Operations on Étale Sheaves of Sets

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IN

DEGREE PROJECT

MATHEMATICS,

SECOND CYCLE, 30 CREDITS

,

STOCKHOLM SWEDEN 2016

Operations on Étale Sheaves

of Sets

ERIC AHLQVIST

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ENGINEERING SCIENCES

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Operations on Étale Sheaves of Sets

E R I C A H L Q V I S T

Master’s Thesis in Mathematics (30 ECTS credits)

Master Programme in Mathematics (120 credits)

Royal Institute of Technology year 2016

Supervisor at KTH: David Rydh

Examiner:

David Rydh

TRITA-MAT-E 2016:26 ISRN-KTH/MAT/E--16/26--SE

Royal Institute of Technology

School of Engineering Sciences

KTH SCI

SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

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Abstract. Rydh showed in 2011 that any unramified morphism f of algebraic spaces (algebraic stacks) has a canonical and universal factorization through an algebraic space (algebraic stack) called the étale envelope of f, where the first morphism is a closed immersion and the second is étale. We show that when f is étale then the étale envelope can be described by applying the left adjoint of the pullback of f to the constant sheaf defined by a pointed set with two elements. When f is a monomorphism locally of finite type we have a similar construction using the direct image with proper support.

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Sammanfattning. Rydh visade 2011 att varje oramifierad morfi f av alge-braiska rum (algealge-braiska stackar) har en kanonisk och universell faktorisering genom ett algebraiskt rum (algebraisk stack) som han kallar den étala omslut-ningen av f, där den första morfin är en sluten immersion och den andra är ´

etale. Vi visar att d˚a f är étale s˚a kan den étala omslutningen beskrivas genom att applicera vänsteradjunkten till tillbakadragningen av f p˚a den konstanta kärven som definieras av en punkterad mängd med tv˚a element. D˚a f är en monomorfi, lokalt av ändlig typ s˚a har vi en liknande beskrivning i termer av framtryckning med propert stöd.

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Acknowledgements

I would like to thank my thesis advisor David Rydh for his support and guid-ance. I am grateful for his commitment and that he always makes time for questions. I think that I still have not been able to ask something that he cannot answer.

I would like to thank Gustav Sæd´en St˚ahl for always taking his time to answer my questions.

I would also like to thank Oliver G¨afvert and Johan W¨arneg˚ard.

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Introduction

There are cases when the Zariski topology is to coarse to work in. For ex-ample if we want to mimic results that are true in the Euclidean topology like the implicit function theorem or cohomology. Hence it may be convenient to work in finer topologies like the étale topology which has properties more like the Eu-clidean topology. The étale topology is an example of a Grothendieck topology and was defined by A. Grothendieck who developed it together with M. Artin and J.-L. Verdier. The aim was to define étale cohomology in order to prove the Weil conjectures [Wei49].

Given a category C we may define a Grothendieck topology on C by assigning a collection of coverings {Ui → U } for each object U in C. A category with a

Grothendieck topology is called a site. An example of a site is the big ´etale site S´_Eton a scheme S, where the underlying category is (Sch/S) and a covering of an

S-scheme U is a jointly surjective family {Ui→ U } of ´etale S-morphisms. Given a

site S with underlying category C, we may consider sheaves on S. That is, functors F : Cop_{→ (Set)}

satisfying a certain gluing condition for each covering {Ui → U }. Every S-scheme

X is a sheaf on S_Et´ when identifying X with the contravariant functor hX =

Hom(Sch/S)(−, X).

If R ⇒ X are ´etale S-morphisms such that the induced map

Hom(Sch/S)(T, R) → Hom(Sch/S)(T, X) × Hom(Sch/S)(T, X)

is injective for every S-scheme T , and gives an equivalence relation ∼ on the set

Hom(Sch/S)(T, X), then we may form the presheaf quotient T 7→ X(T )/ ∼. The

sheafification of this presheaf is an algebraic space over S and is denoted X/R. This generalizes the concept of schemes.

The small site S´et on a scheme (or algebraic space) S has underlying category

(ét/S) (or ét(S)), i.e., the category of étale schemes (algebraic spaces) over S, and coverings as in SEt´. Given a sheaf F on Sét we may construct its espace

´

etalé Fét which is an étale algebraic space over S. This gives an equivalence of categories between sheaves on the small étale site Sét and étale algebraic spaces

over S. In particular, every sheaf on the small ´etale site S´et of an algebraic space

S is representable by an étale algebraic space over S. The espace étalé has the following analogue in classical topology: given a topological space B and a sheaf of sets G on B, the espace étale of G is a topological space E together with a local homeomorphism π : E → B such that G is the sheaf of sections of π (see e.g. [MLM94, Section II.5]).

Every morphism f : T → S of schemes (algebraic spaces), gives rise to a mor-phisms Tét → Sét of sites. Hence we may consider push-forwards f∗: Sh(Tét) →

Sh(Sét) and pullbacks f∗: Sh(Sét) → Sh(Tét). We have that f∗F is just the

restric-tion of the fiber product T ×SF´et to the small ´etale site. In certain cases f∗ has

a left adjoint denoted by f!. For example, in case f : T → S is an object in (´et/S) 9

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10 INTRODUCTION

we get that f!F is the sheaf given by

U 7→a

ϕ

F (U )

for every S-scheme ψ : U → S where the disjoint union is over all S-morphisms U → T . In case, F is a sheaf of pointed sets, we get that f!F is the sheafification

of the presheaf

U 7→_

ϕ

F (U ) .

Rydh shows in [Ryd11] that any unramified morphisms X → Y of algebraic spaces (algebraic stacks) factors as X ,→ EX/Y → Y where the first morphism is

a closed immersion and the second morphism is ´etale. We show that in the case when f is a monomorphism, we get that the restriction EX/Y to the small ´etale site

is naturally isomorphic to the sheaf fc{0, 1}X, where {0, 1}X denotes the constant

sheaf on the small ´etale site on X and fc is the direct image with proper support.

If f is ´etale then EX/Y,´et = f!{0, 1}X, where f! is the left adjoint of the pullback.

Hence we have the following conjecture:

Conjecture. Let X and Y be algebraic spaces and let f : X → Y be a mor-phism locally of finite type. There exists a functor

f#: Sh∗(X´et) → Sh∗(Y´et)

of sheaves of pointed sets such that:

(1) if f is unramified, we have EX/Y = f#{0, 1}X;

(2) if f is ´etale we have f#= f!;

(3) if f is a monomorphism we have f#= fc.

Preliminaries

By a ring we always mean a commutative ring with unity. All rings are assumed to be Noetherian and all schemes are assumed to be locally Noetherian. A morphism of schemes is called proper if it is of finite type, separated, and universally closed. A morphism f : X → Y of schemes is called finite if there is an open covering Y =S Vi

of Y by affine open subschemes Visuch that for every i we have f−1(Vi) is affine and

the induced homomorphism OY(Vi) → OX(f−1(Vi)) is finite. Or equivalently (see

for example [GW10, 12.9]), for every open affine subscheme V ⊆ Y , the inverse image f−1(V ) is affine and OY(V ) → OX(f−1(V )) finite. In particular, every finite

morphism is by definition affine. A morphism is called quasi-finite if it is of finite type and the fiber over each point consists only of finitely many points.

Theorem 0.0.1 (Zariski’s Main Theorem [Mil80, I.1.8]). Let f : X → Y be a morphism of schemes and assume that Y is quasi-compact. The following are equivalent:

(1) f is quasi-finite and separated;

(2) f factors as X−→ Xα −→ Y where α is an open immersion and β is finite.β Lemma 0.0.2 ([GW10, 12.89]). Let f : X → Y be a morphism. The following are equivalent:

(1) f is finite;

(2) f is quasi-finite and proper; (3) f is affine and proper.

For locally Noetherian schemes we have the following topological property: Lemma 0.0.3. Let X be a locally Noetherian scheme and let V ⊆ X be a subset. Then the following are equivalent:

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PRELIMINARIES 11

(1) V is clopen (open and closed) in X;

(2) V is a union of connected components of X.

Proof. (1) ⇒ (2): Let V ⊆ X be a clopen subset intersecting a connected component C of X. Then C ∩ V and (C \ V ) are both open. Thus C = (C ∩ V ) q (C \ V ) and we conclude that C ∩ V = C since C is connected. Hence we see that a clopen subset is always a union of connected components.

(2) ⇒ (1): A connected component is always closed since the closure of a connected subset is connected. We will show that every connected component of X is open. Since X is locally Noetherian it is locally connected (see for example [Sta, Tag 04MF]). But X is locally connected if and only if every connected component of X is open [Bou98, I.11.6.11]. Hence we get that every connected component of X is clopen (open and closed). Now if W is a union of connected components (i.e. clopen subsets) then so is its complement X \ W and hence both W and X \ W are both open, and hence also closed. Thus W is clopen.

The following lemma is trivial but will be useful later on. Lemma 0.0.4. Suppose that we have a cartesian square

T ×SU pr₁ // U f T // S

in the category (Sch) of schemes such that there is a morphism s : S → U satisfying f ◦ s = idS. Then T is the fiber product of s and pr1.

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CHAPTER 1

´

Etale morphisms

An ´etale morphism is the algebraic analogue of a local homeomorphism. For example, a morphism of nonsingular varieties over an algebraically closed field is ´

etale at a point if and only if it induces an isomorphism of the tangent spaces. The main references to this chapter are [Mil80] and [AK70].

