Autumn 2009

Schemes

Denis Garcia

Advisors: Jean Fasel, Donna Testerman

Contents

Introduction 2

1 A categorical introduction 2

2 Sheaves 82.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Schemes 203.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

References 28

1

Introduction

This semester project is an introduction to algebraic geometry. It follows closely [2,Chapter II, 1 and 2] but with an emphasis on the categorical approach. All the proofshave been left out, but there are solutions to some exercises. The first part introducesall the categorical notions which will be needed to define a sheaf, such as functors,natural transformations and direct limits...

The second part follows [2, Chapter II, 1 Sheaves] and contains the solution of someof the exercises. It’s an introduction to sheaf theory with some examples. Sheaves area way to collect local data all into one place, and then see how it behave when wefocus on a single point. The notion of sheaf is necessary to define a scheme.

The third part introduce the notion of ringed spaces and schemes. It follows [2,Chapter II, 1 Sheaves] and also contains some solution to the exercises. Schemes wereintroduced by Alexander Grothendieck in order to broaden the notion of algebraicvariety. A scheme can be seen as a topological space together with a sheaf of commu-tative rings.

1 A categorical introduction

Definition 1.1 (Category). A category C is a class of objects Ob(C ) together with a classof morphisms (or arrows) MorC (A, B) for any objects A, B ∈ Ob(C ) such that:

• there is a composition law:

MorC (A, B)×MorC (B, C) −→ MorC (A, C)( f , g) 7−→ g ◦ f ;

• the composition law is associative;

• for every object C ∈ Ob(C ), there exists a morphism 1C ∈ MorC (C, C) calledidentity morphism, such that for every morphism f ∈ MorC (A, B),

1B ◦ f = f = f ◦ 1A.

Remark. We often write A ∈ C instead of A ∈ Ob(C ), and f : A → B instead off ∈ MorC (A, B).

Example.

1. The class of sets together with the class of functions is a category denoted Set.

2. The class of abelian groups together with the class of morphisms of groups is acategory denoted Ab.

2

3. The class of commutative rings together with the class of morphisms of ring is acategory denoted CRing.

4. The class of topological spaces together with the class of continuous maps is acategory denoted Top.

5. For a topological space X, we define a category Top(X), whose objects are theopen subsets of X and where the only morphisms are the inclusion maps. ThusMor(V, U) = ∅ if V 6⊆ U and Mor(V, U) = {V ↪→ U} if V ⊆ U, where U and Vare open subsets of X.

Definition 1.2 (Dual category). Let C be a category. The dual category C op is definedby:

• Ob(C op) = Ob(C );

• for each A, B ∈ Ob(C op), MorC op(A, B) = MorC (B, A) and f op ◦ gop = (g ◦ f )op.

Definition 1.3 (Initial object). Let C be a category. An initial object of C is an objectI ∈ C such that for every X ∈ C , there exists a unique morphism from I to X.

Example. In the category of commutative rings, Z is an initial object.

Definition 1.4 (Final object). Let C be a category. A final object of C is an object F ∈ C

such that for every X ∈ C , there exists a unique morphism from X to F.

Example. In the category of groups, 0 is an initial and final object.

Definition 1.5 (Isomorphism). Let C be a category. An isomorphism is a morphismf : A → B in C such that there exists a morphism f−1 : B → A in C with f ◦ f−1 = 1Aand f−1 ◦ f = 1B.

Definition 1.6 (Zero morphism). Let C be a category. C has zero morphisms if for everyA, B ∈ C there exists a morphism 0A,B : A → B of C such that for every morphismf : R → S, g : U → V of C the diagram

R

0R,U

��

f // S

0S,V

��U g

// V

commutes.

Example. A category C with a final and initial object I has zero morphisms. ForA, B ∈ C , OA,B : A → B is the composition between A → I and I → B, where I is firstseen as a final object and then as an initial object.

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Definition 1.7 (Kernel). Let C be a category with zero morphisms, and let f : X → Ybe a morphism of C . A kernel of f is a pair (K, k) where K ∈ C and k : K → X is amorphsim of C such that:

• f ◦ k is the zero morphism from K to Y;

• for any morphism k′ : K′ → X such that f ◦ k′ is the zero morphism, there existsa unique morphism u : K′ → K making the following diagram

K

0K,Y

$$k // Xf // Y

K′

∃!u

__??

??

??

??

k′

OO

0K′ ,Y

??���������������

commute.

Example. In the category Ab, K is the usual kernel of morphism of groups, and k is theinclusion morphism.

Definition 1.8 (Product). Let C be a category and let A, B ∈ Ob(C ). A product of A andB is an object of C , denoted A× B, with a pair pA : A× B → A and pB : A× B → Bof morphisms, such that for any pair f : C → A and g : C → B there exists a uniquemorphism u : C → A× B making the diagram

C

f

������

����

����

���

∃!u

��������

g

��???

????

????

????

A A× BpAoo

pB// B

commute.

Example. In the category of sets, Set, the product is the cartesian product of two sets.

Definition 1.9 (Coproduct). Let C be a category and let A, B ∈ Ob(C ). A coproductof A and B is an object of C , denoted A ⊕ B, with a pair iA : A → A ⊕ B andiB : B → A⊕ B of morphisms, such that for any pair f : A → C and g : B → C there

4

exists a unique morphism u : A⊕ B → C making the diagram

AiA //

f

��???

????

????

????

A⊕ B

∃!u

�������� B

iBoo

g

������

����

����

���

C

commute.

Remark. The coproduct is sometimes called the direct sum. The coproduct of A and Bis sometimes denoted A ä B.

Example. In the category of sets, the coproduct is the disjoint union of two sets.

Definition 1.10 (Functor). Let C and D be categories. A covariant functor F from C toD is a mapping that:

• associates to each object C ∈ C an object F(C) ∈ D ;

• associates to each morphism f : A → B of C a morphism F( f ) : F(A) → F(B) ofD ;

satisfying the following two conditions:

• F(1C) = 1F(C) for all C ∈ C ;

• F( f ◦ g) = F( f ) ◦ F(g) for all morphisms g : A → B and f : B → C of C .

Remark. A contravariant functor F : C → D , is a functor F : C op → D .

Example. The identity functor 1C : C → C is defined by:

• 1C (A) = A for all A ∈ C ;

• 1C ( f ) = f for all morphisms f : A → B of C .

Definition 1.11 (Natural transformation). Let C and D be categories, and let F andG be functors from C to D . A natural transformation η from F to G is a mapping thatassociates to every object A ∈ C a morphism ηA : F(A) → G(A) of D called thecomponent of η at A, such that for every morphism f : A → B of C , the followingdiagram

F(A)

ηA

��

F( f ) // F(B)

ηB

��G(A)

G( f )// G(B)

commutes.

