Operations on Étale Sheaves of Sets

DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2016

ERIC AHLQVIST

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

Operations on Étale Sheaves of Sets

ERIC AHLQVIST

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

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 étaleenvelope of f, where the first morphism is a closed immersion and the second is étale. We show that when f is étalethen the étaleenvelope 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. 4

Sammanfattning. Rydh visade 2011 att varje oramifierad morfi f av algebraiska rum (algebraiska stackar) har en kanonisk och universell faktorisering genom ett algebraiskt rum (algebraisk stack) som han kallar den étalaomslut- ningen av f, där den första morfin är en sluten immersion och den andra är étale.Vi visar att d˚a f är étales˚akan den étalaomslutningen beskrivas genom att applicera vänsteradjunkten till tillbakadragningen av f p˚aden konstanta kärven som definieras av en punkterad mängd med tv˚aelement. D˚a f är en monomorfi, lokalt av ändlig typ s˚ahar vi en liknande beskrivning i termer av framtryckning med propert stöd. 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 SædénSt˚ahlfor always taking his time to answer my questions. I would also like to thank Oliver Gäfvert and Johan Wärneg˚ard.

Contents

Acknowledgements 5

Introduction 9 Preliminaries 10

Chapter 1. Etale´ morphisms 13 1.1. Flat morphisms 13 1.2. Unramiﬁed morphisms 15 1.3. Etale´ morphisms 16 1.4. Local structure of ´etalemorphisms 18 1.5. Henselian rings 19

Chapter 2. Representable functors 21 2.1. Deﬁnitions and examples 21 2.2. The Yoneda embedding 22

Chapter 3. Sheaves of sets 25 3.1. Grothendieck topologies and sites 25 3.2. Sheaves of sets 26 3.3. Sieves and elementary topoi 29 3.4. Epimorphisms 30 3.5. Examples of sheaves 33 3.6. Stalks 36 3.7. Sheaﬁﬁcation of a sheaf 37 3.8. Fiber products and pushouts 39

Chapter 4. Operations on sheaves of sets 41 4.1. Morphisms of sites 41 4.2. Direct and inverse image functors 41 4.3. The functor j! of an open immersion 45 ∗ 4.4. The functors f and f! of an object f : T → S in C/S 46 4.5. Operations on sheaves of abelian group/pointed sets 48

Chapter 5. Algebraic spaces 51 5.1. Some descent theory 51 5.2. Algebraic spaces 53 5.3. Some descent theory for algebraic spaces 55 5.4. Etale´ topology on algebraic spaces 58 5.5. The sheaf space (espace étalé) 59 ∗ 5.6. The functors f∗, f , and f! of a morphism of algebraic spaces 62 5.7. The functor f! for sheaves of pointed sets 64 5.8. Direct image with proper support 65 5.9. Connected fibration of a smooth morphism 67 5.10. The functor f! of a non-étalemorphism. 68

7 8 CONTENTS

Chapter 6. The étaleenvelope EX/Y 71 6.1. The sheaf EX/Y 71 6.2. The case when f is a monomorphism locally of finite type 72 6.3. The case when f is étale 73 6.4. Final remark 74 Bibliography 75 Introduction

There are cases when the Zariski topology is to coarse to work in. For example 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 étaletopology 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 étalecohomology 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 étalesite

SEt´ on 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 : C op → (Set) satisfying a certain gluing condition for each covering {Ui → U}. Every S-scheme X is a sheaf on SEt´ 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 Sét on a scheme (or algebraic space) S has underlying category (ét/S) (or ét(S)), i.e., the category of étaleschemes (algebraic spaces) over S, and coverings as in SEt´ . Given a sheaf F on Sét we may construct its espace étalé Fét which is an étalealgebraic space over S. This gives an equivalence of categories between sheaves on the small étalesite Sét and étalealgebraic spaces over S. In particular, every sheaf on the small étalesite Sét of an algebraic space S is representable by an étalealgebraic space over S. The espace étaléhas the following analogue in classical topology: given a topological space B and a sheaf of sets G on B, the espace étaleof 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 morphisms 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- ét ∗ tion of the fiber product T ×S F to the small étalesite. In certain cases f has a left adjoint denoted by f!. For example, in case f : T → S is an object in (ét/S)

9 10 INTRODUCTION we get that f!F is the sheaf given by a U 7→ F(U) ϕ 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→ F(U) . ϕ 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 étale. We show that in the case when f is a monomorphism, we get that the restriction EX/Y to the small étalesite is naturally isomorphic to the sheaf fc{0, 1}X , where {0, 1}X denotes the constant sheaf on the small étalesite on X and fc is the direct image with proper support. If f is étale then EX/Y,ét = 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 morphism 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 étalewe 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 S morphism f : X → Y of schemes is called finite if there is an open covering Y = Vi −1 of Y by affine open subschemes Vi such that for every i we have f (Vi) is affine and −1 the induced homomorphism OY (Vi) → OX (f (Vi)) is finite. Or equivalently (see for example [GW10, 12.9]), for every open affine subscheme V ⊆ Y , the inverse −1 −1 image f (V ) is affine and OY (V ) → OX (f (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; β (2) f factors as X −→α X −→ Y where α is an open immersion and β is finite. 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: 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

pr1 T ×S U / 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.

CHAPTER 1

Etale´ morphisms

An étalemorphism is the algebraic analogue of a local homeomorphism. For example, a morphism of nonsingular varieties over an algebraically closed field is étaleat 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 −⊗A B is exact. It is called faithfully flat if − ⊗A B 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) → Bm 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 m of 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

13 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 ⊗A A[T1,...,Tn]/(f) → A[T1,...,Tn]/(f) a ⊗ g 7→ ϕ(a)g is not injective. Indeed, f may be written as X α f = aαT n α∈N and the non-zero element X α aα ⊗ T n α∈N in m ⊗A A[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 M 0 → M → M 00 of A-modules is exact if and only if the 0 00 sequence M ⊗A B → M ⊗A B → M ⊗A B 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· · ·qX → 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. f g Proof. Let X −→ Y −→ Z be flat morphisms. Take x ∈ X and put y = f(x) 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 f 0 : Y 0 → Y be a morphism. Take any x ∈ X and any y0 ∈ Y 0 such that f(x) = f 0(y0) =: y ∈ Y . We must show that the induced homomorphism

0 0 Oy → Oy ⊗Oy Ox is ﬂat. But again, this follows trivially since

M ⊗ (O 0 ⊗ O ) ∼ M ⊗ O . Oy0 y Oy x = Oy x 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. 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 = myOx and 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 morphismx ¯: Spec Ω → X where Ω is a separably closed field. If Y → X is a morphism then the geometric fiber over a geometric pointx ¯ is the fiber product Y ×X Spec 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 ki are 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 {x} is open in f −1(f(x)) [AK70, VI.2.3], and hence we may assume that X = 16 1. ETALE´ MORPHISMS

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 ∗ 2 ∼ 2 definition we have ΩX/Y = ∆X/Y (J /J ). Hence 0 = (ΩX/Y )x = (J /J )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 −1 ϕ (∆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⊗k A) is an ∼ open immersion we have that A ⊗k A = A. But dimk(A ⊗k A) = 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 immersion.

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 / X ×Y X s×idX 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 étaleat x ∈ X if it is flat and unramified at x. We say that f is étale if it is étaleat every x ∈ X. Remark 1.3.2. Note that Theorem 1.1.14 implies that every étalemorphism is open as a map between topological spaces. 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 ´etaleat 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 ´etaleat 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 étaleand put Ox = OX,x and Oy = OY,y. We have homomorphisms k → Ox → κ(x) and k → Oy → κ(y), which yield exact sequences 2 mx/mx → ΩOx/k ⊗Ox κ(x) → Ωκ(x)/k = 0 , and 2 my/my → 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 ⊗Oy Ox → 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 ⊗Oy κ(x) = ΩOx/k ⊗Ox κ(x) , and hence 2 ∼ 2 my/my = mx/mx . 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 2 2 2 my/my → mx/mx. Let d = dim(my/my). 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 2 2 each i = d, d − 1,..., 0. Hence Ox is flat over Oy. Since my/my → mx/mx is an isomorphism we get from the exact sequence

ΩOy /k ⊗Oy κ(y) → ΩOx/k ⊗Ox κ(x) → ΩOx/Oy ⊗Ox κ(x) → 0 , ∼ that ΩOx/Oy /mxΩOx/Oy = ΩOx/Oy ⊗Ox κ(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 étalemorphisms is étaleand a base change of an étalemorphism 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.

Remark 1.3.7. A morphism is ´etaleif and only if it is smooth and quasi-ﬁnite. 18 1. ETALE´ MORPHISMS

1.4. Local structure of étalemorphisms 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 0 that the formal derivative p (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 étale. A standard étalemorphism is étale. 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 0 the relation p (T )dT = 0 (see e.g. [Mat86, p. 195]). That is ΩB/A is isomorphic 0 ∼ ∼ 0 ∼ to A[T ]/(p , p) as a B-module. But then ΩBb/A = (ΩB/A)b = (A[T ]/(p , p))b = 0 0 (A[T ]/(p))b/(p )b = 0 since p is invertible in Bb. Hence Spec Bb → Spec A is unramified and thus étale. 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 étaleat 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 étale; (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 étalein 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 p0(T ) 0 = ∂(bS − 1)/∂T ∂(bS − 1)/∂S b0/b b is invertible. Since we have already showed that every standard étalemorphism is étale,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 n X ∂pi (1.4.0.1) dp = dT = 0 , 1 ≤ i ≤ n . i ∂T j j=1 j

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 n X ∂f df(T ,...,T ) = dT . 1 n ∂T i i=1 i

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 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 ]/(T n − 1). Consider the morphism f : X → Spec Z given by the canonical homomorphism Z ,→ Z[T ]/(T n − 1). It is clear that f is étalein the open subscheme D(n) = n n−1 n X \ V ((n)) since ∂(T − 1)/∂T = nT and OX (D(n)) = (Z[T ]/(T − 1))n. That n−1 −1 n is, nT has an inverse n T in (Z[T ]/(T − 1))n. Example 1.4.4 (Artin-Schreier cover). Let k be a field of non-zero characteristic p and take f ∈ k[T ]. The morphism Spec k[T, x]/(xp − x − f) → Spec k[T ] is étalesince ∂(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 0 (X,X0) is called a Henselian pair if for every ﬁnite morphism X → X, the induced 0 0 map ClOp(X ) → ClOp(X ×X X0) 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; Q (2) every finite A-algebra B is a direct product of local rings B = Bi; (3) if f : Y → X is a quasi-finite and separated morphism, then

Y = Y0 q Y1 q · · · 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 ´etalemorphism 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 ¯ g0 and h0 coprime, then f factors as gh, where g¯ = g0 and h = h0. 20 1. ETALE´ MORPHISMS

Proof. (1) ⇒ (2): Let f : Spec B → Spec A be ﬁnite. We have

Spec B ×A Spec κ(x) = Spec (B ⊗A κ(x)) . There is a bijective correspondence between idempotents of B and idempotents of ∼ B ⊗A κ(x) = B/mxB. If B is not local, then there exists a non-trivial idempotent e¯ ∈ B/mxB, and hencee ¯ 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 f 0 g Y −→ Y 0 −→ X where f 0 is an open immersion and g is finite. Hence Y 0 = Spec B for some finite Q A-algebra B, and by (2), B = Bi. Each Bi is of the form Bi = OY 0,y0 for some 0 0 ` closed point y ∈ Y . Let Y1 = Spec OY,y where the disjoint union is over all 0 0 closed points y of Y that are contained in Y . Thus Y1 is clopen in Y and hence ` also clopen in Y . Put Y0 = Y \ Y1. Then we have Y = Y0 Y1 and it is clear that 0 Y0 contains no closed points of Y . Since Y1 is finite over X we get that all points in the fiber over x are closed. Hence they are also closed in Y 0 since the preimage 0 of x in Y is closed. Thus x∈ / f(Y0). (3) ⇒ (4): Suppose that f : Y → X is étaleand 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 étaleand hence we have that my = mxOY,y and κ(x) = κ(y) = ∼ OY,y ⊗A κ(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 homomorphism ∼ 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 étale with smooth in Theorem 1.5.4 (4) (see [Gro67, Corollaire 17.16.3]). Example 1.5.6. Any complete local ring is Henselian [Sta, Tag 04GM]. CHAPTER 2

Representable functors

2.1. Deﬁnitions and examples Let C be a category. A functor F : C op → (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 ∈ FX, such that for any Y ∈ C and any element η ∈ FY , 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)) n '{s ∈ Γ(S, OS) : 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 n {s ∈ Γ(S, OS) : 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) n that sends a scheme S to the set Γ(S, OS) is represented by Spec (Z[T1,...,Tn]). Indeed, we have bijections n n Hom(Sch)(S, A ) ' Hom(Ring)(Z[T1,...,Tn], Γ(S, OS)) ' Γ(S, OS) , which are natural in S.

−1 Example 2.1.4 (Gm = Spec Z[T,T ]). 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 isomorphism Z[T,T −1] =∼ Z[T,X]/(TX − 1). Example 2.1.5. (The Grassmannian) Consider the functor op Gk,n :(Sch) → (Set)

21 22 2. REPRESENTABLE FUNCTORS defined by ⊕n ⊕n Gk,n(X) = {F ⊆ OX : OX /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 ⊕n ∗ ∗ ⊕n ⊕n i : F ,→ OX gives a morphism f F → f (OX ) = OY which is injective since i is ⊕n injective and OX /i(F) is locally free [GW10, 8.10]. For any subset I ⊆ {1, . . . , n} we may define a subfunctor GI ⊆ Gk,n by ⊕I ⊕n ⊕n GI (X) = {F ∈ Gk,n(X): OX ,→ OX OX /F is an isomorphism} , ⊕I ⊕n where the morphism OX ,→ OX is induced by the inclusion I,→ {1, . . . , n} and ⊕I by OX we mean OX ⊕ · · · ⊕ OX with one component for each index in I. ⊕n ⊕I For every F ∈ GI (X) we have a morphism OX OX with kernel F, and ⊕n ⊕I ⊕I ⊕n conversely, for every retraction τ : OX OX of the inclusion OX ,→ OX , we get that ker(τ) ∈ GI (X). Hence there is a bijection between the set of retractions ⊕n ⊕I ⊕I ⊕n r : OX → OX of the inclusion OX → OX 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

⊕Ic ⊕I GI (X) ' Hom(OX -mod)(OX , OX ) . But we also have natural bijections

⊕Ic ⊕I c k(n−k) Hom(OX -mod)(OX , OX ) ' Hom(Set)(I × I, Γ(X, OX )) ' Γ(X, OX ) (see [GW10, 7.4.6]), and by Example 2.1.3 we conclude that there is a natural bijection k(n−k) GI (X) ' Hom(Sch)(X, A ) . k(n−k) That is, the functor GI is represented by the aﬃne scheme A . 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 n. PZ

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

τX hX (X) / F(X)

f ∗ f ∗

τY hX (Y ) / F(Y )

∗ ∗ induced by a morphism f : Y → X. We have τY (f) = τY (f (idX )) = f (τX (idX )). This proves the ﬁrst part. 2.2. THE YONEDA EMBEDDING 23

Let f : Y → X be a morphism in C, and let hf : hY → hX be the induced natural transformation. To prove naturality in X, we need to show that the following diagram commutes:

αX,F Hom(hX , F) / F(X)

f # f ∗

αY,F Hom(hY , 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 )).

op Hence we have a functor Hom(h(−), F): C → (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 categories 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 op (CX,Y ) → (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 24 2. REPRESENTABLE FUNCTORS

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

f id Ö 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 S-scheme X. This gives the unique natural transformation F → hS. 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 definition is very similar, keeping in mind that the fiber product Ui ×U Uj 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 ×U Ui → 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 S {ϕi : Ui → U} is jointly surjective if U = 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 subcategory of Sch/S which is closed under taking ﬁber products and such that for any U → S in C/S and any E-morphism U 0 → U, the composition U 0 → U → S is in C/S. Then we get a Grothendieck topology on C/S by taking as coverings: S all collections {ϕi : Ui → U} of E-morphisms over S such that U = 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 morphisms 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.