1.1. Flat morphisms

Recall that a ring homomorphism A → B is called flat, if B is flat when considered as an A-module, i.e., if the functor − ⊗AB is exact. It is called faithfully

flat if − ⊗AB is faithful and exact.

Proposition 1.1.1. Let ϕ : A → B be a ring homomorphism. The following are equivalent:

(1) ϕ is flat;

(2) For every ideal I ⊆ A, the map I ⊗ B → B; a ⊗ b 7→ ϕ(a)b is injective. Proof. If ϕ is flat then clearly I ⊗ B → B is injective since I → A injective implies that I ⊗ B → A ⊗ B = B is injective. For the converse, see [Mil80,

I.2.2].

Proposition 1.1.2. Let ϕ : A → B be a ring homomorphism. The following are equivalent:

(1) ϕ is flat;

(2) For every m ∈ Spm B, the induced homomorphism Aϕ−1_(m)→ B_m is flat.

Definition 1.1.3. Let f : X → Y be a morphism of schemes. Then we say that f is flat at x ∈ X if the induced map OY,f (x) → OX,x is flat. We say that f

is flat if it is flat at every x ∈ X.

Remark 1.1.4. Proposition 1.1.2 implies that a morphism is flat if and only if it is flat at all closed points.

Remark 1.1.5. A flat ring homomorphism induces a flat morphism of spectra. Proposition 1.1.6 ([Mil80, I.2.5]). Let A → B be a ring homomorphisms that makes B a flat A-algebra. Take b ∈ B and suppose that the image of b in B/mB is not a zero-divisor for every maximal ideal m of A. Then B/(b) is a flat A-algebra. Example 1.1.7. Let A be a ring and let f ∈ A[T1, . . . , Tn] be non-zero. Let

V ⊆ Spec A[T1, . . . , Tn] be the closed subscheme given by the ideal (f ), i.e.,

V ∼= Spec A[T1, . . . , Tn]/(f ) .

Suppose that the image of f in (A/m)[T1, . . . , Tn] is non-zero for every maximal ideal

mof A, or equivalently, that the ideal generated by the coefficients of f is A. Then the morphism V → Spec A induced by the morphism ϕ : A → A[T1, . . . , Tn]/(f ) is

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14 1. ´ETALE MORPHISMS

flat by Proposition 1.1.6. The converse is also true since if the coefficients of f is contained in a maximal ideal m of A, then the homomorphism

m⊗AA[T1, . . . , Tn]/(f ) → A[T1, . . . , Tn]/(f )

a ⊗ g 7→ ϕ(a)g is not injective. Indeed, f may be written as

f = X

α∈Nn

aαTα

and the non-zero element

X

α∈Nn

aα⊗ Tα

in m ⊗AA[T1, . . . , Tn]/(f ) will be mapped to zero. Hence by Proposition 1.1.1, ϕ

is not flat.

Proposition 1.1.8 ([Mil80, I.2.7], [AK70, V.1.9]). Let ϕ : A → B be a ring homomorphism. The following are equivalent:

(1) ϕ is faithfully flat;

(2) ϕ is injective and B/ϕ(A) is flat over A;

(3) a sequence M0 _{→ M → M}00 _{of A-modules is exact if and only if the}

sequence M0⊗AB → M ⊗AB → M00⊗AB is exact;

(4) ϕ is flat and the induced morphism Spec B → Spec A is surjective; (5) ϕ is flat and for every maximal ideal m ⊂ A, we have ϕ(m)B 6= B. Hence the following definition agrees with the definition for rings.

Definition 1.1.9. Let f : X → Y be a morphism of schemes. Then we say that f is faithfully flat if it is flat and surjective.

Remark 1.1.10. Proposition 1.1.8 implies that a morphism Spec B → Spec A is faithfully flat if and only if the ring homomorphism A → B is faithfully flat.

Example 1.1.11. For a scheme X, the projection X q · · · q X → X is certainly faithfully flat.

Lemma 1.1.12. Let (A, mA) and (B, mB) be local rings. Then any flat local

homomorphism ϕ : A → B is faithfully flat.

Proof. Since ϕ is local we have ϕ(mA) ⊆ mB and hence mAB 6= B. Hence

the Lemma follows from Proposition 1.1.8. Lemma 1.1.13. A composition of flat morphisms is flat and a base change of a flat morphism is flat.

Proof. Let X −→ Yf −→ Z be flat morphisms. Take x ∈ X and put y = f (x)g and z = g(y). Flatness of g ◦ f follows from the fact that if M is an Oz-module,

then

(M ⊗Oz Oy) ⊗Oy Ox∼= M ⊗Oz Ox.

To show that a base change of a flat map is flat, let f : X → Y be a flat morphism and let f0: Y0 → Y be a morphism. Take any x ∈ X and any y0 _{∈ Y}0

such that f (x) = f0(y0) =: y ∈ Y . We must show that the induced homomorphism Oy0 → O_y0⊗_O

y Oxis flat. But again, this follows trivially since

M ⊗Oy0 (Oy0⊗Oy Ox) ∼= M ⊗OyOx.

Here are some topological properties of flat morphisms.

Theorem 1.1.14 ([Mil80, I.2.12]). Any flat morphism that is locally of finite type is open.

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1.2. UNRAMIFIED MORPHISMS 15

Corollary 1.1.15 ([Mil80, I.3.10]). Any closed immersion which is flat is an open immersion.

Proposition 1.1.16 ([Gro65, 2.3.12]). If f : X → Y is a flat surjective quasi-compact morphism of schemes then Y has the quotient topology induced by f .

Theorem 1.1.17. Let f : X → Y be a morphism of schemes. Then the set flat(f ) = {x ∈ X : f is flat at x}

is open in X.

Proof. See [AK70, V.5.5] 1.2. Unramified morphisms

Definition 1.2.1. Let k be a field and ¯k its algebraic closure. A k-algebra A is called separable if the Jacobson radical of A ⊗k¯k is zero.

Definition 1.2.2. Let f : X → Y be a morphism of schemes which is locally of finite type. Then we say that f is unramified at x ∈ X if mx= myOxand κ(x) is a

finite separable field extension of κ(y), where y = f (x). We say that f is unramified if it is unramified at every x ∈ X.

Definition 1.2.3. A geometric point of a scheme X is a morphism ¯x : Spec Ω → X where Ω is a separably closed field. If Y → X is a morphism then the geometric fiber over a geometric point ¯x is the fiber product Y ×XSpec Ω.

Proposition 1.2.4 ([Mil80, I.3.2]). Let f : X → Y be a morphism which is locally of finite type. The following are equivalent:

(1) f is unramified;

(2) for all y ∈ Y , the projection Xy → Spec κ(y) is unramified;

(3) for all geometric points ¯y : Spec Ω → Y , the projection Xy¯ → Spec Ω is

unramified;

(4) for every y ∈ Y , there is a covering of Xy by spectra of finite separable

κ(y)-algebras;

(5) for every y ∈ Y , we have Xy ∼=` Spec ki, where the kiare finite separable

field extensions of κ(y).

Lemma 1.2.5. A composition of unramified morphisms is unramified and a base change of an unramified morphism is unramified.

Proof. The composition part is trivial. To show the second part, let X → Y be unramified and Z → Y any morphism. By Proposition 1.2.4 it is enough to show that Z ×Y X → Z is unramified after base change to a geometric point. But

a geometric point in Z gives a geometric point in Y and Spec Ω ×Z(Z ×Y X) =

Spec Ω ×Y X and we already know that Spec Ω ×Y X → Spec Ω is unramified.

Proposition 1.2.6 ([AK70, VI.3.3], [Mil80, I.3.5]). Let X and Y be schemes, x a point in X, and f : X → Y a morphism locally of finite type. Let ΩX/Y denote

the sheaf of relative differentials of X over Y . The following are equivalent: (1) f is unramified at x;

(2) we have (ΩX/Y)x= 0;

(3) the diagonal ∆X/Y is an open immersion in a neighborhood of x.

Note that (ΩX/Y)x= ΩOx/Oy.

Proof. (1) ⇒ (2): By base change, we may assume that Y = Spec κ(y) and X = Xy (see [Har77, II.8.10]). The fact that f is unramified at x implies that

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Spec κ(x). Hence we need only show that Ωκ(x)/κ(y) = 0. But this is clear since

κ(x) is a finite separable extension of κ(y).

(2) ⇒ (3): The diagonal ∆X/Y: X → X ×Y X is locally closed and hence

we may choose an open subscheme U of X ×Y X, containing ∆X/Y(X), such that

X → U is a closed immersion. Denote this map i : X → U and let J = i∗OX. By

definition we have ΩX/Y = ∆∗X/Y(J /J

2_{). Hence 0 = (Ω}

X/Y)x∼= (J /J2)i(x) and

by Nakayama’s lemma we have that Ji(x)= 0. Hence there is an open neighborhood

V ⊆ U of i(x) such that J |V = 0. Thus, ∆X/Y|V = i|V is an open immersion.