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Remark.

• If both F and G are contravariant functors, the horizontal arrows in this diagramare reversed.

• If, for every object A in C , the morphism ηA is an isomorphism in D , then η issaid to be a natural isomorphism.

• If η : F → G and ε : G → H are natural transformations between functorsF, G, H : C → D , then we can compose them to get a natural transformationεη : F → H. This is done "componentwise": (εη)X = εX ◦ ηX. This "vertical com-position" of natural transformation is associative and has an identity, and allowsone to consider the collection of all functors from C to D itself as a category.

Example. Let C and D be categories, let F : C → D be a functor. The identity naturaltransformation from F to F, denoted 1F, is the natural transformation such that forevery X in C , the component of 1F at X is 1F(X) : F(X) → F(X).

Definition 1.12 (Counit-unit adjunction). Let C and D be categories. A covariantcounit-unit adjunction between C and D consists of two functors

F : D −→ C

G : C −→ D ,

and two natural transformations

ε : FG −→ 1C

η : 1D −→ GF,

respectively called the counit and the unit of the adjunction, such that the compositions

FFη−−→FGF εF−−→ F

GηG−−→GFG Gε−−→ G

are the identity transformations 1F and 1G on F and G respectively.

Remark.

1. The equations

εF(Y) ◦ F(ηY) = 1F(Y) ∀Y ∈ D

G(εX) ◦ ηG(X) = 1G(X) ∀X ∈ C

are called the first and second counit-unit equations.

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2. If F and G are both contravariant functors, the first natural transformation, ε isdefined from 1C to FG, and the adjunction is called a contravariant counit-unitadjunction.

Proposition 1.1. Let C and D be categories. Let F : D → C and G : C → D be two adjointfunctors and let ε : FG → 1C and η : 1D → GF be a counit-unit adjunction. Then for allX ∈ C and for all Y ∈ D ,

MorC (F(Y), X) ∼= MorD (Y, G(X)).

Proof. [1, Chapter 3, theorem 3.1.5] For each f : F(Y) → X and for each g : Y → G(X),define

Φ( f ) = G( f ) ◦ ηY

Ψ(g) = εX ◦ F(g).

Since ε and η are natural transformations so are Φ and Ψ.We compute the first composition:

Ψ(Φ( f )) = εX ◦ FG( f ) ◦ F(ηY) (F is functor)

= f ◦ εF(Y) ◦ F(ηY) (ε is a natural transformation)

= f ◦ 1F(Y) = f (εF(Y) ◦ F(ηY) = 1F(Y)).

Hence ΨΦ is the identity transformation.Dually, we compute the second composition:

Φ(Ψ(g)) = G(εX) ◦ GF(g) ◦ ηY (G is functor)

= G(εX) ◦ ηG(X) ◦ g (η is a natural transformation)

= 1G(X) ◦ g = g (G(εX) ◦ ηG(X) = 1G(X)).

Hence ΦΨ is the identity transformation, so Φ is a natural isomorphism with inverseΦ−1 = Ψ.

Definition 1.13 (Directed set). A directed set is a pair (I,≤) such that:

• I 6= ∅;

• ≤ is a reflexive and transitive binary relation i.e., ≤ is a preorder;

• for each i, j ∈ I there exists k ∈ I such that i ≤ k and j ≤ k (i.e., k is an upperbound of {i, j}).

Remark. A directed set is a category whose objects are the elements of I and for everyi, j ∈ I, Mor(i, j) = {(i, j)} if i ≤ j, otherwise Mor(i, j) = ∅, with the followingcomposition: (i, j) ◦ (j, k) = (i, k) for all i ≤ j ≤ k in I.

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Definition 1.14 (Direct system). Let (I;≤) be a directed set, and let C be a category.A direct system is a functor from I to C .

Remark. A direct system is denoted {Xi, fi,j}i,j∈I , where {Xi}i∈I is a family of objectsand { fi,j}i,j∈I is a family of morphism such that fi,j : Xi → Xj, for all i ≤ j in I.

Definition 1.15 (Direct limit). Let C be a category. Let {Xi, fi,j}i,j∈I be a direct systemof objects and morphisms of C . The direct limit of this system is an object X ∈ C

together with morphisms {ϕi}i∈I , ϕi : Xi → X, such that ϕi = ϕj ◦ fi,j for all i ≤ j in I.Moreover {X, ϕi}i∈I is universal in the sense that for any other object Y ∈ C togetherwith morphisms {ψi}i∈I , ψi : Xi → Y such that ψi = ψj ◦ fi,j for all i ≤ j in I, thereexists a unique morphism u : X → Y making the following diagram

Xifi,j //

ϕi

��???

????

????

????

ψi

��///

////

////

////

////

////

///

Xj

ϕj

������

����

����

��

ψj

����������������������������

X

∃!u

��������

Y

commute.

Remark. The direct limit may not exist in an arbitrary category.

2 Sheaves

2.1 Theory

Definition 2.1 (Presheaf). Let X be a topological space. A presheaf F of abelian groupson X is a contravariant functor from Top(X) to Ab, such that F (∅) = 0.

Remarks.

(a) Thus to prove that F is a preheaf, one should verify the following properties:

0. F (∅) = 0.

1. For each open set U of X, F (U) is an abelian group.

2. For each open sets V ⊆ U, there is a morphism of groups resV,U from F (U)to F (V).

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Moreover, the morphisms satisfy the following properties:

• for each open set U of X, resU,U = 1F (U);

• for all open sets W ⊆ V ⊆ U, resW,U = resW,V ◦ resV,U .

(b) Let V ⊆ U be two open sets of X. The image by F of the morphism V ↪→ U ofTop(X), denoted resV,U , is called restriction morphism. We often write s|V insteadof resV,U(s), where s ∈ F (U).

(c) The elements of F (U) are called the sections of the presheaf F over the open setU. The elements of F (X) are called the global sections.

(d) We can define sheaves of sets, or sheaves of commutative rings, by changing thetarget of the functor to an appropriate category.

Definition 2.2 (Sheaf). Let X be a topological space. A sheaf of abelian groups F onX is a presheaf of groups on X which satisfies the following conditions.

3. (Uniqueness) If U is an open subset of X, if {Vi}i∈I an open covering of U, and ifs ∈ F (U) is a section such that s|Vi

= 0 for all i ∈ I, then s = 0.

4. (Glueing local sections) If U is an open subset of X, if {Vi}i∈I an open covering ofU, {si}i∈I are such that si ∈ F (Vi) and resVi∩Vj,U(si) = resVi∩Vj,U(sj), then thereexists s ∈ F (U) such that resVi ,U(s) = si (s is unique by condition 3.).

