0

for j < 0.)(6) Verify formula (9.49).(7) Show that

O( j) ⊗OX O(k) O( j + k)and

HomX (O( j),O(k)) O(k − j).(8) Verify that E is indeed an exact sequence. Do this by writing out E|Ui , for i = 0, 1,

explicity as a sequence of free modules over the ring Γ(Ui ,OX ), and show thatE|Ui is split exact.

(9) Use (5) to prove that E is not split in ModOX .

– – –

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Here is a complement on derivations, a follow up to Rem 8.31.

Example 9.51. Take Sp = Mfld. In Rem 8.31 we said that for a manifold X , the sheaf ofsections of the tangent bundle TX is the sheaf TX of derivations of OX .

When considered as a sheaf of X -modules, the sheaf TX has a Lie algebra structure.Namely for every open set U the -module Γ(U, TX ) is a Lie algebra, and the Lie bracketsrespect restriction to open sets. The formula is quite easy – see Exer 9.52 below.

Suppose G is a group object in Mfld. I.e. G is a Lie group. The identity element ise ∈ G. The tanget space TeG is called the Lie algebra of G, with notation g. Let’s see howg gets a Lie algebra structure.

The tangent sheaf TG is trivial. In fact there are two (in general distinct) trivializationsof it. The group G acts on Γ(G, TG) by left translations, and we denote by Γ(G, TG)G the-module of invariant sections, namely the left invariant vector fields on G. The canonicalhomomorphism

Γ(G, TG) → TG,e → ⊗OG,e TG,e TeG = g

induces a bijection

Γ(G, TG)G'−→ g.

In this way we get an embedding

g Γ(G, TG)

that can be viewed as an embedding of sheaves

gG TG

where gG is the constant sheaf g on G. It turns out that the induced OG-module homomor-phism

OG ⊗ gG → TG

is an isomorphism. This shows that TG is a freeOG-module (see Exer 9.37). A calculationshows that the subsheaf gG of TG is closed under the Lie bracket. Under the canonicalisomorphism

g (gG)e

there is a Lie algebra structure on g.The other option is to look at right invariant vector fields on G.

Exercise 9.52. Now (X ,OX ) is a ringed space over , for some commutative ring .By this we mean that OX is a sheaf of -rings, or in other words, there is a given ringhomomorphism X → OX . Consider the sheaf of rings EndX (OX ).

(1) Supposeφ, ψ ∈ Γ(U, EndX (OX ))

are derivations of OU = OX |U , for some open set U. Show that

[φ, ψ] := φ ψ − ψ φ

is also a derivation of OU .(2) Conclude that Der(OX ) is a sheaf of Lie algebras on X .


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10. Vector Bundles and Locally Free Sheaves

Recall that we have three favorite geometries:• Sp = Top. Here = , and OX is the sheaf of continous -valued functions.• Sp = Mfld. Here = , andOX is the sheaf of differentiable-valued functions.• Sp = Var. Here is an algebraically closed field, andOX is the sheaf of algebraic-valued functions.

Given a vector bundle E on X ∈ Sp, its sheaf of sections E = ShSeX (E) is a locally freeOX -module.

We now restate Thm 9.23:

Theorem 10.1. Let X ∈ Sp. The assignment

ShSeX : Vec/X → LFModOX

is an equivalence of categories.

For the proof we shall require the next general concept.

Definition 10.2. Let X be a topological space, let G be a sheaf of groups on X , and letU = Uii∈I be an open covering of X .

(1) A Čech 1-cochain on U with values in G is a collection g = gi, ji, j∈I of sections

gi, j ∈ Γ(Ui ∩Uj , G).

(2) A 1-cochain gi, ji, j∈I is called a Čech 1-cocycle if for every i, j, k ∈ I we have

gj,k ·gi, j = gi,k in the group Γ(Ui ∩Uj ∩Uk , G).

Proof of Theorem 10.1.Step 1. Suppose φ : E → F is a map of vector bundles. So this is a map in Sp/X thatrespects addition and multiplication. But then the corresponding morphism of sheavesφ : E → F also respects the operations, and hence it is a homomorphism of OX -modules.We see that ShSeX is a functor from Vec/X to LFModOX .

Step 2. Here we prove that the functor ShSeX is fully faithful. I.e. the function

ShSeX : HomVec/X (E , F) → HomModOX(E ,F)

is bijective. Nowmaps in Sp/X and inModOX are determined locally. Therefore it sufficesto consider an open set U ⊆ X on which both E and F (and hence also E and F) are trivial,and to prove that the function

(10.3) ShSeU : HomVec/U (E |U , F |U ) → HomModOU(E |U ,F |U )

is bijective.Say the ranks of E and F are m and n. Then

E |U U × Am() and F |U U × An(),

and we get a bijection

HomVec/U (E |U , F |U ) HomSp(U,Matn×m())

We also haveE |U O⊕mU and F |U O⊕nU ,

and we get a bijection

HomModOU(E |U ,F |U ) Γ(U,Matn×m(OX )).