(3) ⇒ (1): By 1.2.4 we may assume that Y = Spec k where k is an algebraically closed field (we may choose Y = Spec κ(y) where y = f (x) and then change base to Spec of the algebraic closure of κ(y)). Since unramified is a local property, we may assume that X = Spec A is affine and that ∆X/Y is an open immersion. Let

z ∈ X be a closed point. Then Hilbert’s nullstellensatz implies that κ(z) = k. Let ϕ : X → X ×Y X be the morphism induced by the identity morphism on X

and the constant morphism X → X with value z. Since the diagonal is open, so is ϕ−1(∆X/Y(X)) = {z}. Hence every closed point of A is open, i.e., every prime ideal

is maximal. Thus, A is Artinian and hence we may assume that A = OX,x with

maximal ideal m and κ(x) = k since x is a closed point. Hence we get that A⊗kA has

a unique maximal ideal m⊗A+A⊗m and since ∆X/Y: Spec A → Spec (A⊗kA) is an

open immersion we have that A ⊗kA ∼= A. But dimk(A ⊗kA) = dimk(A) · dimk(A)

and hence we conclude that A ∼= k. This implies (1). Corollary 1.2.7. Let f : X → Y be a morphism. The following are equivalent:

(1) f is unramified; (2) we have ΩX/Y = 0;

(3) the diagonal ∆X/Y : X → X ×Y X is an open immersion.

Corollary 1.2.8. Let f : X → Y be a morphism of schemes. Then the set unram(f ) = {x ∈ X : f is unramified at x}

is open in X.

Proposition 1.2.9. Any section of an unramified morphism is an open im-mersion.

Proof. If f : X → Y is unramified then the diagonal ∆ : X → X ×Y X is

an open immersion. Given a section s : Y → X of f , we have that Y is the fiber product of the diagram

X ∆ X = Y ×Y X s×idX // X ×Y X

and the projections Y → X both coincide with s. Thus s is obtained by base change from ∆ which is an open immersion, and hence s is an open immersion.

1.3. ´Etale morphisms

Definition 1.3.1. Let f : X → Y be a morphism of schemes. Then we say that f is étale at x ∈ X if it is flat and unramified at x. We say that f is étale if it is étale at every x ∈ X.

Remark 1.3.2. Note that Theorem 1.1.14 implies that every ´etale morphism is open as a map between topological spaces.

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1.3. ´ETALE MORPHISMS 17

Lemma 1.3.3. Let f : X → Y be a morphism of schemes. Then the set ´etale(f ) = {x ∈ X : f is ´etale at x}

is open in X.

Proof. This follows from Proposition 1.2.6 and Theorem 1.1.17. Example 1.3.4. Let X and Y be nonsingular varieties over an algebraically closed field k, and let f : X → Y be a morphism of schemes. Then f is ´etale at x ∈ X if and only if it induces an isomorphism Tf: TX,x → TY,f (x) of the tangent

spaces.

Proof. Indeed, let x ∈ X be a closed point and put y = f (x). Then κ(y) ∼= κ(x) ∼= k. Suppose that f is ´etale and put Ox= OX,x and Oy = OY,y. We have

homomorphisms k → Ox→ κ(x) and k → Oy→ κ(y), which yield exact sequences

mx/m2x→ ΩOx/k⊗Oxκ(x) → Ωκ(x)/k= 0 ,

and

my/m2y → ΩOy/k⊗Oy κ(y) → Ωκ(y)/k= 0

[Mat86, 25.2]. The first map in each of the sequences is an isomorphism [Har77, 8.7]. We also have homomorphisms k → Oy→ Ox, where the last one is faithfully

flat since it is a local homomorphism and f is flat. We get an exact sequence ΩOy/k⊗OyOx→ ΩOx/k → ΩOx/Oy → 0

[Mat86, 25.1], where the first map is an isomorphism by [AK70, VI.4.9]. These may all be viewed as Ox-modules. If we tensor with κ(x) we get that

ΩOy/k⊗Oyκ(x) ∼= ΩOx/k⊗Oxκ(x) ,

and hence

my/m2y∼= mx/m2x.

Since the cotangent spaces are isomorphic and so are the duals.

Conversely, if Tf: TX,x→ TY,y is an isomorphism, then so is the induced map

my/m2y→ mx/mx2. Let d = dim(my/m2y). Then my can be generated by d elements,

t1, . . . , td [AM69, 11.22]. The ring Ox/(t1, . . . , td) is flat over Oy/(t1, . . . , td) =

κ(y) ∼= k and by [Har77, 10.3.A], Ox/(t1, . . . , ti) is flat over Oy/(t1, . . . , ti) for

each i = d, d − 1, . . . , 0. Hence Ox is flat over Oy. Since my/m2y → mx/m2x is an

isomorphism we get from the exact sequence

ΩOy/k⊗Oyκ(y) → ΩOx/k⊗Oxκ(x) → ΩOx/Oy⊗Oxκ(x) → 0 ,

that ΩOx/Oy/mxΩOx/Oy ∼= ΩOx/Oy ⊗Oxκ(x) = 0. Hence, by Nakayama’s lemma,

we conclude that (ΩX/Y)x= ΩOx/Oy = 0, and by Proposition 1.2.6, f is unramified

at x.

Lemma 1.3.5. A composition of étale morphisms is étale and a base change of an étale morphism is étale.

Proof. Follows from Lemma 1.1.13 and Lemma 1.2.5. Definition 1.3.6. A morphism of schemes is called smooth if it is flat, locally of finite presentation, and if the geometric fibers are regular.

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1.4. Local structure of ´etale morphisms

Let A be a ring and p(T ) ∈ A[T ] a monic polynomial. Then A[T ]/(p) is a finitely generated free A-module, and hence flat. Suppose that b ∈ A[T ]/(p) is such that the formal derivative p0(T ) is invertible in (A[T ]/(p))b.

Definition 1.4.1. The morphism of spectra Spec (A[T ]/(p))b→ Spec A

induced by the canonical homomorphism ϕ : A → (A[T ]/(p))b is called standard

´ etale.

A standard ´etale morphism is ´etale. Indeed, it is flat since A[T ]/(p) is a free A-module and (A[T ]/(p))b is a flat A[T ]/(p)-module. Now put B = A[T ]/(p). To

show that Spec Bb → Spec A is unramified, it is enough to prove that the Bb

-module ΩBb/A is zero. We have that ΩB/A is the B-module generated by dT and

the relation p0(T )dT = 0 (see e.g. [Mat86, p. 195]). That is ΩB/A is isomorphic

to A[T ]/(p0, p) as a B-module. But then ΩBb/A ∼= (ΩB/A)b ∼= (A[T ]/(p

0_{, p))}

b ∼=

(A[T ]/(p))b/(p0)b = 0 since p0 is invertible in Bb. Hence Spec Bb → Spec A is

unramified and thus ´etale.

Theorem 1.4.2 (Local structure theorem, [Mil80, I.3.14, I.3.16]). Let f : X → Y be a morphism of schemes. The following are equivalent:

(1) f is ´etale at x;

(2) There exists open affine sets U 3 x and V 3 f (x) such that f (U ) ⊆ V and f |U: U → V is standard ´etale;

(3) There exists open affine sets U = Spec B 3 x and V = Spec A 3 f (x), such that B = A[T1, . . . , Tn]/(p1, . . . , pn) where det (∂pi/∂Tj) is invertible

in B, and f |U: U → V is induced by the canonical homomorphism A → B.

Proof. (1) ⇒ (2): By Lemma 1.3.3, f is ´etale in a neighborhood of x. Now see [Mil80, I.3.14].

(2) ⇒ (3): This follows since (A[T ]/p)b∼= A[T, S]/(p, bS − 1) and

∂p/∂T ∂p/∂S ∂(bS − 1)/∂T ∂(bS − 1)/∂S =p 0_{(T )} ₀ b0_/b _b is invertible.

Since we have already showed that every standard ´etale morphism is ´etale, it is enough to show that (3) implies (2) to finish the proof.

(3) ⇒ (2): Since B is generated as an A-algebra by the elements T1, . . . , Tn,

we have that ΩB/A is generated as a B-module by the elements dT1, . . . , dTn and

the relations (1.4.0.1) dpi= n X j=1 ∂pi ∂Tj dTj= 0 , 1 ≤ i ≤ n .

Indeed, the derivation d : B → ΩB/A is surjective and every element in B may be

written as a polynomial f (T1, . . . , Tn). By the Leibniz rule we have

df (T1, . . . , Tn) = n X i=1 ∂f ∂Ti dTi.

Since the image of det(∂pi/∂Tj) in B is a unit, there is a unique solution to (1.4.0.1),

namely dT1= · · · = dTn= 0. Hence Spec B → Spec A is unramified by Proposition

1.2.6.

To show that B is flat as an A-module, one may use Proposition 1.1.6 and induction on n. We have that A[T1, . . . , Tn] is a free A-module and hence flat over

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1.5. HENSELIAN RINGS 19

A. The idea is to show inductively that A[T1, . . . , Tn]/(p1, . . . , pi) is flat over A as

i ranges from 0 to n. This is done in [Mum99, p. 221]. Example 1.4.3. Let n be a positive integer and X = Spec Z[T ]/(Tn − 1). Consider the morphism f : X → Spec Z given by the canonical homomorphism Z ,→ Z[T ]/(Tn − 1). It is clear that f is ´etale in the open subscheme D(n) = X \ V ((n)) since ∂(Tn_{− 1)/∂T = nT}n−1_{and O}

X(D(n)) = (Z[T ]/(Tn− 1))n. That

is, nTn−1 _{has an inverse n}−1

T in (Z[T ]/(Tn_{− 1))}

n.