Example.

1. For any open subset U ⊆ R, let F (U) be the set of bounded functions from Uto R. The restriction maps are the usual restrictions of functions. Then F isa presheaf, but F is not a sheaf. Consider the following open covering of R,Vi = {x ∈ R : |x| < i} for all i ∈ N, the identity function 1Vi : x 7→ x is boundedfor every i ∈ N but the identity is not bounded on R, and thus the glueingproperty is not satisfied.

2. Let X be a topological space. For any open subset U of X, let F (U) = C0(U, R)the set of continuous map from U to R. The restrictions map are the usualrestriction of functions. Then F is a sheaf.

It is clear that F is presheaf. Let’s prove that F is sheaf. Let U be an open subsetof X and let {Vi}i∈I be an open covering of U.

(a) Let f be an element of C0(U, R) such that f|Vi= 0 for all i ∈ I. Let s ∈ U,

in particular there exists j ∈ I such that s ∈ Vj. As f|Vj= 0, we have

f (s) = fVj(s) = 0, hence f = 0.

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(b) Let { fi}i∈I be a family of maps such that fi ∈ C0(Vi, R) for all i ∈ I andfi|Vi∩Vj

= f j|Vi∩Vjfor all i, j ∈ I. We define f by f (x) = fi(x) if x ∈ Vi. As

fi|Vi∩Vj= f j|Vi∩Vj

for all i, j ∈ I, f is well-defined. Moreover as f|Vi= fi is

continuous for all i ∈ I, and as {Vi}i∈I is an open covering of U, clearlyf ∈ C0(U, R).

Definition 2.3 (Stalk). Let F be a presheaf on a topological space X and let x be apoint in X. The stalk Fx of F at x is the direct limit of the groups F (U) for all theopen neighbourhoods U of x. Thus Fx = lim

x∈UF (U).

Remark.

1. The direct system is given by open sets of X containing x, together with therestriction morphisms, using the following preorder: U ≤ V if V ⊆ U.

2. The elements of the stalk are called germs.

3. It’s not hard to see that an element of Fx is represented by a pair < U, s >,where U is an open neighbourhood of x, and s ∈ F (U). Moreover, two suchpairs < U, s > and < V, t > are equal in Fx if and only if there exists an openneighbourhood W of x, such that W ⊆ U ∩V and s|W = t|W .

Example. Let F be the sheaf of continuous maps from [0, 1] (with the induce topol-ogy) to R. Let x ∈ [0, 1]. The stalk Fx is C([0, 1], R)/ ∼, where ∼ is the followingequivalence relation:

f ∼ g ⇐⇒ ∃U ⊆ [0, 1], x ∈ U open, f|U = g|U .

Definition 2.4 (Morphism of presheaves). Let F and G be two presheaves on a topo-logical space X. A morphism of presheaves ϕ : F → G , is a natural transformation fromF to G .

Remark.

1. Thus a morphism of presheaves consists in, for every open subset U ⊆ X, amorphism of groups ϕ(U) : F (U) → G (U), such that for every open subsetV ⊆ U and for every s ∈ F (U), ϕ(U)(s)|V = ϕ(V)(s|V).

2. The class of presheaves of abelian groups on a topological space X together withthe class of morphisms of presheaves is a category, denoted Prsh(X).

3. If F and G are sheaves, a morphism of presheaves from F to G is called amorphism of sheaves.

4. The class of sheaves of abelian groups on a topological space X together with theclass of morphisms of sheaves is a category, denoted Ab(X).

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Proposition 2.1. Let F , G be sheaves, and let ϕ : F → G be a morphism of sheaves. Thenfor all x ∈ X, the following morphism

ϕx : Fx −→ Gx

< U, s > 7−→< U, ϕ(U)(s) >,

where U is an open subset of X and s ∈ F (U), is well-defined.

Proof. Let U and V be two open subsets of X. Let s ∈ F (U) and t ∈ F (V) such that< U, s >=< V, t >. Thus there exists W ⊆ U ∩V such that s|W = t|W . Let’s show thatϕx(< U, s >) = ϕx(< V, t >):

(ϕ(U)(s))|W = ϕ(W)(s|W) (ϕ is a morphism of presheaves)

= ϕ(W)(t|W) (s|W = t|W)

= (ϕ(V)(t))|W . (ϕ is a morphism of presheaves)

Thus ϕx(< U, s >) = ϕx(< V, t >), and ϕx is well-defined.

Proposition 2.2. Let F and G be sheaves on a topological space X, and let ϕ : F → G be amorphism of sheaves. Then ϕ is an isomorphism of sheaves if and only if the induced map onthe stalks is an isomorphism of groups for all x ∈ X.

Proof. See [2, Chapter II, Proposition 1.1].

Definition 2.5 (Sheaf associated to a presheaf). Let F be a presheaf on a topologi-cal space X. The sheaf associated to F is a sheaf F+ together with a morphism ofpresheaves θ : F → F+, such that for every sheaf G , and for every morphism ofpresheaves ϕ : F → G there exists a unique morphism of sheaves ψ : F+ → G suchthat the following diagram

Fθ //

ϕ

��

F+

∃!ψ

����

��

��

��

G

commutes.

Proposition 2.3. Let F be a presheaf on X. Then the sheaf F+ associated to F exists and isunique up to isomorphism. Moreover, θx : Fx → F+

x is an isomorphism for every x ∈ X.

Proof. See [3, Chapter 2, Proposition 2.15]

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Definition 2.6 (Presheaf kernel). Let F and G be presheaves on a topological space X.Let ϕ : F → G be a morphism of presheaves. The presheaf kernel of ϕ is the presheafU 7→ ker(ϕ(U)).

Proposition 2.4. Let F , G be sheaves and ϕ : F → G a morphism of sheaves. Thenker(ϕ) : U → ker(ϕ(U)) is a sheaf.

Proof. First we show that ker(ϕ) is a presheaf. As ϕ(U) is a morphism of groups, itis clear that ker(ϕ(U)) is an abelian group for all open subsets U of X. Let V ⊆ Ube an open subset. We define ρV,U : ker(ϕ(U)) → ker(ϕ(V)) by ρV,U(s) = s|V for alls ∈ ker(ϕ(U)). Using the fact that ϕ is a morphism of presheaves, we see that ρV,U iswell-defined, i.e., im(ρU,V) ⊆ ker(ϕ(V)). We now show that ker(ϕ) is a presheaf:

0. ker(ϕ)(∅) = ker(ϕ(∅)) = ker(0) = {0}

1. Let s ∈ ker(ϕ(U)), ρU,U(s) = s|U = s = idker(ϕ(U))(s).

2. Let W ⊆ V ⊆ U be open subsets of X and s ∈ ker(ϕ)(U). Then

ρW,V ◦ ρV,U(s) = ρW,V(s|V)

= (s|V)|W= s|W (because F is presheaf)

= ρW,U(s). (because s ∈ ker(ϕ)(U))

We now show that ker(ϕ) is a sheaf. Let U be an open subset of X and {Vi}i∈I an opencovering of U.