But there is a canonical bijection

HomSp(U,Matn×m()) Γ(U,Matn×m(OX )),

45 | file: notes-190201-d2


and it is compatible with the function (10.3).

Step 3. In this last step we will prove that ShSeX is essentially surjective on objects. Takesome locally free OX -module E of rank n. Let U = Uii∈I be an open covering of X thattrivializes E , with trivializations

φi : E |Ui

'−→ O⊕nUi

.

Now for each i, j we have an isomorphism

φ j |Ui∩Uj (φi |Ui∩Uj )−1 : O⊕nUi∩Uj

'−→ O⊕nUi∩Uj

in ModOUi∩Uj . This isomorphism is left multiplication by a matrix

gi, j ∈ Γ(Ui ∩Uj ,GLn(OX )).

An easy calculation shows that gi, ji, j∈I is a 1-cocycle with values in the sheaf of groupsG := GLn(OX ).

We are now going to construct a vector bundle E on X , by gluing local pieces. For eachi let

Ei := An(Ui) = Ui × An().

This is a trivial rank n vector bundle on Ui . We define an isomorphism

Ei |Ui∩Uj

'−→ Ej |Ui∩Uj

in Vec/(Ui ∩Uj) by left multiplication by the matrix gi, j , this time viewed as a map

gi, j : Ui ∩Uj → GLn()

in Sp. These isomorphisms agree on triple intersections, by the cocycle condition. Hence(see the beginning of Section 7) we obtain a vector bundle

E ∈ Vec/X .

Let E ′ := ShSeX (E). We claim that E ′ E in ModOX . This is Exer 10.4(2) below.

Exercise 10.4. Finish the proof of the theorem, i.e.(1) Show that gi, ji, j∈I is a 1-cocycle.(2) Show that E ′ E .

In general, given a category C, let us denote by Ob(C) the set of isomorphism classes ofobjects of C.

Remark 10.5. Here is a way to think about Ob(C). Define C× to be the subcategory of Con all the objects, whose morphisms are the isomorphisms (the invertible morphisms) ofC. We can view C× as a graph; and then Ob(C) is the set of connected components of C×.

Corollary 10.6. The functor ShSeX induces a bĳection

Ob(Vecn/X)'−→ Ob(LFnModOX ).

11. Nonabelian First Cohomology

In Definition 10.2 we introduced Čech 1-cocycles. Given a topological space X , an opencovering U = Uii∈I of X , and a sheaf of groups G on X , we denote by Z1

(U , G) the setof 1-cocycles.

A 0-cochain f = fii∈I is a collection of sections

fi ∈ Γ(Ui , G).

46 | file: notes-190201-d2


Thus f = fii∈I is an element of the group

C0(U , G) :=∏i∈I

Γ(Ui , G).

Suppose g = gi, ji, j∈I is a 1-cocycle and f = fii∈I is a 0-cochain. Define the1-cochain

(11.1) D( f )(g) :=

fj ·gi, j · f −1i

i, j∈I

.

More precisely, the (i, j)-component of D( f )(g) is

fj |Ui∩Uj ·gi, j · f−1i |Ui∩Uj ∈ Γ(Ui ∩Uj , G).

Lemma 11.2. Formula (11.1) is a left action of the group C0(U , G) on the set Z1(U , G).

Exercise 11.3. Prove Lemma 11.2.

Definition 11.4. Given a topological space X , an open covering U = Uii∈I of X and asheaf of groups G on X , the 1-st nonabelian Čech cohomology of G on U is the quotient set

H1(U , G) :=

Z1(U , G)C0(U , G)

for the action D of the group C0(U , G) on the set of 1-cocycles.The set H1

(U , G) is pointed – the special element is the class of the trivial cocyclegi, ji, j∈I with gi, j = 1.

The theory of 1-st nonabelian Čech cohomology is very general. We will only study itin a particular setting: a ringed space (X ,OX ), and the sheaf of groups G = GLn(OX ) forsome n ≥ 1.

Definition 11.5. Let (X ,OX ) be a ringed space, let U = Uii∈I be an open covering of X ,and let E be a rank n locally free OX -module.

(1) A trivialization datum for E on U is a collection φ = φii∈I of isomorphisms

φi : E |Ui

'−→ O⊕nUi

in ModOUi .(2) If E admits some trivialization datum on U , then we say that E is trivialized on U .

Lemma 11.6. In the situation of Definition 11.5, let

gi, j ∈ Γ(Ui ∩Uj ,GLn(OX ))

be the matrix of the isomorphism

φ j φ−1i : O⊕nUi∩Uj

'−→ O⊕nUi∩Uj

.

Then gi, ji, j∈I is a 1-cocycle.


We call g = gi, ji, j∈I the cocycle associated to the trivialization datum φ = φii∈I .