Example 1.4.4 (Artin-Schreier cover). Let k be a field of non-zero character-istic p and take f ∈ k[T ]. The morphism

Spec k[T, x]/(xp− x − f ) → Spec k[T ]

is ´etale since ∂(xp− x − f )/∂x = pxp−1_{− 1 = −1. If p does not divide the degree}

of f then this covering is non-trivial.

1.5. Henselian rings

For a scheme X, we denote by ClOp(X) the collection of clopen subsets of X. Definition 1.5.1. Let X be a scheme and X0 a closed subscheme. The pair

(X, X0) is called a Henselian pair if for every finite morphism X0→ X, the induced

map ClOp(X0) → ClOp(X0×XX0) is bijective.

Definition 1.5.2. A local ring (A, m) is called Henselian if (Spec A, Spec A/m) is a Henselian pair.

Lemma 1.5.3. There is a bijective correspondence ClOp(Spec A) 'idempotents of A .

Proof. If e ∈ A is idempotent and p ∈ Spec A, then e ∈ p if and only if 1 − e /∈ p. Hence we get that V (e) = D(1 − e) is clopen with complement D(e) = V (1 − e). Conversely, if U ⊆ Spec A is clopen, then U ∪ (Spec A \ U ) is an open cover of Spec A and hence, by the sheaf property, there is a unique element

a ∈ OSpec A(Spec A) = A ,

such that a|U = 1 and a|Spec A\U = 0. Hence (1 − a)|U = 0 and (1 − a)|Spec A\U = 1.

Thus a(a − 1) = 0 since the restrictions to U and Spec A \ U are zero and hence a is idempotent. We get that U = D(a). If f is a polynomial with coefficients in a local ring A with maximal ideal m, then we use the notation ¯f for its image in (A/m)[x].

Theorem 1.5.4 ([Mil80, I.4.2]). Let (A, m) be a local ring, X = Spec A, and let x be the closed point in X. The following are equivalent:

(1) A is Henselian;

(2) every finite A-algebra B is a direct product of local rings B =Q Bi;

(3) if f : Y → X is a quasi-finite and separated morphism, then Y = Y0q Y1q · · · q Yn,

where x /∈ f (Y0) and for i ≥ 1, Yi = Spec Bi is finite over X where each

Bi is a local ring;

(4) if f : Y → X is an ´etale morphism then every morphism γ : Spec κ(x) → Y , such that f ◦ γ(Spec κ(x)) = x, factors through a section s : X → Y of f ;

(5) if f ∈ A[x] is a monic polynomial such that ¯f factors as ¯f = g0h0, with

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Proof. (1) ⇒ (2): Let f : Spec B → Spec A be finite. We have Spec B ×ASpec κ(x) = Spec (B ⊗Aκ(x)) .

There is a bijective correspondence between idempotents of B and idempotents of B ⊗Aκ(x) ∼= B/mxB. If B is not local, then there exists a non-trivial idempotent

¯

e ∈ B/mxB, and hence ¯e lifts to some non-trivial idempotent e ∈ B. Hence

B ∼= eB × (1 − e)B where eB 6= 0 and (1 − e)B 6= 0. Iterating this process yields the desired splitting.

(2) ⇒ (3): Let f : Y → X be quasi-finite and separated. According to Theorem 0.0.1, f factors as

Y f

0

−→ Y0 g−→ X

where f0 is an open immersion and g is finite. Hence Y0 = Spec B for some finite A-algebra B, and by (2), B =Q Bi. Each Bi is of the form Bi= OY0_,y0 for some

closed point y0 ∈ Y0_{. Let Y}

1 = ` Spec OY,y where the disjoint union is over all

closed points y of Y0 that are contained in Y . Thus Y1 is clopen in Y0 and hence

also clopen in Y . Put Y0= Y \ Y1. Then we have Y = Y0` Y1and it is clear that

Y0 contains no closed points of Y0. Since Y1 is finite over X we get that all points

in the fiber over x are closed. Hence they are also closed in Y0 since the preimage of x in Y0 is closed. Thus x /∈ f (Y0).

(3) ⇒ (4): Suppose that f : Y → X is ´etale and we have a morphism Spec κ(x) → Y

with image y ∈ Y such that f (y) = x. Then we have embeddings κ(x) ,→ κ(y) ,→ κ(x) and hence κ(y) = κ(x). Since OY,y is a flat A-module, it is free [Mil80,

I.2.9]. But f is ´etale and hence we have that my = mxOY,y and κ(x) = κ(y) =

OY,y ⊗Aκ(x). That is, OY,y has rank 1, i.e., OY,y ∼= A. By (3) we may assume

that Y = Spec B where B is a local ring. That is, B = OY,y ∼= A. Hence (4) holds.

(4) ⇒ (5): See the proof of (d) ⇒ (d’) ⇒ (d) in [Mil80, I.4.2].

(5) ⇒ (1): It is enough to show that for every finite A-algebra B, the homo-morphism

B → B ⊗A(A/m) ∼= B/mB

gives a bijection of idempotents. But this follows immediately from (5) since every nilpotent in B/mB lifts to a unique idempotent in B. Remark 1.5.5. One may actually replace ´etale with smooth in Theorem 1.5.4 (4) (see [Gro67, Corollaire 17.16.3]).

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CHAPTER 2

Representable functors

2.1. Definitions and examples

Let C be a category. A functor F : Cop _{→ (Set) is called representable if it}

is isomorphic to the functor hX = HomC(−, X) for some object X in C. We also

say that X represents the functor F . Note that for a morphism ϕ : Y → Z in C, the map ϕ∗: hX(Z) → hX(Y ) is given by sending a morphism ψ : Z → X to the

morphism ψ ◦ ϕ : Y → X. Furthermore, we have a natural transformation hϕ: hY → hZ

defined by sending a morphism β : W → Y to the composition ϕ ◦ β : W → Z. Remark 2.1.1. A functor F : C op→ (Set) is representable if and only if it has a universal object, that is, if there exists a pair (X, ξ), where X is an object in C and ξ ∈ F X, such that for any Y ∈ C and any element η ∈ F Y , there exists a unique f ∈ Hom(Y, X) such that f∗(ξ) = η.

Example 2.1.2. Let A be a ring, let f1, . . . , fm∈ A[T1, . . . , Tn] be polynomials,

and put R = A[T1, . . . , Tn]/(f1, . . . , fm). Let S be a scheme over Spec A and put

X = Spec R. We have

Hom(Sch)(S, X) ' Hom(A-alg)(R, Γ(S, OS))

' {s ∈ Γ(S, OS)n: f1(s) = · · · = fn(s) = 0} ,

where the last isomorphism is given by sending an A-algebra homomorphism ϕ to the tuple (ϕ(T1), . . . , ϕ(Tn)) (clearly fi(ϕ(T1), . . . , ϕ(Tn)) = ϕ(fi(T1, . . . , Tn)) for

all 1 ≤ i ≤ n). Hence we see that the functor (Sch/A) → (Set) that sends a scheme S over Spec A to the set

{s ∈ Γ(S, OS)n: f1(s) = · · · = fn(s) = 0}

is represented by Spec (A[T1, . . . , Tn]/(f1, . . . , fn)).

Example 2.1.3 (Affine n-space). In particular, the functor (Sch)op→ (Set)

that sends a scheme S to the set Γ(S, OS)n is represented by Spec (Z[T1, . . . , Tn]).

Indeed, we have bijections

Hom(Sch)(S, An) ' Hom(Ring)(Z[T1, . . . , Tn], Γ(S, OS)) ' Γ(S, OS)n,

which are natural in S.

Example 2.1.4 (Gm = Spec Z[T, T−1]). As another special case of Example

2.1.2, we get that the functor (Sch)op _{→ (Set) that sends a scheme S to the set}

Γ(S, OS)∗of units in Γ(S, OS) is represented by Gm. This follows from the

isomor-phism Z[T, T−1] ∼_{= Z[T, X]/(T X − 1).}

Example 2.1.5. (The Grassmannian) Consider the functor Gk,n: (Sch)op→ (Set)

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22 2. REPRESENTABLE FUNCTORS

defined by

Gk,n(X) = {F ⊆ O⊕nX : O ⊕n

X /F is locally free of rank n − k}

and which takes a morphism f : Y → X to the map Gk,n(f ) : Gk,n(X) → Gk,n(Y )

which takes F to the pullback f∗F . To see that f∗F ∈ Gk,n(Y ) note first that

i : F ,→ O_X⊕ngives a morphism f∗F → f∗_(O⊕n X ) = O

⊕n

Y which is injective since i is

injective and O⊕n_X /i(F ) is locally free [GW10, 8.10]. For any subset I ⊆ {1, . . . , n} we may define a subfunctor GI ⊆ Gk,nby

GI(X) = {F ∈ Gk,n(X) : O⊕IX ,→ O ⊕n

X O

⊕n

X /F is an isomorphism} ,

where the morphism O_X⊕I,→ O⊕n_X is induced by the inclusion I ,→ {1, . . . , n} and by O_X⊕I we mean OX⊕ · · · ⊕ OX with one component for each index in I.