25 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 étalesite on X). Let (ét/X) be the category whose objects are étalemorphisms 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 étalesite over S). The site with underlying category (Sch/S) and coverings which are jointly surjective families {Ui → U} of étale S- morphisms is denoted by SEt´ . Example 3.1.9 (Big flat site on S). Consider the category (Sch/S) with coverings 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 / SEt´ / SZar

Sét / Szar induced by the identity morphism S → S. Proposition 3.1.13 ([FGI+05, Proposition 2.33]). Let f : X → Y be a surjective 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; S (2) there is a covering Y = 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 Y F(U) → F(Ui) i∈I 3.2. SHEAVES OF SETS 27 is injective. A presheaf F is called a sheaf if the diagram Y pr1 Y (3.2.0.2) F(U) → F(Ui) ⇒ F(Uj ×U Ul) pr i∈I 2 (j,l)∈I×I is an equalizer diagram for every covering {Ui → U}i∈I in S, where the parallel arrows are deﬁned as follows: Y Y prk : F(Ui) → F(Uj ×U Ul) , k ∈ {1, 2} i j,l ∗ sends (ai)i to the element with component at index (j, l) equal to pr1aj if k = 1 ∗ and pr2al 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 Q Q F(Ui) is an equalizer of the diagram F(Ui) ⇒ F(Ui ×U Uj) is to say that for Q Q Q each arrow A → F(Ui) such that the composites h: A → F(Ui) ⇒ F(Ui ×U 0 Uj) coincides, there is a unique arrow h : A → F(U) such that h is the composition 0 Q of h with the arrow F(U) → F(Ui). To say that there always exists such an arrow h0 is to say that F(U) maps Q surjectively onto the subset of F(Ui) consisting of all elements whose images Q under the two maps to F(Ui ×U Uj) coincide. To say that such a map h0 (whenever it exists) is unique, is to say that the map Q F(U) → F(Ui) is injective, i.e., that F is separated. Remark 3.2.3. Note that if we have morphisms Y

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

hZ×X Y = 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 ×X Y 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×X Y (W ).

That is, hZ×X Y = 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. 28 3. SHEAVES OF SETS

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)E of a set M). Let M be a set and S a scheme. For every S-scheme X, deﬁne

π0(X) M(X) = M = 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 S Hom(Top)(−,X) where X is a topological space. Let Ui be 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 SZar is a sheaf. Indeed, let X be an S-scheme and let {Ui → U} be a covering in SZar. The ﬁber product Ui ×U Uj may be identiﬁed with the intersection Ui ∩ Uj in U. It is a well known fact that S if U = 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 ` Q ` and each Ui affine. If F satisfies (1) then F( Ui) = F(Ui) since {Uj → Ui}j is a Zariski open covering. We have that a a a Ui ×U Ui = (Ui ×U Uj) , ` and hence if the index set I is finite then Ui is affine, and hence the upper row in the diagram Q Q F(U) / F(Ui) / F(Ui ×U Uj)

` ` ` F(U) / F( Ui) / F(( Ui) ×U ( Ui)) ` is an equalizer diagram by (2) (this is the diagram arising from the cover { Ui → U}). 0 Now let {g : V → V } be any covering in SFpqc. By Proposition 3.1.13 there 0 S 0 is a covering V = Vi of open quasi-compact subschemes such that the image 0 0 Vi := g(Vi ) is open and aﬃne for each i. Hence we may write each Vi as a ﬁnite 3.3. SIEVES AND ELEMENTARY TOPOI 29

0 0 union of open subschemes Vik ⊆ Vi . The Vi’s form an open aﬃne covering of V . Now consider the following diagram: 0 / 0 0 F(V ) γ / F(V ) / F(V ×V V )

α δ

β Q QQ 0 / QQ 0 0 F(Vi) / F(Vik) / F(Vik ×V Vil) i i k i k,l

Q QQ 0 0 F(Vi ∩ Vj) / F(Vik ∩ Vjl) i,j i,jk,l

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(V 0) and suppose that s maps to the same element via the 0 0 two maps to F(V ×V V ). Then δ(s) maps to the same element via the two maps Q Q 0 0 Q to i k,lF(Vik ×V Vil), which implies that δ(s) ∈ im(β). Let t ∈ i F(Vi) be the element such that β(t) = δ(s). We have that δ(s) maps to the same element Q Q 0 0 in i,j k,lF(Vik ∩ Vjl) and since the bottom row is injective t must map to the Q same element via the two maps to i,j F(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. 30 3. SHEAVES OF SETS

Definition 3.3.1. Let U be an object in a category C. Then a subfunctor of op hU : C → (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 compositions 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 β W ⇒ X → Y α 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 β W ⇒ X → V α 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

β W ⇒ X → Y α 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:

pr1 X ×Y X ⇒ X → Y. pr2 We say that X → Y is a universally eﬀective epimorphism if for every morphism 0 0 0 Y → Y , the morphism X ×Y Y → Y 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

HomC(X, −): C → (Set) . 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 Y X W h → h ⇒ h 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 X X W F → h such that the compositions F → h ⇒ h 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 X Y X is commutative and hence F → h factors through h → h . 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 ∗ pr1 F(U) → F(V ) F(V ×U V ) ⇒∗ pr2 is an equalizer diagram in (Set) and by Yoneda’s lemma and Remark 3.2.3, this is isomorphic to ∗ pr1

Hom(hU , F) → Hom(hV , F) Hom(hV ×hU hV , F) . ⇒∗ pr2

If we have a morphism τ : hV → F such that the composites hV ×hU hV ⇒ hV → F ∗ ∗ coincides, then pr1(τ) = pr2(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 ∗ we have that ϕi (y) ∈ G(Ui) is in the image of αUi : F(Ui) → G(Ui). 32 3. SHEAVES OF SETS

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 ϕi (x) ∈ G(Ui) for every i, then x ∈ G(U). Q Q Proof. (2) ⇒ (1) : G(U) → G(Ui) is injective since F(U) → F(Ui) is Q injective. But (2) says exactly that G(U) → G(Ui) is surjective onto the set Q of points y ∈ G(Ui) such that p1(y) = p2(y) where p1, p2 are the morphisms Q Q G(Ui) ⇒ 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 ϕi (x) ∈ im(αUi ) for all i. Then G0 is a sheaf by Lemma 3.4.7. Consider the characteristic morphisms

χG, χG0 : G ⇒ Ω . 0 Then since αU (F(U)) ⊆ G (U), we have χG ◦α = χG0 ◦α and if α is an epimorphism, 0 then we have χG0 = χG, i.e., G = G 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 ϕi (y) ∈ im(αUi ) for every index i. Thus we have a diagram

fU G(U) / H(U) gU Q ∗ Q ∗ ϕi ϕi Q f Q Ui Q G(Ui) / H(Ui) Q / gUi Q ∗ Q Q ∗ Q Q ∗ Q ∗ where ( ϕi )(fU (y)) = ( fUi )(( ϕi )(y)) = ( gUi )(( ϕi )(y)) = ( ϕi )(gU (y)). Q ∗ Q But ( ϕi ): H(U) → 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 ϕi (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 Y Y Y Q F (Ui) × G(Ui) F (Ui) ⇒ F (Ui) → H (Ui) Q coincide is to say that whenever two elements a, b ∈ F(Ui) map to the same Q Q element in G(Ui), they also map to the same element in H(Ui). That is, 3.5. EXAMPLES OF SHEAVES 33

∗ Q Q every preimage of (ϕi (x))i ∈ G(Ui) in F(Ui) will map to the same element Q ` ` z ∈ H(Ui). But this argument also holds if we replace Ui by Ui ×U Uj and ∗ Q since the two images of (ϕi (x))i in 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 ×G F ⇒ F → G is a coequalizer diagram. Corollary 3.4.10. Any morphism τ : F → G of sheaves which is an epimorphism and a monomorphism is an isomorphism. Proof. Lemma 3.4.9 implies that

F ×G F ⇒ F → G is a coequalizer diagram. But τ is a monomorphism, and hence the two projections F ×G F ⇒ 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,SEt´ , 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: ϕ ψ 0 → A −→ B −→ B ⊗A B 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

ϕ⊗idB ψ⊗idB B −−−−→ B ⊗A B −−−−→ B ⊗A B ⊗A B 0 0 0 is exact. Define a map r : B⊗AB → B; r(b⊗b ) = bb . Now take b⊗b ∈ 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 0 0 0 0 b ⊗ b = (idB ⊗ r)(b ⊗ 1 ⊗ b ) = (idB ⊗ r)(1 ⊗ b ⊗ b ) = 1 ⊗ bb , 0 which is clearly in the image of ϕ ⊗ bb . 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

Õ 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 XEt´ is the same as the site ((Spec Z)Et´ /X) since Spec Z is the ﬁnal object in (Sch). 34 3. SHEAVES OF SETS

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 Ui comes 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) Q / Q HomX (U, Z) / i HomX (Ui,Z) / i,j HomX (Ui ×U Uj,Z) is an equalizer diagram. We have a commutative diagram Q / Q HomX (U, Z) / i HomX (Ui,Z) / i,j HomX (Ui ×U Uj,Z) _ _ _

Q / Q hZ (U) / i hZ (Ui) / i,j hX (Ui ×U Uj) Q from which it is clear that HomX (U, Z) → HomX (Ui,Z) is injective. If an Q element α ∈ HomX (Ui,Z) is mapped to the same element via the two maps Q Q HomX (Ui,Z) // HomX (Ui ×U Uj,Z) Q then the image of α in 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 Q pZ ◦γ◦ϕi = pUi = pU ◦ϕi. But hX is a sheaf and hence the map hX (U) → 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 SFpqc is a sheaf. In- deed, hX satisfies the sheaf condition for Zariski open coverings by Example 3.2.8. op By Proposition 3.5.3, we may assume that hX is just a presheaf (Sch) → (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 ⊗A B) . But this sequence is isomorphic to a sequence

0 → hX (U) → hX (V ) → hX (V ×U V ) , which is exactly the sheaf condition for hX . S If X is any scheme, write X = 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 Ui be the preimages of the Xi under any of these maps and let Vi be the preimages of the Ui in V . Then for each i the maps Vi → Ui ⇒ Xi are 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

pr1 V ×U V ⇒ V → X pr2 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. −1 −1 Now deﬁne sets Ui = g (Xi) and Vi = f (Xi) for all i. Then the morphisms

f|Vi Vi ×U Vi ⇒ Vi −−−→ Xi are equal and since Xi is aﬃne, f|Vi factors uniquely through some morphism gi : Ui → Xi. Since hX is separated, we have that gi and gj agree 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,SEt´ , 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 ⊗A B) and taking global sections of U, V, and V ×U V gives exactly A, B, and B ⊗A B. 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 FFpqc is a sheaf on SFpqc, it remains to show that

FFpqc(U) → FFpqc(V ) ⇒ FFpqc(V ×U V ) 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 / V 1 θ $ pr2 f S : f ϕ V / U 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 ⊗A B) . ∗ ∼ ∼ Similarly we get that (ϕ ◦ f ◦ pri) F = (M ⊗A B ⊗A B) . Let α: A → B be the homomorphism corresponding to f. Then we only need to show that the following diagram is exact:

idM ⊗α idM ⊗ψ 0 → M −−−−−→ M ⊗A B −−−−−→ M ⊗A B ⊗A B, 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. 0 0 As in the proof of Lemma 3.5.1, let r : B ⊗A B → B be the map b ⊗ b 7→ bb . 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 − ⊗A B to it. Take 36 3. SHEAVES OF SETS

0 0 0 m⊗b⊗b ∈ ker(idM ⊗ψ⊗idB) ⊆ M ⊗AB⊗AB. Then m⊗b⊗1⊗b = m⊗1⊗b⊗b and 0 0 applying idM ⊗ idB ⊗r to both sides of the equality we get that m⊗b⊗b = m⊗1⊗bb which is in the image of idM ⊗α ⊗ idB : M ⊗A B → M ⊗A B ⊗A B. This proves that the sequence is exact and hence FFpqc is a sheaf on SFpqc.

3.6. Stalks Similarly as for sheaves of topological spaces we may define the stalk of a sheaf at a point. In this case, a ”point” will be a geometric point Spec Ω → X and the colimit will be over étaleneghborhoods. Definition 3.6.1. An étaleneighborhood of a geometric pointx ¯: Spec Ω → X is a commutative diagram

Spec Ω / U

# X where U → X is ´etale.

Definition 3.6.2. Let X be a scheme and F a presheaf on X´et. The stalk of F at a geometric pointx ¯: Spec Ω → X is deﬁned as

Fx¯ =−→ lim F(U) , where the colimit is over all ´etaleneighborhoods U ofx ¯.

Proposition 3.6.3. Let τ : F → G be a morphism of sheaves on X´et. The following are equivalent: (1) τ is an isomorphism; (2) τx¯ : Fx¯ → Gx¯ is a bijection for each geometric point x¯ in X.

Proof. (1) ⇒ (2): We have that τ is an isomorphism if and only if τU : F(U) → G(U) is a bijection for each étale U → X and hence (2) holds. (2) ⇒ (1): By Corollary 3.4.10, it is enough to show that τ is an epimorphism and a monomorphism. If τU : F(U) → G(U) is injective for each étale U → X, then τ is a monomorphism. Hence it is enough to show that τ is locally surjective and that τU : F(U) → G(U) is injective for each étale U → X. Locally surjective: Take any s ∈ G(U). Letx ¯ be a geometric point in U and let sx¯ be the image of s in Gx¯. Since τx¯ is surjective, there is a tx¯ ∈ Fx¯ mapping to sx¯. Let ϕ: V → U be an étaleneighborhood ofx ¯ and let t ∈ F(V ) be an element with ∗ image tx¯ in Fx¯. Then t and ϕ (s) both have image tx¯ in Fx¯. This implies that we ∗ may choose V such that ϕ (s) = τV (t). Injectivity: Suppose that t1, t2 ∈ F(U) satisfies τU (t1) = τU (t2). Letx ¯ be a 0 0 geometric point in U. Let sx¯ be the image of τU (t1) = τU (t2) in Gx¯ and let tx¯ be 0 its preimage in Fx¯. Then t1 and t2 must have image tx¯ in Fx¯ and hence there is ∗ ∗ an étaleneighborhood ψ : Wx¯ → X ofx ¯, such that ψ (t1) = ψ (t2). Now we may choose geometric pointsx ¯ such that the Wx¯’s form an étalecovering of U and since F is a sheaf, we conclude that t1 = t2.

