FREIE UNIVERSITÄT BERLIN ÉCOLE NORMALE SUPÉRIEUREDE RENNES

Low-degree covers in algebraic geometry

Gabriel LEPETIT

Thesis of first year of Master, written under the supervision of Fabio TONINI

MAGISTÈRE 2 2015-2016 This document has been redacted from May 2nd to July 1st 2016 in the :

Freie Universität Berlin - Mathematisches Institut Arnimallee 3 14195 Berlin

My advisor was Fabio TONINI (http://www.mi.fu-berlin.de/users/tonini/), re- searcher in the Arithmetische Geometrie team directed by Hélène ESNAULT (http://www. mi.fu-berlin.de/users/esnault/).

1 Contents

Introduction 3

1 Sheaves and schemes 4 1.1 Sheaves of rings ...... 4 1.2 Schemes ...... 7 1.3 Quasi-coherent sheaves of modules ...... 9 1.4 Locally free sheaves ...... 11 1.5 Relative spectrum of a quasi-coherent sheaf of algebras ...... 13 1.6 Symmetric algebra and exterior powers of algebra ...... 18

2 Double covers 20 2.1 Basic facts about covers ...... 20 2.2 Double covers via line bundles ...... 23

3 Triple covers 28 3.1 The local case ...... 29 3.2 The global case ...... 31 3.2.1 Equivalence between Cov3(Y ) and e3(Y ) ...... 31 D 3.2.2 Equivalence between e3(Y ) and 3(Y ) ...... 34 D D Bibliography 38

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Acknowledgements

First of all, I would like to address special thanks to my advisor, Fabio Tonini, who patiently answered my questions and enthusiastically introduced me to the world of algebraic geom- etry. I thank also Hélène Esnault who accepted to receive me in Berlin, and all the members of the team and guests with whom I had the chance to have fruitful discussions, in particular Lei Zhang, Marta Pieropan and Marco d’Addezio. I am also grateful to Lars Kindler and his wonderful coffee machine which deliciously boosted my productivity.

2 INTRODUCTION

This thesis aims to define a simple notion of cover in algebraic geometry that would rea- sonably look like the one we know in topology and that is central in differential geometry and algebraic topology. In this context, if Y is a topological space, a cover of Y is a continu- ous map f : X Y such that for all y Y , there exists an open neighbourhood V of y and a non-empty set→I such that f 1 V ∈F U and f is a homeomorphism from U onto V . y − ( ) = i Ui i i I y | We know that if Y is connected, and if the∈ fibers are finite sets, then they all have the same number of elements which is the common cardinality of the sets I y , y Y . This is what we call an unramified cover : the copies of Y produced by the cover don’t∈ cross. We would like to have a similar algebraic notion, that would allow ramification, as in the example 2.1.5 : the cardinality of the fibers would still be the same, but only if we count the elements with multiplicity.

The notion of cover plays an important role in the current research in algebraic geometry, for several reasons. First, it is a convenient way to build new spaces – for example algebraic varieties – out of the ones we already know. Moreover, in the theory of moduli spaces, the study of the Hurwitz spaces (see [12]), which are parametrizing covers of curves, have been an active field of research for the last years. The Hurwitz spaces are not simply studied as sets of covers, but as schemes whose closed points are the covers of curves. The covers are often studied together with group actions, especially in the theory of Galois covers (cf [11]). For instance, Z/2Z acts on the double covers, but this situation does not recur for higher-degree covers, as Z/3Z does not act on every 3-cover. A cover of a scheme Y is simply defined as an affine morphism of schemes f : X Y such that f X is a locally free Y -algebra of finite rank (cf 2.1.1). We will first present→ briefly in the∗O first chapter the prerequisitesO of algebraic geometry necessary to understand the study of covers, in particular the quasi-coherent sheaves of algebras. In the second and third chapters, we give a full description of the categories of covers of degree 2 and 3. The goal is to extract the essential data of the multiplication map, and the trace map for sheaves of algebras (proposition 2.1.6) play a central role in this study. It happens that the covers of degree 1 are the isomorphisms (example 2.1.4), the ones of degree 2 correspond classically to the pairs ( , σ) where is a locally free sheaf of rank 1 and σ is a morphism from to Y (theoremL 2.2.1),L and the 3-covers, as described in L ⊗ L O 3 2 [9], are similar to the ( , δ) where is a locally free Y -module of rank 2 and δ : S Λ is a morphism (theoremE 3.0.7). E O E → E

For higher degree, it is not possible anymore to have a general description of the covers with locally free sheaves and maps without further relations, but G. CASNATI and T. EKDAHL ([3] and [2]) developed the theory of Gorenstein covers, leading to a particular study of the 4 and 5-covers, while in [7], an analysis of the quadruple covers of algebraic varieties was done. Rita PARDINI ([10]) studied the triple covers in characteristic 3, a case where our method doesn’t work. The work of M. BOLOGNESI and A. VISTOLI ([1]) is another relevant article involving the triple covers.

3 Chapter 1

Sheaves and schemes

In all the thesis, I will assume that the basic facts of commutative algebra are known, and use freely the vocabulary of the theory of categories. The main results we use are summed up in [6] pp 541-545 (categories), [8], chapter 1 (tensor product and localization). The main reference for this chapter is the book of Qing Liu ([8]), chapter 2.

1.1 Sheaves of rings

Definition 1.1.1 A presheaf of sets on the topological space X is a contravariant functor from the category of the open subsets of X into the category of sets. More concretely, itF is the data of a set (U), for every U open subset of X , and, for V U open subsets, restriction F ⊂ maps ρUV : (U) (V ) satisfying the following axioms : F → F For all W V U, ρUW = ρVW ρUV . • ⊂ ⊂ ◦ ρUU = idU . •

The elements of (X ) are called global sections. Example 1.1.2. TheF most intuitive example of a presheaf is the presheaf of rings on X 0 0 : U (U, R), endowed with the restrictions ρUV : f f V , where (U, R) is the set | ofF the7→ continuous C fonctions from U to R. 7→ C Because of this example, we denote, for an arbitrary presheaf, ρUV (s) = s V . | Definition 1.1.3 A sheaf on X is a presheaf on X satisfying the following axioms : F S (Uniqueness)For U open of X and U = Ui open covering of U, if s, t (U) are • i I ∈ F such that i I, s t , then s t. ∈ Ui = Ui = ∀ ∈ | | (Glueing) For U open of X and U = i I Ui open covering of U, if for every i I, • s U and i, j I, s s ∪ ∈ , then there exists a s U such that∈ i ( i) i Ui Uj = j Ui Uj ( ) i∈ FI, s s . ∀ ∈ | ∩ | ∩ ∈ F Ui = i ∀ ∈ |

Definition 1.1.4

α(U) A morphism of presheaves of sets α : is a family of morphisms (U) (U) compatible with the restrictions : for everyF →V G U X , the following diagramF commutes.−→ G ⊂ ⊂

4 α(U) (U) (U) F G ρUVF ρUVG α(V ) (V ) (V ) F G If and are sheaves, we say that α is a morphism of sheaves. F G Remark. We can also consider presheaves of groups, abelian groups, rings, A-modules or A-algebras where A is a fixed ring. In each case, the restriction maps are morphisms in the considered category, as well as the morphisms of presheaves. We see that a sheaf if a way to pass from the local data to the global one. Definition 1.1.5 Let be a sheaf on X and x X . The stalk of at x is F ∈ F x = lim (U) F x U F −→∈ where the limit is taken on the inductive system (for the inclusion) of the open subsets of X containing x.

Concretely, in x , any element is defined on a open neighbourhood of x and two ele- ments are identifiedF if they coincide on an open neighbourhood of x.

We can also take the stalk at x of a morphism α : , by setting αx : sx (α(U)(s))x F → G → if s (U). αx is a morphism. Proposition∈ F 1.1.6 Let be a sheaf on X , U X an open subset, and s, t (U). Then s = t x F ⊂ ∈ F ⇔ ∀ ∈ U, sx = t x . Proposition 1.1.7

If α : is a morphism of sheaves on X , then α is an isomorphism if and only if αx is an isomorphismF → G for all x X . ∈ The following proposition allows us to define a sheaf starting with a presheaf, and this operation, called sheafification, preserve the stalks and the morphisms. Proposition 1.1.8 (sheafification) Let be a presheaf on X . There exists a unique sheaf † on X , together with a mor- phismFθ : † satisfying the following universal propertyF : For everyF →u : F morphism, where is a sheaf, there exists a unique morphism † F → G G ue : such that u = ue θ. F → G ◦ † Moreover, for all x X , θx is an isomorphism between x and x . ∈ F F If Psh(X ) (resp. Sh(X )) is the category of the presheaves (resp. sheaves) on X , sheafifi- † cation defines a functor : Psh(X ) Sh(X ) . If u is a morphism of presheaves from · −→ † to , u† is the unique morphismeF making 7−→ F this diagram commute : F G u

θ F Gθ F u† G † † F G 5 Let us remark that the family of morphisms (θ ) define a natural transformation between i † i X X F idPsh(X ) and where : Sh( ) Psh( ) is the inclusion functor. We can keep◦ · in mind for the rest→ of the thesis that, since the morphisms θ preserve the stalks, it is not a problem to define a morphism between presheaves if we want to study the sheafified morphism.