For every F ∈ GI(X) we have a morphism OX⊕n O ⊕I

X with kernel F , and

conversely, for every retraction τ : O⊕n_X _O⊕I_X of the inclusion O⊕I_X ,→ O_X⊕n, we get that ker(τ ) ∈ GI(X). Hence there is a bijection between the set of retractions

r : O_X⊕n→ O_X⊕Iof the inclusion O_X⊕I→ O⊕n_X and elements of GI(X). It is not hard

to see that this bijection is functorial in X. Such a retraction r must be the identity on the indices in I and hence r is completely determined by its values on the index set Ic= {1, . . . , n} \ I. Hence we conclude that we have a functorial bijection

GI(X) ' Hom(OX-mod)(O

⊕Ic

X , O

⊕I

X ) .

But we also have natural bijections

Hom(OX-mod)(O

⊕Ic

X , O

⊕I

X ) ' Hom(Set)(Ic× I, Γ(X, OX)) ' Γ(X, OX)k(n−k)

(see [GW10, 7.4.6]), and by Example 2.1.3 we conclude that there is a natural bijection

GI(X) ' Hom(Sch)(X, Ak(n−k)) .

That is, the functor GI is represented by the affine scheme Ak(n−k). This may

be used to show that the functor Gk,n is representable (see [GW10, Proposition

8.14]). In particular, one may show that G1,n+1 is represented by the projective

space Pn

Z.

2.2. The Yoneda embedding

Lemma 2.2.1 (Yoneda’s lemma). For any object X in C, the map αX,F: Hom(hX, F )

∼

−→ F (X) τ 7→ τX(idX) ,

is a bijection which is natural in X and F .

Proof. The first part follows from the fact that any natural transformation τ : hX → F is completely determined by the image of idX ∈ hX(X) in F (X).

Indeed, consider the commutative diagram hX(X) τX // f∗ F (X) f∗ hX(Y ) τY // F(Y )

induced by a morphism f : Y → X. We have τY(f ) = τY(f∗(idX)) = f∗(τX(idX)).

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2.2. THE YONEDA EMBEDDING 23

Let f : Y → X be a morphism in C, and let hf: hY → hXbe the induced natural

transformation. To prove naturality in X, we need to show that the following diagram commutes: Hom(hX, F ) αX,F // f# F (X) f∗ Hom(hY, F ) αY,F // F(Y )

where f#is the map defined by taking hX→ F to the composition hY → hX → F .

Let τ : hX→ F be a natural transformation. Then

f∗◦ αX,F(τ ) = f∗(τX(idX))

= τY(f )

= τY(f ◦ idY)

= (τ ◦ hf)Y(idY)

= αY,F◦ f#(τ ) .

Naturality in F is trivial since if η : F → G is a natural transformation of functors, then Hom(hX, F ) → Hom(hX, G) is just given by composition with η and

by definition we have (η ◦ τ )X(idX) = ηX(τX(idX)).

Hence we have a functor Hom(h(−), F ) : Cop→ (Set); X 7→ Hom(hX, F ) and

Lemma 2.2.1 says that there is an isomorphism of functors Hom(h(−), F ) ∼= F .

Remark 2.2.2. Note that Yoneda’s lemma implies that any map F (X) → F (Y ) given by a morphism Y → X is exactly the map given by left composition by hY → hX.

A morphism f : X → Y in a category C gives a natural transformation hX → hY

by composing with f . Thus, the assignment X 7→ hX is a functor C → PreSh(C)

from C to the category PreSh(C) of functors C op_{→ (Set) (that is, the category of}

presheaves on C). Yoneda’s lemma implies that

HomPreSh(C)(hX, hY) ' HomC(X, Y ) ,

i.e., the functor X 7→ hX is fully faithful.

Definition 2.2.3. The embedding C → PreSh(C); X 7→ hX is called the

Yoneda embedding.

Remark 2.2.4. Yoneda’s lemma implies that there is an equivalence of cate-gories between C and the category of representable functors F : C op_{→ (Set) given}

by sending an object X to hX. Hence if Y is an object in C, we write

X(Y ) = hX(Y ) = HomC(Y, X) .

Remark 2.2.5. Note that two objects X and Y in a category C are isomorphic if and only if hX and hY are isomorphic as functors

(CX,Y)op→ (Set)

where CX,Y is the full subcategory of C with only two objects X and Y .

Remark 2.2.6. Note that since id : S → S is the final object in Sch/S, we have that hS is the final object in the category PreSh(Sch/S). Indeed, every S-scheme

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24 2. REPRESENTABLE FUNCTORS

X comes with a morphism f : X → S. Morphisms ϕ : X → S are commutative diagrams X ϕ // f S id S

Obviously, we must have f = ϕ and hence hS(X) consists of a single point. Thus,

if F is a presheaf on (Sch/S) then there is a unique map F (X) → hS(X) for each

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CHAPTER 3

Sheaves of sets

In the following chapter we will discuss sheaves on sites, which is a generaliza-tion of the concept of a sheaf on a topological space. The definigeneraliza-tion is very similar, keeping in mind that the fiber product Ui×UUj in the category Open(X) of open

subsets of a topological space X, with morphisms given by inclusions, is just the intersection Ui∩ Uj taken in U .

3.1. Grothendieck topologies and sites

The main references to the following section are [FGI+05, Mil80, Mil21, Tam94].

Definition 3.1.1. Let C be a category with fiber products. A Grothendieck topology on C is defined by the following data: for each object U in C we have a collection Cov(U ) of coverings of U . A covering is a set of arrows {Ui → U }i∈I.

The coverings satisfy the following axioms:

(1) if V → U is an isomorphism then {V → U } is a covering;

(2) if {Ui→ U } is a covering and V → U is a morphism, then {V ×UUi→ V }

is a covering;

(3) if {Ui→ U } is a covering and for every index i we have a covering {Vij →

Ui}, then {Vij → Ui → U } is a covering of U .

A category together with a Grothendieck topology is called a site. If S is a site then the underlying category is denoted by Cat(S).

For any family of maps {φi: Ui → U } between spaces of any kind, we say

{ϕi: Ui→ U } is jointly surjective if U =S ϕi(Ui).

Remark 3.1.2. We will sometimes also use the following notation, as in [Mil80, II.1]: Let E be a class of morphisms of schemes such that

(1) every isomorphism is in E,

(2) any composition of morphisms in E is in E, and (3) any base change of a morphism in E is in E.

Let S be a scheme and E a class of morphisms as above. Let C/S be a full sub-category of Sch/S which is closed under taking fiber products and such that for any U → S in C/S and any E-morphism U0 → U , the composition U0 _{→ U → S}

is in C/S. Then we get a Grothendieck topology on C/S by taking as coverings: all collections {ϕi: Ui → U } of E-morphisms over S such that U =S ϕi(Ui). The

resulting site will be denoted by SE or (C/S)E.

Example 3.1.3 (Small classical topology on a topological space X). Consider the category Open(X) of open subsets of a topological space X, where the mor-phisms are given by inclusions. A covering of an open subset U ⊆ X is a jointly surjective family {Ui→ U }.

Example 3.1.4 (The big classical topology on (Top)). Consider the category (Top) of topological spaces. A covering of a topological space U is a jointly surjective family of open embeddings Ui→ U .

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26 3. SHEAVES OF SETS

Example 3.1.5 (Small Zariski site on X). Consider the category ZarOp(X) of Zariski-open subsets of a scheme X with morphisms that are inclusion maps. A covering is a jointly surjective family {Ui→ U }. This site is denoted Xzar.

Example 3.1.6 (Big Zariski site on S). Consider the category (Sch/S) of schemes over S. A covering of a scheme U → S is a jointly surjective family of open immersions Ui→ U over S. The corresponding site is denoted by SZar.

Example 3.1.7 (Small étale site on X). Let (ét/X) be the category whose objects are étale morphisms U → X and whose arrows are X-morphisms V → U of schemes. The coverings are jointly surjective families of morphisms {Ui → U }

in (ét/X) (i.e., étale X-morphisms). By Lemma 1.3.5, the property of being étale is stable under base change and composition and hence this defines a Grothendieck topology on (ét/X). The corresponding site is denoted Xét.

Example 3.1.8 (Big ´etale site over S). The site with underlying category (Sch/S) and coverings which are jointly surjective families {Ui → U } of ´etale

S-morphisms is denoted by S´_Et.

Example 3.1.9 (Big flat site on S). Consider the category (Sch/S) with cov-erings which are jointly surjective families {Ui → U } of flat S-morphisms which

are locally of finite presentation. Note that this implies that the induced map ` Ui→ U is flat and surjective, i.e., faithfully flat. This site is denoted by SFl.

Definition 3.1.10. A morphism X → Y of schemes is called an fpqc morphism if it is faithfully flat and every quasi-compact open subset of Y is the image of a quasi-compact open subset of X.

Example 3.1.11 (Big fpqc site on S). The site with underlying category (Sch/S) and coverings which are jointly surjective families {Ui→ U } of S-morphisms

such that the induced map` Ui→ U is fpqc is denoted by SFpqc.

Remark 3.1.12. For a scheme S, we have continuous morphisms (see Definition 4.1.1) SFpqc // SFl // S´Et // SZar S´et // Szar

induced by the identity morphism S → S.