Example 3.6.4. Let M be a set and X a scheme, and consider the sheaf MX on X´et. Letx ¯: Spec Ω → X be a geometric point in X. Then we have

(MX )x¯ = M. Indeed, for every étaleneighborhood U → X ofx ¯ we have a morphism M π0(U) → M π0(Spec Ω) = M, 3.7. SHEAFIFICATION OF A SHEAF 37 and hence we have a map ϕ:(MX )¯x → M. To see that ϕ is a bijection, let a be an element in M and let U → X be any étaleneighborhood ofx ¯. Let U 0 be the connected component in U containingx ¯. Then U 0 → U is étaleand the map 0 π0(Spec Ω) → π0(U ) is clearly a bijection since it is just a point going to a point. Thus 0 M π0(U ) → M π0(Spec Ω) = M is bijective. Thus ϕ:(MX )¯x → M is a bijection. 3.7. Sheafification of a sheaf Definition 3.7.1. Let F be a presheaf on a site S.A sheafification of F is a sheaf F a together with a morphism F → F a, such that any morphism F → G from F to a sheaf G on S factors uniquely through F → F a, i.e., a ∼ HomSh(S)(F , G) = HomPreSh(S)(F, G) . Note that if F and G are presheaves and there exists sheafifications of F and G, then a morphism F → G of presheaves gives a morphism F → Ga which factors uniquely through a morphism F a → Ga. Hence, if we find a consistent way to assign to each presheaf F a specific sheafification F → F a, then this assignment F 7→ F a becomes a functor, which is a left adjoint of the forgetful functor from sheaves to presheaves. Proposition 3.7.2. Given a presheaf F on a site S, there exists a sheafification F → F a, which is unique up to canonical isomorphism. Proof. For each object U in Cat(S), define a relation ∼ on F(U) by saying ∗ ∗ that a ∼ b if there exists a covering {ϕi : Ui → U}i∈I such that ϕi (a) = ϕi (b) for all i ∈ I. It is clear that this is an equivalence relation. Let f : V → U be a morphism in Cat(S). Then {V ×U Ui → V } is a covering and we get, for each i, a commutative square

F(V ×U Ui) o F(Ui) O O

f ∗ F(V ) o F(U) . Hence it is clear that f ∗(a) ∼ f ∗(b) whenever a ∼ b in F(U) and we get a well- deﬁned map F(U)/ ∼ → F(V )/ ∼. This implies that U 7→ F(U)/ ∼ gives a presheaf on S and it is clear from its deﬁnition that this presheaf is separated. Denote this presheaf by F + and the canonical morphism F → F + by α. If τ : F → G is a morphism to a sheaf G, we get for each i a commutative square τ F(U) U / G(U)

∗ ∗ ϕi ϕi F(Ui) / G(Ui) τUi ∗ ∗ If ϕi (a) = ϕi (b) then Gϕi(τU (a)) = Gϕi(τU (b)), and it follows from the deﬁnition + of a sheaf that τU (a) = τU (b). Hence F → G factors uniquely through α: F → F . Now let U be the collection of pairs ({Ui → U}, {si}) where {Ui → U} is a + ∗ ∗ + covering and si ∈ F (Ui) is such that pr1(si) = pr2(sj) ∈ F (Ui ×U Uj) for all indices i, j. We deﬁne a relation on U by saying that ({Ui → U}, {si}) ∼ ({Vj → + U}, {tj}) if, for all i, j, the elements si and tj have the same image in F (Ui ×U Vj) under the restriction maps corresponding to the two projections. To see that this is an equivalence relation we only need to check transitivity. This follows from 38 3. SHEAVES OF SETS

+ the fact that F is separated. Indeed, if ({Vj → U}, {tj}) ∼ ({Wk → U}, {zk}), consider the following diagram

+ + F (Ui) / F (Ui ×U Vj) 9 7

+ + + ' ) F (U) / F (Vj) F (Ui ×U Wk) / F(Ui ×U Vj ×U Wk) 7 5

% + + ' F (Wk) / F (Vj ×U Wk)

+ Since F is separated, it follows that all arrows are injective and since si and zk + pulls back to the same element in F (Ui ×U Vj ×U Wk), they must agree already + in F (Ui ×U Wk). Hence ∼ is an equivalence relation. Deﬁne F a(U) = U/ ∼. If V → U is a morphism, we deﬁne a map U → V by ∗ sending a pair ({Ui → U}, {si}) to the pair ({Ui ×U V → V }, {pr1(si)}), where pr1 : Ui ×U V → Ui is the projection. We have a commutative diagram + + + F (Ui) / F (Ui ×U Vj) o F (Vj)

+ + + F (Ui ×U V ) / F (Ui ×U Vj ×U V ) o F (Vj ×U V ) from which it follows that two pairs are equivalent in U only if their images in V are equivalent. Hence we have a well-defined map F a(U) → F a(V ) and we conclude that F a is a presheaf. We have a map F +(U) → F a(U) defined by sending s ∈ F +(U) to the pair {U −→id U, {s}}. Hence we have a morphism F → F a. a To show that F is separated, let {ϕi : Ui → U} be any covering. The map a a F (U) → F (Ui) is defined by sending the class represented by ({Vj → U}, {tj}) to the class represented by ∗ ({Vj ×U Ui → Ui}, {pr1(tj)}) . 0 0 Suppose that ({Vk → U}, {tk}) is such that ∗ 0 ∗ 0 ({Vj ×U Ui → Ui}, {pr1(tj)}) ∼ ({Vk ×U Ui → Ui}, {pr1(tk)}) ∗ ∗ 0 for all indices i. Then pr1(tj) and pr1(tk) will pull back to the same element in + 0 ∗ + 0 tjk ∈ F (Vj ×U Vk ×U Ui) for all indices j, k. But then both pr1(tj) ∈ F (Vj ×U Vk) ∗ 0 + 0 + 0 and pr2(tk) ∈ F (Vj ×U Vk) will pull back to tjk in F (Vj ×U Vk ×U Ui) and since F + is separated, it follows that ∗ ∗ 0 + 0 pr1(tj) = pr2(tk) ∈ F (Vj ×U Vk) . 0 0 a Q a That is, ({Vj → U}, {tj}) ∼ ({Vk → U}, {tk}). Thus F (U) → F (Ui) is injective, i.e., F a is separated. a a ∗ ∗ Now suppose that si ∈ F (Ui) and sj ∈ F (Uj) satisfies pr1(si) = pr2(sj) ∈ a F (Ui ×U Uj) for all i, j ∈ I. But si and sj are represented by pairs ({Uik → Ui}, {sik}) and ({Ujl → Uj}, {sjl}), and thus the pairs ({Uik ×U Uj → Ui ×U ∗ ∗ Uj}, {pr1(sik)}) and ({Ui ×U Ujl → Ui ×U Uj}, {pr2(sjl)}) are equivalent. That is, ∗ ∗ pr1(sik) and pr2(sjl) pulls back to the same element in + + F ((Uik ×U Uj) ×Ui×U Uj (Ui ×U Ujl)) = F (Uik ×U Ujl) .

But the pairs ({Uik → Ui}, {sik}) and ({Ujl → Uj}, {sjl}) gives pairs ({Uik → Ui → U}, {sik}) and ({Ujl → Uj → U}, {sjl}) in U and hence sik and sjl must + pull back to the same element in F (Uik ×U Ujl), namely the same element as the ∗ + a one pr1(sik) and F prjl(sjl) pulls back to. The element in F (U) represented by 3.8. FIBER PRODUCTS AND PUSHOUTS 39

({Uik → Ui → U}, {sik}) and ({Ujl → Uj → U}, {sjl}) will then pull back to si in a a a Q a F (Ui) and to sj in F (Uj). This shows that F (U) → F (Ui) is an equalizer Q a Q a a of F (Ui) ⇒ F (Ui ×U Uj) and hence F is a sheaf. By construction, F → F a is a sheafification of F, and by the universal property of the sheafification F a is unique up to isomorphism. Given the morphism F → F a, a this isomorphism is unique, i.e., F is unique up to canonical isomorphism. a Lemma 3.7.3. Let X be a scheme. Let F be a presheaf on Xét and F its a sheafification. If x¯: Spec Ω → X is a geometric point then Fx¯ ' (F )x¯. Proof. See [Mil80, II.2.14.(c)].

Corollary 3.7.4. Let P be a presheaf on Xét and F a sheaf on Xét. Sup- pose that we have a natural transformation τ : P → F such that the induced map τx¯ : Px¯ → Fx¯ is a bijection for every geometric point x¯ in X. Then τ : P → F is a sheafification of P.

3.8. Fiber products and pushouts If F, G, and H are presheaves on a site S, then it is not hard to see that the ﬁber product F ×H G in PreSh(S) is the presheaf deﬁned by

X 7→ F(X) ×H(X) G(X) , where the ﬁber product is taken in (Set). Indeed, to give a natural transformation τ : M → F ×H G of presheaves is to give (in a natural way), for every object X in S, a commutative diagram M(X) / G(X)

F(X) / H(X) . Lemma 3.8.1. If F, G, and H are sheaves on a site S, then the presheaf deﬁned by X 7→ F(X) ×H(X) G(X) is a sheaf.

Proof. Easy. Consider a covering {Ui → U} and draw a big diagram. Corollary 3.8.2. If F → H and G → H are morphisms of sheaves on a site S, then the presheaf fiber product F ×H G is a sheaf. Remark 3.8.3. It is not true in general that the presheaf pushout of sheaves is a sheaf. However, if we take the sheafification of the presheaf pushout, we get a pushout in the category of sheaves.

CHAPTER 4

Operations on sheaves of sets

4.1. Morphisms of sites Let f : X → Y be a morphism of schemes and let U → Y be an étalemorphism. By Lemma 1.3.5, the projection U ×Y X → X is étale. Furthermore, if V is an étale Y -scheme and U → V is a Y -morphism (hence étale),we get a morphism U ×Y X → V . Hence the universal property of the fiber product implies that we get an X-morphism U ×Y X → V ×Y X, which again is étalesince it is given by base change from U → V . Hence we have a functor f × :(ét/Y ) → (ét/X)

U 7→ U ×Y X.

Moreover, if {Ui → U} is a covering then {Ui ×Y X → U ×Y X} is a covering since × Ui ×Y X = Ui ×U U ×Y X. That is, the functor f takes coverings to coverings. Furthermore, if T → U is a morphism, then × f (T ×U Ui) = T ×U Ui ×Y X

= T ×U (U ×Y X) ×U×Y X (Ui ×Y X)

= (T ×Y X) ×U×Y X (Ui ×Y X) × × = f (T ) ×f ×(U) f (Ui) . We say that the functor f × is continuous. Definition 4.1.1. Let S and S0 be sites. A functor F : Cat(S) → Cat(S0) is a called continuous if for every covering {Ui → U} in S, F satisfies the following properties: 0 (1) {F (Ui) → F (U)} is a covering in S ; (2) for any morphism T → U, the morphism F (T ×U Ui) → F (T )×F (U) F (Ui) is an isomorphism. Definition 4.1.2. Let S and S0 be sites. A morphism of sites f : S → S0 is a continuous functor f cat : Cat(S0) → Cat(S). Note that a sheaf on a site S in the diagram in remark 3.1.12 will also be a sheaf on any site to the right of S in the diagram. Remark 4.1.3. Let X be a scheme. If S and S0 are sites on X such that every 0 0 object in S may also be considered an object in S (for example if S = Xét and 0 S = XEt´ ), and every covering in S is also a covering in S, then we get a morphism of sites π : S → S0. We call such a morphism π a restriction morphism. If F is a 0 sheaf on S then the direct image π∗F (see below) is the restriction of F to S . 4.2. Direct and inverse image functors Definition 4.2.1. Let f : S → S0 be a morphism of sites given by a continuous functor f cat : Cat(S0) → Cat(S). Let F be a presheaf on S. Then we define a 0 presheaf fpF on S by cat fpF = F ◦ f .

41 42 4. OPERATIONS ON SHEAVES OF SETS

Definition 4.2.2. Let F be a presheaf on Xét and let f : X → Y be a morphism ¯ of schemes. Let f : Xét → Yét be the morphism of sites given by the functor × f : Cat(Yét) → Cat(Xét). We define the direct image (push-forward) presheaf fpF on Yét by ¯ fpF = fpF .

That is, for every ´etale V → Y we have fpF(V ) = F(V ×Y X). Remark 4.2.3. This notation may sometimes be confusing. For a functor F : C → D and a presheaf F on D, a common notation is F pF := F ◦ F . Hence we × p would have fpF = (f ) F. 0 Let f : S → S be a morphism of sites. It is clear that fp is a functor PreSh(S) → 0 p PreSh(S ) and we will show that fp has a left adjoint f . That is, there exists a functor f p : PreSh(S0) → PreSh(S) such that whenever F and G are presheaves on S and S0 respectively, we have a natural bijection p ∼ HomPreSh(S)(f G, F) −→ HomPreSh(S0)(G, fpF) . Definition 4.2.4. Let f : S → S0 be a morphism of sites given by a continuous functor f cat : Cat(S0) → Cat(S), and let G be a presheaf on S0. We deﬁne a presheaf f pG on S by p f G(U) =−→ lim G(V ) , for every object U in S, where the colimit is over all morphisms ϕ: U → f cat(V ) (V an object in S0) in Cat(S) and a morphism (V, ϕ) → (V 0, ϕ0) is a morphism V → V 0 such that the following diagram commutes: U ϕ ϕ0 { # f cat(V ) / f cat(V 0)

Then f pG is indeed a presheaf since for a morphism ψ : U 0 → U in S, we map every morphism ϕ: U → f cat(V ) to the morphism ϕ ◦ ψ : U 0 → f cat(V ). In this way we get a natural map f pG(U) → f pG(U 0). Remark 4.2.5. Note that we have canonical morphisms p p G → fpf G , f fpF → F if F and G are presheaves on sites S and S0 respectively, and f : S → S0 is a morphism of sites. Indeed, for every object V in S0 we have p p cat 0 fpf G(V ) = f G(f (V )) = lim−→ G(V ) where the colimit is over morphisms f cat(V ) → f cat(V 0). One such object is of cat cat p course id: f (V ) → f (V ) and hence we have a map G(V ) → fpf G(V ). It is not hard to see that this map is natural in V . Let U be an object in S. Then p cat f fpF(U) =−→ lim fpF(V ) =−→ lim F(f (V )) where the limit is over morphisms U → f cat(V ). Every such morphism gives a map cat p F(f (V )) → F(U) and hence we get a map f fpF(U) → F(U). It is not hard to p see that this gives a morphism f fpF → F. Proposition 4.2.6. Let f : S → S0 be a morphism of sites. The functor f p is a left adjoint of fp. 4.2. DIRECT AND INVERSE IMAGE FUNCTORS 43