We will now explain how it is possible to define a sheaf on a basis of open subsets (see [5], pp. 25-28, for more details). Let X be a topological space, and a basis of open subsets of X (i.e. for all open U of B S X , there exist elements Ui of such that U = i Ui). A -presheaf of sets (resp.B groups, rings) on X is a contravariant functor from , seen as a subcategoryB of the category of the open subsets of X , to the category of setsB (resp. groups, rings). Definition 1.1.9 A -sheaf on X is a -presheaf satisfying the two following axioms : B B F S (Uniqueness) If U is an element of such that U = Ui, Ui , and s (U) • B i I ∈ B ∈ F is such that s 0 i I, then s 0. ∈ Ui = = | ∀ ∈ S (Glueing) If U is an element of such that U = Ui, Ui , if for every i I, • B i I ∈ B ∈ ∈ si (Ui) and i, j I, W such that W Ui Uj, we have si W = sj W , then there∈ F exists a s ∀ ∈U such∀ ∈ that B i I, s s⊂. ∩ | | ( ) Ui = i ∈ F ∀ ∈ |

We can define stalks of a -presheaf the same way we did with presheaves, as well as morphisms of -presheaves.B Proposition 1.1.10B Let 0 be a -sheaf on X . Then there exists a unique (up to isomorphism) sheaf on 0 X suchF that B(U) = (U) for every U in . The sheaf is defined by F F F B F V X , (V ) = lim (U) ∀ ⊂ F U F ←−U∈BV ⊂ 0 Moreover, if x X , then x = x . ∈ F F Remark. Thanks to the universal property of the projective limit, we can also extend a morphism of -sheaves in a unique way in a morphism between the corresponding sheaves. Moreover, theB stalks of these two morphisms will coincide. Therefore, this construction gives an equivalence between the categories of the - sheaves on X and of the sheaves on X . B

If f : X Y is a continuous map, the inverse and direct image functors are used to pass from a sheaf→ on X to a sheaf on Y .

If ( , ρ) is a sheaf on X , the direct image f of is the sheaf defined by f (V ) = 1 ∗ ∗ • Ff V V Y F F 1 1 F ( − ( )) for all with the restrictions µVV = ρf (V )f (V ). F ⊂ 0 − − 0 If α : is a morphism of sheaves on X , then f α : f f is the morphism F → G 1 ∗ ∗F → ∗G of sheaves defined by (f α)(V ) = α(f − (V )) for all V Y . ∗ ⊂

6 We have a morphism : (f )f (x) x but it isn’t injective in general. But for ∗F s −→ Fs f (x) x instance if f induces a homeomorphism7−→ from X into its image, then this morphism will be an isomorphism.

1 If is a sheaf on Y , the inverse image f − of is the sheaf defined by : • G G G 1
U X , f − (U) = lim (U) ∀ ⊂ G f (U) V G −→⊂ We define the restrictions using the action of the inductive limits on the morphisms, and likewise taking the inverse image of a morphism of sheaves makes sense.

x X f 1 i V Y What we should keep in mind is that if , then ( − )x = f (x), and if : , 1 ∈ G G → is an inclusion, then i− is the restricted sheaf V : V 0 V (V 0). G G| ⊂ 7→ G

Finally, we will quite often use exact sequences of sheaves. We should underline the fact that if γ is a morphism of sheaves of groups, U Im (γ(U)) is only a presheaf and not a sheaf. That is why Im γ is defined as the sheafification7→ of this presheaf. Besides, U ker(γ(U)) is a sheaf called ker γ. Definition7→ 1.1.11

α β A sequence of morphisms of sheaves is said to be exact if ker β = Im α. F −→ G −→ H Proposition 1.1.12 A sequence of sheaves on X is exact if and only if for all x X , the

sequence xF →x G →x His exact. ∈ F → G → H

1.2 Schemes Definition 1.2.1

A locally ringed space is a pair (X , X ), where X is a topological space and X is a sheaf O O of rings on X such that for all x X , X ,x := ( X )x is a local ring. If its maximal ideal is ∈ O O mx , then the field k(x) := X ,x /mx is called the residue field of X at x. O Definition 1.2.2 # A morphism of locally ringed spaces f : X Y is f = (f , f ) where f is a continuous # → map between X and Y , and f : Y f X is a morphism of sheaves such that for every can x X f # f O → ∗O , x = Y,f (x) ( X )f (x) X ,x is a local ring morphism. ∈ O −→ ∗O −→ O

We will omit the X in the notation for simplicity. Remark. If f :OX Y and g : Y Z are morphisms of locally ringed spaces, then # # f g is•(f g, f g→ f ). → ◦ ◦ ∗ ◦ If U is an open subset of X and i : (U, X U ) (X , X ) is the inclusion morphism, with • O | → O # V X , i (V ) : X (V ) (i X U )(V ) = X (U V ) ∗ | ∀ ⊂ O s −→ s UOV O ∩ 7−→ | ∩

then the restriction of a morphism f : (X , X ) Y to U is f U = f i. O → | ◦ 7 Let A be a ring. We want to put a structure of locally ringed space on X = Spec(A), set of the prime ideals of A. Let us recall briefly that we can define a topology on X , called the Zariski topology, whose closed subsets are the V (I) = p X : I p , I ideal of A. The open subsets D(f ) = X V (f A), f A are called principal open{ ∈ subsets⊂ and} form a basis of open subsets of X . \ ∈ B We set X (D(f )) = Af , localization of A at f . This definition makes sense, since if O D(f ) = D(g), Af Ag . If D(f g) D(f ), the restriction map is given by Af Af g = (Af )g . The association 'X is a -sheaf and⊂ therefore can be extended to a sheaf on→X . We call X the structural sheafO of XB. O

Furthermore, if x = p X , then X ,p = Ap, localization of A by A p. See [8], p.17, for a more∈ detailedO proof. \ Definition 1.2.3

An affine scheme is a locally ringed topological space that is isomorphic to some • X = Spec A endowed with his structural sheaf.

A scheme is a locally ringed space (X , X ) such that there exists a open covering • X S U of X such that for all i I, UO , is an affine scheme. = i ( i X Ui ) i I ∈ O | ∈

If X is a scheme, an open subset U such that the open subscheme (U, X U ) is affine is said to be open affine. O |

Finally, we will study the morphisms of schemes, i.e. the morphisms of locally ringed spaces between schemes. Definition 1.2.4 An open embedding is a morphism i : X Y such that i is a topological open embedding i.e. X→ i X x X i# ( induces an homeomorphism from onto ( ) open), and , x : Y,i(x) X ,x is an isomorphism. ∀ ∈ O → O

Remark. A morphism i : X Y is an open embedding if and only if there exists V open → subset of Y such that i induces an isomorphism of schemes between (X , X ) and (V, Y V ). O O | Proposition 1.2.5

If u : A B is a morphism of rings, then there exists a morphism fu : Spec B Spec A → # → such that fu (Spec A) = u.

Theorem 1.2.6 Let Y be an affine scheme. Then for every scheme X , there exists a canonical bijection

ψX : Homsch.(X , Y ) Homrings( Y (Y ), X (X )) ϕ −→ ϕ(X ) O O 7−→

This family of maps defines a natural transformation between the functors X 7→ Hom(X , Y ) and X Hom( Y (Y ), X (X )) : if α : Z X is a morphism of schemes, then we have 7→ O O →

8 ψX Hom(X , Y ) Hom( Y (Y ), X (X ))

O # O α α (Z) ·◦ ◦· ψZ Hom(Z, Y ) Hom( Y (Y ), Z (Z)) O O The key argument of the proof is to use the previous proposition when X is an affine scheme, and to reduce to this case by taking an affine covering when X is an arbitrary scheme. Remark. In particular there is an anti-equivalence of categories between the category of the affine schemes and the category of the rings. Definition 1.2.7 If X and Y are schemes, Y is said to be a X -scheme if it is given a morphism α : Y X , called structural morphism. If X = Spec A is affine, we also say that Y is a A-scheme.→ A morphism f : Y Y 0 of X -schemes is a morphism of schemes such that : → f Y Y 0 α α0 X

commutes. The set of the morphism of X -schemes between Y and Y 0 is denoted by HomX (Y, Y 0). It is easy to see with the theorem 1.2.6, that if A is a ring, then the category of affine schemes over A is equivalent to the category of A-algebras.

1.3 Quasi-coherent sheaves of modules

For this section, we refer to [8], chapter 5, pp. 158-163. Definition 1.3.1

Let X be a scheme. A X -module is a sheaf on the topological space X such that for all O F U X open, (U) is a X (U)-module and if V U, the restriction ρUV is a morphism ⊂ F O ⊂ of X (U)-modules for the structure of X (U)-module on (V ) defined by a.s = a V .s. O O F |

Remark. A morphism of X -modules is simply a morphism of sheaves ϕ such that for every O U X , ϕ(U) is a morphism of X (U)-modules. ⊂ O As for the modules over a ring, we can define the tensor product and the direct sum of sheaves of X -modules : O If and are two -modules, we define a presheaf : U U U X ( ) X (U) ( ) • F G O H 7→ F ⊗O G endowed, for V U with the restrictions ρUVF ρUVG : s t s V t V . Clearly, if x X , the stalk of⊂ at x is . ⊗ ⊗ 7→ | ⊗ | x = x X ,x x ∈ H H F ⊗O G The tensor product of and is the sheafification of . According to the X proposition 1.1.8, weF have ⊗O G F G H

( X )x = x = x X ,x x F ⊗O G H F ⊗O G Using the universal property of the sheafification, we will often rather give the de- scription of morphisms on than define them on . X H F ⊗O G 9 L If ( i)i I is a family of X -modules, then the X -module i is the sheafification of • F ∈ O O i I F the presheaf defined by ∈

  M M U X , i (U) = i(U) ∀ ⊂ i I F i I F ∈ ∈ the direct sum on the right being a direct sum of X (U) modules. O Definition 1.3.2

Let X be a scheme and be a X -module. We say that is quasi-coherent if for all x X , there exists an openF neighbourhoodO U of x and an exactF sequence of X -modules ∈ O (J) (I) X U X U [U 0 O | → O | → F → This concept is very convenient, as there is an equivalence between the category of quasi-coherent Spec A-modules and the category of A-modules, given by the operation of sheafification ofO a module. Proposition 1.3.3

Let A be a ring, and M be a A-module. Then, if X = Spec A, we can define a X -module O Me by setting, for every principal open subset D(f ) X , Me (D(f )) = Mf ⊂ Moreover, if x = p Spec A, then Me x = Mp. ∈ Remark. The operation of sheafification is similar to the construction of the affine schemes• p. 8.