Proposition 3.1.13 ([FGI+05, Proposition 2.33]). Let f : X → Y be a sur-jective morphism of schemes. The following are equivalent:

(1) every quasi-compact open subset of Y is the image of a quasi-compact open subset of X;

(2) there is a covering Y =S Vi of Y by open affine subschemes Vi such that

each Vi is the image of a quasi-compact open subset of X;

(3) for every x ∈ X, there is an open neighborhood U of x, such that the restriction f : U → f (U ) is quasi-compact and f (U ) is open in Y ; (4) for every x ∈ X there is a quasi-compact open neighborhood U of x such

that f (U ) is open in Y and affine.

3.2. Sheaves of sets

Definition 3.2.1. A presheaf (of sets) on a site S is a functor F : Cat(S)op→ (Set). A presheaf F is called separated if for every covering {Ui→ U }, the map

F (U ) →Y

i∈I

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3.2. SHEAVES OF SETS 27

is injective.

A presheaf F is called a sheaf if the diagram (3.2.0.2) F (U ) →Y i∈I F (Ui) pr1 ⇒ pr₂ Y (j,l)∈I×I F (Uj×UUl)

is an equalizer diagram for every covering {Ui → U }i∈I in S, where the parallel

arrows are defined as follows: pr_k: Y i F (Ui) → Y j,l F (Uj×UUl) , k ∈ {1, 2}

sends (ai)i to the element with component at index (j, l) equal to pr∗1aj if k = 1

and pr∗

2al if k = 2.

Given a presheaf F and a morphism U → V , we call the induced map F (V ) → F (U ) a restriction map.

Remark 3.2.2. To say that (3.2.0.2) is an equalizer diagram, or that F (U ) → Q F (Ui) is an equalizer of the diagramQ F (Ui) ⇒Q F (Ui×UUj) is to say that for

each arrow A →Q F (Ui) such that the composites h : A →Q F (Ui) ⇒Q F (Ui×U

Uj) coincides, there is a unique arrow h0: A → F (U ) such that h is the composition

of h0 with the arrow F (U ) →Q F (Ui).

To say that there always exists such an arrow h0 is to say that F (U ) maps surjectively onto the subset of Q F (Ui) consisting of all elements whose images

under the two maps toQ F (Ui×UUj) coincide.

To say that such a map h0(whenever it exists) is unique, is to say that the map F (U ) →Q F (Ui) is injective, i.e., that F is separated.

Remark 3.2.3. Note that if we have morphisms Y

α

Z β // X in some category C with fiber products, then

hZ×XY = hZ×hX hY

in the category PreSh(C). Indeed, let F be a presheaf and suppose that we have a commutative diagram: F // hY hZ // hX

Let W be an object in C. An element in F (W ) will be mapped to some ϕ : W → Y and some ψ : W → Z such that α ◦ ϕ = β ◦ ψ. But this gives a unique map γ : W → Z ×XY such that ϕ = prY ◦ γ and ψ = prZ◦ γ. Hence we conclude that

every F (W ) → hY(W ) and F (W ) → hZ(W ) factors uniquely through hZ×XY(W ).

That is, hZ×XY = hZ×hX hZ.

Remark 3.2.4. To define a presheaf we do not need a topology on the category, and hence we may not only speak of presheaves on sites, but also presheaves on categories.

A morphism of sheaves is just a natural transformation of presheaves. Given a site S, we get a category of sheaves on S, i.e., a category where the objects are sheaves on S and the morphisms are morphisms of sheaves on S.

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Definition 3.2.5. Given a site S, let PreSh(S) denote the category of presheaves on S and let Sh(S) denote the category of sheaves on S.

Example 3.2.6 (Sheaf on (C/S)Eof a set M ). Let M be a set and S a scheme.

For every S-scheme X, define

M (X) = Mπ0(X)_{= Hom}

(Set)(π0(X), M ) ,

where π0(X) denotes the set of connected components of X. A morphism f : Y → X

of S-schemes maps a connected component of Y into a connected component of X, and hence f defines a map σ : π0(Y ) → π0(X) . Hence we get a map

Hom(Set)(π0(X), M ) → Hom(Set)(π0(Y ), M ) .

This defines a presheaf and it is not hard to see that this presheaf is also a sheaf. This sheaf will be denoted by MS or just M .

Example 3.2.7. A representable functor (Top)op → (Set) is a sheaf in the big classical topology (defined in Example 3.1.4). Indeed, consider the functor

Hom(Top)(−, X) where X is a topological space. LetS Uibe an open covering of U

and suppose that we have continuous maps fi: Ui→ X for each i ∈ I, such that fi

and fj agree on Ui∩ Uj for each i, j ∈ I. Then there is a unique continuous map

f : U → X such that f |Ui = fi.

Example 3.2.8. Any representable presheaf F on SZaris a sheaf. Indeed, let X

be an S-scheme and let {Ui→ U } be a covering in SZar. The fiber product Ui×UUj

may be identified with the intersection Ui∩ Uj in U . It is a well known fact that

if U =S Ui and we have S-morphisms fi: Ui→ X that agree on each intersection

Ui∩ Uj, then there is a unique S-morphism f : U → X such that f |Ui= fi.

Lemma 3.2.9 ([FGI+05, 2.60]). A presheaf F on SFpqc is a sheaf if and only

if it satisfies the following two conditions:

(1) F satisfies the sheaf condition for Zariski open coverings;

(2) for any cover {V → U } in SFpqc with U and V affine, we have that

F (U ) → F (V ) ⇒ F(V ×U V )

is an equalizer diagram.

Proof. It is clear that the two conditions are necessary for F to be a sheaf on SFpqc. For the converse, suppose that {Ui → U }i∈I is a covering in SFpqc, with U

and each Ui affine. If F satisfies (1) then F (` Ui) =Q F (Ui) since {Uj →` Ui}j

is a Zariski open covering. We have that a Ui ×U a Ui =a(Ui×UUj) ,

and hence if the index set I is finite then` Ui is affine, and hence the upper row

in the diagram

F (U ) // Q F(Ui) // Q F(Ui×UUj)

F (U ) // F(` Ui) // F((` Ui) ×U(` Ui))

is an equalizer diagram by (2) (this is the diagram arising from the cover {` Ui →

U }).

Now let {g : V0 → V } be any covering in SFpqc. By Proposition 3.1.13 there

is a covering V0 = S V0

i of open quasi-compact subschemes such that the image

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3.3. SIEVES AND ELEMENTARY TOPOI 29

union of open subschemes V_ik0 ⊆ V0

i. The Vi’s form an open affine covering of V .

Now consider the following diagram: F (V ) _γ // α F (V0₎ _//// δ F (V0_× V V0) Q i F (Vi) β _// Q i Q k F (V0 ik) //// Q i Q k,l F (V0 ik×V Vil0) Q i,j F (Vi∩ Vj) // Q i,j Q k,l F (V0 ik∩ Vjl0)

The first two columns are equalizer diagrams by (1) and the second row is an equalizer diagram by (2). The maps α and β are injective and hence so is γ. Thus F is a separated presheaf on SFpqc and hence the bottom row is injective. Now

take an element s ∈ F (V0) and suppose that s maps to the same element via the two maps to F (V0×V V0). Then δ(s) maps to the same element via the two maps

to Q

i

Q

k,lF (Vik0 ×V Vil0), which implies that δ(s) ∈ im(β). Let t ∈

Q

iF (Vi) be

the element such that β(t) = δ(s). We have that δ(s) maps to the same element in Q

i,j

Q

k,lF (Vik0 ∩ Vjl0) and since the bottom row is injective t must map to the

same element via the two maps to Q

i,jF (Vi ∩ Vj). Thus t ∈ im(α) and since δ

is injective, we see that s ∈ im(γ). Thus the top row is an equalizer diagram and hence F is a sheaf on SFpqc.

3.3. Sieves and elementary topoi

Let X be a set and A a subset of X. Then A is completely determined by a characteristic map χA: X → {0, 1}, where χA(x) = 0 if x ∈ A and χA(x) = 1 if

x /∈ A. This gives a pullback diagram A //

{0}

X // {0, 1}

One says that the inclusion (monomorphism) {0} → {0, 1} is a ”subobject classi-fier”. In general, if C is a category with terminal object 1, then a subobject classifier for C is a monomorphism 1 → Ω such that for every monomorphism A → X in C, there is a unique pullback square

A // 1 X // Ω (see [ML98, p. 105]).

A category C is called an elementary topos, if it satisfies the following three properties:

(1) C has all finite limits; (2) C has a subobject classifier;

(3) C is cartesian closed (see [ML98, p. 97]).

The category Sh(S) of sheaves on a site S is an elementary topos [MLM94, III.7.4]. The subobject classifier is defined in terms of sieves.

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Definition 3.3.1. Let U be an object in a category C. Then a subfunctor of hU: Cop→ (Set)

is called a sieve.

The subobject classifier Ω in Sh(S) is defined by taking Ω(U ) to be the set of all so called ”closed sieves” on U for any object U in S. Let F be a sheaf on S. Every subsheaf G of F is determined by its characteristic morphism χG: F → Ω.

See [MLM94, Section III.7] for details. We will only use the fact that distinct subsheaves gives distinct characteristic morphisms.