Proof. A natural transformation τ : f pG → F is given by the following data: for every U in S and every morphism U → f cat(V ), where V is an object in S0, we have maps G(V ) → F(U) that are compatible with morphisms V → V 0 such that the following diagram commutes U ϕ ϕ0 { # f cat(V ) / f cat(V 0) In particular, since τ is natural in U, we get that G(V ) → F(U) factors through F(f cat(V )) → F(U), the pullback of U → f cat(V ). Hence we have a map G(V ) → cat F(f (V )) = fpF(V ) for every V , which is natural in V . That is, we have a natural transformation η : G → fpF. Conversely, suppose that we start with a natural transformation η : G → fpF. Each morphism U → f cat(V ) yields a map F(f cat(V )) → F(U) and hence we get maps G(V ) → F(U) that are compatible with morphisms V → V 0 as before. Hence we have a unique morphism f pG(U) → F(U). It is clear that if we start with τ, constructs η, and then apply the second construction we get τ back and vice versa. Hence these two constructions are inverses of each other. It is not hard to see that the constructions are natural in F and G. We will now see what the pullback construction looks like for a sheaf on the small étalesite of a scheme. Let f : X → Y be a morphism of schemes and G a ¯ presheaf on Yét. Let f : Xét → Yét be the morphism of sites given by the functor × f : Cat(Yét) → Cat(Xét). Let V → Y be étale.To give a morphism × ϕ: U → f (V ) = V ×Y X is to give a commutative square

g U / V

f X / Y and if we have a Y -morphism h: V → V 0 such that the following diagram commutes: U ϕ ϕ0

{ 0$ V ×Y X / V ×Y X then we have g0 = h ◦ g. p Definition 4.2.7. We define the inverse image (pullback) presheaf f G on Xét by f pG = f¯pG . p That is, for U → X étale,we have f G(U) =−→ lim G(V ) , where the colimit is over all commutative diagrams

g U / V

f X / Y where a morphisms between two such diagrams (V, g) and (V 0, g0) is given by a Y -morphism h: V → V 0 such that g0 = h ◦ g. 44 4. OPERATIONS ON SHEAVES OF SETS

Corollary 4.2.8. Let f : X → Y be a morphism of schemes. The functor f p is a left adjoint of fp. Proof. Just to enlighten things, we prove this without using Proposition 4.2.6. Given a commutative square g U / V

f X / Y (which we again denote by (V, g)) we get a morphism U → V ×Y X. If (V, g) → 0 0 (V , g ) is a morphism between such diagrams, we get morphisms U → V ×Y X → 0 V ×Y X. This means that if we have a morphism G → fpF, we get compatible maps G(V ) → fpF(V ) = F(V ×Y X) → F(U) . That is, we get a morphism f pG → F. Conversely, since g factors through V ×Y X, any map G(V ) → F(U) factors p through F(V ×Y X) = fpF(V ), and hence a morphism f G → F gives a morphism G → fpF. These constructions are clearly inverses of each other and hence we conclude that there is a bijection p HomPreSh(Xét)(f G, F) ' HomPreSh(Yét)(G, fpF) which is natural in F and G. Lemma 4.2.9. Let f : S0 → S be a morphism of sites given by a continuous functor f cat. If X is an object in S, then p ∼ f hX = hf cat(X) . p In particular, f hX is a sheaf. 0 p Proof. Let U be an object in S . We have f hX (U) = lim−→ hX (V ) where the colimit is over morphisms ϕ: U → f cat(V ). Every morphism V → X yields a morphism U → f cat(V ) → f cat(X) by applying f cat and composing with U → cat f (V ). This gives a map hX (V ) → hf cat(X)(U) which is functorial in V . Hence p we get a canonical map αU : f hX (U) → hf cat(X)(U). Consider a morphism ϕ: U → f cat(X). Then ϕ is the image of the identity X → X under the morphism hX (X) → hf cat(X)(U). Hence αU is surjective. To see that αU is injective, suppose that we have morphisms V → X, V 0 → X, U → f cat(V ), and U → f cat(V 0) such that the compositions β : U → f cat(V ) → f cat(X) and γ : U → f cat(V 0) → f cat(X) coincide. Then β is the image of the identity X → X under the map hX (X) → hf cat(X)(U), and γ is the image of the identity X → X under the morphism hX (X) → hf cat(X)(U). That is, β and γ will map to the p same element in f hX (U). Thus αU is injective. The αU ’s gives an isomorphism p of functors α: f hX → hf cat(X). Corollary 4.2.10. Let Z → Y be étaleand let f : X → Y be any morphism of schemes. If Z represents a sheaf G on Yét, then Z ×Y X represents the presheaf p p f G on Xét. In particular, f G is a sheaf. p We will now consider the restrictions of the operations fp and f to the categories of sheaves. Lemma 4.2.11. Let f : S → S0 be a morphism of sites given by a continuous cat 0 functor f . If F is a sheaf on S then fpF is a sheaf on S . In particular, if F is a sheaf on Xét and f : X → Y is a morphism of schemes, then fpF is a sheaf on Yét. 4.3. THE FUNCTOR j! OF AN OPEN IMMERSION 45

Proof. This is obvious since f cat takes coverings to coverings and commutes with fiber products. It is not true in general for a morphism f : S → S0 of sites that f pG is a sheaf when G is a sheaf. Hence we make the following definitions for sheaves: Definition 4.2.12. Let f : S → S0 be a morphism of sites and let F be a sheaf on S. Then we define the direct image (push-forward) f∗F to be the sheaf f∗F = fpF. Definition 4.2.13. Let f : S → S0 be a morphism of sites and let G be a sheaf on S0. Then we define the inverse image (pullback) f ∗G to be the sheafification of the presheaf f pG, i.e., f ∗G = (f pG)a. Remark 4.2.14. Note that the names direct image (push-forward) and inverse image (pullback) will easily lead to confusion since the direct image (push-forward) along a morphism of sites is really the inverse image (pullback) along the corresponding continuous functor and vice versa. ¯ When f : X → Y is a morphism of schemes we will also write f∗F = f∗F and ∗ ¯∗ ¯ × f G = f G where f : Xét → Yét is the morphism of sites induced by the functor f , and F and G are sheaves on Xét and Yét respectively. Remark 4.2.15. From the universal property of the sheafification of a presheaf, ∗ it follows that f∗ and f are adjoints and we have ∗ HomSh(S)(f G, F) ' HomSh(S0)(G, f∗F) .

Example 4.2.16 (The sheaf Gm). Let Gm be the presheaf on SFpqc deﬁned by ∗ Gm(U) = Γ(U, OU ) for each S-scheme U, i.e., Gm(U) is the multiplicative group of Γ(U, OU ). We will show that Gm is representable, and hence a sheaf on SFpqc. We have ∗ −1 Γ(U, OU ) ' Hom(Ring)(Z[T,T ], Γ(U, OU )) −1 ' Hom(Sch)(U, Spec Z[T,T ]) −1 ' Hom(Sch/S)(U, Spec Z[T,T ] ×Spec Z S) −1 and hence Gm is represented on (Sch/S) by the scheme Spec Z[T,T ] ×Spec Z S and hence Gm is a sheaf on SFpqc.

Example 4.2.17 (Sheaf µn of n:th roots of unity in OU ). Let µn be the presheaf n on SFpqc deﬁned by µn(U) = {x ∈ Γ(U, OU ): x = 1} for each S-scheme U. We have that n µn(U) = {x ∈ Γ(U, OU ): x = 1} n ' Hom(Ring)(Z[T ]/(T − 1), Γ(U, OU )) n ' Hom(Sch/S)(U, Spec (Z[T ]/(T − 1)) ×Spec Z S) and hence µn is represented on (Sch/S) by the scheme n Spec (Z[T ]/(T − 1)) ×Spec Z S.

In particular, µn is a sheaf on SFpqc.

4.3. The functor j! of an open immersion Definition 4.3.1. Let j : U → X be an open immersion of schemes and let F be a presheaf on Uét. For any étalemorphism f : V → X define ( F(V ) if f(V ) ⊆ U; jp!F(V ) = ∅ otherwise. 46 4. OPERATIONS ON SHEAVES OF SETS

Note that for any presheaf G on Xét, we have a bijection ∗ HomPreSh(Xét)(jp!F, G) ' HomPreSh(Uét)(F, j G) .

It may happen that jp!F is not a sheaf even though F is. Hence we make the following definition: Definition 4.3.2. In the situation of Definition 4.3.1, we define a j!F = (jp!F) . Remark 4.3.3. The universal property of the sheafification of a presheaf implies that we have bijections ∗ HomSh(Xét)(j!F, G) ' HomPreSh(Xét)(jp!F, G) ' HomPreSh(Uét)(F, j G) , ∗ ∗ and since F and j G are both sheaves, we see that j! is a left adjoint of j .

∗ 4.4. The functors f and f! of an object f : T → S in C/S Let S be a scheme, and choose a class of morphisms E and a category C/S as in Remark 3.1.2. Let f : T → S be an object in C/S. Then we get two sites SE and TE. Every object U → T in C/T may be considered as an object in C/S by composing with f and every morphism over T may also be considered a morphism over S. We get a functor

fcat : C/T → C/S which takes an object g : U → T to the object f ◦ g : U → T → S and which takes a morphism U 0 → U over T to the same morphism U 0 → U considered as a morphism over S. It is clear that fcat is a continuous functor with respect to the topologies on C/T and C/S and hence it defines a morphism of sites ˆ f : SE → TE . p Now consider the functor f : PreSh(SE) → PreSh(TE) defined as in the case where SE = Sét and TE = Tét. That is, if F is a sheaf on SE then for every p object U → T we have f F(U) = lim−→ F(V ) where the colimit is over commutative diagrams U / V

ϕ ψ f T / S with ϕ and ψ in C/T and C/S respectively. But since f ◦ ϕ: U → S is an object in ˆ p ˆ C/S, this colimit is just F(U → T → S) = f∗F(U → T ). Hence f F = fpF and ˆ p since fpF is a sheaf (Lemma 4.2.11), we get that f F is a sheaf. That is ∗ ˆ f = f∗ . Remark 4.4.1. Note that if f : X → Y is ´etaleand g : T → Y is any morphism, we get a diagram p X ×Y T / T

q g f X / Y

If F is a sheaf on T´et then we have an isomorphism ∗ ∼ ∗ f g∗F = q∗p F . ∗ 4.4. THE FUNCTORS f AND f! OF AN OBJECT f : T → S IN C/S 47

Remark 4.4.2. Given an object g : U → T in TE and f : T → S in C/S as ˆ before, we have that elements in f∗F(U) is the same as commutative diagrams

U / F

g f T / S in PreSh(SE). On the other hand, T (U) is the set of morphisms h: U → T such that f ◦ h = f ◦ g, and S(U) consists only of the morphism f ◦ g. Thus, the set of such diagrams is exactly the set of elements in T (U) ×S(U) F(U). Hence we conclude that ∗ ˆ f F = f∗F = (T ×S F)|Tét , where (T ×S F)|Tét is the restriction of T ×S F to étaleschemes over S which factors through T .

ˆ ˆ∗ By Proposition 4.2.6 we know that f∗ has a left adjoint f .

ˆ∗ ˆ Definition 4.4.3. The left adjoint f of f∗ is denoted by f!.

∗ Hence f! is a left adjoint of f . Recall that if F is representable we have that fˆpF is a sheaf when F is a sheaf. Using Deﬁnition 4.2.4 we may describe the ˆp functor f explicitly. Let F be a sheaf on TE. According to Deﬁnition 4.2.4, we ˆp have for every object g : U → S in C/S that f F(U) =−→ lim F(V ) where the colimit is over all commutative diagrams

U / V

g f STo in C/S. If we ﬁx some ϕ: U → T and take the composition f ◦ ϕ: U → S, then the colimit lim−→ F(U) over commutative diagrams

U / V

f◦ϕ f STo is exactly F(U). On the other hand, if we take some U → S which does not factor through f : T → S then there are no diagrams as above and we have fˆpF(U) = ∅. Hence we conclude that for every object g : U → S we have a fˆpF(U) = F(U) ϕ where the coproduct is over all morphisms ϕ: U → T such that f ◦ ϕ = g. If SE = Sét and f is an open immersion then there is a unique ϕ: U → T such that f ◦ ϕ = g and hence f! is the sheafification of the presheaf ( F(U) if U → S factors through f : T → S ; U 7→ ∅ otherwise.

This proves that the deﬁnition of f! agrees with the deﬁnition of f! in Section 4.3. 48 4. OPERATIONS ON SHEAVES OF SETS

4.5. Operations on sheaves of abelian group/pointed sets It is also natural to talk about sheaves of abelian groups and sheaves of pointed sets. There are forgetful functors

(Ab) → (Set∗) → (Set) where the first one forgets the group structure but remembers the zero, and the second one forgets the zero. The sheaves constructed in Example 3.5.5 and Example 3.5.6 are examples of sheaves of abelian groups since they are even sheaves of rings. ∗ Given a morphism f : X → Y of schemes, we may define functors f∗ and f between categories of sheaves of abelian groups/pointed sets exactly as before. In case f is an E-morphism, i.e., an object in C/Y (see Remark 3.1.2), it is easy to see that the functor f ∗ will be the same as in the case of sheaves of sets. More generally, this holds when the colimit f p(U) is filtered for every X-scheme U. Let f : X → Y be an object in C/Y and let fˆ be defined as in 4.4. Let F be a sheaf of abelian groups/pointed sets on (C/Y )E. Since coproducts are direct sums in (Ab) and wedge products in (Set∗) we get that f!F is the sheafification of the presheaf which takes g : U → Y to M fˆpF(U) = F(U) ϕ in the case of abelian groups and the sheafification of _ fˆpF(U) = F(U) ϕ in the case of pointed sets, where the direct sum/wedge product is over all morphisms ϕ: U → X, such that f ◦ ϕ = g. Remark 4.5.1. The distinguished point of a pointed set will often be denoted by ∗ or 0. Lemma 4.5.2. Let f : X → Y be a morphism of schemes and let F be a sheaf (of sets, pointed sets, or abelian groups) on Yét. Let x¯: Spec Ω → X be a geometric point and put y¯ = f ◦ x¯. Then we have an isomorphism ∗ ∼ (f F)x¯ = Fy¯ .

∗ ∗ Proof. We have that (f F)x¯ = lim−→ f F(U) = lim−→ −→lim F(V ) where the ﬁrst colimit is over ´etaleneighborhoods U ofx ¯ and the second colimit is over commutative diagrams U / V ;

Spec Ω / X / Y where V → Y is étale.Thus, V is an étaleneighborhood of Spec Ω. On the other 0 0 hand, Fy¯ = lim−→ F(V ) where the colimit is over étaleneighborhoods V ofy ¯, i.e., over commutative diagrams

0 0 X ×Y V / V 9

Spec Ω / X / Y 0 ∗ ∼ where V → Y is ´etale.Hence we see that (f F)x¯ = Fy¯. 4.5. OPERATIONS ON SHEAVES OF ABELIAN GROUP/POINTED SETS 49

Lemma 4.5.3. Let F be a sheaf (of sets, pointed sets, or abelian groups) on the small étalesite of a scheme X. Let f : X → Y be étaleand let g : U → Y be any morphism. Consider the fiber product

q X ×Y U / U

p g f X / Y Then we have an isomorphism ∗ ∼ ∗ g f!F = q!p F . ∗ ∼ ∗ Proof. For any sheaf G on Uét we have an isomorphism f g∗G = p∗q G (Re- mark 4.4.1). Hence, by the adjoint properties, we have natural bijections ∗ HomSh∗(Uét)(g f!F, G) ' HomSh∗(Yét)(f!F, g∗G) ∗ ' HomSh∗(Xét)(F, f g∗G) ∗ ' HomSh∗(Xét)(F, p∗q G) ∗ ∗ ' HomSh∗((X×Y U)ét)(p F, q G) ∗ ' HomSh∗(Uét)(q!p F, G) . ∗ ∼ ∗ This implies that g f!F = q!p F .