This transformation is compatible with the direct sum and the tensor product : • M M ãM M and M N M N i = fi åA = e X e i I i I ⊗ ⊗O ∈ ∈ Proposition 1.3.4 With the same notations, a sequence of A-modules L M N is exact if and only if → → the sequence of X -modules eL Me Ne is exact. Thus if M is a A-module, then Me is quasi-coherent. O → →

Theorem 1.3.5

If X is a scheme and is a X -module, then is quasi-coherent if and only if for all U F O F open affine, U = á(U). F| F Corollary 1.3.6 Let QCoh(Y ) be the category of quasi-coherent sheaves on the scheme Y . If Y = Spec A is affine, then there is a equivalence of categories between QCoh(Y ) and ModA given by the functors ∆ : (Y ) and Λ : M Me . F 7→ F 7→

10 1.4 Locally free sheaves

Let X be a scheme. Definition 1.4.1

n A free sheaf of rank n on X is a sheaf on X that is isomorphic to X . • F O A sheaf is said to be a locally free sheaf of rank n if there exists a covering • X S XFof X by open subsets such that for every i I, is free of rank n. = i Xi i I ∈ F| ∈ S A sheaf is locally free of finite rank if there exists a covering X = X i together • F i I with integers n such that is locally free of rank n . ∈ i Xi i F|

A locally free sheaf of rank 1 is called an invertible sheaf, or a line bundle.

Remark. If is a free sheaf of rank n on X , then if (α1,..., αn) is a basis of the X (X )- F O module (X ), for every U X , (α1 U ,..., αn U ) is a basis of (U). We often say that (αi) is a basis ofF . ⊂ | | F An immediateF consequence of this fact is that we can check the equality of two mor- phisms , free of finite rank, on (X ). Moreover,F → G a morphismF α between twoF free sheaves on X is entirely determined by the matrix of α(X ), also denoted by M(α). Definition 1.4.2

Let be a quasi-coherent X -module. It is said to be finitely presented if for every U F O open affine subset of X , (U) is a finitely presented X (U)-module. F O Proposition 1.4.3

Let be a finitely presented quasi-coherent sheaf on X . Then, if n N∗, is locally free ofF rank n if and only if x X , x is a free X ,x -module of rank n∈. F ∀ ∈ F O Lemma 1.4.4 Let A be a ring, M, N two finitely presented A-modules and p Spec A. If u : M N is up ∈ uf → such that Mp Np for p Spec A, then there exists f A p such that Mf Nf . ' ∈ ∈ \ ' Proof. First, let us notice that coker u = N/Im u is finitely generated since N is of finite type. u The exact sequence 0 ker u M N coker u 0 yields, by flatness of the → u → → → → localization, 0 ker up Mp Np coker up 0. → → → → → But, by hypothesis, coker up = 0, so x coker u, fx A p : fx x = 0. Taking generators x ,..., x of coker u and setting f f∀ ...∈f , we obtain∃ ∈ \x coker u, f x 0, f A p. 1 n = x1 xn = ∀ ∈ ∈ \ Finally, coker uf = (coker u)f = 0 implies that uf is surjective. Thanks to [13, Tag 0517], also ker u is finitely generated and likewise, there exists f 0 ∈ A p such that ker uf = 0. Therefore, uf f : Mf f Nf f is an isomorphism. \ 0 0 0 → 0

Proof (of 1.4.3). Let us assume that x is free of rank n for all x X . We can assume that X is affine since we want to prove a localF statement. ∈ ψ We have Me , where M is a finitely presented X (X )-module. Therefore we have an F ' O isomorphism x Me x . F '

11 n n Composing with an isomorphism between x and X ,x , we obtain ϕx : X ,x Me x . Since n F O O → X is free, and replacing X by an open affine V if necessary, we can find γ morphism from O n X to Me such that γx = ϕx . O n Thus, γx : X ,x Mp and γx is the localization at x of γ(X ) as X is affine. According O ' n to the lemma, there exists f X (X ) such that γ(X )f : X (X )f Mf is an isomorphism. Therefore, as D f Spec ∈X O is an open affine, M O n → . ( ) X ( )f e D(f ) X D(f ) ' O | → O |

Remark. If is a locally free sheaf of finite rank, then we call rank of at x X the integer rankF rank . The function x rank is locally constant.F Therefore,∈ if x = X ,x x x X is connected,F it is constantO F of value d, so 7−→is locally freeF of rank d. F Definition 1.4.5

Let , be two sheaves (of abelian groups, rings, of X -modules...) on X . The sheaf HomF( G, ) is defined by O F G

U X , Hom( , )(U) = Hom( U , U ) ∀ ⊂ F G F| G|

In particular, if is a X -module, the dual sheaf of is ∨ := Hom( , X ). F O F F F O Proposition 1.4.6

Let , 0 be locally free sheaves of finite rank. Then Hom( , 0) is a locally free sheaf of finiteE rank.E E E

Theorem 1.4.7 If Y is a scheme and is a locally free sheaf on Y of finite rank, and are quasi- coherent sheaves on YH, then we have a canonical bijection F G

u Hom( , ) H Hom( , ∨) F ⊗ H G −→ F G ⊗ H

Proof. First of all, the canonical morphism p : ∨ End( ) turns out to H ⊗s Hϕ −→ t ϕH(t)s be an isomorphism, since is locally free of finite rank⊗ and7−→ the corresponding7→ map for a free module on a ring isH classically an isomorphism. So, composing with the structural morphism ι : Y End( ) , we define : O1 −→ id H 7−→ H u Y ∨

ι Ý O H ⊗ Hp End( ) H

We also have the evaluation morphism ev : ∨ Y . H ⊗s Hϕ −→ Oϕ(s) u ⊗ α7−→id If α : , we set u (α) : ∨ ⊗ ∨. H F ⊗ H → G F −→ F ⊗ H ⊗ H −→ Gβ ⊗id H id ev ⊗ Given a morphism β : ∨, we set v (β) = ∨ ⊗ and we want to prove thatFv →is G the ⊗ H inverse of u H. F ⊗ H −→ G ⊗ H ⊗ H −→ G H H Given α : and β = u (α), then, if U is an open subset of Y and u(U)(1) = P F ⊗ H → G H j hj ϕj, hj , ϕj ∨, γ = v (β) is, on U : ⊗ ∈ H ∈ H H

12 id u id α id id id ev ⊗ ⊗ ⊗ ⊗ ⊗ ∨ ∨ F ⊗ H F ⊗ H ⊗ H ⊗ H G ⊗ H ⊗ H G P P P f h j f hj ϕj h j α(f hj) ϕj h j ϕj(h)α(f hj) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ P We want to show that for every f (U), h (U), j ϕj(h)α(f hj) = α(f h). This statement can be checked on the∈ basis F of open∈ H subsets constituted⊗ by the U ⊗open B affine such that U is free. To simplify, let us assume that U = Y , and let us fix a basis v YH| v Y ( j)j=1...r of ( ). Then the dual family ( ∗j )j=1...r is a basis of ∨( ). H H € Š € Š r p Y P v v h P v h v h u Y P v v As ( ) j j ∗j = j ∗j ( ) j = = id (Y ), we have ( )(1) = j ∗j . ⊗ 7→ H j=1 ⊗ Consequently, X f (Y ), i 1; r , γ(Y )(vi) = α(Y )(f vj)v∗j (vi) = α(Y )(f vi) ∀ ∈ F ∀ ∈ J K j ⊗ ⊗ and by linearity, we have the desired equality and v u = id. H ◦ H We show that u v = id using a similar method. H ◦ H

1.5 Relative spectrum of a quasi-coherent sheaf of alge- bras Definition 1.5.1

Let Y be a scheme. A sheaf of algebras on Y , or Y -algebra, is a Y -module, where A O O (U) is endowed with a structure of ring that is compatible with the structure of Y (U)- Amodule. O

There are several useful equivalent ways to see a Y -algebra : O It is a sheaf of rings together with a morphism of sheaves, called structural mor- • A phism, α : Y . The structure of Y (U)-module on (U) is given by a O → A O A ∀ ∈ Y (U), f (U), a.f = α(a)f . O ∀ ∈ A It is a Y -module , endowed with a multiplication morphism of Y -modules m : • O whichA has a neutral element 1 and satisfies the commutativityO and Aassociativity ⊗ A → A diagrams (the swap map is just s A t t s): ⊗ 7→ ⊗

swap id ( ) ( ) A ⊗ A ⊗ A A ⊗ A ⊗ A A ⊗ A A ⊗ A m id id m m m ⊗ ⊗ A ⊗m A Am ⊗ A and A A

Remark. Seeing the commutativity and associativity of m as the commutativity such dia- grams makes clear the fact that these properties are equivalent to the corresponding local diagrams (i.e. on the stalks).