Lemma 11.8. In the situation of Definition 11.5, suppose we have another trivializationdatum φ′ = φ′ii∈I of E on U , with associated cocycle g′ = g′i, ji, j∈I . Then the cocyclesg and g′ are cohomologous; i.e. they are equivalent by the action of the group C0(U , G).


Suppose E ∈ LFnModOX trivializes on a covering U . By definition there is sometrivialization datum φ; and by Lemma 11.8 the cohomology class of the associated cocycleg is independent of φ. This allows the next definition.

47 | file: notes-190201-d2


Definition 11.10. Let E ∈ LFnModOX be a sheaf that trivializes on a covering U . Thecohomology class of E is the element

clU (E) := [g] ∈ H1(U ,GLn(OX ))

represented by the cocycle g that’s associated to some trivialization datum φ of E on U .

Lemma 11.11. Suppose E ,F ∈ LFnModOX are isomorphic. If E trivializes on a coveringU , then so does F ; and moreover clU (F) = clU (E).


Definition 11.13. Let (X ,OX ) be a ringed space, and let U = Uii∈I be an open coveringof X . We denote by Ob(LFnModOX )

U the set of isomorphism classes of rank n locallyfree OX -modules that trivialize on U .

This definition makes sense by Lemma 11.11. Note that Ob(LFnModOX )U is a pointed

set: the special element is the class of the trivial module O⊕nX .

Theorem 11.14. Let (X ,OX ) be a ringed space, and let U = Uii∈I be an open coveringof X . The function

clU : Ob(LFnModOX )U → H1

(U ,GLn(OX ))

is an isomorphism of pointed sets.

Proof.Step 1. Let us prove that the function clU is injective. Suppose E and F are objectsof LFnModOX that trivialize on U , and clU (E) = clU (F). Choose trivialization dataφ = φii∈I and ψ = ψii∈I for E and F respectively. The corresponding cocycles areg = gi, ji, j∈I and h = hi, ji, j∈I . The equality clU (E) = clU (F) means that there is a0-cochain f = fii∈I such that

D( f )(g) = h.

Nowφi : E |Ui

'−→ O⊕nUi

andψi : F |Ui

'−→ O⊕nUi

are isomorphisms in ModOUi , and

fi ∈ Γ(Ui ,GLn(OX )) EndOUi(O⊕nUi

).

Let us defineθi := ψ−1

i fi φi : E |Ui

'−→ F |Ui .

On Ui ∩Uj we have

θi |Ui∩Uj = ψ−1i |Ui∩Uj fi |Ui∩Uj φi |Ui∩Uj

= (ψ−1j |Ui∩Uj hi, j) fi |Ui∩Uj (g

−1i, j φ j |Ui∩Uj )

= ψ−1j |Ui∩Uj fj |Ui∩Uj φ j |Ui∩Uj )

= θ j |Ui∩Uj .

Therefore the locally defined isomorphisms θi glue to a global isomorphism

θ : E '−→ F .

We see that E and F are isomorphic.

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Step 2. Now we prove that function cl is surjective. Consider a 1-cocycle g = gi, ji, j∈I .We are going to construct a sheaf E by gluing the sheaves

Ei := O⊕nUi

using the isomorphisms

φi, j := gi, j : Ei |Ui∩Uj

'−→ Ej |Ui∩Uj .

This is analogous to step 3 in the proof of Theorem 10.1. The cocycle condition ensuresthat gluing is possible (this is Theorem 7.4). The resulting locally free sheaf E comesequipped with trivialization datum, and the associated cocycle is g.

Step 3. The trivial sheaf O⊕nX can be equipped with the identity trivialization on U . Thecorresponding cocycle g is the trivial one. We see that the function cl preserves the specialpoints.

Corollary 11.15. Let (X ,OX ) be a ringed space, let U be an open covering of X , and letE be a rank n locally free OX -module that trivilaizes on U . TFAE:

(i) The sheaf E is trivial, i.e. E O⊕nX in ModOX .(ii) The cohomology class

clU (E) ∈ H1(U ,GLn(OX ))

is the trivial class.

Proof. An immediate consequence of Theorem 11.14.

Corollary 11.16. Let X ∈ Sp (one of our three chosen geometries), let U be an opencovering of X , and let E be a rank n vector bundle on X that trivilaizes on U . TFAE:

(i) The vector bundle E is trivial, i.e. E X × An() in Vec/X .(ii) The cohomology class

clU (ShSeX (E)) ∈ H1(U ,GLn(OX ))

is the trivial class.

Proof. Combine Corollaries 11.16 and 10.6, noting that Corollary 10.6 is an isomorphismof pointed sets.

Lecture 11, 2 Jan 2019

In hindsight I realized that we did not talk enough about gluing X-spaces. The nexttheorem should close this gap.