Lemma 4.5.4. Let F be a sheaf of pointed sets on Tét and let f : T → S be étale. Let s¯: Spec Ω → S be a geometric point and let t¯1,..., t¯n be the geometric points of T over s¯. Then we have an isomorphism of pointed sets n ∼ _ (f!F)s¯ = Ft¯i . i=1 Proof. Suppose first that S = Spec Ω. Then an étaleneighborhood of id: Spec Ω → Spec Ω is just an étalemorphism U → Spec Ω with a section Spec Ω → U. The geometric points t¯1,..., t¯n corresponds exactly to the sections S → T of f. An étaleneigh- borhood of t¯i gives an étaleneighborhood ofs ¯ ∈ Spec Ω and conversely, any étale U → Spec Ω with a section Spec Ω → U, yields for a section Spec Ω → T with image t¯i, an étaleneighborhood U ×S T → T of t¯i. Furthermore, if there is no section Spec Ω → T and if U → Spec Ω is an étaleneighborhood, then U → Spec Ω does not factor through T → Spec Ω. Hence we conclude that n ∼ _ (f!F)s¯ = Ft¯i . i=1 Now we use the previous lemma to finish the proof. Lets ¯: Spec Ω → S be a geometric point and consider the diagram

q T ×S Spec Ω / Spec Ω

p s¯ f T / S. ∗ From Lemma 4.5.2 we have (f!F)s¯ = (¯s (f!F))x¯ wherex ¯ = id: Spec Ω → Spec Ω, ∗ and by Lemma 4.5.3 this equals (q!(p F))x¯. From the ﬁrst part of the proof we have 50 4. OPERATIONS ON SHEAVES OF SETS

∗ W ∗ (q!(p F))x¯ = i(p F)x¯i where the wedge is over all sections Spec Ω → T ×S Spec Ω W ∗ W of q. But again, by Lemma 4.5.2 we have i(p F)x¯i = i Fp(¯xi), and hence n _ (f!F)s¯ = Fp(¯xi) . i=1 CHAPTER 5

Algebraic spaces

Algebraic spaces are geometric objects that are more general than schemes. A scheme is obtained by gluing together affine schemes in the Zariski topology, whereas an algebraic space is obtained by gluing together affine schemes in the étaletopology (or even finer topologies). Since the étaletopology is finer than the Zariski topology, this gives a ”bigger” category than the category of schemes. The main references to this chapter are [Art71, Knu71], and [Sta]. Before we give the definition of an algebraic space we introduce some descent theory. This will later be used to extend properties of schemes to properties of sheaves (in particular, algebraic spaces), and to extend properties of morphisms of schemes to properties of morphisms of sheaves (in particular, algebraic spaces).

5.1. Some descent theory

Definition 5.1.1. A morphism F → G in the category (Sch\/S) of presheaves (Sch/S)op → (Set) is called representable if for all S-schemes X and all morphisms hX → G in (Sch\/S), the functor F ×G hX is representable. Remark 3.2.3 implies the following lemma:

Lemma 5.1.2. If X → Y is a morphism of schemes then hX → hY is representable. Let P be a property of morphisms in a category C with fiber products, let T be a Grothendieck topology on C, and let S be the resulting site. In order to extend properties of morphisms of schemes to properties of morphisms of presheaves, we make the following definitions: Definition 5.1.3. We say that P is T -local on the base if for every covering {Ui → U} with respect to T and any morphism f : X → U, the following are equivalent: (1) f has P ; (2) for every i, the morphism X ×U Ui → Ui has P . Lemma 5.1.4. Let ϕ: A → R be a faithfully flat homomorphism of rings and let ψ : B → A be a ring homomorphism such that the composition ϕ ◦ ψ is flat. Then ψ is flat. Proof. Let M 0 → M → M 00 be any sequence of B-modules. Since the composition B → A → R is flat, we get that 0 00 M ⊗B R → M ⊗B R → M ⊗B R is exact. But this sequence equals 0 00 M ⊗B A ⊗A R → M ⊗B A ⊗A R → M ⊗B A ⊗A R

51 52 5. ALGEBRAIC SPACES and since A → R is faithfully flat, Proposition 1.1.8.(3) implies that 0 00 M ⊗B A → M ⊗B A → M ⊗B A is exact. That is B → A is flat. Lemma 5.1.5. The property ”flat” is fpqc local on the base. Proof. We only need to prove that (2) implies (1) since a base change of a flat morphism is flat. Let f : X → U be any morphism of schemes. Suppose that {Ui → U} is a flat covering and that X ×U Ui → Ui is flat for each i. Take x ∈ X and put u = f(x). The question is local and hence we may assume that 0 ` X = Spec B and U = Spec A where A = OU,u and B = OX,x. Since U := Ui → U is surjective, there is a u0 ∈ U 0 mapping to u ∈ U. The induced morphism 0 Spec OU 0,u0 → Spec OU,u is faithfully flat by Lemma 1.1. Put A = OU 0,u0 . 0 0 We have that B → A ⊗A B is faithfully flat since A → A is faithfully flat. 0 We also have that the composition A → B → B ⊗A A is flat since it equals the 0 0 composition A → A → B ⊗A A which is a composition of flat homomorphisms. Hence A → B is flat by Lemma 5.1.4. Lemma 5.1.6. The property ”étale”is fpqc local on the base. Proof. By the previous lemma we only need to show that the property ”unramified” is fpqc local on the base. A morphism X → U is unramified if and only if it is locally of finite type and the diagonal ∆: X → X ×U X is an open 0 0 immersion. But if U → U is an fpqc cover, then U ×U X is the fiber product corresponding to the morphisms ∆: X → X ×U X and the induced morphism 0 U ×U X ×U X → X ×U X which is also an fpqc cover. Hence it suffices to show that ”locally of finite type” and ”open immersion” are properties that are fpqc local on the base. This follows from [Sta, Tag 02KX] and [Sta, Tag 02L3]. Definition 5.1.7. We say that a stable property P of morphisms in C is T -local on the domain if for any morphism X → Y in C and any T -covering {Xi → X}, the following are equivalent: (1) X → Y has P ; (2) Xi → X → Y has P for every i. Lemma 5.1.8. The property ”flat” is fpqc local on the domain. 0 Proof. We need only prove that if {Xi → X} is a covering such that X := ` 0 Xi → X is faithfully flat, then a morphism f : X → Y if X → X → Y is flat. Take any x ∈ X and put y = f(x). The question is local and hence we may assume ` that X = Spec OX,x and Y = Spec OY,y. Since Xi → X is surjective, there is 0 0 ` an x ∈ X in the fiber over x. Since Xi → X is flat, the local homomorphism OX,x → OX0,x0 is faithfully flat by Lemma 1.1. Hence OY,y → OX,x is flat by Lemma 5.1.4. Lemma 5.1.9. The property ”étale”is étalelocal on the domain. Proof. We only need to prove the implication (2) ⇒ (1) of Definition 5.1.7. Suppose that f : X0 → X is an étalecover and that X → Y is any morphism such that X0 → Y is unramified. By [Sta, Tag 036O] we have that X → Y is locally of finite type. There is an exact sequence ∗ 0 → f ΩX/Y → ΩX0/Y → ΩX0/X → 0 0 [Har77, II.8.11], [AK70, VI.4.9]. We have ΩX0/Y = 0 since X → Y is unramified. ∗ Hence f ΩX/Y = 0. This implies that the stalk ∗ 0 0 0 (f ΩX/Y )x = (ΩX/Y )x ⊗OX,x OX ,x 5.2. ALGEBRAIC SPACES 53 is zero for all x0 ∈ X0 and x = f(x0). But f is surjective and since f is flat, the induced homomorphism OX,x → OX0,x0 is faithfully flat. Thus we conclude that (ΩX/Y )x = 0 for all x ∈ X and hence ΩX/Y = 0. That is, X → Y is unramified. By Lemma 5.1.8 we get that X → Y is étale. Definition 5.1.10. Let α: F → G be a representable morphism in (Sch\/S) and let P be a property of morphisms over S which is preserved under base change and is fpqc local on the base. Then we say that α has P if for every S-scheme U and every morphism hU → G, with V an S-scheme representing hU ×G F we have ∼ that the morphism V → U (corresponding to the morphism hV = hU ×G F → hU ) has P .

5.2. Algebraic spaces Definition 5.2.1. Let C be a category with ﬁber products, and S an object in C. We call a diagram R ⇒ U in C/S a categorical equivalence relation if for every object V in C/S, the map

HomC/S(V,R) → HomC/S(V,U) × HomC/S(V,U) = HomC/S(V,U ×S U) is injective and gives an equivalence relation on HomC/S(V,U). In particular, the induced morphism R → U ×S U is a monomorphism.

Definition 5.2.2. Let S be a scheme. A categorical equivalence relation R ⇒ U in (Sch/S) is called an ´etale equivalence relation if the two projection morphisms p1, p2 : R,→ U ×S U ⇒ U are ´etale. Definition 5.2.3. Let S be a scheme. An algebraic space over S is a sheaf on

SEt´ such that there exists an S-scheme U and a representable morphism hU → A, such that for any S-scheme X and morphism hX → A, the morphism hU ×A hX → hX is induced by a surjective and ´etalemorphism of schemes.

Definition 5.2.4. We call a morphism hU → A, as in Definition 5.2.3, a representable étalecovering. If R is a scheme representing hU ×A hU , we say that the diagram hR ⇒ hU → A is a presentation of A. Remark 5.2.5. An S-scheme X becomes an algebraic space when identifying it with its representable sheaf hX . Indeed, it is a sheaf in the étaletopology, and for the S-scheme U we can just choose U = X.

0 Remark 5.2.6. If hU → A is a representable étalecovering and U → U is an étalecovering of schemes, then hU 0 → A is a representable étale covering. Indeed, ∼ for any morphism hX → A we have that hU ×A hX = hZ for some scheme Z and 0 0 ∼ 0 ∼ 0 hU ×A hX = hU ×hU hZ = hU ×U Z . Since U → U is an étalecovering, so is 0 0 U ×U Z → Z, and hence U ×U Z → Z → X is an étalecovering. Definition 5.2.7. A morphism of algebraic spaces A → B is just a morphism A → B of presheaves (i.e., a natural transformation of functors).

We denote the category of algebraic spaces over a schemes S by (AlgS) or just (Alg) if no base scheme is speciﬁed. Also, we denote the category of algebraic spaces over S over a base space B by (AlgS/B) (or just (Alg/B)). We may also consider the subcategory (Sch/B) of schemes over an algebraic space B.

Remark 5.2.8. By Remark 2.2.6 we have that hS (given by id : S → S) is the ﬁnal object in (AlgS). 54 5. ALGEBRAIC SPACES

Proposition 5.2.9 ([Knu71, 1.3]). Let A be an algebraic space over a scheme S and hR ⇒ hU → A a presentation of A. Then (1) R ⇒ U is an étaleequivalence relation in the category (Sch/S), and (2)h R ⇒ hU → A is a coequalizer diagram in Sh(SEt´ ), i.e., hU → A is an effective epimorphism. Proof. To prove (1), note first that the morphisms R ⇒ U are étaleby the definition of an algebraic space. Since hR = hU ×A hU , it follows that for any S-scheme V , the morphism

Hom(hV , hR) → Hom(hV , hU ) × Hom(hV , hU ) is injective. Yoneda’s lemma now implies that R ⇒ U is a categorial equivalence relation. This proves (1). To prove (2), Lemma 3.4.9 and Lemma 3.4.8 implies that it is enough to show that hU → A is locally surjective. Let X be a scheme. An element ϕ ∈ A(X) is by Yoneda’s lemma a morphism ϕ: hX → A and by deﬁnition, there is an S-scheme V such that hV = hX ×A hU and V → X is surjective and ´etale.Thus {V → X} is a covering in Sh(SEt´ ). Since hV = hX ×A hU we have that the composition hV → hX → A is equal to the composition hV → hU → A. That is, the pullback of ϕ to A(V ) is the image of an element in hU (V ), and hence hU → A is locally surjective. This proves (2).

From now on we will just write X instead of hX . If R ⇒ U is a categorial equivalence relation, then for every scheme T , the maps R(T ) ⇒ U(T ) gives an equivalence relation ∼ on U(T ). Let U/R be the sheaf associated to the presheaf T 7→ U(T )/ ∼. Theorem 5.2.10. Let A be an algebraic space over a scheme S. Then

(1) the diagonal ∆ : A → A ×S A is representable; (2) for every S-scheme U we have that every S-morphism f : U → A is representable. Proof. Statement (1) is [CLO12, A.1.1]. We show that (1) and (2) are equivalent. (1) ⇒ (2): Suppose that U and V are schemes and we have morphisms U → A and V → A. Then U ×S V is a scheme and we have a morphism U ×S V → A×S A. But then

U ×A V = A ×A×S A (U ×S V ) which is a scheme by (1). (2) ⇒ (1): Suppose that Z is a scheme and let f : Z → A ×S A be any morphism. Then f factors as Z → Z ×S Z → A ×S A and we have

A ×A×S A (Z ×S Z) = Z ×A Z, which is a scheme by (2). Hence

A ×A×S A Z = (Z ×A Z) ×Z×S Z Z which again is a scheme and hence ∆ is representable. Note that Theorem 5.2.10 says that if U and V are S-schemes and we have S-morphisms U → A and V → A, then the ﬁber product U ×A V is a scheme. Theorem 5.2.11 ([CLO12, A.1.2]). Let U be a scheme over S. If R ⇒ U is an ´etaleequivalence relation, then the quotient sheaf U/R is an algebraic space over S and R ⇒ U → U/R is a presentation of U/R. Corollary 5.2.12. Let A be an algebraic space and let R ⇒ U → A be a presentation of A. Then A is isomorphic to the quotient sheaf U/R. 5.3. SOME DESCENT THEORY FOR ALGEBRAIC SPACES 55

Proof. Follows from the universal property of a coequalizer. Definition 5.2.13. Let A be an algebraic space over a scheme S given by an ´etaleequivalence relation R ⇒ U. Then we say that (1) A is locally separated if the morphism R → U ×S U is an immersion; (2) A is separated if R → U ×S U is a closed immersion.

Proposition 5.2.14. If A1 → B and A2 → B are morphisms of algebraic spaces, then the presheaf ﬁber product A1 ×B A2 is an algebraic space. As a result, we have that the category (Alg) of algebraic spaces has ﬁber products, products, and coproducts.