3. Let s ∈ ker(ϕ)(U) such that ρVi ,U(s) = 0 for all i ∈ I. As ker(ϕ)(U) ⊆ F (U),s ∈ F (U) and as ρVi ,U(s) = resVi ,U(s) = 0 for all i ∈ I, and since F is a sheaf,using the local identity condition, we have s = 0.

4. Let s ∈ ker(ϕ)(U) and {si}i∈I be a family of sections such that si ∈ ker(ϕ)(Vi)for all i ∈ I and ρVi∩Vj,U(si) = ρVi∩Vj,U(sj) for all i, j ∈ I. Since F is a sheaf, bythe glueing condition there exists s ∈ F (U) such that s|Vi

= si for all i ∈ I. Wenow show that s is in ker(ϕ)(U),

ϕ(Vi)(s|Vi) = ϕ(Vi)(si) = 0 ∀i ∈ I

= (ϕ(U)(s))|Vi= 0 ∀i ∈ I.

And thus as ϕ(U)(s) ∈ G (U) and as G is a sheaf, by the local identity conditionϕ(U)(s) = 0, i.e., s ∈ ker(ϕ)(U).

Remark. Let F , G be sheaves and ϕ : F → G a morphism of sheaves. Then thepresheaf kernel of ϕ is called kernel and denoted ker ϕ.

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Definition 2.7 (Presheaf image). Let F and G be two presheaves on a topologicalspace X. Let ϕ : F → G be a morphism of presheaves. The presheaf image of ϕ is thepresheaf defined by U 7→ im(ϕ(U))

Definition 2.8 (Image). Let F and G be two sheaves on a topological space X. Letϕ : F → G be a morphism of sheaves. The image of ϕ, denoted im(ϕ) is the sheafassociated to the presheaf image of ϕ.

Definition 2.9 (Injective morphism of sheaves). Let F and G be two sheaves on atopological space X. A morphism of sheaves ϕ : F → G is injective if ker(ϕ) = 0. Thatis ker(ϕ)(U) = 0 for all open subset U ⊆ X.

Definition 2.10 (Surjective morphism of sheaves). Let F and G be two sheaves on atopological space X. A morphism of sheaves ϕ : F → G is surjective if im(ϕ) = G .

Remark. If ϕ : F → G is surjective, the morphisms ϕ(U) : F (U) → G (U) need not tobe surjective for all open subset U ⊆ X.

Definition 2.11 (Exact sequence of sheaves). A sequence of sheaves and morphisms

. . . −→ Fi−1ϕi−1−−−→ Fi

ϕi−−→ Fi+1 −→ . . .

is exact if ker ϕi = im ϕi−1 for all i.

Definition 2.12 (Direct image sheaf). Let X and Y be two topological spaces, let F bea sheaf on Y, and let f : X → Y be a continuous map. The direct image sheaf, denotedf∗F , is the sheaf on Y defined by ( f∗F )(V) = F ( f−1(V)) for all open subsets V ⊆ Y.

Definition 2.13 (Inverse image sheaf). Let X and Y be two topological spaces, let G bea sheaf on Y, and let f : X → Y be a continuous map. The inverse image sheaf, denotedf−1G , is the sheaf associated to the presheaf on X, defined by, U 7→ lim

V⊇ f (U)G (V) for

all open subsets U ⊆ X.

Definition 2.14 (Restriction of sheaves). Let F be a sheaf on a topological space X.Let Z ⊆ X be an open set with the induced topology. If i : Z → X is the inclusionmap, the restriction of F to Z, denoted F |Z, is the sheaf i−1F . Thus F |Z is the sheafU 7→ F (U) where U is an open subset of the topological space Z.

2.2 Exercises

Exercise. [2, Chapter II, Exercise 1.2]

(a) Let ϕ : F → G be a morphism of presheaves and x ∈ X, then

ker ϕx = (ker ϕ)x.

andim ϕx = (im ϕ)x.

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(b) Let ϕ : F → G be a morphism of presheaves and x ∈ X. ϕ is injective (respectivelysurjective) if and only if the induced map on the stalk ϕx is injective (respectivelysurjective) for all x ∈ X.

(c) Show that a sequence . . . −→ Fi−1ϕi−1−−−→ Fi

ϕi−−→ Fi+1 −→ . . . of sheaves andmorphisms is exact if and only if for each x ∈ X the corresponding sequence ofstalks is exact as a sequence of abelian groups.

Solution. (a) Let < U, s >∈ Fx be an element of ker ϕx. Then

ϕx(< U, s >) =< U, ϕ(U)(s) >=< X, 0 >,

so there exists an open set W ⊆ U such that (ϕ(U)(s))|W = 0 = ϕ(W)(s|W). Thuss|W ∈ ker ϕ(W) = (ker ϕ)(W) and then < U, s >=< W, s|W) >∈ (ker ϕ)x.

Let < U, s >∈ Fx be an element of (ker ϕ)x, i.e. U is an open neighbourhood of xand s ∈ (ker ϕ)(U). Thus ϕ(U)(s) = 0 and so,

ϕx(< U, s >) =< U, ϕ(U)(s) >

=< U, 0 > .

Hence < U, s >∈ ker ϕx.

Let < V, t >∈ im ϕx, i.e., V is an open neighbourhood of x, t ∈ G (V) and thereexists < U, s >∈ Fx such that ϕx(< U, s >) =< V, t >. Thus there exists anopen set W ⊆ U ∩V such that x ∈ W and (ϕ(U)(s))|W = t|W . Moreover as ϕ is amorphism of sheaves, (ϕ(U)(s))|W = ϕ(W)(s|W), and thus

< V, t >=< W, ϕ(W)(s|W) >,

clearly ϕ(W)(s|W) ∈ im ϕ(W), thus < V, t >∈ (im ϕ)x because the stalks of thepresheaf image are in bijection with the stalks of the image.

Let < V, t >∈ (im ϕ)x, i.e. V is an open neighbourhood of x and t ∈ im(ϕ(V)).Thus there exists s ∈ F (V) such that ϕ(V)(s) = t, thus

< V, t > =< V, ϕ(V)(s) >

= ϕx(< V, s >).

Thus < V, t >∈ im ϕx.

(b) The result follows from (a) and the definition of an injective (respectively, surjec-tive) morphism of sheaves.

(c) A sequence is exact if ker(ϕi+1) = im(ϕi), hence the result follows from (a).