Theorem 11.17. Let Sp be one of the two geometries Top orMfld. Given X ∈ Sp, an opencovering X =

⋃i∈I Ui , for every i a Ui-space

πi : Vi → Ui ,

and for every i, j an isomorphism

fi, j : π−1i (Ui ∩Uj)

'−→ π−1

j (Ui ∩Uj)

of X-spaces. The condition is that

fj,k |π−1j (Ui∩Uj∩Uk )

fi, j |π−1i (Ui∩Uj∩Uk )

= fi,k |π−1i (Ui∩Uj∩Uk )

for all i, j, k. Then there is an X-space

π : Y → X ,

49 | file: notes-190201-d2


with isomorphisms of X-spaces

fi : π−1(Ui)'−→ Vi

such thatfi, j fi |π−1(Ui∩Uj )

= fj |π−1(Ui∩Uj )

for every i, j.Moreover, the X-space (Y , π), with the collection of isomorphism fi, is unique up to a

unique isomorphism.

Exercise 11.18. Prove this theorem. Note that in Mfld the spaces must be Hausdorff andhave a countable topology (“the second axiom of countability”). Also note that f −1

i (Vi)

is an open covering of Y .

Remark 11.19. The geometry Var was excluded from the theorem above because of thefollowing technicality: we stipulated that varieties are always quasi-projective. I am notsure that the gluing process yields a quasi-projective variety. It is possible to generalize thenotion of variety to handle this. But in Sch this difficutly will disappear anyhow, so thereis no reason to make this effort for varieties.

What will be relevant in Sch is that the glued X-scheme Y is separated over X; hence ifX is a separated scheme then so is Y . This is analogous to the Hausdorff condition in theproof above for Mfld.

Exercise 11.20. LetV0,V1,W ∈ Top, and let gi : W → Vi open embeddings (i.e. gi(W) ⊆ Vi

is an open subspace, and gi : W → gi(W) is an isomorphism in Top). Prove that the gluing,or fibered coproduct,

Y := U0 tW U1

exists in Top. This is a space Y with maps hi : Ui → Y such that

h0 g0 = h1 g1 : W → Y ,

and (Y , h0, h1) are universal for this property.Find an example of the above in which V0,V1,W are Hausdorff, but Y is not.Conclude that the fibered coproduct does not exists in Mfld. (It also does not exist in

Var, but is does exist in Sch.)

– – –

We now analyze the Möbius band as a manifold.

Example 11.21. We work with Sp = Mfld, so = . Let Z := (−1, 1) ⊆ A1(), let

X := S1 × Z

be the trivial (open) band, and let Y be the Möbius band. We claim that Y and Z are notdiffeomorphic. We do this by showing that the tangent bundle TX is a trivial rank 2 bundleon X , whereas the tangent bundle TY is not a trivial rank 2 bundle on Y .

The tangent bundle of the circle S1 is trivial – it has a basis consisting of the vector field∂∂θ , where θ is the angle. And the tangent bundle of the line segment Z is trivial – it hasa basis consisting of the vector field ∂

∂t , where t is the linear coordinate. Hence tangentbundle

TX = TS1 × TZ

of the band X is trivial, with basis ( ∂∂θ ,∂∂t ).

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Figure 3. Open covering of the circle

Recall how we constructed the Möbius band in Example 7.5. Let ψ : Z → Z be thediffeomorphism ψ(z) := −z. Let I := 0, 1, 2. We have the open covering

S1 =⋃i∈I

Ui

shown in Figure 3. For i ∈ I we define the differentiable manifold

Vi := Ui × Z .

The gluing data φi, j is

φi, j := id× id : (Ui ∩Uj) × Z → (Ui ∩Uj) × Z

for (i, j) ∈ (0, 1), (1, 2), and

φ0,2 := id×ψ : (U0 ∩U2) × Z → (U0 ∩U2) × Z .

These are extended (i.e. for j ≤ i) by φi, j := φ−1j,i and φi,i := id. We obtain the S1-space Y ,

with open covering V = Vi.The tangent bundle TY trivializes on the covering V , and as basis for

TVi = TUi × TZ

we take ( ∂∂θ ,∂∂t ). The corresponding matrices

gi, j ∈ Γ(Vi, j ,GL2(OY ))

are

g0,1 =

[1 00 1

], g1,2 =

[1 00 1

], g0,2 =

[1 00 −1

].

(They happen to be constant matrices.)How can we prove that the cohomology class

cl(TY ) = [g] ∈ H1(V ,GL2(OY ))

is not trivial?Given any cocycle

h = hi, j ∈ Z1(V ,GL2(OY ))

the functiondet(hi, j) : Vi, j → GL1()

51 | file: notes-190201-d2


is continous. Since Vi, j is connected, the sign of det(hi, j)(y) is constant for y ∈ Vi, j . Thusdet(hi, j) ∈ ±1. So we get a function

sign : Z1(V ,GL2(OY )) → ±1,

sign(h) :=∏

0≤i< j≤2sign(det(hi, j)) ∈ ±1.