Proof. By Corollary 3.8.2 we only need to show that the sheaf A1 ×B A2 is an algebraic space. Let U1 → A1, U2 → A2, and V → B be representable étale coverings. Then we have a morphism U1 ×B U2 → A1 ×B A2. Also, U1 ×B U2 is representable by Theorem 5.2.10. Suppose that we have a morphism X → A1 ×B A2 where X is a scheme. This means that we have morphisms X → A1 → B and X → A2 → B and hence we have ∼ X ×A1×BA2 (U1 ×B U2) = (U1 ×A1 X) ×X (U2 ×A2 X) , where the last fiber product is representable since it is a fiber product of schemes over a scheme. Since U1 ×A1 X → X and U2 ×A2 X → X are both induced by surjective étalemorphisms of schemes, we conclude that

(U1 ×A1 X) ×X (U2 ×A2 X) → X is also induced by a surjective ´etalemorphism of schemes. Proposition 5.2.15 ([Knu71, Proposition 1.4]). Let A and B be algebraic spaces. Every morphism A → B is induced by a diagram

pr1 U ×A U / U pr2 g f pr1 V ×B V // V pr2 where α : U → A and β : V → B are representable étalecoverings such that pri ◦ g = f ◦ pri for i = 1, 2. Proof. First, it is clear that such a diagram yields a unique morphism γ : A → B such that β ◦ f = γ ◦ α since U → A is a coequalizer of U ×A U ⇒ U in Sh(SEt´ ). Let A → B be a morphism and let U 0 → A and V → B be representable étale coverings. The composition U 0 → A → B is an element y ∈ B(U 0) and since V → B is locally surjective, there is an étalecovering ϕ: U → U 0 such that ϕ∗(y) ∈ B(U) is in the image of V (U) → B(U). Hence U → U 0 → A is also a representable étale covering. A preimage of ϕ∗(y) is a morphism U → V of algebraic spaces and for every choice of preimage we get a morphism U ×A U → V ×B V by the universal property of V ×B V . 5.3. Some descent theory for algebraic spaces In this section we will extend some properties of schemes (morphisms of schemes) to properties of algebraic spaces (morphisms of algebraic spaces). A good definition will give us the usual properties of schemes (morphisms of schemes) when restricting to the subcategory of schemes inside the category of algebraic spaces. 56 5. ALGEBRAIC SPACES

Let A → B be a morphism of algebraic spaces over a scheme S and suppose that we have an S-scheme X and a morphism X → B. Let U → A and V → B be representable ´etalecoverings. Then X ×B V is representable, say by Z. Thus we have a commutative diagram

Z ×B A / X ×B A

% # V ×B A / A

Z / X

& V /$ B

If A → B is representable, then X ×B A and V ×B A are representable. If P is a property of morphisms of schemes which is stable under base change and étale-local on the base, then X ×B A → X has P if V ×B A → V has P . Indeed, this follows since Z → X is an étalecovering. Hence we have proved: Lemma 5.3.1. Let A → B be a representable morphism of algebraic spaces and let V → B be a representable étalecovering. Let P be a property of morphisms of schemes which is stable under base change and étalelocal on the base. Then the following are equivalent:

(1) V ×B A → V has P ; (2) X ×B A → X has P for every scheme X and every morphism X → B. Lemma 5.3.1 implies that the following definition agrees with Definition 5.1.10: Definition 5.3.2. Let A → B be a representable morphism of algebraic spaces and let P be a property of morphisms of scheme which is stable under base change and étalelocal on the base. Then we say that A → B has P if there is a representable étalecovering V → B such that the induced morphism V ×B A → V of schemes has P . Remark 5.3.3. Hence we may speak of morphisms of algebraic spaces being open immersions, closed immersions, etc., keeping in mind that we are considering representable morphisms only. For the non-representable case we need to impose stronger conditions on P to be able to extend it to morphisms of algebraic spaces. A good definition will agree with Definition 5.3.2, when restricting to the class of representable morphisms. Let X be a scheme and suppose that we have a morphism X → B. Then X ×B A is an algebraic space. Let U → X ×B A be a representable étalecovering.

U / X ×B A / X

# A / B

If X ×B A is representable, say by a scheme Z, then U → Z is an étalecovering. Let P be a property of morphisms of schemes which is étalelocal on the domain. Then X ×B A → X has P if and only if the composition U → X ×B A → X has P . Lemma 5.3.4. Let ϕ: A → B be a morphism of algebraic spaces and let V → B 0 and U → A×BV be representable étalecoverings. Let P be a property of morphisms 5.3. SOME DESCENT THEORY FOR ALGEBRAIC SPACES 57 of schemes which is stable under base change, étalelocal on the base, and étalelocal on the domain. Then the following are equivalent: (1) U 0 → V has P ; (2) for every scheme X, every morphism X → B, and every representable étalecovering U → X ×B A, the composition U → X ×B A → X has P . Proof. We obviously need only show that (1) implies (2). Let X → B and U → A ×B X be as in (2). By definition of a representable étalecovering, we have 0 0 that X ×B V is representable and hence U ×V (V ×B X) = U ×B X is a scheme. Suppose that U 0 → V has P . Since P is stable under base change, we get that 0 U ×B X → V ×B X has P . 0 The morphism U ×B X → V ×B X factors through

(V ×B A) ×A (A ×B X) = A ×B V ×B X → V ×B X since 0 0 U ×B X = U ×A×BV (A ×B V ×B X) . 0 Hence we have a morphism U ×B X → A ×B X and we may consider the fiber product 0 0 (U ×B X) ×A×BX U = U ×A U. 0 By Theorem 5.2.10 we have that U ×A U is a scheme and since U → A is a 0 0 representable étalecovering we get that U ×A U → U ×B X is an étalecovering. Since P is étalelocal on the domain, we get that the composition 0 0 U ×A U → U ×B X → V ×B X 0 0 has P . But U ×A U is the fiber product of U ×B X and U ×B V over A×B V ×B X, which implies that the morphism 0 0 U ×A U → U ×B X → V ×B X is the same as the morphism 0 U ×A U → U ×B V → A ×B V ×B X → V ×B X. 0 We have that U ×B V is a scheme and U ×A U → U ×B V is an étalecovering. Again, since P is étalelocal on the domain, we get that U ×B V → V ×B X has P . But P is étalelocal on the base and since V ×B X is an étalecovering and U ×B V = U ×X (V ×B X) we get that U → X has P . This proves the lemma. The argument is summarized in the following diagram:

U ×B V / U / A ×B X / X 8 7 ;

0 ' # U ×A U A ×B V ×B X / V ×B X A / B 7 ; ?

& 0 0 ' # U ×B X / U / A ×B V / V Hence we make the following definition: Definition 5.3.5. Let α: A → B be a morphism of algebraic spaces and let P be a property of morphisms over S which is stable under base change, étalelocal on the base, and étalelocal on the domain. Then we say that α has P if for some some representable étalecovering V → B, and some representable étalecovering U → X ×B A, the composition U → X ×B A → X has P . 58 5. ALGEBRAIC SPACES

Remark 5.3.6. Hence we may speak of morphisms of algebraic spaces being, surjective, étale,flat, faithfully flat, locally of finite type, unramified, etc.

Lemma 5.3.7. Let f : A → B be a morphism in (AlgS). The following are equivalent: (1) f is ´etaleand surjective;

(2) f is ´etaleand an epimorphism in Sh(SEt´ );

Proof. (1) ⇒ (2): We will show that f is locally surjective in Sh(SEt´ ). Let X be an S-scheme and suppose that we have a morphism X → B. Let U → A ×B X be a representable étalecovering. By Lemma 5.3.4, the induced morphism U → X is surjective and étaleand hence an étalecovering of schemes. Thus f is locally surjective and hence an epimorphism in Sh(SEt´ ). (2) ⇒ (1): If f is an étaleepimorphism then f is locally surjective and hence, for every representable étale covering V → B, there is an étalecovering V 0 → V such that the diagram V 0 / V

A / B 0 commutes. Hence V → V factors through A ×B V . Let U → A ×B V be a representable étalecovering. Then U → A ×B V → V is étale,since f is étale. 0 Hence V q U → A ×B V is a representable étalecovering and the composition 0 V q U → A ×B V → V is surjective and étale.

5.4. Etale´ topology on algebraic spaces All examples of sites of schemes in Section 3.1 may be extended to algebraic spaces. Here are some examples. Example 5.4.1 (Big ´etalesite on (Alg/B) or (Alg)). The underlying category has morphisms X → B of algebraic spaces as objects and commutative diagrams

X1 / X2

Õ B as morphisms. The coverings are collections {Ui → U} of étale morphisms over B ` such that Ui → U is étaleand surjective. This site is denoted BEt´ . Note that if U and the Ui are schemes, then {Ui → U} is an étalecovering of schemes. Example 5.4.2 (Small étalesite on A). The underlying category is ét(A), i.e., it has étalemorphisms X → A of algebraic spaces as objects and commutative diagrams

X1 / X2

Õ A as morphisms. The coverings are collections {Ui → U} of étalemorphisms over A ` such that Ui → U is étaleand surjective. This site is denoted Aét. Note that if U and the Ui are schemes, then {Ui → U} is an étalecovering of schemes. 5.5. THE SHEAF SPACE (ESPACE ETAL´ E)´ 59

Remark 5.4.3. Let X be a scheme. Every sheaf on Xét extends uniquely to a sheaf on étalealgebraic spaces over X. Conversely, every sheaf on étalealgebraic spaces over X is completely determined by its restriction to étaleschemes over X. Indeed, if F is a sheaf on étalealgebraic spaces and R ⇒ U → A is an algebraic space, then F(A) is completely determined by F(U) ⇒ F(R). If F is a sheaf on étaleschemes over X, then we may define F(A) to be the equalizer of F(U) ⇒ F(R) in (Set). op This implies that the presheaf (AlgS/A) → (Set) defined by an algebraic space X over A is a sheaf on AEt´ . Indeed, X is a sheaf when restricted to S-schemes over A since a covering {Ui → U} in Aét where U and the Ui are schemes is also a covering in Sét when considered as morphisms of S-schemes. By the discussion above, X extends uniquely to a sheaf on Aét.

Each scheme X has a structure sheaf OXét defined in Example 3.5.6. The structure sheaf OAét on Aét is defined by OAét (U) := Γ(U, OU ) when U is an étale scheme over A. If B is an étalealgebraic space over A and R ⇒ V → B is a presentation of B, then OAét (B) is completely determined by OAét (V ), OAét (R), and the sheaf criterion.

Example 5.4.4 (The big fpqc site on A). The underlying category is (Alg/A) ` and coverings are collections {Ui → U} such that Ui → U is fpqc.

5.5. The sheaf space (espace ´etal´e)

We will show that every sheaf F on Sét is represented on (ét/S) by a unique ét (up to isomorphism) étalealgebraic space F over S, i.e., every sheaf F on Sét is ét the restriction of a unique (up to isomorphism) étale algebraic space F to Sét. First we will show that the fiber product of two étale S-schemes over a sheaf on Sét is a scheme.

Lemma 5.5.1. Let F be a sheaf on S´et and let ∆: F → F ×S F be the diagonal. Then

(1) for every ´etale S-scheme Z, and any S-morphism Z → F ×S F, we have

that F ×F×S F Z is represented by an open subscheme of Z; (2) for every étale S-schemes U and V , and any elements α ∈ F(U) and β ∈ F(V ), the sheaf fiber product U ×F V in Sh(Sét) is represented by an open subscheme of U ×S V .

Proof. We ﬁrst show that (1) and (2) are equivalent. It is clear that (1) implies (2) since U ×F V = F ×F×S F (U ×S V ). To show that (2) implies (1), note that a morphism Z → F ×S F factors through the diagonal Z → Z ×S Z. By (1) we have that Z ×F Z → Z ×S Z is an open immersion, and since

F ×F×S F Z = (Z ×F Z) ×Z×S Z Z and the property ”open immersion” is preserved under base change, we get that

F ×F×S F Z is represented by an open subscheme of Z. Hence (1) and (2) are equivalent. Now we prove (1). To give a morphism Z → F ×S F is to give two morphisms α, β : Z ⇒ F such that the compositions Z ⇒ F → S coincide. Let W ⊆ Z be the subset of Z consisting of all points z such that αz¯ = βz¯. The claim is that W is open in Z. Indeed, if αz¯ = βz¯ then there is an ´etaleneighborhood ϕ: U → Z of z¯ such that ϕ∗(α) = ϕ∗(β). The image ϕ(U) is open in Z and W is the union of all such ϕ(U) as z ranges over the points of Z. Thus we have an open immersion 60 5. ALGEBRAIC SPACES i : W,→ Z and i ◦ α = i ◦ β. This gives a commutative diagram

i W / Z

i◦α ∆ F / F ×S F . Now suppose that X is any ´etale S-scheme and suppose that we have a commutative diagram ψ X / Z

∆ F / F ×S F . ∗ ∗ Then ψ (α) = ψ (β) and hence Lemma 4.5.2 implies that αz¯ = βz¯ for all points z ∈ ψ(X). That is, ψ factors uniquely through i. Furthermore, ϕ = α ◦ ψ = β ◦ ψ and since ψ factors through i, so does ϕ. Hence we conclude that F ×F×S F Z is represented by W . This proves the lemma. Lemma 5.5.2. Let S be a scheme and let ét(S) be the category of étalealgebraic spaces over S. The functor ét(S) → Sh(Sét) that takes an étalealgebraic space A to the restriction A|Sét is fully faithful. Proof. Let A and B be étalealgebraic spaces over S and let U → A be a representable étalecovering. Since A is étaleover S, we have that U is étale over S, and hence A(U) = (A|Sét )(U) and B(U) = (B|Sét )(U). Suppose that we have a morphism A|Sét → B|Sét . Since A(U) = (A|Sét )(U) we get a morphism

U → A|Sét → B|Sét and since B(U) = (B|Sét )(U) we get a morphism U → B such that the compositions U ×A U ⇒ U → B coincide. Since U ×A U ⇒ U → A is a coequalizer diagram in Sh(SEt´ ), we get a unique morphism A → B through which U → B factors. Theorem 5.5.3 ([Mil80, V.1.5]). Let S be a scheme and let F be a sheaf on the site Sét. Let π : SEt´ → Sét be the restriction morphism (see Remark 4.1.3). Then there exists a unique locally separated algebraic space Fét which is étaleover ét S, and such that F = π∗(F ). Proof. Define a U = V, (V,ϕ) where the disjoint union is over all pairs (V, ϕ) where V is an étale S-scheme and ϕ ∈ F(V ). Hence we get a canonical element s ∈ F(U). By Lemma 5.5.1, U ×F U is represented by an open subscheme R of U ×S U. The morphisms U ×S U ⇒ U are étalesince U is étaleover S, and R → U ×S U is an open immersion. Hence R ⇒ U is an étaleequivalence relation which gives us a locally separated algebraic space Fét. ét Since U → F is an epimorphism in Sh(SEt´ ) it is also an epimorphism in ét Sh(Sét). To show that F and F are equal as sheaves on Sét, Lemma 3.4.9 implies that it is enough to show that U → F is an epimorphism in Sh(Sét), i.e., locally surjective. But this is obvious since every morphism T → F from an étale S-scheme T to F, factors through U. Uniqueness follows from Lemma 5.5.2. Corollary 5.5.4. Let X be an algebraic space over a scheme S and let F be ét a sheaf on Xét. Then F is represented by an algebraic space F over S which is étaleover X. 5.5. THE SHEAF SPACE (ESPACE ETAL´ E)´ 61

Proof. As in Theorem 5.5.3 we define a U = V (V,ϕ) where the disjoint union is over all pairs (V, ϕ), where V is an étalescheme over X and ϕ ∈ F(V ). The idea is to show that U ×F U is represented by an algebraic space which in term is represented by a scheme R. Then we may define Fét to be ét the algebraic space defined by R ⇒ U and check that F represents F and is étale over X. By Theorem 5.2.10 we have that U ×X U is a scheme and hence Lemma 5.5.1 implies that U ×F U is represented by an open subscheme R ⊆ U ×X U. ét Define F to be the algebraic space given as the quotient of R ⇒ U. We know that F is completely determined by its restriction to the category of étaleschemes over X. It is obvious that the canonical morphism U → F is locally surjective as a morphism of sheaves on schemes over X, since any morphism ϕ: V → F is part of the disjoint union defining U. Thus R ⇒ U → F is a coequalizer diagram in the category of sheaves on the the restriction of the site Xét to S-schemes that are étaleover X. This means that Fét and F agrees as sheaves on étaleschemes over X, and by Remark 5.4.3 we conclude that they also agree as sheaves on Xét.