3.4. Epimorphisms

Recall that an arrow X → Y between objects in a category C is called an epimorphism if whenever we have two arrows f, g : Y ⇒ Z such that the compo-sitions X → Y ⇒ Z agree, we have f = g. Equivalently we can say that the map HomC(Y, Z) → HomC(X, Z) is injective for every object Z. Recall also that a

diagram W β ⇒ α X → Y

is a coequalizer diagram (X → Y is a coequalizer of W ⇒ X) if every morphism X → V such that the two maps W ⇒ X → V coincide factors uniquely through X → Y . Now, if we have a pair of morphisms α, β : W → X, then every morphism X → V such that the maps

W

β

⇒

α

X → V

coincide factors through X → Y if and only if HomC(Y, V ) maps surjectively onto

the subset {f ∈ Hom(X, V ) : f ◦ α = f ◦ β}. Hence we conclude that W

β

⇒

α

X → Y

is a coequalizer diagram in a small category C if and only if the diagram HomC(Y, V ) → HomC(X, V ) ⇒ HomC(W, V )

is an equalizer diagram in (Set) for every object V in C. This means in particular that HomC(Y, V ) → HomC(X, V ) is injective for every Y → V and thus that X → Y

is an epimorphism.

Definition 3.4.1. A morphism X → Y in a category C with fiber products is called an effective epimorphism if the following diagram is a coequalizer diagram:

X ×Y X

pr₁

⇒

pr₂

X → Y .

We say that X → Y is a universally effective epimorphism if for every morphism Y0 → Y , the morphism X ×Y Y0→ Y0 is an effective epimorphism.

Remark 3.4.2. Thus, to say that any presheaf hX on SFpqc (given by an

S-scheme X) is a sheaf, (it is a sheaf in the Zariski topology) is to say that any fpqc morphism V → U , with U and V affine, is an effective epimorphism.

Definition 3.4.3. For an object X in a category C, let hX be the covariant functor

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3.4. EPIMORPHISMS 31

There is also a covariant version of Yoneda’s lemma which states that for any covariant functor F : C → (Set) there is a bijection

Hom(hX, F ) ' F (X) . In particular, we have

Hom(hX, hY) ' Hom(Y, X) .

Note that the functor X 7→ hX is contravariant, since a morphism Y → X gives a morphism hX→ hY _{by precomposing with Y → X.}

Lemma 3.4.4. Let X, Y , and W be objects in a category C. Then the diagram W ⇒ X → Y

is a coequalizer diagram in C if and only if the induced diagram hY → hX _{⇒ h}W

is an equalizer diagram in the category Funct(C, (Set)) of functors C → (Set). Proof. The ”if part” is an easy consequence of the covariant Yoneda lemma. To prove the converse, suppose that we have a natural transformation of functors F → hX such that the compositions F → hX _{⇒ h}W coincide. By the discussion above, we know that for every object Z in C, F (Z) → hX(Z) factors through hY(Z) → hX(Z). If Z → Z0 is a morphism then we get a diagram

F (Z) // hY(Z) // hX(Z) F (Z0) // hY(Z0) // hX(Z0)

where the right square commutes and the big square commutes. Since the map hY(Z0) → hX(Z0) is injective by the discussion above, we get that the left square is commutative and hence F → hX factors through hY → hX. Lemma 3.4.5. Let S be a site on which representable presheaves are sheaves. If {V → U } is a covering in S then the induced morphism hV → hU is an effective

epimorphism in the category Sh(S).

Proof. Let F be a sheaf on S. Then the diagram F (U ) → F (V ) pr∗₁ ⇒ pr∗ 2 F (V ×UV )

is an equalizer diagram in (Set) and by Yoneda’s lemma and Remark 3.2.3, this is isomorphic to Hom(hU, F ) → Hom(hV, F ) pr∗₁ ⇒ pr∗ 2 Hom(hV ×hU hV, F ) .

If we have a morphism τ : hV → F such that the composites hV×hUhV ⇒ hV → F

coincides, then pr∗₁(τ ) = pr∗₂(τ ) and hence τ is in the image of Hom(hU, F ) ,→ Hom(hV, F ) .

That is, there is a unique morphism hU → F such that hV → F factors through

hV → hU.

Definition 3.4.6. Let α : F → G be a morphism of sheaves on a site S. Then we say that α is locally surjective if for every object U in S, the following is true: for every y ∈ G(U ), there is a covering {ϕi: Ui → U } such that for every index i

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A subsheaf of a sheaf F is a subfunctor/subpresheaf of F which is a sheaf. Lemma 3.4.7. Let F be a sheaf on a site S and let G ⊆ F be a subpresheaf. Then the following are equivalent:

(1) G is a sheaf;

(2) for every object U in S and every element x ∈ F (U ), the following holds: for every covering {ϕi: Ui → U }, if ϕ∗i(x) ∈ G(Ui) for every i, then

x ∈ G(U ).

Proof. (2) ⇒ (1) : G(U ) → Q G(Ui) is injective since F (U ) → Q F (Ui) is

injective. But (2) says exactly that G(U ) → Q G(Ui) is surjective onto the set

of points y ∈ Q G(Ui) such that p1(y) = p2(y) where p1, p2 are the morphisms

Q G(Ui) ⇒Q G(Ui×U Uj). Hence G is a sheaf.

(1) ⇒ (2): This is clear from the definition of a sheaf. Lemma 3.4.8. Let S be a site. For a morphism α : F → G in Sh(S), the following are equivalent:

(1) α is an epimorphism; (2) α is locally surjective.

Proof. Define a presheaf G0 by defining G0(U ) to be the set of x ∈ G(U ) such that there exists a covering {ϕi: Ui → U } such that ϕ∗i(x) ∈ im(αUi) for all i.

Then G0 _{is a sheaf by Lemma 3.4.7. Consider the characteristic morphisms}

χG, χG0: G ⇒ Ω .

Then since αU(F (U )) ⊆ G0(U ), we have χG◦α = χG0◦α and if α is an epimorphism,

then we have χG0 = χ_G, i.e., G = G0 and hence α is locally surjective. Another way

to say this is that since α factors through the monomorphism i : G0 → G and α is an epimorphism, we have that i is also an epimorphism. But in a topos, every monomorphism which is also an epimorphism is an isomorphism [MLM94, IV.2.2]. Conversely, suppose that α is locally surjective and let f, g : G ⇒ H be two morphisms of sheaves such that the compositions F → G ⇒ H agree. Then for every object U ∈ S the maps F (U ) → G(U ) ⇒ H(U ) agree. For every y ∈ G(U ) there is a covering {ϕi: Ui→ U } such that ϕ∗i(y) ∈ im(αUi) for every index i. Thus

we have a diagram G(U ) gU // fU // Q ϕ∗ i H(U ) Q ϕ∗ i Q G(Ui) Q g_Ui// Q f_Ui // Q_H(U i) where (Q ϕ∗ i)(fU(y)) = (Q fUi)((Q ϕ ∗ i)(y)) = (Q gUi)((Q ϕ ∗ i)(y)) = (Q ϕ∗i)(gU(y)). But (Q ϕ∗

i) : H(U ) → Q H(Ui) is injective since H is a sheaf, and hence we have

that fU(y) = gU(y). Thus f = g and α is an epimorphism.

Lemma 3.4.9. Let S be a site. Any epimorphism α : F → G in Sh(S) is effec-tive.

Proof. If α is an epimorphism, then it is locally surjective. That is, for every object U in site and every element x ∈ G(U ), there is a covering {ϕi: Ui → U }

such that ϕ∗_i(x) ∈ im(αUi) for all i. To show that α is effective, let H be a sheaf

and suppose that we have a morphism F → H. To say that the compositions Y F (Ui) ×Q G(Ui) Y F (Ui) ⇒ Y F (Ui) → Y H (Ui)

coincide is to say that whenever two elements a, b ∈ Q F (Ui) map to the same

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3.5. EXAMPLES OF SHEAVES 33

every preimage of (ϕ∗_i(x))i ∈ Q G(Ui) in Q F (Ui) will map to the same element

z ∈Q H(Ui). But this argument also holds if we replace` Ui by` Ui×U Uj and

since the two images of (ϕ∗_i(x))i in Q G(Ui×U Uj) coincide, we get that the two

images of z in H(Ui×U Uj) coincide. Since H is a sheaf, this means that z has

a (unique) preimage w ∈ H(U ). Define a map γU: G(U ) → H(U ) by γU(x) = w.

Since α is a natural transformation, these γU will patch together to a natural

transformation γ : G → H such that F → H factors through γ. Since F → G is an epimorphism, γ is the only morphism with this property. Hence F ×GF ⇒ F → G

is a coequalizer diagram. Corollary 3.4.10. Any morphism τ : F → G of sheaves which is an epimor-phism and a monomorepimor-phism is an isomorepimor-phism.

Proof. Lemma 3.4.9 implies that

F ×GF ⇒ F → G

is a coequalizer diagram. But τ is a monomorphism, and hence the two projections F ×GF ⇒ F coincide. This implies that the identity idF: F → F factors through

τ : F → G. Hence we have a morphism η : G → F such that η ◦ τ = idF. Thus

τ = τ ◦ η ◦ τ and since τ is an epimorphism, we get that τ ◦ η = idG.

3.5. Examples of sheaves

To show that a representable presheaf on SFpqc (or SFl, S´_Et, X´et) is a sheaf, we

will use the following lemma.

Lemma 3.5.1. Let ϕ : A → B be a faithfully flat ring homomorphism. Then the following diagram is exact:

0 → A−→ Bϕ −→ B ⊗ψ AB

b 7→ b ⊗ 1 − 1 ⊗ b .