A little calculation (exercise) shows that if

D( f )(h) = h′

for a cochian f then

(11.22) sign(h) = sign(h′).

Therefore we get a function

(11.23) sign : H1(V ,GL2(OY )) → ±1.

For the trivial cohomology class the sign is 1. But for [g] = cl(TY ) we get

(11.24) sign(cl(TY )) = −1.

So by Corollary 11.16 the bundle TY is not trivial.

Exercise 11.25. Prove that (11.22) is true.

Let X ∈ Mfld of dimension n. The cotangent bundle of X is the rank n vector bundleT∗X that’s dual to TX . I.e. the locally free sheaves

TX = ShSeX (TX)

andΩ

1X = ShSeX (T∗X)

satisfyΩ

1X = HomOX (TX ,OX ).

The n-th exterior power sheafΩ

nX :=

∧nOX(Ω1

X )

is locally free of rank 1, and it is called the canonical sheaf (by algebraic geometers).

Definition 11.26. Let X ∈ Mfld. The manifold X is called orientable if the sheaf ΩnX is

trivial.If X is orientable then there exists a global section

ω ∈ Γ(X ,ΩnX )

which is a basis of this sheaf. Such a section ω is called a volume form.

Example 11.27. For X = An() with linear coordinates t1, . . . , tn there is a standardvolume form

ω := d(t1) ∧ · · · d(tn).

Example 11.28. A volume form ω lets us perform Riemann integration on the manifoldX . Given a function f : X → with suitable properties (e.g. continous with compactsupport) the integral ∫

X

f ·ω ∈

exists. It is amusing to prove this.

Proposition 11.29. The Möbius band Y ∈ Mfld is not an orientable manifold.

Exercise 11.30. Prove this proposition. (Hint: modify Example 11.21.)

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Exercise 11.31. Find a manifold X whose tangent bundle TX in not trivial, yet X isorientable.

– – –

Let (X ,OX ) be a ringed space. According to Exercise 9.36, ifM andN are locally freeOX -modules then so is M ⊗OX N . It is easy to see that

M ⊗OX N N ⊗OX M.

Exercise 11.32. Let M be a rank 1 locally free OX -module. Show that

N := HomOX (M,OX )

is also a rank 1 locally free OX -module, and

M ⊗OX N OX .

This justifies the next definition.

Definition 11.33. The Picard group of a ringed space (X ,OX ) is the abelian groupPic(X ,OX ), whose elements are the isomorphism classes of the rank 1 locally free OX -modules, with multiplication induced by (− ⊗OX −).

As before, given an open covering U = Uii∈I of X we denote by Pic(X ,OX )U the

subgroup of Pic(X ,OX ) consisting of the sheaves that trivialize on U .It is not hard to see that when G is a sheaf of abelian group on X , the cohomology set

H1(U , G) is an abelian group.

Theorem 11.34. Let (X ,OX ) be a ringed space. Given an open covering U of X , there isa canonical group isomorphism

cl : Pic(X ,OX )U '−→ H1

(U ,GL1(OY )).

Exercise 11.35. Prove the theorem.

To get all of Pic(X ,OX ) we need to go to the direct limit on all open coverings.

Exercise 11.36. Here is a toy version of Pic(X) in algebraic geometry. Let A be a commu-tative local ring, and define the ringed space (X ,OX ) as follows: X := x, a single point,and

Γ(X ,OX ) := A.

Calculate the group Pic(X ,OX ).

Example 11.37. Let X := Pn() ∈ Var. We will (hopefully) prove in the next semesterthat every rank 1 locally free OX -module L is isomorphic to O(i) for some i ∈ . Seeformula (9.46) for the definition of O(i) in the case n = 1. Hence

Pic(X) .

12. Operations on Sheaves

We fix a nonzero commutative base ring .

Definition 12.1. Let f : Y → X be a map of topological spaces. Given a Y -module N ,we define the presheaf

f∗(N ) ∈ ModpreX

by the formulaΓ(U, f∗(N )) := Γ( f −1(U),N ).

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Lemma 12.2. The presheaf f∗(N ) is a sheaf.

Exercise 12.3. Prove the lemma.

Definition 12.4. Let f : Y → X be a map of topological spaces. Given a X -module M,we define the sheaf

f −1(M) ∈ ModY

to be the sheaf associated to the presheaf

V 7→ limU→

Γ(U,M),

where U runs over the open sets of X that contain f (V).

Proposition 12.5. Let f : Y → X be a map of topological spaces.(1) Given M ∈ ModX and N ∈ ModY , there is an isomorphism

HomModX(M, f∗(N ))

'−→ HomModY

( f −1(M),N ).

This isomorphism is functorial in M and N .(2) The functors

f∗ : ModY → ModX

andf −1 : ModX → ModY

are adjoint.


Proposition 12.7. The functor

f∗ : ModY → ModX

is left exact, and the functor

f −1 : ModX → ModY

is exact.