13 Definition 1.5.2

A morphism of Y -algebras φ : is, in a equivalent way : O A → B φ φ ⊗

A ⊗m A B ⊗n B φ A morphism φ of Y -modules such that φ(1 ) = 1 and • O A B A B φ A B α β A morphism of sheaves of rings such that Y O • A quasi-coherent sheaf of algebras is simply a sheaf of algebras that is quasi-coherent as a sheaf of modules. The category of quasi-coherent Y -algebras is denoted by QAlg(Y ). O Definition 1.5.3 Let X and Y be two schemes. A morphism f : X Y is called affine if for every V open 1 → affine of Y , f − (V ) is affine. We call Aff(Y ) the category of affine morphisms f : X Y , X scheme, in which a → morphism between f : X Y and f 0 : X 0 Y is u : X X 0 such that f 0 u = f . → → → ◦ Proposition 1.5.4 Let f : X Y be a morphism of schemes. The following are equivalent : → 1. The map f is affine.

S 1 2. There exists a covering Y = Yi of Y such that i I, f − (Yi) is affine. i I ∀ ∈ ∈

Proof. Omitted, see [14], p. 208-210

Example 1.5.5. A morphism between affine schemes is affine.

We will eventually introduce the notion of relative spectrum, which play for a quasi- coherent algebra a similar role as Spec A for a ring A. This construction generalises the statement of the theoremA 1.2.6 : if A is a ring, B a A-algebra and X a A-scheme, there is a canonical bijection HomA(X , Spec B) HomA algebras(B, X (X )). Moreover, the categories of the A-algebras is equivalent to the' category− of affineO schemes over Spec A ; here, the categories of the quasi coherent Y -algebras and of the affine morphisms X Y will be equivalent through the Spec functor.O → Theorem 1.5.6

Let Y be a scheme and be a quasi-coherent Y -algebra. We can find a Y -scheme Spec , with structural morphismA f such that f Ois affine and f . Y SpecY = Moreover,A if π : Z Y is a Y -scheme, ∗O A A → : Hom Z, Spec Hom , ϕZ Y ( Y ) Y ( π Z ) Au −→ f u# O A ∗O 7−→ ∗ is a functorial bijection.

14 This means that if κ : W Y is a Y -scheme and α : W Z is a morphism of Y -schemes, then the following→ diagram commutes : →

ϕ Z Hom , π HomY (Z, Spec Y ) Y ( Z ) A O A ∗O v v α β π α# β 7→ ◦ 7→ ∗ ◦ ϕ W Hom , κ HomY (W, Spec Y ) Y ( W ) A O A ∗O

See [14], page 444 for less informations ; another construction can be found in [4], p. 41. Lemma 1.5.7 (Glueing schemes)

Let (X i)i I a family of schemes. If, for all i I, there exist a family (X i j)j I such that ∈ ∈ ∈ X i j is an open subset of X i, and for all i, j I, an isomorphism of schemes fi j : X i j ' X ji such that : ∈ →

1. f id ii = Xi

2. i, j, k I, fi j(X i j X ik) = X ji X jk ∀ ∈ ∩ ∩ 3. (cocycle condition) The following diagram commutes :

fik X i j X ik X ki X k j ∩ ∩

fi j f jk X ji X jk ∩

Then there exists a unique scheme X together with open embeddings ψi : X i X S → such that, for i, j I, ψi = ψj fi j on X i j, and X = ψ(X i). ∈ ◦ i

Proof. Omitted, see [8], pp. 49-50.

Remark. If φi : X i Y are Y -schemes, and the fi j is compatible with the φi, then the glueing X of the X i →is a Y -scheme and its structural morphism φ satisfies, for every i I, . ∈ φ Xi = φi | In the next proof, we will use some basic facts about the fibered products of schemes that can be found in the third chapter [8] (pp. 78-85). Notation : A commutative diagram of the form above is called cartesian if Z satisfies the universal property of the fibered product. We indicate it with the square in the center of the picture.

Z X ƒ

X 0 Y

Proof (of 1.5.6). Let I be the set of open affine of Y . For i I, we denote by Ui the associated ∈ open affine. Let us set X i = Spec (Ui), and fi : X i Ui the morphism induced by the A → 1 structural morphism Y (Ui) (Ui). We define, for all j I, X i j = fi− (Ui Uj). If U U , we haveO the→ following A cartesian diagram, because∈ U ∩ U k i ( k) = ( i) Y (Ui ) ⊂ A A ⊗O Y (Uk) (it follows from [8], proposition 1.14 (b) p. 163 and proposition 1.12 (b) p. 162). O 15 αki X k X i

fk ƒ fi U inc U k i (1.1)

As open embeddings are stable by base change, αki is an open embedding. Moreover, 1 αki(X k) = fi− (Uk). As αki is compatible with fk and fi, it is a morphism of Y -schemes.

Let i, j I. We want to define a family of isomorphisms βi j : X i j X ji. Let us first∈ notice that if k I is such that Uk Ui Uj, then the→ diagram 1.1 factors through X i j – and likewise through∈ X ji : ⊂ ∩

αki

inc. X k X i j X i αk,i,j fk fi

Uk Ui Uj Ui ∩

It is then easy to see that the family of the Im αk,i,j when Uk Ui Uj is a covering of X i j ; therefore, we can define βi j by glueing the morphisms : ⊂ ∩

βi j X i j X ji

αk,i,j αk,j,i X k

1 1 under the condition that if U , U U U , α α 1 α α 1 , k l i j ( k,j,i −k,i,j) fi− (Uk Ul ) = ( l,j,i −l,i,j) fi− (Uk Ul ) ⊂ ∩1 ◦ 1 | ∩ ◦ | ∩ since the image of αk,i,j (resp. αl,i,j) is fi− (Uk) (resp. fi− (Ul )). Let us first consider the case where Ul Uk. We have the following diagram : ⊂

αlk αk,i,j X l X k X i j fl ƒ fk ƒ Ul Uk Ui Uj ∩ the whole rectangle being cartesian. Therefore, by uniqueness of the fibered product, we have αl,i,j = αk,i,j αlk, and likewise for X ji. Considering X l as an open subset of X k, and passing to the inverse,◦ we get α 1 α 1 and on the other hand α α which − 1 = −l,i,j k,j,i Xl = l,j,i k,i,j fi− (Ul ) gives the result. | |

Now, in the general case, it suffices to show that for every Ur Uk Ul , the maps ⊂ ∩

βi j βi j X i j X ji X i j X ji

αk,i,j Xr αk,j,i Xr αl,i,j Xr αl,j,i Xr | X | | X | r and r coincide. But according to the first case, , with the same result for so αk,i,j X r = αr,i,j = αl,i,j X r αr,j,i we have the wished equality. | |

16 The reason why the cocycle condition is true can be seen on this diagram (with Ui jk = Ui Uj Uk): ∩ ∩

X v αvi αvk

β 1 ik 1 fi− (Ui jk) fk− (Ui jk) βi j βjk αvi αvk 1 X v f j− (Ui jk) X v αv j αv j

fi Furthermore, the structural morphism f : X Y is obtained by glueing the X i Ui Y , so f 1 U f 1 U X and f is affine. Likewise,→ for every i I, f U → X→ − ( i) = i− ( i) = i X ( i) = Xi ( i) = ∈ ∗O O (Ui) so as the open affine form a basis of open subsets of Y , f X = . A ∗O A Let us prove the second part of the theorem. The commutativity of the diagram is clear.

If π : Z Y is a Y -scheme, we want to show that ϕZ is a bijection. Let β : π Z → # A → ∗O is a morphism ; we are looking for a u : Z X = Spec Y such that f u = β. → A 1 ∗ For every i I, β(Ui) yields, thanks to 1.2.6 a morphism υi : π− (Ui) Spec (Ui) = ∈ 1 → A X i, which gives by composition with X i X a morphism ui : π− (Ui) X . Moreover, since the following diagram commutes, for U→k Ui, → ⊂ β U ( i ) 1 (Ui) Z (π− (Ui)) A O β U ( k) 1 (Uk) Z (π− (Uk)) A O we have υ 1 k X π− (Uk) k X

υ 1 i X π− (Ui) i X

and thus u 1 u , which, similarly to the construction of the β above, proves that i π (Uk) = k i j | − we can glue the ui into a morphism of Y -schemes u : Z X . → We define ψZ (β) = u. If u : Z X is a morphism of Y -schemes, let us take β = ϕZ (u) = υ # → 1 0i inclusion f u and u ψZ ϕZ u . We know that u 1 π Ui X i where υ is 0 = ( )( ) 0π (Ui ) = − ( ) 0i ∗ ◦ | − −→ 1 −→ 1 induced by β(Ui). Moreover, as u is a morphism of Y -schemes, u(π− (Ui)) f − (Ui) = X i υ 1 i inclusion # ⊂ and u 1 can be decomposed as π U X X . Since u X β U , we have π (Ui ) − ( i) i ( i) = ( i) | − −→ −→ υi = υ0i and finally u = u0. Furthermore, if β : π Z is a morphism of Y -algebra, β 0 = (ϕZ ψZ )(β) is A → ∗O O # 1 # ◦ actually β. Indeed, if u = ψZ (β), then for all i I, β 0(Ui) = u (f − (Ui)) = u (X i) = β(Ui) by construction of u. Therefore, ϕZ and ψZ are∈ inverses of each other.