14

Exercise. [2, Chapter II, Exercise 1.5] Show that a morphism of sheaves is an isomor-phism if and only if it is both injective and surjective.

Solution. It’s a consequence of the previous exercise part (b), and of the Proposition2.2.

Exercise. [2, Chapter II, Exercise 1.8] For any open subset U ⊆ X, show that the functorΓ from sheaves on X to abelian groups (defined by Γ(U, F ) = F (U)) is a left exactfunctor, i.e., if 0 → F ′ → F → F ′′ is an exact sequences of sheaves, then 0 →F ′(U) → F (U) → F ′′(U) is an exact sequence of groups.

Solution. Let 0 −→ F ′ ϕ′−−→ Fϕ−→ F ′′ be an exact sequence of sheaves. We first

need to show that ϕ′(U) is injective for all open subsets U ⊆ X. But by definition ofan injective morphism of sheaves, ker(ϕ′(U)) = ker(ϕ′)(U) = {0}. Moreover, as thesequence is exact, ϕ ◦ ϕ′ = 0, thus ϕ(U) ◦ ϕ′(U) = 0, and thus im(ϕ′(U)) ⊆ ker(ϕ(U))for all open subsets U ⊆ X. As ϕ′ is injective, for all open subsets U ⊆ X, F ′(U) andim(ϕ′(U)) are isomorphic, thus there is a bijection between F and the presheaf imageof ϕ′, U 7→ im(ϕ′(U)), thus as F is a sheaf, the presheaf image of ϕ′ is a sheaf too andso im(ϕ′(U)) = im(ϕ′)(U) for all open subsets U ⊆ X.

Exercise (Direct sum). [2, Chapter II, Exercise 1.9] Let F and G be two sheaves on atopological space X. Show that the presheaf U 7→ F (U)⊕ G (U) is a sheaf. It is calledthe direct sum of F and G , and is denoted by F ⊕ G . Show that it plays the role ofdirect sum and of direct product in the category of sheaves of abelian groups on X.

Solution. We first show that F × G : U 7→ F (U)× G (U) is a sheaf and then that itplays the role of direct product and of direct sum in the category of sheaves. Let U bean open subset of X. We define the restrictions maps as follow:

ρV,U : F (U)× G (U) −→ F (V)× G (V)(s, t) 7−→ (s|V , t|V).

Let (s, t) ∈ F (U)× G (U). We now verify the sheaf definition:

0. ∅ 7→ F (∅)× G (∅) = 0× 0 = 0.

1. ρU,U(s, t) = (s|U , t|U) = (s, t) = 1U×U(s, t).

2. Let W ⊆ V ⊆ U be open subsets of X,

ρW,V ◦ ρV,U(s, t) = ((s|V)|W , (t|V)|W) = (s|W , t|W) = ρW,U(s, t).

3. Let {Vi}i∈I be an open covering of U, and let (s, t) be an element of F (U)×G (U)such that ρVi ,U(s, t) = 0 for all i ∈ I. Thus ρVi ,U(s, t) = (s|Vi

, t|Vi) = (0, 0) for all

i ∈ I, and as F and G are sheaves, s = t = 0.

15

4. Let (si, ti) ∈ F (Vi)× G (Vi) for all i ∈ I such that ρVi∩Vj,U(si, ti) = ρVi∩Vj,U(sj, tj)for all i, j ∈ I. Thus, as

(si)|Vi∩Vj= (sj)|Vi∩Vj

and as F is a sheaf, there exists s ∈ F (U) such that s|Vi= si for all i ∈ I. Using

the same argument, there exists t ∈ G (U) such that t|Vi= ti for all i ∈ I. And

thus ρVi ,U(s, t) = (si, ti) for all i ∈ I.

We now show that this sheaf plays the role of direct product. Let U ⊆ X be anopen set. We define the following morphisms of sheaves

pF :F × G −→ F

pF (U) :(s, t) 7−→ s,

and

pG :F × G −→ G

pG (U) :(s, t) 7−→ t,

for every (s, t) ∈ F(U)× G(U). Let C be a sheaf, and let f : C → F and g : C → G betwo morphisms of sheaves. We define u : C → F × G by

u(U)(c) = ( f (U)(c), g(U)(c))

for all c ∈ C(U). Clearly pF ◦ u = f and pG ◦ u = g. Let u′ : C → F × G be amorphism of sheaves such that pF ◦ u′ = f and pG ◦ u′ = g. As u′ is determined byits components at F and at G , clearly u′ = u and thus by Definition 1.8, F × G playsthe role of direct product

We now show that the direct product of two sheaves plays also the role of directsum. Let U ⊆ X be an open set. We define the following morphisms of sheaves

iF :F −→ F × G

iF (U) :s 7−→ (s, 0),

and

iG :G −→ F × G

iG (U) :t 7−→ (0, t),

for every s ∈ F(U) and every t ∈ G(U). Let C be a sheaf, and let f : F → C andg : G → C be two morphisms of sheaves. We define u : F × G → C by

u(U)(s, t) = f (U)(s) + g(U)(t)

for all (s, t) ∈ F (U) × G (U), where "+" is the operation law in the abelian groupC(U). Clearly u ◦ iF = f , and u ◦ iG = g. Let u′ : F × G → C be a morphism of

16

sheaves such that u′ ◦ iF = f and u′ ◦ iG = g. Let (s, t) ∈ F(U)× G(U), as u′(U) is amorphism of groups:

u′(U)(s, t) = u′(U)((s, 0) + (0, t))= u′(U)(s, 0) + u′(U)(0, t)= (u′ ◦ iF )(U)(s, t) + u′ ◦ iG (U)(s, t)= f (U)(s) + g(U)(t)= u(U)(s, t).

Hence u′ = u and thus by Definition 1.9 F × G plays also the role of direct sum.

Exercise (Adjoint property). [2, Chapter II, Exercise 1.18] Let X and Y be two topologi-cal spaces, and let f : X → Y be a continuous map. Show that for any sheaves F on Xthere is a natural map f−1 f ∗F → F , and for any sheaf G on Y there is natural mapG → f∗ f−1G . Use these maps to show that there is a natural bijection of sets, for anysheaf F on X and G on Y,

MorAb(X)( f−1G , F ) ∼= MorAb(Y)(G , f∗F ).

Hence we say that f−1 is a left adjoint of f∗, and that f∗ is a right adjoint of f−1.

Solution. In the following diagram,

MorPrsh(X)( f−1G , F ) //

��

MorPrsh(Y)(G , f∗F )

MorAb(X)( f−1G , F ) // MorAb(Y)(G , f∗F )

the left vertical arrow is an isomorphism (by definition of the associated sheaf), thus ifwe prove that the top horizontal arrow is an isomorphism then the bottom one will bean isomorphism too. Thus we will only prove the adjunction in the case of presheaves,and so we will only consider morphisms of presheaves.