Proof. We have that ϕ is injective by Proposition 1.1.8. It is clear that ϕ(A) is contained in the kernel of ψ and hence we need only prove the reverse inclusion.

Again, by Proposition 1.1.8, it is enough to check that the sequence B ϕ⊗idB

−−−−→ B ⊗AB

ψ⊗idB

−−−−→ B ⊗AB ⊗AB

is exact. Define a map r : B⊗AB → B; r(b⊗b0) = bb0. Now take b⊗b0 ∈ ker(ψ⊗idB).

Then b ⊗ 1 ⊗ b0 = 1 ⊗ b ⊗ b0 since (b ⊗ 1 − 1 ⊗ b) ⊗ b0 = b ⊗ 1 ⊗ b0− 1 ⊗ b ⊗ b0_{. Hence}

we have that

b ⊗ b0= (idB⊗ r)(b ⊗ 1 ⊗ b0) = (idB⊗ r)(1 ⊗ b ⊗ b0) = 1 ⊗ bb0,

which is clearly in the image of ϕ ⊗ bb0_{. This finishes the proof.}

If S is a site and X is an object in S, then we may define a Grothendieck topology on (Cat(S)/X) by taking the covers to be collections of commutative diagrams (morphisms over X)

Ui //

U

X

such that {Ui → U } is a cover in S. It is not hard to see that this defines a

Grothendieck topology on (Cat(S)/X) and we denote the resulting site by (S/X). Example 3.5.2. The site X´_Et is the same as the site ((Spec Z)_Et´/X) since

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Proposition 3.5.3. Let S be a site on which every representable presheaf is a sheaf and let X be an object in S. Then every representable presheaf on (S/X) is a sheaf on (S/X).

Proof. Let {ϕi: Ui→ U } be a covering in S/X. That is, U and each Uicomes

with a morphism pU and pUi (respectively) to X such that for each i, Ui→ U is a

morphism over X. Let Y be an object in (S/X). For objects V, W in (S/X), we write HomX(V, W ) for the set of morphisms V → W in Cat(S/X) (i.e., morphisms

over X) and we write hW(V ) for the set of morphisms V → W in Cat(S). We need

to show that for any object pZ: Z → X in (S/X)

HomX(U, Z) // QiHomX(Ui, Z) //// Qi,jHomX(Ui×UUj, Z)

is an equalizer diagram. We have a commutative diagram HomX(U, Z) _ // Q iHomX _(Ui, Z) //// Qi,jHomX(U_{_}i×UUj, Z) hZ(U ) _{// Q} ihZ(Ui) //// Qi,jhX(Ui×UUj)

from which it is clear that HomX(U, Z) → Q HomX(Ui, Z) is injective. If an

element α ∈Q HomX(Ui, Z) is mapped to the same element via the two maps

Q HomX(Ui, Z) //// Q HomX(Ui×UUj, Z)

then the image of α inQ hZ(Ui) is mapped to the same element via the two maps

Q hZ(Ui×U Uj) and hence has a preimage γ ∈ hZ(U ). For each i we have that

pZ◦γ ◦ϕi= pUi = pU◦ϕi. But hXis a sheaf and hence the map hX(U ) →Q hX(Ui)

is injective. That is, pZ◦ γ = pU and hence γ is a morphism over X.

Example 3.5.4 (Sheaf defined by an S-scheme X). Consider the category (Sch/S) and the presheaf hX given by an S-scheme X. This is a sheaf on SFpqc

[FGI+_{05, Theorem 2.55], i.e., every representable presheaf on S}

Fpqc is a sheaf.

In-deed, hX satisfies the sheaf condition for Zariski open coverings by Example 3.2.8.

By Proposition 3.5.3, we may assume that hXis just a presheaf (Sch)op→ (Set),

i.e., we don’t need to worry about any base scheme. Let {V → U } be a fpqc-covering with V = Spec B and U = Spec A.

We first show that hX satisfies the property (2) of Lemma 3.2.9 in case X is

affine. Suppose that X = Spec R. From the exact sequence of Lemma 3.5.1 and left exactness of the functor Hom(R, −), we have an exact sequence

0 → Hom(R, A) → Hom(R, B) → Hom(R, B ⊗AB) .

But this sequence is isomorphic to a sequence

0 → hX(U ) → hX(V ) → hX(V ×UV ) ,

which is exactly the sheaf condition for hX.

If X is any scheme, write X = S Xi as a union of affine open subschemes.

Suppose that f, g : U → X are maps such that the compositions V → U ⇒ X are equal. Since V → U is surjective, this implies that f and g are equal as maps of topological spaces. Let Uibe the preimages of the Xi under any of these maps and

let Vi be the preimages of the Uiin V . Then for each i the maps Vi→ Ui⇒ Xiare

equal and since Vi, Ui, and Xi are affine, the previous part implies that the maps

Ui⇒ Xi are equal as morphisms of schemes. Hence hX is separated.

Suppose that f ∈ hX(V ) is a morphism such that the compositions

V ×UV

pr₁

⇒

pr2

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3.5. EXAMPLES OF SHEAVES 35

coincide. This implies that if a1, a2 ∈ V maps to the same element in U , then

they map to the same element in X. Since V → U is surjective, this implies that g factors through U as a function of sets. Proposition 1.1.16 says that U has the quotient topology induced by the map V → U and hence g factors through U as a continuous map. Denote the resulting map by g : U → X.

Now define sets Ui= g−1(Xi) and Vi= f−1(Xi) for all i. Then the morphisms

Vi×UVi⇒ Vi

f |_Vi

−−−→ Xi

are equal and since Xi is affine, f |Vi factors uniquely through some morphism

gi: Ui→ Xi. Since hXis separated, we have that giand gjagree on the intersection

Ui∩ Uj. Hence we may glue the gi to a map U → X through which f factors. This

proves the fact that hX is a sheaf on SFpqc (and hence also on SFl, S´_Et, and S´et).

Example 3.5.5 (The structure sheaf on SFpqc). Let OSFpqc be the presheaf on

SFpqc defined by U 7→ Γ(U, OU) for any U → S in SFpqc. It is clear that OSFpqc

satisfies the sheaf condition for Zariski open coverings and hence it remains to check the second condition of Lemma 3.2.9. If {V → U } is an fpqc covering with U = Spec A and V = Spec B, it is in particular flat and surjective, i.e., the corresponding homomorphism A → B is faithfully flat. Condition (2) of Lemma 3.2.9 now follows from Lemma 3.5.1 since V ×U V = Spec (B ⊗AB) and taking

global sections of U, V, and V ×UV gives exactly A, B, and B ⊗AB. Hence OSFpqc

is a sheaf on SFpqc. This example is actually a special case of the next example.

Example 3.5.6 (Sheaf defined by a quasi-coherent OS-module). Let F be a

coherent OS-module. Then we have a presheaf FFpqc on SFpqc defined by U 7→

Γ(U, ϕ∗F ) for every morphism ϕ : U → S and the obvious restriction maps. We have that FFpqc satisfies the sheaf condition for Zariski coverings since ϕ∗F is a

quasi-coherent OU-module [Har77, II.5.8]. To show that FFpqcis a sheaf on SFpqc,

it remains to show that

FFpqc(U ) → FFpqc(V ) ⇒ FFpqc(V ×UV )

is an equalizer diagram for each fpqc cover {f : V → U } with U and V affine. We have a commutative diagram

V ×U V _pr 1 // pr2 V f θ $$ S V f // U ϕ ::

Since θ = ϕ ◦ f , it follows that θ∗F = (ϕ ◦ f )∗_{F = f}∗_(ϕ∗_{F ). If U = Spec A and}

V = Spec B, then ϕ∗F = M∼ _{for some A-module M. Hence}

θ∗F = f∗(M∼) ∼= (M ⊗AB)∼.

Similarly we get that (ϕ ◦ f ◦ pr_i)∗F ∼= (M ⊗AB ⊗AB)∼. Let α : A → B be the

homomorphism corresponding to f . Then we only need to show that the following diagram is exact:

0 → M idM⊗α

−−−−−→ M ⊗AB

idM⊗ψ

−−−−−→ M ⊗AB ⊗AB ,

where ψ is as in Lemma 3.5.1, that is, ψ = e1 − e2 where e1 and e2 are the

ring homomorphisms corresponding to the projections pr1, pr2: V ×U V ⇒ V

respectively.

As in the proof of Lemma 3.5.1, let r : B ⊗AB → B be the map b ⊗ b0 7→ bb0. It

is clear that the sequence is exact at M and hence, by Proposition 1.1.8 it is enough to check that the sequence is exact after applying the functor − ⊗AB to it. Take

Operations on Étale Sheaves of Sets

IN

DEGREE PROJECT

MATHEMATICS,

SECOND CYCLE, 30 CREDITS

,

STOCKHOLM SWEDEN 2016

Operations on Étale Sheaves

of Sets

ERIC AHLQVIST

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ENGINEERING SCIENCES

Operations on Étale Sheaves of Sets

E R I C A H L Q V I S T

Master’s Thesis in Mathematics (30 ECTS credits)

Master Programme in Mathematics (120 credits)

Royal Institute of Technology year 2016

Supervisor at KTH: David Rydh

Examiner:

David Rydh

Acknowledgements

Contents

Introduction

´

Etale morphisms

Representable functors

Sheaves of sets