Definition 12.9. Let (X ,OX ) and (Y ,OY ) be -ringed spaces. A map of -ringed spaces

( f , ψ) : (Y ,OY ) → (X ,OX )

is a map of topological spacesf : Y → X

together with a homomorphism of sheaves of X -rings

ψ : OX → f∗(OY ).

If (Z ,OZ ) is another -ringed space, and if

(g, θ) : (Z ,OZ ) → (Y ,OY )

is a map of -ringed spaces, then there’s a canonical isomorphism

f∗(g∗(OZ )) ( f g)∗(OZ )

of sheaves of rings on X , and

ψ f∗(θ) : OX → ( f g)∗(OZ )

is a homomorphism of sheaves of X -rings. We let

( f , ψ) (g, θ) : (Z ,OZ ) → (X ,OX )

54 | file: notes-190201-d2


be( f , ψ) (g, θ) := ( f g, ψ f∗(θ)).

In this way we get a category RSp/.By Proposition 12.5 (adjunction) the homomorphism ψ induced a homomorphism

ψ ′ : f −1(OX ) → OY

of Y -modules. It is in fact a homomorphism of Y -rings.

Definition 12.10. Let( f , ψ) : (Y ,OY ) → (X ,OX )

be a map in RSp/. Given M ∈ ModOX we let

f ∗(M) := OY ⊗ f −1(OX )f −1(M) ∈ ModOY .

Here f −1(M) is an f −1(OX )-module on Y , and we use the ring homomorphism ψ :f −1(OX ) → OY in the tensor product.

Proposition 12.11. Let( f , ψ) : (Y ,OY ) → (X ,OX )

be a map in RSp/.(1) Given M ∈ ModOX and N ∈ ModOY , there is an isomorphism

HomModOX(M, f∗(N ))

'−→ HomModOY

( f ∗(M),N ).

This isomorphism is functorial in M and N .(2) The functors

f∗ : ModOY → ModOX

andf ∗ : ModOX → ModOY

are adjoint.


Proposition 12.13. The functor

f ∗ : ModOX → ModOY

is right exact.


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Lecture 12, 9 Jan 2019

13. Locally Ringed Spaces

We fix a nonzero commutative base ring .

Definition 13.1. A ringed space (X ,OX ) ∈ RSp/ is called a locally ringed space if forevery point x ∈ X the stalk OX,x is a local ring.

Definition 13.2. Let (X ,OX ) be a locally ringed space over . Given a point x ∈ X , themaximal ideal of OX,x is denoted by mx , and the residue field is

k(x) := OX,x/mx .

Suppose A and B are local -rings (commutative), with maximal ideals m and n

respectively. A -ring homomorphism ψ : A → B is called a local homomorphism ifψ(m) ⊆ n. In this case there is an induced ring homomorphism on the residue fields

ψ : A/m→ B/n.

Definition 13.3. Let (X ,OX ) and (Y ,OY ) be locally ringed spaces over. Amap of locallyringed spaces over is a map of -ringed spaces

( f , ψ) : (Y ,OY ) → (X ,OX )

such that for every point y ∈ Y the -ring homomorphism

ψy : OX, f (y) → OY,y

between the stalks is a local homomorphism.The category of locally ringed spaces over, withmaps as above, is denoted by LRSp/.

Definition 13.4. If (X ,OX ) is a locally ringed space and U ⊆ X is an open subset, then(U,OX |U ) is also a a locally ringed space. It is called an open subspace of (X ,OX ), andthe inclusion map

(U,OX |U ) → (X ,OX )

is called an open embedding.

Proposition 13.5. Consider one of our three favorite geometries Sp, with correspondingbase field . For X ∈ Sp the sheaf of functions is OX .

(1) The -ringed space (X ,OX ) is a locally ringed space.(2) Let f : Y → X be a map in Sp. Define the homomorphism of sheaves ofX -rings

f ∗ : OX → f∗(OY ),

called pullback, as follows: for an open set U ⊆ X and g ∈ Γ(U,OX ),

f ∗(g) := g f ∈ Γ( f −1(U),OY ) = Γ(U, f∗(OY )).

Then( f , f ∗) : (Y ,OY ) → (X ,OX )

is a map of locally ringed spaces over .


The next theorem is supposed to tell us how powerful the concept of locally ringedspaces is.

57 | file: notes-190201-d2


Theorem 13.7. Consider one of our three favorite geometries Sp, with corresponding basefield . The functor

LRS : Sp→ LRSp/,that sends a space X ∈ Sp to the locally ringed space

LRS(X) := (X ,OX )

from Proposition 13.5(1), and sends a map f : Y → X in Sp to the maps of locally ringedspaces

LRS( f ) := ( f , f ∗) : (Y ,OY ) → (X ,OX )

from Proposition 13.5(2), is fully faithful.

The theorem means that the geometry Sp is completely encoded in the abstract notionof “map of locally ringed spaces”.