Corollary 1.5.8 The categories QAlg(Y ) and Aff(Y ) are equivalent through Aff(Y ) QAlg(Y ) G f : X Y f X ∗ → F O Spec Y Y A → A 17 Proof. An affine morphism is quasi compact and separated (see [6], p. 321), so [8], p. 163 implies that f X is quasi-coherent. If QAlg∗O Y , then G F f , according to the theorem 1.5.6, ( ) ( )( ) = SpecY = ∗ A whereAf : ∈ Spec Y Y is the◦ structuralA morphism.O A A → If f : X Y is an affine morphism, then X = Spec Y f X . Indeed, with the same → ∗1O 1 notations as in the proof of 1.5.6, if i I, X i = Spec ( X (f − (Ui)) = f − (Ui), since f is 1 ∈1 O affine. If i, j I, βi j : f − (Ui Uj) f − (Uj Ui) is the identity. Thus it is clear that X is the glueing of∈ the X i. ∩ → ∩

Let us study the action on the morphisms : if u : X X 0 is a morphism between two → affine morphisms f : X Y and f 0 : X 0 Y , then X = Spec Y , X 0 = Spec Y 0, where f X , f X and→ G u is given by→ the bijection ϕ : HomAY Spec Y , SpecA Y = 0 = 0 0 ( ) ( 0 ) HomA ∗O ,A above.∗ O The diagram of the theorem 1.5.6 shows the functoriality.A A → Y ( 0) O A A Likewise, the action of F an a morphism β : 0 is given by the inverse of ϕ. Hence, F and G are quasi inverses of each other forA the→ morphisms. A

1.6 Symmetric algebra and exterior powers of algebra

For proofs and details, see [6], pp. 196-199 (exterior powers) and 287-288 (symmetric algebra). n Let A be a ring, and M be a A-module. We set n ¾ 0, Tn(M) = M ⊗ and T(M) = L ∀ Tn(M). This module endowed with the product : n¾0

(m1 mn) (m10 m0l ) = m1 mn m10 m0l ⊗ · · · ⊗ × ⊗ · · · ⊗ ⊗ · · · ⊗ ⊗ ⊗ · · · ⊗ A i.e. p q T M T M T M is a graded (non-commutative) -algebra ( , N, p( ) q( ) p+q( )). ∀ ∈ ⊂ Definition 1.6.1 If I is the ideal of T(M) generated by the x y y x, x, y M, then the symmetric algebra of M is S(M) = T(M)/I. Endowed with⊗ − the product⊗ induced∈ by the product of T(M), it is a commutative A-algebra. L n As I is generated by homogeneous elements, we know that S(M) = S M, where n N n n n ∈ S M = Tn(M)/(I Tn(M)) = M ⊗ /In, In being the submodule of M ⊗ generated by the m m m ∩ m n 1 n σ(1) σ(n), σ permutation of 1, . . . , . ⊗ · · · ⊗ − ⊗ · · · ⊗ { } The module Sn M has the following universal property, that comes from the universal n n n property of the quotient : it is endowed with a morphism π : M ⊗ S M, and if f : M ⊗ N NA A i.e. f m → m f m → , -module is a -linear symmetric morphism ( ( 1 n) = ( σ(1) m f ⊗ · · ·S ⊗n M ⊗ · · · ⊗ σ(n)) for every permutation σ), then factors uniquely through :

f n M ⊗ N π f Sn M e

A consequence of this property is that the construction of Sn is functorial : if u : M N is a morphism, then we can build a morphism Snu : Sn M SnN· such that → → Snu Sn M SnN

πM πN u n n ⊗ n M ⊗ N ⊗

18 Definition 1.6.2 If J is the ideal of T(M) generated by the x x, x M, then the exterior algebra of M is the A-module Λ(M) = T(M)/J. ⊗ ∈ L n n n n As previously, Λ(M) = Λ M, where Λ M = M ⊗ /Jn, Jn = J M ⊗ being the submod- n N ∩ ∈ n n ule generated by the m1 mn M ⊗ such that i = j : mi = mj. The module Λ M is called the nth exterior power⊗ · · · of ⊗ M. ∈ ∃ 6 n n The universal property of Λ M is that every alternating morphism f : M ⊗ N, N n → A-module, (i.e. i = j : mi = mj f (m1 mn) = 0) factors through Λ M. Again, Λn acts∃ 6 also on the morphisms⇒ ⊗ and · · · ⊗ therefore define a functor. · n Notation : The class of m1 mn in Λ M is denoted by m1 mn, while its class n in S M is denoted by m1 ... mn⊗. · · · ⊗ ∧ · · · ∧

Proposition 1.6.3

Let M be a free A-module of rank r and (e1,..., er ) a basis of M. Then, for n , the family of the e e when J j < < j 1, . . . , n N ( j1 jr )J = 1 r is a basis of the∈ free A-module Λn M. ∧· · ·∧ { ··· } ⊂ { }

r In particular, if n = r, then Λ M is of rank 1, and is also called det M. Proposition 1.6.4 n n n n Given n N, M a A-module, and p Spec A, (S M)p = S Mp and (Λ M)p = Λ Mp ∈ ∈

Let us now generalize this construction to X -modules, when X is a scheme. O Definition 1.6.5

Let be a X -module. F O The nth exterior power Λn of , is the sheafification of the presheaf U Λn U , X (U) ( ) • endowed with the restrictionsF F 7→ O F

n n n U V X , Λ ρUV : Λ U Λ U ( ) X (U) ( ) X (V ) ( ) ∀ ⊂ ⊂ O F −→ O F s1 sn s1 V ... sn V ∧ · · · ∧ 7−→ | ∧ | n n n S is the sheafification of U S U , endowed with the restriction S ρUV . X (U) ( ) • F 7→ O F

n n Once again, Λ and S are functors from the category of X -modules into itself. Proposition 1.6.6 · · O n n Let be a quasi-coherent X -module. Then Λ and S are quasi-coherent. More- F O F n F n n n over, if U is an open affine of X and U = Me , then (Λ ) U = ΛßM and (S ) U = SßM. F| F | F | n n Remark. This proposition together with 1.6.4 shows that if x X , (S )x = S x and n n ∈ F F (Λ )x = Λ x . F F

19 Chapter 2

Double covers

2.1 Basic facts about covers

In this section, we fix a base scheme Y . Definition 2.1.1

A d-cover of a scheme Y is an affine morphism f : X Y , where f X is a quasi-coherent ∗ locally free Y -algebra of rank d. The number d is called→ the degreeO of the cover. O Definition 2.1.2

We denote, for d N∗, the category of the d-covers of Y by Covd (Y ). ∈ A morphism between f : X Y and f 0 : X 0 Y in this category is p : X X 0 isomorphism of schemes such that→ → →

p X X 0

f f Y 0

Proposition 2.1.3 The restriction of the functors in 1.5.8 yields an equivalence of categories between the

category of d-covers of Y and the category of quasi-coherent locally free Y -algebras of degree d, with isomorphisms as arrows. O

Thus, considering a cover f : X Y is the same as taking QAlg(Y ) which is locally → A ∈ free. In this situation, f is the structural morphism SpecY ( ) Y . For this reason, we will essentially consider the d-covers as locally free algebras ofA rank→d in the next two chapters.

Example 2.1.4. The 1-covers are the isomorphisms. • Indeed, if is a quasi-coherent locally free algebra of rank 1, endowed with his

structural morphismA α : Y , then for all x Y , αx : Y,x x is a morphism of O → A ∈ O → A algebras and x is a Y,x -module of rank 1. As 1 / mx , (1) is a basis of the k(x)-vector space of dimensionA 1O x /mx x . Thus, using Nayakama’s∈ lemma, it is a generator of A A x . Thanks to the lemma 2.2.5, we know that (1) is a basis of x . A A Therefore αx is an isomorphism since αx (1) = 1, and so is α.

If K is a number field of degree d, then we know that the ring of the integers K is a • O free Z-module of degree d. Therefore, Spec K Spec Z is a d-cover. O →

20 2 Example 2.1.5. We know that p : z z is a topological cover of C∗, but not of C because 1 → 1 p− (0) has one element, whereas the cardinality of p− (λ) is 2 if λ = 0. So 0 is a "double point". We want to find an algebraic equivalent to p and see if the same6 phenomenon occurs in 0.

Let k be a field, and consider the morphism of k-algebras ϕ : k[X ] k[Z] . It X −→ Z 2 induces an affine morphism of schemes f : S T, where S = Speck[Z] and 7−→T = Speck[X ]. → The quasi-coherent T -module (for the structure given by f ) f S is kß[Z]. But there is a surjectiveO morphism of k[X ]-algebras α : k[X ][∗OY ] k[Z] , whose Y −→ Z Y 2 X k Z k X Y Y 2 X k X k X Y 7−→ k X kernel is ( ). Hence [ ] [ ][ ]/( ) = [ ] k[X ] [ ] is a free [ ]-module of rank 2 and− f is a cover of degree' 2. − ⊕

Let us now replace k by C, or any algebraically closed field. We know by Hilbert’s Null- stellensatz that the maximal ideals of C[X ] are the (X λ), λ C. Moreover (cf [8], p. 1 − ∈ 83), if λ C, and p = (X λ), f − (p) is Sλ = S T Spec k(p), where k(p) = T,p/mp ∈ − × O ' C[X ]/(X λ) C is the residue field. Sλ satisfies the following base change diagram : − ' Sλ Spec C[Z] ƒ f Spec k(p) Spec C[X ]

So  X ‹  Z ‹  Z ‹ S Spec Z C[ ] Spec C[ ] Spec C[ ] λ = C[ ] C[X ] = = 2 ⊗ (X λ) (ϕ(X λ)) (Z λ) − − − Z If λ 0, and µ2 λ, we deduce from the Chinese remainder theorem that C[ ] = = 2 = 6 (Z λ) C[Z] C[Z] − C C, hence Sλ = Spec C Spec C has two elements. (Z µ) × (Z + µ) ' × t −  Z ‹ If λ 0, S Spec C[ ] Z is a singleton, but it is the spectrum of a -vector = 0 = 2 = ( ) C (Z ) { } space of dimension 2, which is a kind of multiplicity. We call this phenomenon ramification.