We define the following natural transformation,

ε : Prsh(X) −→ Prsh(X)

εF : f−1 f∗F −→ F ,

such that for all open subsets U ⊆ X,

εF (U) : ( f−1 f∗F )(U) −→ F (U)

< f−1(V), s > 7−→ s|U ,

where V is an open subset of Y such that f (U) ⊆ V, and s ∈ F ( f−1(V)). This iswell-defined as f (U) ⊆ V implies U ⊆ f−1(V), and thus s|U exists. Let < f−1(V), s >

17

be in f−1 f∗F , where f (U) ⊆ V and s ∈ F ( f−1(V)). Let W ⊆ U be two open subsetsof X. We now show that εF is compatible with the restriction morphisms:

(εF (U)(< f−1(V), s >))|W = (s|U)|W (by definition of ε)

= s|W (because F is a sheaf)

= εF (W)(< f−1(V), s >|W)

= εF (W)(< f−1(V), s >).

Thus εF is a morphism of presheaves. We now show that for every morphism ofpresheaves ϕ : F → F ′, the following diagram

f−1 f∗F

εF

��

f−1 f∗(ϕ)// f−1 f∗F ′

εF ′

��F ϕ

// F ′

commutes. Let ϕ : F → F ′ be a morphism of presheaves, let V be an open subset ofY such that f (U) ⊆ V and let s ∈ F ( f−1(V)), using, in order, the definition of ε, thatϕ is a morphism of presheaves, and again the definition of ε, we obtain:

(εF ′ ◦ ( f−1 f∗ϕ))(U)(s) = εF ′(U)(( f−1 f∗ϕ)(U)(< f−1(V), s >))

= εF ′(U)(< f−1(V), ( f∗ϕ)(V)(s) >)

= εF ′(U)(< f−1(V), ϕ( f−1(V))(s) >)

= ϕ( f−1(V))(s)|U= ϕ(U)(s|U)

= ϕ(U)(εF (U)(< f−1(V), s >))

= (ϕ ◦ εF )(U)(< f−1(V), s >).

Thus the diagram is commutative.We now define the following natural transformation,

η : Prsh(Y) −→ Prsh(Y)

ηG : G −→ f∗ f−1G ,

such that for all open subsets U ⊆ Y,

ηG (U) : G (U) −→ ( f∗ f−1)G (U)s 7−→< U, s > .

18

This is well-defined since f ( f−1(U)) ⊆ U. Let W ⊆ U be two open subsets of X, andlet s ∈ G (U). We now show that ηG is compatible with the restriction morphisms:

ηG (W)(s|W) =< W, s|W >

= (< U, s >)|W .

Thus ηG is a morphism of presheaves. Using the same arguments as before, we showthat for each morphism of presheaves ψ : G → G ′, the following diagram

G

ηG

��

ψ // G ′

ηG ′

��f∗ f−1G

f∗ f−1(ψ)// f∗ f−1G ′

commutes.We compute the first counit-unit equation, f∗ε ◦ η f∗ . The morphism of presheaves

η f∗F is defined by:

η f∗F (U) : ( f∗F )(V) −→ ( f∗ f−1 f∗F )(V)s 7−→< V, s > .

Notice that ( f∗ f−1 f∗F )(V) = limf ( f−1(V))⊆U

F ( f−1(U)) and f ( f−1(V)) ⊆ V. The mor-

phism of presheaves f∗εF is defined by:

f∗εF (V) : ( f∗ f−1 f∗F )(V) −→ ( f∗F )(V)

< f−1(U), s > 7−→ s|V .

Notice that s|V = s| f−1(V) where s is first seen as an element of f∗F (V) (sheaf over Y)and then as an element of F ( f−1(V)) (sheaf over X). Thus it is clear that f∗ε ◦ η f∗ =1 f∗ .

We now compute the second counit-unit equation, ε f−1 ◦ f−1η. The morphism ofpresheaves f−1ηG is defined by:

f−1ηG (U) : ( f−1G )(U) −→ ( f−1 f∗ f−1G )(U)< V, s > 7−→< V, s > .

The first < V, s > is seen as an element of

( f−1G )(U) = limf (U)⊆V

G (V)

19

where f (U) ⊆ V and s ∈ G (V). The second < V, s > is seen as an element of

( f−1 f∗ f−1G )(U) = limf (U)⊆V

( f∗ f−1G )(V)

as f (U) ⊆ V and s ∈ ( f∗ f−1G )(V) = limf ( f−1(V))⊆W

G (W), because f ( f−1(V)) ⊆ V. The

morphism of presheaves ε f−1G is defined by:

ε f−1G (U) : ( f−1 f∗ f−1G )(U) −→ ( f−1G )(U)

< V, s > 7−→ s|U .

Notice that f (U) ⊆ V implies s|U exists. Thus it is clear that ε f−1 ◦ f−1η = 1 f−1 .Hence we have a counit-unit adjunction between Prsh(X) and Prsh(Y), thus by

Proposition 1.1,MorPrsh(X)( f−1G , F ) ∼= MorPrsh(Y)(G , f∗F )

for all F ∈ Prsh(X) and for all G ∈ Prsh(Y).

3 Schemes

In this section, a ring will stand for a commutative ring with unit, 1 6= 0.

3.1 Theory

We first define the algebraic notions needed to study schemes.

Definition 3.1 (Local ring). A commutative ring A, is a local ring if A has exactly onemaximal ideal, denoted mA.

Definition 3.2 (Local morphism of local rings). Let A and B be local rings with maxi-mal ideal mA and mB respectively. A morphism of rings g : A → B is a local morphismof local rings if g−1(mB) = mA.

Definition 3.3 (Spectrum of a ring). Let A be a commutative ring. The spectrum of A,denoted Spec A, is the set of all proper prime ideals of A.

Definition 3.4. Let A be a commutative ring and let a be an ideal of A. We define thesubset V(a) ⊆ Spec A to be the set of all prime ideals which contain a. For an elementf ∈ A we define D( f ) = Spec A−V(( f )).

Proposition 3.1. Let A be a commutative ring.

1. If a and b are ideals of A, then V(ab) = V(a) ∪V(b).

2. If {ai}i∈I is a family of ideals of A, then V(∑i∈I

ai) =⋂i∈I

V(ai).

20

3. V(A) = ∅ and V((0)) = Spec A.

Proof. See [2, Chapter II, Lemma 2.1].

Definition 3.5 (Zariski topology). Let A be a commutative ring. We define the Zariskitopology on Spec A by taking the subsets of the form V(a) to be the closed subsets,where a is an ideal of A.