Definition 5.5.5. The algebraic space Fét of Theorem 5.5.3 and Corollary 5.5.4 is called the espace étalé or sheaf space of F.

Remark 5.5.6. Theorem 5.5.3 and Lemma 5.5.2 implies that there is an equivalence of categories between the category ét(S) of étalealgebraic spaces over S and the category Sh(Sét) of sheaves on Sét.

Let π : SEt´ → Sét be the restriction morphism. Let F be a sheaf on Sét. The ∗ ∗ 0 pullback π F is defined by π F(V ) = lim−→ F(V ) where the limit is over commutative diagrams V / V 0

ét Õ S Given a morphism ϕ: V → F from an étale S-scheme V to F, we have π∗F(V ) = F(V ) and we get a canonical morphism ϕ: V → π∗F. Thus, we get a canonical morphism ψ : U → π∗F where U is defined as in Theorem 5.5.3. We will show that ψ is an epimorphism and hence that π∗F = Fét.

Lemma 5.5.7. With the setup in Theorem 5.5.3, we have F´et = π∗F.

Proof. It suﬃces to show that ψ : U → π∗F is locally surjective. Let T be an S-scheme and take α ∈ π∗F(T ). Then there exists a commutative diagram

f T / V

´et Ö S and a β ∈ F(V ) = π∗F(V ) such that α is the image of β under the induced map f ∗ : π∗F(V ) → π∗F(T ). But f ∗ is the map deﬁned by taking a morphism V → π∗F to the composition T → V → π∗F, i.e., α: T → π∗F factors through β : V → π∗F. On the other hand, β may be considered as a morphism V → F and hence we have 62 5. ALGEBRAIC SPACES

V,→ U and by deﬁnition of ψ : U → π∗F, we also have a commutative diagram

V / U

β ψ Õ π∗F Hence we conclude that α: T → π∗F factors through U → π∗F. That is, ψ is locally surjective.

Definition 5.5.8. Let S be a scheme and let π : SEt´ → Sét be the restriction morphism (see Remark 4.1.3). Then we make the following definitions: ∼ ∗ (1) A sheaf F on SEt´ is called locally constructible if F = π π∗F; ét (2) A sheaf F on Sét is called construcitble if F is of finite type over S; (3) A sheaf F on SEt´ is called construcitble if it is locally constructible and ét (π∗F) is of finite type. Lemma 5.5.7 implies the following corollary: Corollary 5.5.9. Let S be a scheme. The following are equivalent:

(1) F is a locally constructible sheaf on SEt´ ; (2) F is an ´etalealgebraic space over S.

∗ 5.6. The functors f∗, f , and f! of a morphism of algebraic spaces Let f : X → Y be a morphism of algebraic spaces. The discussion in the beginning of Section 4.1 also holds for algebraic spaces instead of schemes and hence we get a continuous functor f × : ´et(Y ) → ´et(X)

(U → Y ) 7→ (U ×Y X → X) . ∗ Thus we may deﬁne functors f∗ : Sh(XE) → Sh(YE) and f : Sh(YE) → Sh(XE) using the recipe of Section 4.2. When f is an E-morphism we may show, in the exact same manner as before, ∗ ˆ ˆ that f = f∗ where f : YE → XE is the morphism given by the continuous functor which takes an ´etale U → X to the composition

f U → X −→ Y. ˆ∗ ˆ ∗ Again we define f! to be the left adjoint f of f∗ = f . Let G be a sheaf on the small étalesite Sét of a scheme S and let f : T → S be any morphism of schemes (or algebraic spaces). Then f is an object in SEt´ and we may consider the functors in Section 4.4. We showed in Remark 4.4.2 (or Lemma 4.2.9) that ∗ ét ét f (G ) = T ×S G . Now assume that S and T are algebraic spaces that are not necessarily schemes ét and that f : T → S is any morphism. If G is a sheaf on Sét then G is an algebraic space which represents G on ét(S). Lemma 5.6.1. Let f : T → S be any morphism of algebraic spaces and let X → S be étale. Let π be the restriction morphism from the big étalesite to the small étalesite, and let hX be the representable sheaf of X on the big étalesite. Then we have ∗ ∼ ∗ f π∗hX = π∗f hX . ∗ 5.6. THE FUNCTORS f∗, f , AND f! OF A MORPHISM OF ALGEBRAIC SPACES 63

∗ ∗ Proof. We have π∗f hX = π∗hT ×S X . The sheaf f π∗hX is the sheaﬁﬁcation of the presheaf P given by

P(U) =−→ lim π∗hX (V ) for every ´etale U → T , where the colimit is over all commutative diagrams

U / V

T / S with V → S ´etale.Thus, an element in such a hX (V ) is given by a diagram

U / V / X

~ T / S

and hence we have compatible maps hX (V ) → hT ×S X (U). This gives a morphism

τ : P → π∗hT ×S X . Since X is ´etaleover S we get that τU : P(U) → π∗hT ×S X (U) is surjective for each ´etale U → T . On the other hand, if we have two morphisms α, β : V ⇒ X such that the compositions U → V ⇒ X coincides, then we get a commutative diagram

# V ×X V /) V

α V / X where V ×X V → V is ´etalesince β is ´etale. Hence we conclude that τU is also injective and hence τ is an isomorphism. In particular, P is a sheaf and hence ∗ ∼ ∗ f π∗hX = P = π∗f hX .

Corollary 5.6.2. Let S and T be algebraic spaces. Let G be a sheaf on S´et and let f : T → S be any morphism. Then we have an isomorphism

∗ ∼ ét f G = π∗(T ×S G ) of sheaves on Tét, where the fiber product is taken in (Alg).

Let f : T → S be an étalemorphism of algebraic spaces and let F be a sheaf ét on Tét. The espace étalé F is étaleover T and if we compose with f we get an étalemorphism Fét → T → S. We know that the functor f ∗ exists and has a left ét ét adjoint f!. The claim is that F → T → S is equal to (f!F) → S.

ét ét Lemma 5.6.3. The étalealgebraic space (f!F) → S is equal to F → T → S.

Proof. We show that the functor ét(T ) → ét(S) which takes Fét → T to Fét → T → S is a left adjoint to f ∗. For an étalealgebraic space Gét over S we 64 5. ALGEBRAIC SPACES

∗ ´et ´et have f G = T ×S G , and hence this is obvious from the following diagram:

F´et

$ ´et ´et T ×S G /) G

T / S

ét ét There is a one-to-one correspondence between morphisms F → T ×S G over T ét ét and morphisms F → G over S. This proves the lemma.

With the same setup as above, Lemma 4.2.9 implies that fˆpF is a sheaf since it is represented by the algebraic space Fét. Indeed, we have that fˆpFét = (fˆ)catFét = (Fét → T → S). Let ψ : V → S be an étalealgebraic space over S. A morphism V → Fét gives a commutative diagram V

F´et / T /' S and hence we conclude that

ét a ét a Homét(S)(V, (f!F) ) = F (V ) = F(V ) ϕ ϕ where the disjoint unions are over all morphisms ϕ: V → T such that ψ = f ◦ ϕ. ˆp ét ét But this is just f F(V ). Since Homét(S)(V, (f!F) ) = f!F(V ) and (f!F) → S is just the étalealgebraic space Fét → T → S we get the following explicit description of f!F: Lemma 5.6.4. Let f : T → S be an étalemorphism of algebraic spaces (or just schemes) and let F be a sheaf on Tét. Then for any étale ψ : V → S, we have a f!F(V ) = F(V ) , ϕ where the disjoint union is over all morphisms ϕ: V → T such that ψ = f ◦ ϕ.

5.7. The functor f! for sheaves of pointed sets Again, let f : T → S be an étalemorphism of algebraic spaces. In case F is a sheaf of pointed sets, we have the presheaf construction _ (ψ : U → S) 7→ F(U) ϕ where the wedge is over all morphisms ϕ: U → T such that f ◦ ϕ = ψ. Denote this presheaf by P. The presheaf P need not be a sheaf and hence we also need to take the sheafification of P to get the sheaf f!F of pointed sets. The following lemma gives a more explicit description of f!F. Lemma 5.7.1. Let f : T → S be an étalemorphism of algebraic spaces and let F be a sheaf of pointed sets on Tét. Let {0}T and {0}S be the constant sheaves of the set {0} on Tét and Sét respectively. Let f! denote the functor of pointed sets 5.8. DIRECT IMAGE WITH PROPER SUPPORT 65

∗ 0 defined as the left adjoint of f and let f! be the corresponding functor of sheaves of sets. Then f!F ∈ Sh∗(Sét) is the pushout in Sh(Sét) of the diagram

0 f! {0}T / {0}S

0 f! F

0 Proof. For any sheaf of sets G on Tét we have the explicit description f! G(U) = ` 0 0 ϕ G(U) as in Lemma 5.6.4. The morphism f! {0}T → f! F is induced by the unique morphism {0}T → F and can be described as follows for each étale ψ : U → S: we 0 ` have f! {0}T (U) = ϕ{0}T (U) and for each factorization U → T → S the unique point in {0}T (U → T ) will be mapped to the distinguished point in F(U → T ). The pushout is a sheaf of pointed sets by taking the image of   0 a {0}S(U) → f! F {0}S (U) 0 f! {0}T as the distinguished point for each étale U → S. Let P be the presheaf on Sét defined by _ P(U) = F(U) , ϕ for each étale ψ : U → S, where the wedge is over all morphisms ϕ: U → T such that f ◦ ϕ = ψ. The lemma now follows from the fact that the diagram

0 f! {0}T / {0}S

0 f! F / P is a pushout diagram in the category of presheaves (´et/S)op → (Set). Indeed, for every ´etale U → S we have that for any set M, a commutative diagram

0 f! {0}T (U) / {0}S(U)

0 f! F(U) / M sends the distinguished point in F(U → T ) to the image of {0}S(U) in M for each factorization U → T → S. If there is no factorization U → T → S then 0 f!{0}T (U) = f! F(U) = ∅ and {0}S(U) = P(U) = {0}. Hence both maps to M factors uniquely through P(U).

5.8. Direct image with proper support Let f : T → S be a representable, separated, and locally of ﬁnite type morphism of algebraic spaces. Deﬁne a functor

fc : Sh∗((Sch/T )Et´ ) → Sh∗((Sch/S)Et´ ) called the direct image with proper support. If ϕ:(M, p) → (N, q) is a map of pointed sets, then we deﬁne its kernel ker(ϕ) to be the sets of points x ∈ M such that ϕ(x) = q, i.e., ker(ϕ) = {x ∈ M : ϕ(x) = q} . 66 5. ALGEBRAIC SPACES

Definition 5.8.1. Let F be a sheaf of pointed sets on (Sch/T )Et´ . For any ´etale U → S where U is a scheme, we deﬁne [ fcF(U) = ker (F(U ×S T ) → F((U ×S T ) \ Z)) , Z where the union is over all closed subschemes Z ⊆ U ×S T such that the restriction Z → U is proper.

Definition 5.8.2. Let F be a sheaf of pointed sets on Sét and let ϕ: U → S be an étale S-scheme. For every s ∈ F(U) and every geometric pointx ¯: Spec Ω → U, ∗ we define sx¯ to be the image of s in (ϕ F)x¯ = Fx¯. We define the support of s to be the subspace supp(s) = {x ∈ U : sx¯ 6= ∗} , ∗ where ∗ is the distinguished point in (ϕ F)x¯ = Fx¯.

Remark 5.8.3. Note that the subset supp(s) is closed in U. Indeed, if sx¯ = ∗ then there exists an étaleneighborhood ψ : V → U ofx ¯ such that ψ∗(s) = ∗. The image ψ(V ) is open in U and the restriction V → ψ(V ) is an étalecovering. Hence we get that s maps to ∗ in F(ψ(V )) since F(ψ(V )) → F(V ) is injective. Hence we get that supp(s) is just the complement of the union of all open subschemes i: W,→ U such that i∗(s) = ∗. Lemma 5.8.4. Let ϕ: V → U be an étalemorphism of schemes and let F be a 0 ∗ sheaf of pointed sets on Uét. Take any s ∈ F(U) and let s = ϕ (s) ∈ F(V ). Then we have that supp(s0) = ϕ−1(supp(s)) . Proof. Suppose that we have a point u ∈ supp(s) and v ∈ V such that ϕ(v) = u. Letv ¯: Spec Ω → V be a geometric point with image v and letu ¯ = ϕ ◦ v¯. ∗ 0 ∗ ∗ ∗ Since Fv¯ =v ¯ F(Spec Ω) it follows that sv¯ =v ¯ ϕ (s) =u ¯ (s) = su¯. This proves the lemma. Lemma 5.8.5. We have that

fcF(U) = {s ∈ F(U ×S T ) : supp(s) → U is proper} , where supp(s) is given the reduced induced closed subscheme structure [Har77, Example 3.2.6].

Proof. Let M = {s ∈ F(U ×S T ) : supp(s) → U is proper}. Clearly M ⊆ fcF(U) since s ∈ ker (F(U ×S T ) → F(U ×S T \ supp(s))) .

Conversely, if s ∈ fcF(U) then there is a closed subscheme Z ⊆ U ×S T which is proper over U, such that s ∈ ker(F(U ×S T ) → F(U ×S T \ Z)). But then supp(s) ⊆ Z and since supp(s) is closed in Z we get that supp(s) is proper over U.

Lemma 5.8.6. The presheaf fcF is a sheaf on (Sch/S)Et´ .

Proof. It is clear that fcF is separated since f∗F is a sheaf and we have an injective map fcF(U) → f∗F(U) for each U → U ×Y X, which is natural in U. Q Now suppose that we have an element (si)i ∈ fcF(Ui) such that the two images Q of (si)i in fcF(Ui ×U Uj) coincide. Since f∗F is a sheaf we get that (si)i comes from a unique element s ∈ f∗F(U). We need to prove that supp(s) is proper over T . 0 ` If si has proper support for each i then the element s ∈ F(( Ui) ×S T ) with ` ` 0 image (si)i has proper support over Ui. Let V = Ui and let Z = supp(s ). The induced morphism ϕ: V ×S T → U ×S T is an étalecovering since V → U 5.9. CONNECTED FIBRATION OF A SMOOTH MORPHISM 67 is an étalecovering. Hence Lemma 5.8.4 implies that Z = ϕ−1(ϕ(Z)). Hence we conclude that Z = V ×U ϕ(Z). The property ”proper” is étalelocal on the base [Sta, Tag 02L1] and hence if Z,→ V ×S T is proper over V then ϕ(Z) is proper over U. But ϕ(Z) is the support of s.

Lemma 5.8.7. If f is proper then fc = f∗. Proof. Let U → S be ´etale.The restriction of a proper morphism to a closed subset is again proper and hence every section s ∈ F(U ×S T ) has proper support over U.

5.9. Connected fibration of a smooth morphism Here we follow the construction in [LMB00, 6.8]. Let T and S be schemes and let π : T → S be a smooth morphism of finite type. For every S-scheme X we define a relation ∼ on HomSch/S(X,T ) as follows: Let α, β : X → T be two morphisms over S. We say that α ∼ β if for every algebraically closed field k and every morphism ξ : Spec k → X, the two images of Spec k in Spec k ×S T are in the same connected component. The relation ∼ is an equivalence relation.