Proposition 2.1.6

Let be a cover of degree d. There exists a unique morphism Tr : Y verifying A A A → O the following property : for every U affine open subset of Y such that the Y (U)-module (U) is free of rank d, Tr (U) is the standard trace application on (UO). A Moreover, Tr (1 ) = dA. A A A Actually this theorem results from a more general principle that we describe in this lemma : Lemma 2.1.7

Let be a locally free Y -module of rank d. There exists a unique morphism Tr : E O End( ) Y verifying the following property : for every U affine open subset of Y such E → O that the Y (U)-module (U) is free of rank d, Tr (U) is the standard trace application O E End( (U)) Y (U). E → O

21 d Proof. Let be the set of the affine open subsets V of Y such that V = à(V ) V . Since is locallyB free, is a basis of open subsets of Y . We can thereforeE| defineE a morphism'O of

EY -modules on B. O B d Let V , and λ be an isomorphism of V -modules between V and V . The map λ is ∈ B O E| O entirely determined by the data of λ(V ) Iso( (V ), Y (V )). ∈ E O We can define Tr λ,V on the free sheaf End( ) V by

E | ϕ :f λÝ f λ 1 λ − d Tr 7→ End( V ) End( V ) = Md ( V (V )) V (V ) E| O O O s Let λ0 : V V be another isomorphism. The diagram

E| → O ϕλ Ý 0 End( V ) Md ( V (V )) | Ý E O ϕλ Tr ϕλ λ 1 0 − Md ( V (V )) V (V ) O Tr O 1 commutes because the ordinary trace map for matrices satisfies Tr (ABA− ) = Tr (B), A, B ∀ ∈ Md V V , A invertible, and the diagonal map is an isomorphism. Hence Tr λ,V Tr λ ,V . ( ( )) = 0 O d Therefore, we can set Tr (V ) = Tr λ,V (V ) for every V and λ : V V . In order to see that Tr is well-defined as a map of sheaves, let us∈ check, B for W E| V'O , that the following diagram is commutative : ⊂ ∈ B

Tr (V ) End( V ) Y (U) E| O restriction restriction Tr (W) End( W ) Y (V ) E| O d Let us consider an isomorphism λ : V V . Then λ W is an isomorphism between W d E| 'O | E| and W . O

ϕλ Tr V End( V ) Md ( V (V )) Y (V ) E| O O ρ restriction ψ restriction Tr End M W W W ( W ) ϕ d ( W ( )) Y ( ) | λ W E | O O

where ψ is the morphism of restriction coefficient by coefficient. Indeed, if M Md ( Y (V )), then : ∈ O

ψ M ϕλ ρ ϕλ 1 M ( ) = ( W − )( ) | ◦ 1◦ 1 = λ W (λ− M λ) W λ−W | ◦ 1 ◦ ◦ | | 1 = λ W λ−W M W λ W λ−W = M W | ◦ | ◦ | ◦ | ◦ | |

But the restriction M W as a morphism of sheaves correspond to the restricted matrix, | since, if (e1,..., ed ) is the canonical basis of Y (V ), then (e1 W ,..., es W ) is the canonical basis O P | P| of Y (W) and i = 1 . . . d, M W (ei W ) = M(ei) W = ( j mi,j ej) W = j mi,j W ej W . O ∀ | | | | | | With this ψ, it is clear that the diagram from the right commutes. Thus, the big rectangle commutes. As Tr (W) doesn’t depend on the isomorphism we choose, the horizontal arrows

22 correspond to Tr (V ) and Tr (W). Hence, Tr is well defined on and coincides with the actual trace map when V is free. B E|

Proof (of 2.1.6). Let ϕ : End( ) a UA −→ t aA a a ( ) ( a : 0 (U) 0) A Composing ϕ with the trace∈ A map built7−→ in the lemma7→ gives· us a morphism Tr : Y that satisfies the conditions of the theorem. A A → O Moreover, Tr (1 ) = d since they coincide locally : if U is an open affine such that A A (U) is free of rank d, Tr (U)(1 U ) = d. A A A |

Remark. If is a locally free -algebra of rank d, then for every x Y , Tr Tr . Y ( )x = x / Y,x A O ∈ A A O Proposition 2.1.8 u v If are morphisms of sheaves such that v u = id , then if j : ker v is theF −→ inclusion, G −→u F j : ker v is an isomorphism and◦ its inverseF is v (id →uv) G: ker v. ⊕ F ⊕ → G ⊕ − G → F ⊕ In particular, if is a d-cover and d Y (Y )∗, then Y ker Tr . A ∈ O A'O ⊕ A Proof. A direct computation show the first assertion. Tr Here, A f = id. d ◦ f Tr /d Y A Y O A O Therefore, β = f inclusion : Y ker Tr is an isomorphism. ⊕ O ⊕ A → A

Remark. The first statement is not specific to the sheaves, it is also true for modules for instance.

◊ ◊ ◊ ◊

Our goal in the next section and the next chapter is to find a concrete description of Cov2(Y ) and Cov3(Y ). Essentially, given ( , m) a locally free algebra of rank 2 or 3, we will try to find the "smallest" component of theA multiplication m that contain all the information.

2.2 Double covers via line bundles

Let Y be a scheme with 2 Y (Y )∗. ∈ O Let 2(Y ) the category whose objects are the pairs ( , σ) where is an invertible sheaf onDY and σ : Y is a morphism of Y -modules.L In this category,L a morphism L ⊗L → O O between ( , σ) and ( 0, σ0) is an isomorphism p : 0 such that L L L → L p p ⊗ 0 0 L ⊗ L L ⊗ L σ σ0 Y O

23 Consider he following associations :

Λ from Cov2(Y ) to 2(Y ) which associates with a locally free algebra ( , m) the • D inclusion m Tr /2 A object (ker Tr , σ), where σ : A Y . A L ⊗ L −→ A ⊗ A −→ A −→ O If u : 0 is an isomorphism of Y -algebras, wet Λ(u) := u ker Tr . A → A O | A ∆ from 2(Y ) to Cov2(Y ) that associates with ( , σ) the Y -algebra Y , endowed • D L O L ⊕O with the multiplication m : ( Y Y ) ( Y ) ( Y ) ( ) Y whose last component is σ O 0⊗ : O ⊕ O ⊗ LY ⊕ Land ⊗ O the⊗ othersL ⊗ L are−→ the O natural⊕ L ones following from the desired⊕ compatibilityL ⊗ L → O with⊕ L the Y -module structure. More precisely, for every open subset U : O

m(U)((a l) (a0 l0)) = (aa0 + σ(U)(l l0)) (al0 + a0l) ⊕ ⊗ ⊕ ⊗ ⊕

If v : 0 is a morphism in 2(Y ), we set ∆(v) = v id. L → L D ⊕ The main result of this chapter is : Theorem 2.2.1

The functors Λ and ∆ are well-define and quasi-inverses of each other. Thus, Cov2(Y ) and 2(Y ) are equivalent categories. D

We will see that we can actually study a "local" simplified problem corresponding to the situation we get if we localize the global problem by interesting ourselves to the stalks. Solving the local case is only a matter of commutative algebra. For the double covers as well as the triple ones, the only difficulty in the global case is therefore to define global maps that we can localize to use the local case.

Λ is well-defined

Proposition 2.2.2

If is a locally free Y -module of finite rank, and 1, 2 quasi-coherent Y -modules E O E E O such that = 1 2, then 1, 2 are locally free of finite rank. E E ⊕ E E E We admit the following lemma : Lemma 2.2.3 S If is a quasi-coherent Y -module and Y = Ui, where Ui is open and affine, then F O i I F is finitely presented (i.e. for every U open affine,∈ (U) is finitely presented) if and only F if for every i I, (Ui) is finitely presented. ∈ F

Proof (of the proposition 2.2.2). Let us first prove that 1 and 2 are finitely presented. S ni Let Y Ui be an open covering of Y such that UE E. We have = i Y Ui i I E| 'O | ∈ injection projection 0 2 1 0 E E E And injection 2 id projE E

E

24 so that 1 0, and by restricting this exact sequence to Ui we get a finite presentationE → of E → E. The→ lemma ensures us that is finitely presented. Symetrically, it’s 1 Ui 1 | also the case forE 2. E E According to the proposition 1.4.3, we know that, for i = 1, 2, i is locally free of finite rank if and only if x Y , i,x is a free Y,x -module of finite rank.E We are brought back to a local situation that∀ is∈ solvedE in the propositionO 2.2.4.

Proposition 2.2.4 If (R, m) is a local ring and M a free R-module of finite rank such that M splits in M = N L, N, LR-modules. Then N and L are free of finite rank. ⊕ Lemma 2.2.5 If (R, m) is a local ring, a surjective morphism between two free R-modules of same finite rank is injective.

n n Proof. Let ϕ : R R be such a surjective morphism and K = ker ϕ. Let ψ a morphism→ such that ϕ ψ = id. The proposition 2.1.8 gives an isomorphism n n ◦ n n ψ : R R K, which pass to the quotient in ψ : k ' k K/mK, where k = R/m. n n Hence K→/mK⊕= 0. Moreover, K is of finite type since the−→ projection⊕ R R K K is surjective, so by Nakayama’s lemma, K = 0. ' ⊕ →

n Proof (of the proposition). To simplify, we can assume that M = R . Let us tensorise by R/m n : if k = R/m, k = N/mN L/mL as k-vector spaces. Let s = dimk N/mN, then L/mL is of dimension n s. ⊕ − s By Nakayama’s lemma, if (x1,..., xs) is a basis of N/mN, then ϕ1 : R N is ei −→ xi s 7−→ surjective, where (ei) is the canonical basis of R . n s s n s n Symetrically, we have ϕ2 : R −  L and thus R R −  N L = R . hence, according to the lemma 2.2.4, this morphism is injective and so⊕ are ϕ1 and⊕ϕ2.