Remark.

1. Proposition 3.1 ensures that the Zarisky topology is well-defined.

2. The sets of the form D( f ), f ∈ A, are a base of open subsets of Spec A.

Definition 3.6 (Multiplicatively closed set). Let A be a commutative ring. A multiplica-tively closed set of A, is a subset S ⊆ A verifying the following conditions:

• 1 ∈ S and 0 6∈ S;

• S is closed under multiplication, i.e., for all x, y ∈ S, xy ∈ S.

Example. Let f ∈ A, f not nilpotent, then {1, f , f 2, . . . } = { f i}i∈N is a multiplicativelyclosed set.

Definition 3.7 (Localization of a ring). Let A be a commutative ring and let S be amultiplicatively closed set of A. The localization of A at S, denoted S−1A, is the ringdefined as S× A/ ∼, where ∼ is the following equivalence relation:

(s1, a1) ∼ (s2, a2) ⇐⇒ ∃t ∈ S, t(s2a1 − s1a2) = 0.

Remark. An element (s, a) ∈ S−1A is often denoted as .

Example.

1. Let p be a prime ideal of A, then A− p is a multiplicatively closed set of A andwe denote Ap = (A− p)−1 A.

2. Let f ∈ A, f not nilpotent, and let S be the multiplicatively closed set { f i}i∈N.The localization of A at S is denoted A f , and A f = S−1A.

Definition 3.8 (Sheaf of rings on Spec A). Let A be a commutative ring. We define asheaf of rings, denoted OSpec A, over Spec A. For an open set U ⊆ Spec A we defineOSpec A(U) to be the set of functions s : U → äp∈U Ap such that s(p) ∈ Ap for eachp, and such that s is "locally the quotient of elements of A". To be precise, we requirethat for each p ∈ U, there is a neighbourhood V of p, contained in U, and elementsa, f ∈ A, such that for each q ∈ V, f 6∈ q, and s(q) = a

f in Aq.Now it is clear that sums and products of such functions are again such functions,

and that the element 1 which gives 1 in each Ap is an identity. Thus OSpec A(U) is acommutative ring. If V ⊆ U are two open sets of Spec A, the natural restriction mapof functions is a morphism of rings. It is then clear that OSpec A is a presheaf. Finally,it is clear from the local nature of the definition that OSpec A is a sheaf.

21

Proposition 3.2. Let A be a commutative ring.

1. For any p ∈ Spec A, the stalk (OSpec A)p is isomorphic to the local ring Ap.

2. For any element f ∈ A, the ring OSpec A(D( f )) is isomorphic to the localised ring A f .

3. In particular OSpec A(Spec A) ∼= A.

Proof. See [2, Chapter II, Proposition 2.2].

Definition 3.9 (Ringed space). A ringed space is a pair (X, OX), where X is a topologicalspace, and OX is a sheaf of commutative rings on X.

Definition 3.10 (Morphism of ringed spaces). Let (X, OX) and (Y, OY) be ringed spaces.A morphism of ringed spaces from (X, OX) to (Y, OY) is a pair ( f , f ]), where f : X → Yis a continuous map and f ] : OY → f∗OX is a morphism of sheaves (of commutativerings on Y) .

Definition 3.11 (Locally ringed space). A locally ringed space is a ringed space (X, OX)such that for all x ∈ X, the stalk OX,x is a local ring.

Definition 3.12 (Morphism of locally ringed spaces). A morphism of locally ringed spacesis a morphism ( f , f ]) of ringed spaces, such that for all x ∈ X, the induced map oflocal rings f ]

x : OY, f (x) → OX,x is a local morphism of local rings.

Definition 3.13 (Isomorphism of locally ringed spaces). An isomorphism of locally ringedspaces is a morphism of locally ringed spaces with a two-sided inverse. Thus a mor-phism of locally ringed spaces ( f , f ]) is an isomorphism if and only if f is a homeo-morphism, and f ] is an isomorphism of sheaves.

Proposition 3.3. Let A be a commutative ring. Then (Spec A, OSpec A) is a locally ringedspace.

Proof. Clear from Proposition 3.2.

Definition 3.14 (Affine scheme). An affine scheme is a locally ringed space (X, OX)which is isomorphic to some (Spec A, OSpec A) where A is a commutative ring.

Definition 3.15 (Scheme). A scheme is a locally ringed space (X, OX) in which everypoint has an open neighbourhood U such that (U, OX|U) is an affine scheme.

Remark. The class of schemes together with the class of morphisms of schemes is acategory denoted Sch.

Example. Let k be a field, then Spec k is an affine scheme whose topological spaceconsists of one point (0) and whose sheaf consists only of the field k.

22

3.2 Exercises

Exercise. [2, Chapter II, Exercise 2.1] Let A be a ring, let X = Spec A, let f ∈ A (notnilpotent), and let D( f ) ⊆ X be the open complement of V(( f )). Show that the locallyringed space (D( f ), OX|D( f )) is isomorphic to Spec(A f ).

Solution. Let f ∈ A be a non nilpotent element. First notice that D( f ) is the set of allprime ideals p of A such that f 6∈ p. We define the following morphisms of rings:

i : A −→ A f

a 7−→ a1

,

and

α : Spec A f −→ Spec A

q 7−→ i−1(q).

As i−1(q) is prime, α is well-defined. Moreover im α ⊆ D( f ). Suppose (ad absurdum)that there exists q ∈ Spec A f such that i−1(q) 6∈ D( f ), i.e. f ∈ i−1(q), thus f

1 ∈ q, hence,

11

=ff

=1f

f1

(as f is not nilpotent and as q is an ideal of Spec A f )

is an element of q, and thus q = A f 6∈ Spec A f .We now show that the following map is a homeomorphism:

β : D( f ) −→ Spec A f

p 7−→ i(p)A f ,

where i(p)A f is the ideal of A f generated by i(p). We can verify that

i(p)A f ={

af n : n ∈ N, a ∈ p

}is prime in A f , so β is well-defined. Let p ∈ D( f ), we compute β ◦ α:

i−1(i(p)A f ) = i−1{

af n : n ∈ N, a ∈ p

}⊇ p.

To show the other inclusion, let b ∈ i−1{

af n : n ∈ N, a ∈ p

}, thus b

1 = af n , where a ∈ p

and n ∈ N thus f n+mb = f ma where m ∈ N and as f 6∈ p, b ∈ p. Thus i−1(i(p)A f ) = p.Let q ∈ Spec A f . The other composition, α ◦ β:

i(i−1(q))A f = i({

a :a1∈ q

})A f

={ a

1:

a1∈ q

}A f ⊆ q

23

To show the other inclusion, let b = cf m ∈ q. As q is an ideal of A f , f mb = c

1 ∈ q. Thusi(i−1(q))A f = q. Moreover as β(V(a)) = V(i(a)A f ), where a is an ideal of A and asβ−1(V(b)) = V(i−1(b)), where b is an ideal of A f , β is a homeomorphism.