Proof. The faithfulness of the functor LRS is easy to see: the forgetful functor Sp→ Setis faithful ( f = g iff they are equal as functions between sets), and it factors through LRS.

The challenge is to prove fullness. Let

( f , ψ) : (Y ,OY ) → (X ,OX )

be a morphism in LRSp/ between

(X ,OX ) = LRS(X)

and(Y ,OY ) = LRS(Y )

for two spaces X ,Y ∈ Sp. We must prove that the continous map f : Y → X is a map inSp, and that ψ = f ∗.

Step 1. We prove that ψ = f ∗. Consider an open set U ⊆ X and a function g ∈ Γ(U,OX ).We must prove that

ψ(g) = f ∗(g) ∈ Γ(U, f∗(OY )).

In other words, letting V := f −1(U) ⊆ Y , we must prove that the functions

ψ(g), g f : V →

are equal.Take an arbitrary point y ∈ V . Let x := f (y) ∈ U and a := g(x) ∈ . So (g f )(y) = a,

and it remains to prove that ψ(g)(y) = a too. Let us introduce some temporary notationfor some canonical ring homomorphisms, that will help keep track of things: the germhomomorphism

germx/U : Γ(U,OX ) → OX,x

and the residue homomorphism

resx : OX,x → k(x).

We already talked about the evaluation homomorphism

evx/U : Γ(U,OX ) → , evx/U (g) = g(x).

These fit into a commutative diagram

(13.8)

id

&&Γ(U,OX )

germx/U//

evx/U

77OX,xresx // k(x) //

58 | file: notes-190201-d2


There are similar homomorphisms for y ∈ V .Because ( f , ψ) is a morphism of locally ringed spaces, we have a local -ring homo-

morphism ψy : OX,x → OY,y . There is a commutative diagram of -rings

(13.9) Γ(U,OX )ψ

//

germx/U

evx/U

&&

Γ(V ,OY )

germy/V

evy/V

xx

OX,x

ψy//

resX

OY,y

resy

k(x)ψy

//

k(y)

id

//

in which the columns are (13.8) and itsY version. The function g in the top left corner goesto g(x) = a in the bottom left corner, and hence to a in the the bottom right corner. Butgoing first to the top right corner and then down, g goes to ψ(g)(y). Therefore ψ(g)(y) = a,as claimed.

Step 2. It remains to prove that f : Y → X is a map in Sp. For SP = Top it is automatic. Forthe other two geometries this is a local question. Take y ∈ Y and x := f (y) ∈ X . Choose anopen neighborhoodU of x in X that embeds intoAn(), as an open subspace for SP = Mfld,and as a closed intersect open subspace for SP = Var. Take V := f −1(U) ⊆ Y , which is anopen neighborhood of y. Let t1, . . . , tn ∈ Γ(An(),O) be the coordinate functions. Themap

f |V : V → U

is in Sp if the functionsti f = f ∗(ti) : V →

are in Sp, i.e. iff ∗(ti) ∈ Γ(V ,OY ).

But by step 1 we know that

f ∗(ti) = ψ(ti) ∈ Γ(V ,OY ).

14. Schemes

All rings are assume to be commutative.We fix a base ring , which is some nonzero ring. The general choice is = ; but

sometimes other choices are made (cf. Example 14.16).Let A be a-ring. The prime spectrum of A is the set Spec(A) of prime ideals of A. As

we all know, this is a topological space with the Zariski topology. By definition the closedsets of Spec(A) are the sets

Z(a) := p ∈ Spec(A) | a ⊆ p,

where a is some ideal in A.We sometimes refer to Z(a) as the zero locus of the ideal a. Here is the reason: to a

prime ideal p we associate the local ring Ap and the residue field

k(p) := Ap/pp.

59 | file: notes-190201-d2


An element a ∈ A has a class a(p) ∈ k(p), coming from the canonical ring homomorphismA→ k(p). Then

(14.1) Z(a) = p ∈ Spec(A) | a(p) = 0 for all a ∈ a.

Exercise 14.2. Prove the formula above.

For an element s ∈ A we define

(14.3) D(s) = p ∈ Spec(A) | s < p.

This is an open set: it it the complement of the closed set Z(a), where a := (s), the idealgenerated by s. We call such an open set a principal open set. Analogously to (14.1) wehave

(14.4) D(s) = p ∈ Spec(A) | s(p) , 0.

Proposition 14.5. The principal open sets are a basis of the topology of Spec(A). Namelyevery open set set U is a union U =

⋃i D(si) for a suitable collection si of elements of A.

Exercise 14.6. Prove the proposition above.

We now define a sheaf of ring on Spec(A). We use an auxilliary Godement sheaf. Let’swrite X := Spec(A) as a topological space. The sheaf of rings GSh(OX ) is this: for an openset U ⊆ X we let

Γ(U,GSh(OX )) :=∏p∈U

Ap.