Here, the proposition 2.1.8 states that = ker Tr Y and ker Tr is quasi-coherent since it is the kernel of a morphism betweenA quasi-coherentA ⊕ O sheaves (cf A[8], p. 162). As is locally free of finite rank, so is ker Tr and passing to the stalks, we see it is of degreeA 1, i.e. invertible. A

If u is a morphism between and 0 in Cov2(Y ), then we have to check that v = Λ u u ker Tr is an isomorphismA onto kerA Tr , and preserves σ, σ . ( ) = 0 0 These| threeA statements can be checked locally,A i.e. on the stalks. So let us consider this situation : R is a local ring where 2 R∗, A = ker Tr A R and A0 = ker TrA0 R are two free ∈ ⊕ ⊕ R-algebras of rank 2, σ and σ0 are defined as in the global case, and u is an isomorphism of R-algebras between A and A . We set v u . 0 = ker Tr A |

First, let (ei)i be a basis of A, and a A. Then (u(ei))i is a basis of A0, and tu(a) : b0 u a b u e u ae ∈ t u e 7→t ( ) 0 sends ( i) to ( i). Thus, the matrix of u(a) in ( ( i)) is the same as the one of a in ei and Tr A u a Tr A a . This proves the first fact. 0 ( ( )) = ( ) On the other hand, v is an isomorphism because it is injective and if a0 ker Tr A , there ∈ 0 exists a A such that a0 = u(a), so 0 = Tr (a0) = Tr (a) and a ker Tr A. ∈ ∈

25 Finally, the compatibility of v with σ, σ0 results from the fact that u is a morphism of R-algebras, combined with Tr A u = Tr A. 0 ◦

∆ is well-defined

We want to prove two things : first that m is commutative and associative. We noticed in the section 1.5 that this was equivalent to the commutativity and associativity of the map on the stalk mx for every x Y . Furthermore, if v : 0 is a morphism of Y -modules ∈ L → L O and u = v id : Y 0 Y , we want u to be a morphism of Y -algebras. This is true if and⊕ only if,L for ⊕ O all →x LY the⊕ O map on the stalks ux is a morphismO of Y,x -algebra ∈ O That’s why we can assume that Y = Spec R, with R local ring. We are brought back to the study of (L, σ), where L is a free R-module of rank 1, (R, m) being a local ring in which 2 is invertible, and σ : L L R is a morphism of R-modules. The multiplication map on A = L R we are interested⊗ in→ is m induced by σ 0. Lemma⊕ 2.2.6 ⊕ 2 Let z be a generator of L and σˆ = σ(z z) Then A R[X ]/(X σˆ) as a R-module, with respect to the multiplications. ⊗ ' −

2 Proof. Let ψ : R[X ]/(X σˆ ) = R Rx A = R Rz , defined as a R-linear map. ψ − ⊕ −→ ⊕ X = x z is clearly bijective. Moreover, it preserves the7−→ multiplication, since :

ψ((a + bx)(c + d x)) = ψ(ac + bdσˆ + (ad + bc)x) = ac + bdσˆ + (ad + bc)z

And m(ψ(a + bx) ψ(c + d x)) = m((a + bz) (c + dz)) = ac + bdσˆ + adz + bcz. ⊗ ⊗

2 So the map m has the same properties as the multiplication on R[X ]/(X σˆ ) and therefore is commutative and associative. −

Let v be an isomorphism L L0 that preserve σ, σ0. We want to show that u = v id is a morphism of R-algebras and→ an isomorphism. ⊕ u can be seen as a morphism from R X / X 2 σˆ R Rx to R X / X 2 σˆ R Rx . [ ] ( ) = [ ] ( 0) = 0 − 2 ⊕ − ⊕ Moreover, the compatibility with σ, σ0 means that u(x) = σ(x x) = σˆ . ⊗ So u((a + bx)(c + d x)) = u(ac + bdσˆ + (ad + bc)x) = ac + bdσˆ + (ad + bc)u(x), and

2 u(a + bx) u(c + d x) = (a + bu(x))(c + du(x)) = ac + bdu(x) + (ad + bc)u(x) × = ac + bdσˆ + (ad + bc)u(x)

Thus u is indeed a morphism of R-algebras, and obviously an isomorphism.

∆ and Λ are quasi inverses of each other.

1. ∆ Λ 1 . Cov2(Y) ◦ ' Let ( , m) be a locally free Y -algebra of rank 2, ( , σ) = Λ( , m) (so = ker TrA ,m). O L A L A Let us denote the multiplication induced by σ on = Y by n. We want to show that m = n, which can be proved locally. A L ⊕O

26 So let us take these notations : (R, m) is a local ring such that 2 R∗ (A, m) is a ∈ free R-algebra of rank 2, L = ker Tr A,m is endowed with σ induced by m, n is the multiplication induced by σ on A. 2 We know that A = L R = Rz R, and (A, n) R[X ]/(X σˆ), σˆ = σ(z z). We set m(z z) = a + bz, a,⊕b R. ⊕ ' − ⊗ ⊗ ∈ 0 a‹ Therefore 0 Tr z Tr b. = A,m( ) = 1 b =  ‹ Tr (A,m) Moreover, Tr A,m(m(z z)) = 2a. But σ = m , so actually, a = σˆ , which ⊗ 2 ◦ L L means that m(z z) = σ(z z) = n(z z). Thus, m = n| .⊗ ⊗ ⊗ ⊗

Finally, if u is a morphism from ( , m) to ( 0, m0), then (∆ Λ)(u) = u id, A A ◦ |L ⊕ = ker Tr ,m. The decomposition = Y combined with the fact that u is a L A A O ⊕ L morphism of Y -algebras implies that (∆ Λ)(u) = u. O ◦

2. Λ ∆ 1 . 2(Y) ◦ ' D Let us consider ( , σ) 2(Y ). If ∆( , σ) = ( , m), then (Λ ∆)( , σ) = ( , ν), L ∈ D L A ◦ L M where = ker Tr ,m, and ν is induced by m. M A Once again, the equality = and σ = ν can be checked on the stalks, so we go back to the local situationL as above.M 2 We have A = L R = Rz R and we know that A R[X ]/(X σˆ) as R-algebras, σˆ = σ(z z). ⊕ ⊕ ' − ⊗ 0 σˆ‹ Tr A,m(z) = Tr = 0 (matrix in the basis (1, z)), so L M. 1 0 ⊂ Moreover, if w is a basis of M, then there exists a, b R such that w = a + bz and a bσˆ‹ ∈ 0 = Tr (w) = Tr = 2a, hence a = 0 because 2 R∗. So w L and L = M. b a ∈ ∈ Tr Tr Finally, if l, l0 L, σ(l l0) R, so σ(l l0) = (σ(l l0)) = (m((0 l) (0 l0))) = ∈ ⊗ ∈ ⊗ 2 ⊗ 2 ⊕ ⊗ ⊕ ν(l l0) by construction, because M = L. ⊗

If v is a morphism between , σ and , σ , then Λ ∆ v v id ker Tr ( ) ( 0 0) ( ( )) = ( ) Y = L L ⊕ | L ⊕O (v id) according to the computation above, so Λ(∆(v)) = v. ⊕ |L

27 Chapter 3

Triple covers

In this chapter, we will mainly refer to the article of Rick Miranda ([9]). Let us first describe the framework of the construction. We consider in all this part Y a scheme such that 6 Y (Y )∗. ∈ O Let 3(Y ) be the category whose objects are pairs ( , δ) where is a locally free sheaf of 3 2 rank 2 onD Y and δ : S E Λ E is a morphism of Y -modules.E In thisE category, a morphism → O between ( , δ) and ( 0, δ0) is an isomorphism u : 0 such that E E E → E S3u 3 3 S S 0 δ E E δ0 Λ2u 2 2 Λ Λ 0 E E commutes. Consider the following associations :

If ( , m) is a locally free Y -algebra of rank 3, then thanks to the proposition 2.1.8 • A O 2 (3 Y (Y )∗), we have = Y , = ker Tr , and we can define β : S as m projection ∈ O A 2 E ⊕ O E A E → E the factorisation through S of |E⊗E Y . We will prove in 3.2.2 that there exists a unique δ :ES3 E ⊗det E −→such E ⊕ that O this−→ diagam E commutes : E → E

β id S2 ⊗ E ⊗ E E ⊗ E δ 2 S det (3.1) E E

We define H from Cov3(Y ) to 3(Y ) which associates with ( , m) the object ( , δ). D A E If u : 0 is an isomorphism of Y -algebras, we set H(u) = u . A → A O |E If ( , δ) is an object of 3(Y ), then thanks to the proposition 3.2.5, we can find a • uniqueE β : S2 satisfyingD the diagram 3.1. Moreover, we will show in the propo- E → E 2 sition 3.2.3 that there exists a unique η : S Y = such that endowed with E → O A A m induced by β η is a Y -algebra. We set K( , δ) = ( , m) and for v : 0 ⊕ O E A E → E morphism in 3(Y ), K(v) = v id. D ⊕

The main result of this chapter is :

28 Theorem 3.0.7

The functors K and H are quasi inverses of each other and therefore 3(Y ) and Cov3(Y ) are equivalent categories. D

To make the proof clearer, we add a third category that will be equivalent to the first ones : the objects of e3(Y ) are the pairs ( , β), where is a locally free sheaf of rank 2 2 and β : S is a morphismD of Y -modulesE such thatE there exists a unique morphism δ making theE diagram → E 3.1 commutative.O