We now show that β] : OSpec A f → β∗OX|D( f ) is an isomorphism of sheaves. Usingthe stalks, we need to show that

(OSpec A f )q∼= (β∗OX|D( f ))q, ∀q ∈ Spec A f .

Let q ∈ Spec A f and let p = β−1(q). Using Proposition 3.2 (2.) and that β is a bijection,we have

(OSpec A f )q∼= (A f )q

∼= (A f )β(p)∼= (A f )i(p)A f

.

Using again that β is a bijection and Proposition 3.2 (1.), we have

(β∗OX|D( f ))q∼= (OX|D( f ))β−1(q)

∼= (OX|D( f ))p∼= Ap.

Thus we only need to show that (A f )i(p)A fis isomorphic to Ap. But this is clear using

the following maps:

Ap −→ (A f )i(p)A f

ab7−→

a1b1

,

and

(A f )i(p)A f−→ Ap

af n

bf m

7−→ a f m

b f n .

Exercise. [2, Chapter II, Exercise 2.2] Let (X, OX) be a scheme, and let U ⊆ X be anopen subset. Show that (U, OX|U) is a scheme. We call this the induced scheme structureon the open set U, and we refer to (U, OX|U) as an open subscheme of X.

Solution. Let x ∈ X, and let V be an open neighbourhood of x such that (V, O|V) is anaffine scheme. As U ∩ V is a non-empty open set in V, there exists f ∈ OV(V) suchthat D( f ) ⊆ U ∩ V and x ∈ D( f ). By Proposition 3.2 (2.),(D( f ), OX|D( f )) is an affinescheme, and thus (U, OX|U) is a scheme.

Exercise. [2, Chapter II, Exercise 2.4] Let A be a ring and let (X, O|X) be a scheme. Givena morphism f : X → Spec A, we have the associated map on sheaves f ] : OSpec A →f∗OX. Taking global sections we obtain a homomorphism A → OX(X). Thus there isa natural map

α : MorSch(X, Spec A) −→ MorCRing(A, OX(X)).

Show that α is bijective.

24

Solution. We will show that there is a counit-unit adjunction between these two con-travariant functors:

Γ : Sch −→ CRing

(X, OX) 7−→ OX(X),

and

Spec : CRing −→ Sch

A 7−→ (Spec A, OSpec A).

The first natural transformation is

ε : 1CRing −→ Γ ◦ Spec

εA : A −→ OSpec A(Spec A)

a 7−→ (sa : p 7→ a1

, ∀p ∈ Spec A).

The second natural transformation is

η : 1Sch −→ Spec ◦Γ

η(X,OX) : (X, OX) −→ (Spec OX(X), OSpec OX(X))

η(X,OX) = (ϕ, ϕ]),

where ϕ is the following map:

ϕ : X −→ Spec OX(X)

x 7−→ i−1x (mx),

where mx is the maximal ideal of OX,x (which is unique as (X, OX) is scheme), andix is the following morphism of rings ix : OX(X) → OX,x. We now show that ϕ is acontinuous map (on a basis of open subset of Spec OX(X)), using the following lemma.

Lemma 3.1. Let U ⊆ X, U is open if and only if there exists on open covering {Vi}i∈I of Xsuch that U ∩Vi is an open subset in Vi for all i ∈ I.

Let f ∈ Spec OX(X),

ϕ−1(D( f )) = X f = {x ∈ X :f16∈ mx}.

Covering X by open affine subschemes, and using Lemma 3.1, we see that ϕ is contin-uous. We define the morphism of sheaves

ϕ] : OSpec OX(X) −→ ϕ∗OX.

25

By definition of OSpec OX(X) and ϕ∗OX, it suffices to define ϕ] on stalks. We can use thelocal morphisms OX(X)i−1

x (mx) → OX,x induced by ix : OX(X) → OX,x.Using the following isomorphism

OSpec OX(X)(Spec OX(X)) −→ ϕ∗OX(Spec OX(X))

we can easily compute the first and second counit-unit equations. The first equation

Γ(η(X,OX)) ◦ εΓ(X,OX= 1Γ(X,OX)

can be decomposed into two parts:

εΓ(X,OX: OX(X) −→ OSpec OX (Spec OX)

a 7−→ sa,

and

Γ(X, OX) : OSpec OX (Spec OX) −→ OX(X)

s 7−→ ϕ](s).

Thus it is clear that Γη ◦ εΓ = 1Γ The second equation

Spec(εA) ◦ ηSpec(A) = 1Spec(A)

can be decomposed into two parts:

ηSpec(A) : (Spec A, OSpec A) −→ Spec(OSpec A(Spec A))

p 7−→ i−1(mp) = p,

and we can verifiy that

Spec(εA) : Spec(OSpec A(Spec A)) −→ (Spec A, OSpec A)

is the identity. Thus it is clear that η Spec ◦ Spec ε = 1Spec, and we have the counit-unitadjunction.

Exercise. [2, Chapter II, Exercise 2.5] Describe Spec Z, and show that it is a final objectfor the category of schemes, i.e., each scheme X admits a unique morphism to Spec Z.

Solution. The only prime ideals of Z are (0) and the ideals of the form (p) where pis a prime number. As the ideals of Z are (n) where n ∈ N, the closed subsets areV((n)) = {(p) : p prime, p|n}.

As Z is an initial object in the category CRing, using Exercise 3.2 with A = Z, it isclear that (Spec Z, OSpec Z) is a final object in the category of schemes, Sch.

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Exercise. [2, Chapter II, Exercise 2.10] Describe Spec R[X]. How does its topologicalspace compare to R? To C?

Solution. The prime ideals of R[X] are (0), all the ideals generated by X − a wherea ∈ R, and all the ideals generated by (X − α)(X − α) where α ∈ C and =α 6= 0. Thusthe following application:

R −→ Spec R[X]a 7−→ (X − a)

is an injection, while the following application

C −→ Spec R[X]α 7−→ ((X − α)(X − α)) if =α 6= 0

α 7−→ (X − α) if =α = 0

is a surjection.

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References

[1] Francis Borceux, Handbook of categorical algebra. 1, Encyclopedia of Mathematicsand its Applications, vol. 50, Cambridge University Press, Cambridge, 1994, Basiccategory theory.

[2] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977, GraduateTexts in Mathematics, No. 52.

[3] Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Math-ematics, vol. 6, Oxford University Press, Oxford, 2002, Translated from the Frenchby Reinie Erné, Oxford Science Publications.

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