Definition 14.7. Let A be a ring and write X := Spec(A) as a topological space. Thestructure sheaf of Spec(A) is the subsheaf OX ⊆ GSh(OX ) satisfying this condition: asection

a = ap ∈ Γ(U,GSh(OX )) =∏p∈U

Ap

belongs to Γ(U,OX ) if there is an open covering U =⋃

i Ui , and pairs of elements (ai , si)in A, such that Ui ⊆ D(si), and

a|Ui = (ai/si)|Ui in Γ(Ui ,GSh(OX )).

That is to say,ap = (ai/si)p in Ap

for all p ∈ Ui .

Proposition 14.8. Let A be a ring and write X := Spec(A). The sheaf OX is the sheafifi-cation of the presheaf

U 7→ AS(U),

where for an open set U ⊆ X we let S(U) ⊆ A be the multiplicatively closed set

S(U) := s ∈ A | s(p) , 0 for all p ∈ U,

and AS(U) is the localization.


Proposition 14.10. Let A be a ring and write X := Spec(A). For every point x = p ∈ Xthe stalk is

OX,x = Ap.

Thus (X ,OX ) is a locally ringed space.


60 | file: notes-190201-d2


Definition 14.12. An affine-scheme is a locally ringed space (X ,OX ) ∈ LRSp/ whichis isomorphic to

(Spec(A),OSpec(A))

for some -ring A.

Definition 14.13. A -scheme is a locally ringed space (X ,OX ) ∈ LRSp/ that has anopen covering X =

⋃i Ui , such that each open subspace (Ui ,OX |Ui ) is an affine-scheme.

Definition 14.14. Suppose (X ,OX ) and (Y ,OY ) are -schemes. A map of -schemes

( f , ψ) : (Y ,OY ) → (X ,OX )

is by definition a map of locally ringed -spaces.Thus the category of -schemes is the full subcategory of LRSp/ on the -schemes.

Example 14.15. Suppose is a nonzero noetherian ring.For every n ≥ 0 there is an affine scheme

An := Spec([t1, . . . , tn])

called n dimensional affine space.There is also the scheme Pn

, called n dimensional projective space. It has an open

covering Pn=

⋃ni=0 Ui , where each Ui An

.

Example 14.16. Now suppose is an algebraically closed field. A point x ∈ Pnis called

-rational if its residue field is . The set of -rational points is denoted by Pn(). Thisset has an induced topology and a sheaf of rings on it, making it the projective space fromthe classical Var/.

More generally, we can consider an irreducible closed subset X ⊆ Pn, and endow it

with the induced reduced sheaf of rings, making (X ,OX ) into a closed subscheme of Pn.

This is called a reduced projective scheme. The set X() of -rational points is dense inX , by the Hilbert Nullstellensatz. It is a classical projective variety.

For an open subscheme U of X we get a quasi-projective variety U(). All objects ofVar/ arise in this way.

By Theorem 13.7, Var/ is a full subcategory of Sch/.

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References[Gro] A. Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. 9 (1957), 119-221.[Har] R. Hartshorne, “Algebraic Geometry”, Springer-Verlag, New-York, 1977.[HltSt] P.J. Hilton and U. Stammbach, “A Course in Homological Algebra”, Springer, 1971.[Lee] John M. Lee, “Introduction to Smooth Manifolds”, LNM 218, Springer, 2013.[KaSc] M. Kashiwara and P. Schapira, “Sheaves on manifolds”, Springer-Verlag, 1990.[Mac1] S. Maclane, “Homology”, Springer, 1994 (reprint).[Mac2] S. Maclane, “Categories for the Working Mathematician”, Springer, 1978.[Rot] J. Rotman, “An Introduction to Homological Algebra”, Academic Press, 1979.[Row] L.R. Rowen, “Ring Theory” (Student Edition), Academic Press, 1991.[We] C. Weibel, “An introduction to homological algebra”, Cambridge Studies in AdvancedMath. 38, 1994.[Ye1] A. Yekutieli, “Derived Categories”, prepublication, eprint https://arxiv.org/abs/1610.09640.[Ye2] A. Yekutieli, “Commutative Algebra”, Course Notes, http://www.math.bgu.ac.il/~amyekut/teaching/2017-18/

comm-alg/course_page.html.[Ye3] A. Yekutieli, “Homological Algebra”, Course Notes, http://www.math.bgu.ac.il/~amyekut/teaching/2017-18/

hom-alg/course_page.html.

Department of Mathematics, Ben Gurion University, Be’er Sheva 84105, Israel.Email: [email protected], Web: http://www.math.bgu.ac.il/~amyekut.

63 | file: notes-190201-d2

https://arxiv.org/abs/1610.09640

http://www.math.bgu.ac.il/~amyekut/teaching/2017-18/comm-alg/course_page.html

http://www.math.bgu.ac.il/~amyekut/teaching/2017-18/comm-alg/course_page.html

http://www.math.bgu.ac.il/~amyekut/teaching/2017-18/hom-alg/course_page.html

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