Lemma 5.9.1. There exists an open subscheme R ⊆ T ×S T such that α ∼ β : X → T if and only if the morphism X → T ×S T induced by α and β factors uniquely through R,→ T ×S T . Furthermore, the induced morphisms R ⇒ T are smooth.

Proof. Let p1, p2 : T ×S T → T be the two projections and let ∆: T → T ×S T be the diagonal. We have that p1 is smooth of finite presentation since p1 is obtained by a base change from π. Let α, β : X → T be morphisms over S and let x be a point in X. Then we have a morphism ξ : Spec κ(x) → X. −1 For every t ∈ T , let Rt be the connected component in p1 (t) containing ∆(t) and define [ R = Rt ⊆ T ×S T. t∈T But the compositions α ◦ ξ and β ◦ ξ maps Spec κ(x) into T and hence we have a morphism Spec κ(x) → T ×S T . Hence we conclude that α ∼ β ⇔ α(x) and β(x) are in the same connected component in π−1(π ◦ α(x)) ⊆ T ⇔ the image of x in T ×S T is mapped to the same component in T via the two projections. But this happens if and only if the image of x is in R. Indeed, the κ(x)-point z ∈ T ×S T −1 with p1(z) = α(x) and p2(z) = β(x) is in the same component in p1 (α(x)) as ∆(α(x)). By [Gro67] we have that R is an open subscheme of T ×S T and hence we conclude that X → T ×S T factors uniquely through R → T ×S T . Furthermore, since f : T → S is smooth, so are T ×S T ⇒ T . An open immersion is smooth and hence we conclude that the morphisms R → T ×S T → T are smooth. Lemma 5.9.2 ([Sta, Tag 0464, Tag 01TX]). Let f : X → Y be a morphism of schemes where Y is locally Noetherian. Then the following are equivalent: (1) f is locally of finite type; (2) f is locally of finite presentation. Any open immersion into a locally Noetherian scheme is quasi-compact [Sta, Tag 01OX]. The following result is due to Artin, and implies that the sheaf quotient T/R is an algebraic space. 68 5. ALGEBRAIC SPACES

Proposition 5.9.3 ([LMB00, Corollaire 10.4]). Let R and T be schemes over S. Suppose that R ⇒ T is a categorial equivalence relation such that the morphisms R ⇒ T are ﬂat and locally of ﬁnite presentation and the morphism R,→ T ×S T is quasi-compact. Then T/R is an algebraic space.

Definition 5.9.4. The algebraic space T/R of Lemma 5.9.1 is denoted by π0(T/S). Lemma 5.9.5. Let ϕ: U → S be any morphism. Then we have ∼ U ×S π0(T/S) = π0(U ×S T/U) .

0 Proof. Let R be the open subscheme of (T ×S U)×U (T ×S U) representing the equivalence relation on T ×S U giving rise to π0(T ×S U/U). Since T → π0(T/S) is a representable smooth covering, so is T ×S U → π0(T/S)×S U. Hence it is enough to show that 0 R = (T ×S U) ×π0(T/S)×S U (T ×S U)

= (T ×π0(T/S) T ) ×S U

= R ×S U.

Since R,→ T ×S T is an open immersion, we have that

R ×S U → (T ×S U) ×U (T ×S U) = (T ×S T ) ×S U

0 is an open immersion. The image of this open immersion is exactly R .

Lemma 5.9.6. The algebraic space π0(T/S) is ´etaleover S.

Proof. We have that π0(T/S) → S is flat and locally of finite presentation since T → S is flat and locally of finite presentation and T → π0(T/S) is a smooth cover. By Lemma 5.9.2, it is enough to check that π0(T/S) → S is unramified. By Proposition 1.2.4, it is enough to check this after base change to a geometric point. By the previous lemma, we need only show that π0(T ×S Spec Ω/Spec Ω) → Spec Ω is unramified, where Ω is a separably closed field. But π0(T ×S Spec Ω/Spec Ω) is the scheme where we have identified all points in each connected component of T ×S Spec Ω. That is, |π0(T ×S Spec Ω)| π0(T ×S Spec Ω/Spec Ω) = Spec Ω .

Hence π0(T ×S Spec Ω/Spec Ω) is unramiﬁed over Spec Ω and we conclude that π0(T/S) is ´etaleover S.

5.10. The functor f! of a non-étalemorphism. Wise showed in [Wis15] that f ∗ has a left adjoint also when f is flat, locally of finite presentation, and with reduced geometric fibers. Recall Remark 5.5.6, i.e., that there is an equivalence of categories between the category of étalealgebraic spaces over the base and the category of sheaves on the small étalesite on the base. Existence of the algebraic space π0(U/V ) when U → V is flat, locally of finite presentation, and with reduced geometric fibers follows form [Rom11, Théorème 2.5.2].

Theorem 5.10.1 ([Wis15, Theorem 4.5]). Let f : T → S be a morphism of algebraic spaces which is flat, locally of finite presentation, and with reduced geometric ∗ fibers. Then the functor f : Sh(Sét) → Sh(Tét) has a left adjoint. 5.10. THE FUNCTOR f! OF A NON-ETALE´ MORPHISM. 69

Proof in the case of schemes. Suppose that T,S are schemes, ψ : Z → T an ´etale T -scheme, and ϕ: Y → S an ´etale S-scheme. Let R ⇒ Z → π0(Z/S) be a presentation of π0(Z/S). We have canonical bijections

Hom(Sch/S)(Z,Y ) ' Hom(Sch/T )(Z,T ×S Y ) ∗ ' Hom´et(T )(Z, f Y ) , where the last bijection follows from Lemma 4.2.9. Hence it is enough to show that there is a natural bijection

Hom(Sch/S)(Z,Y ) ' Homét(S)(π0(Z/S),Y ) . Let g : Z → Y be an S-morphism. Since Y is étaleover S, it has discrete fibers and hence Z ×Y Spec k is both open and closed in Z ×S Spec k for each Spec k → Y . This implies that the compositions R ⇒ Z → Y agree and hence g factors uniquely through Z/R = π0(Z/S).

CHAPTER 6

The ´etale envelope EX/Y

Similarly as in Chapter 3, one may define sheaves of pointed sets. In [Ryd11] Rydh shows that every unramified morphism X → Y of algebraic spaces (algebraic stacks) has a canonical and universal factorization X,→ EX/Y → Y , where X,→ EX/Y is a closed immersion and EX/Y → Y is étale.

6.1. The sheaf EX/Y Definition 6.1.1. Let f : X → Y be an unramiﬁed morphism of algebraic op spaces. We deﬁne a presheaf EX/Y :(Sch/Y ) → (Set) as follows: For any Y - scheme U, let EX/Y (U) be the set of commutative diagrams

U ×Y X :

V / U where V → U ×Y X is an open immersion and V → U is a closed immersion. If g : U 0 → U is a Y -morphism then for each V as above, then we get a commutative diagram 0 U ×Y X 8

0 0 V ×U U / U and hence EX/Y is indeed a presheaf. For any Y -scheme U we may choose V = ∅ to get a diagram as in Deﬁnition 6.1.1, and hence EX/Y is a presheaf of pointed sets.

Proposition 6.1.2. EX/Y is a sheaf on the the big fpqc site on (Sch/Y ).

Proof. First it is clear that if {Ui → U} is an fpqc covering then the compositions Y Y EX/Y (U) → EX/Y (Ui) ⇒ EX/Y (Ui ×U Uj) 0 ` 0 coincide. Put U := Ui. Furthermore, since g : U → U is surjective we get Q that preimages of distinct sets are distinct, and hence EX/Y (U) → EX/Y (Ui) is injective. Finally, suppose that for every i we have an element

Ui ×Y X :

Vi / Ui in EX/Y (Ui) such that −1 −1 pr1 (Vi) = pr2 (Vj) ∈ EX/Y (Ui ×U Uj)

71 ´ 72 6. THE ETALE ENVELOPE EX/Y for every pair i, j. We want to show that there exists an open subset of U ×Y X with S 0 0 preimage V = Vi ⊆ U ×Y X. Since f : U ×Y X → U ×Y X is fpqc, Proposition 1.1.16 says that U ×Y X has the quotient topology induced by f. Then we only need to show that f −1(f(V )) = V . Clearly V ⊆ f −1(f(V )). Suppose to derive a contradiction that there is an element v0 ∈ f −1(f(V )) \ V . Then there is a v ∈ V 0 0 −1 such that f(v) = f(v ). But then v ∈ Vi for some i and (v, v ) is in pr1 (Vi) but −1 not in pr2 (Vj) for any j. This is a contradiction and hence we conclude that f(V ) is the unique open subset of U ×Y X with preimage V . −1 0 Since f (f(V )) = V we have that V = U ×U f(V ) and since the property ”closed immersion” is ´etalelocal on the base, we get that f(V ) → U is a closed immersion.

The restriction of EX/Y to the big ´etalesite extends uniquely to a sheaf on YEt´ which we also denote by EX/Y . Rydh shows in [Ryd11] that EX/Y is locally constructible. That is, EX/Y is an ´etalealgebraic space over Y .

6.2. The case when f is a monomorphism locally of ﬁnite type

When the morphism f : X → Y is a monomorphism, we may relate EX/Y to the constant sheaf {0, 1} via the functor f!. Note that a base change of a monomorphism is again a monomorphism. Indeed, given a morphism U → Y where U is a scheme, suppose that we have two morphisms T ⇒ U ×Y X such that the compositions T ⇒ U ×Y X → U coincide. Since X → Y is a monomorphism we get that the compositions T ⇒ U ×Y X → X also coincide. Hence, by the universal property of the ﬁber product, we get that the morphisms T ⇒ U ×Y X coincide. Note also that a section s: S → T of a monomorphism g : T → S is always an isomorphism since the equality g ◦ s ◦ g = g implies that s ◦ g = idT . Proposition 6.2.1. Let f : X → Y be a monomorphism locally of ﬁnite type of algebraic spaces. Then there is a natural isomorphism ∼ EX/Y = fc{0, 1}X where fc is the direct image with proper support.

Proof. Let U be an ´etalescheme over Y . We have that U ×Y X is a scheme since f is a monomorphism (see [Sta, Tag 0463] and [Sta, Tag 0418]). The support π0(U×S T ) of a section s ∈ f∗{0, 1}X (U) = {0, 1} is just the union of the connected components where s takes the value 1. Hence we may deﬁne a function

τU : fc{0, 1}X (U) → EX/Y (U) s 7→ supp(s) , where supp(s) is given the open subscheme structure with respect to the open and closed inclusion supp(s) ,→ |U ×Y X|. Any scheme V,→ U ×Y X which is proper over U and which is a union of connected components in U ×Y X deﬁnes a unique section s ∈ fc{0, 1}X (U), namely the one marking the connected components of V with a 1 and the connected components of U ×Y X \ V with a 0. Suppose that we have a diagram

U ×Y X j : p

i V / U representing an element in EX/Y (U). Then j : V → U ×Y X is an open immersion. But since j and p are injective, we have that j(V ) = p−1(i(U)) which is closed in 6.3. THE CASE WHEN f IS ETALE´ 73

U ×Y X since i(V ) is closed in U. Hence V is both open and closed in U ×Y X. That is, V is a union of connected components in U ×Y X. Furthermore, since U ×Y X → U is a monomorphism, so is i and hence i is a closed immersion if and only if i is proper. Hence we get that τU is a bijection and it remains to prove that τU is natural in U. 0 0 A Y -morphism U → U yields a map EX/Y (U) → EX/Y (U ) taking V to the 0 0 scheme V which is the disjoint union of the connected components in U ×Y X map- 0 π0(U×Y X) π0(U ×Y X) ping to the image of V inside U ×Y X. The map {0, 1} → {0, 1} takes the element (a1, . . . , a|π0(U×Y X)|), marking the connected components of 0 π0(U ×Y X) U ×Y X that are in V ⊆ U ×Y X with a 1, to the element in {0, 1} mark- 0 ing the components of U ×Y X that are mapping to components in V ⊆ U ×Y X with a 1. Hence we conclude that we have a natural isomorphism ∼ τ : fc{0, 1}X −→ EX/Y . Corollary 6.2.2. Let f : X → Y be a closed immersion of algebraic spaces. There is a natural isomorphism ∼ EX/Y = f∗{0, 1}X , as sheaves on the big (small) ´etalesite on Y , where {0, 1}X denotes the constant sheaf of the set {0, 1} on the big (small) ´etalesite on X.

6.3. The case when f is ´etale

When f : X → Y is étalewe know that there exists a functor f! : Sh∗(Xét) → ∗ Sh∗(Yét) which is a left adjoint of f . Let EX/Y,ét denote the restriction of EX/Y to the small étalesite on Y . Proposition 6.3.1. Let f : X → Y be an étale morphism of algebraic spaces. There is a natural isomorphism ∼ EX/Y,ét = f!{0, 1}X , where {0, 1}X denotes the constant sheaf of the pointed set {0, 1} on the small étale site on X.

Proof. Let P be the presheaf on Yét defined by _ P(U) = {0, 1}π0(U) ϕ for all étale ψ : U → Y , where the wedge product is over all morphisms ϕ: U → X such that f ◦ ϕ = ψ. Then f!{0, 1}X is the sheafification of P. It is clear that if U → Y factors through f, then we get a section U → U ×Y X π0(U) which is open since U ×Y X → U is unramified. An element in {0, 1} gives a collection of connected components in U and if we take the union we get a subscheme V and a clopen immersion V → U. Since U → U ×Y X is open, so is the restriction to V and hence we get an element in EX/Y,ét(U). If we argue as in the end of the proof of Proposition 6.2.2, we conclude that this gives a natural transformation τ : P → EX/Y,ét. Hence it suffices to show that τy¯ : Py¯ → (EX/Y,ét)y¯ is an isomorphism for each geometric pointy ¯: Spec Ω → Y . Letx ¯1,..., x¯n be the geometric points in X overy ¯. Then by Lemma 4.5.4 and Example 3.6.4, we have that n ∼ _ ∼ _ ∼ Py¯ = ({0, 1}X )x¯i = {0, 1} = |Xy¯| ∪ {∗} .

i=1 |Xy¯| ´ 74 6. THE ETALE ENVELOPE EX/Y

∗ By Lemma 4.5.2 we have that (EX/Y,ét)y¯ = (¯y EX/Y,ét)(Spec Ω). Since EX/Y is locally constructible we have that EX/Y is an étalealgebraic space over Y and by Lemma 5.6.1 we get that ∗ ∼ y¯ EX/Y,ét = π∗(Spec Ω ×Y EX/Y ) , where π : (Spec Ω)Et´ → (Spec Ω)ét is the restriction morphism. Since id: Spec Ω → Spec Ω is étale,this implies that ∼ (EX/Y,ét)y¯ = EX/Y (Spec Ω) = |Xy¯| ∪ {∅} . 6.4. Final remark Propositions 6.2.1 and 6.3.1 gives us reason to believe that that one may extend this construction to the general case. Hence we have the following conjecture: Conjecture. Let X and Y be algebraic spaces and let f : X → Y be a morphism locally of finite type. There exists a functor

f# : Sh∗(X´et) → Sh∗(Y´et) of sheaves of pointed sets such that:

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