3.1 The local case

Let (R, m) be a local ring such that 6 R∗. ∈ Let LCov3(R) be the set of the free algebras (A, m) of rank 3. We know that if A ∈ LCov3(R), then A = ker Tr A R and ker Tr A is a free R-module of rank 2 (theorem 2.1.8 and proposition 2.2.4). ⊕

Let d3(R) be the set of the pairs (E, φ2) where E is a free R-module of rank 2 and φ2 2 is a morphism from S E to E such that, if (z, w) is a basis of E, the matrix of φ2 in the  ‹ 2 2 2 a d c 4 basis (z , zw, w ) of S E is of the form − , (a, b, c, d) R . This condition doesn’t b a d ∈ − depend on the basis we choose, thanks to the following lemma, which also shows that d3(R) is the local equivalent of 3(Y ). Lemma 3.1.1 D 2 Let R be a ring, E a free R-module of rank 2 and β : S E E a morphism. If (z, w) is a basis of E, then the existence of the factorisation →

β id S2 E E ⊗ E E ⊗µ ⊗ν δ S3 E det E

a d c‹ is equivalent to : M 2 2 β where a, b, c, d R. (z ,zw,w )( ) = − b a d ∈ − Proof. A direct check shows that the existence of the factorisation is equivalent to ker µ ker ψ, with ψ = ν (β id). ⊂ 2 2 But the kernel of◦ µ ⊗is generated by (z w zw z, zw w w z).  ‹ a e c⊗ − ⊗ ⊗ − ⊗ 2 M 2 2 z w Moreover, if (z ,zw,w )(β) = , a direct computation shows that ψ( b f d ⊗ − 2 zw z) = (a + f )z w and ψ(zw w w z) = (d + e)z w. ⊗So ∧ ⊗ − ⊗ ∧ §  ‹ a = f a d c ker µ ker ψ M 2 2 β − (z ,zw,w )( ) = − ⊂ ⇔ e = d ⇔ b a d − −

We are going to explain why taking a free R-algebra of rank 3 is the same as considering an object (E, φ2) of d3(R).

29 Let (A, m) LCov3(R), A = E R, where E = ker Tr A. As m is commutative, we get φ such that : ∈ ⊕

m E E E E | ⊗ A ⊗ φ S2 E

2 2 And we define φ = (φ1 : S E R)+(φ2 : S E E). For the same reasons as in the case of the double covers, the interesting→ part of m : (R→R) (R E) (E R) (E E) E R is expressed by φ, since the other components are⊗ imposed⊕ ⊗ by the⊕ fact⊗ that⊕A is⊗ a R-algebra.→ ⊕ 9 Let (z, w) be a basis of E. We can find (a, b, c, d, e, f , g, h, i) R such that ∈  2  φ(z ) = g + az + bw φ(zw) = h + ez + f w 2  φ(w ) = i + cz + dw

m is associative so in particular m(z m(z w) w) = m(m(z z) w) (condition (1)) and m(z m(w w)) = m(m(z w) w⊗) (condition⊗ ⊗ (2)). ⊗ ⊗ Using⊗ the expression⊗ of φ, and⊗ the⊗ compatibility of m with the structure of R-module, we get for (1) gw + a(h + ez + f w) + b(i + cz + dw) = hz + e(g + az + bw) + f (h + ez + f w), i.e.

2 (ah + bi) + (ae + bc)z + (a f + bd + g)w = (eg + f h) + (h + ae + e f )z + (be + f )w

and for (2) :

2 (cg + dh) + (i + ac + de)z + (bc + d f )w = (eh + f i) + (e + c f )z + (h + e f + d f )w

As (1, z, w) is a basis of A, we finally obtain :

 2  g = f + be bd a f h = bc − e f − (3.2) 2  i = e + c f −ac de − −

In other words, φ1 is determined by φ2.   0 g h Moreover, E = ker Tr A so 0 = Tr (z) = Tr 1 a e = a + f and likewise 0 = Tr (w) = 0 b f e + d. So a d c‹ M 2 2 φ (3.3) (z ,zw,w )( 2) = b −a d − i.e. (E, φ2) is in d3(R). Furthermore :

2 2
M 2 2 2 a bd ad bc 2 d ac (z ,zw,w )(φ1) = ( ) ( ) ( ) (3.4) − − − − We have reduced the number of parameters of the problem from 9 to 4.

30 Reciprocally, let us take (E, φ2) d3(R). If (z, w) is a basis of E, the matrix of φ2 in the a ∈ d c‹ basis z2, zw, w2 of S2 E is φ . We can define ( ) 2 = b −a d 2 − 2
2 φ1 = 2(a bd) (ad bc) 2(d ac) morphism from S E to R A = E R, and thus φ : S2 E −A defining− a multiplication− − map n on A. ⊂ ⊕ → n is associative, since it satisfies the conditions 3.2 which are clearly sufficient. Furthermore :

E = ker Tr A,n (3.5)

 2  0 2(a bd) (ad bc) − − − because Tr A,n(z) = Tr 1 a d  = 0 and likewise Tr A,n(w) = 0 ; 0 b −a − moreover, if x = p + qz + rw A is such that Tr A,n(x) = 0, then 3p = 0, hence p = 0 since 3 is invertible in R, and x E.∈ ∈

In conclusion, the way we have constructed φ2 out of m and n out of φ2 allows us clearly to state the following result : Theorem 3.1.2

The two maps we have built, ∆ : d3(R) LCov3(R) and Λ : LCov3(R) d3(R) −→ −→ (E, φ2) (E R, n) (A, m) (E, φ2) are inverses of each other. 7−→ ⊕ 7−→

Proof. The fact that (∆ Λ)(A, m) = (A, m) is obvious by construction, and (Λ ∆)(E, φ2) = ◦ ◦ (E, φ2) principally because if n is the multiplication induced by φ2, then E = ker Tr E R,n. ⊕

3.2 The global case

Let Y be a scheme such that 6 Y (Y )∗. ∈ O

3.2.1 Equivalence between Cov3(Y ) and e3(Y ) Theorem 3.2.1 D

There is an equivalence of categories between Cov3(Y ) and e3(Y ) given by : D

Cov3(Y ) e3(Y ) Θ D ( , m) ( = ker Tr , β) A A Ω E ( Y , n) ( , β) E ⊕ O E m projection where in the first line, β = |E⊗E Y . E ⊗ E −→ E ⊕ O −→ E

31 Θ is well-defined Proposition 3.2.2

Given = Y , locally free Y -algebra of rank 3, with = ker Tr A, whose multipli- cation inducesA O β⊕: ES2 , thereO exists a unique δ : S3 Edet such that E → E E → E β id S2 ⊗ E ⊗µ E E ⊗ν E δ S3 det E E Proof (of the proposition 3.2.2). As in 3.1.1, we immediately see that the existence of the factorisation is equivalent to ker µ ker ψ where ψ = ν (β id). This is the same as saying ⊂ ◦ ⊗ that ker µx ker ψx for all x Y , that is to say ( x , βx ) d3( Y,x ) which is true. Indeed βx ⊂ ∈ E ∈ O corresponds in the local case, with R = Y,x to the map φ2 and the formula 3.3 shows the result. O

Construction of Ω

Let ( , β) be an object of e3(Y ). PropositionE 3.2.3 D 2 2 There exists a unique η : S Y such that β η : S Y = induces a structure of Y -algebra. E → O ⊕ E → O ⊕ E A O Proof. We will proceed in two steps : first, proving the theorem when Y = Spec (R) is affine and = Ee, E free R-module of rank 2. Then we will prove the general case. FirstE step : Let us fix (z, w) basis of E. As in the proof of 3.2.2, the matrix of β in a d c‹ z2, zw, w2 is of the form . ( ) b −a d − Existence : Keeping in mind the local computation leading to 3.4, we define η by the 2 2
• matrix 2(a bd) (ad bc) 2(d ac) and we know that, as the stalks of β η satisfies the conditions− − 3.2,− the induced− multiplication is associative. ⊕

2 Uniqueness : If η0 : S Y is such that φ0 = β η0 induces an associative mul- • tiplication map on EY →, then O by looking at the stalks⊕ and using the computation E ⊕ O leading to the system 3.2, we show that the matrix of η0 is necessarily 2 2
2(a bd) (ad bc) 2(d ac) , whence η = η0. − − − − Second step : Let be the basis of open subsets of Y composed by the U open affine of

Y such that U is freeB of rank 2. Thanks to the first step, for U , we have built a unique E| ∈ B ηU corresponding to βU = β U . | Now, if V U are two elements of , the multiplication induced by βU ηU restricted to V induces a⊂ structure of Y -algebraB on V . ⊕ O E| In other terms, (ηU ) V satisfies the property characterising ηV , which implies that they are equal. In particular,| this result gives the compatibility with the restriction maps which allows us to define a morphism η of sheaves by setting U , η(U) = ηU (U). ∀ ∈ B The uniqueness in the general case ensues from the uniqueness in the first step because if η0 is a morphism such that η β yields a structure of Y -algebra on , then for U , ⊕ O E ∈ B η0U β U induces a structure of Y -algebra on U and thus η0U = η U . Hence η0 = η. | ⊕ | O E| | | 32 Proposition 3.2.4

If u : ( , β) ( 0, β 0) is a morphism in e3(Y ), then v = Ω(u) = u id : Y 0 Y is an isomorphismE → E of Y -algebras. D ⊕ E ⊕O → E ⊕O O

Proof. Let us take ηβ , ηβ as in proposition 3.2.3. If v = id u = Ω(v), then, as u is compat- 0 ⊕ ible with β, β 0, we only have to prove :

S2u 2 2 S S 0 Eηβ Eηβ 0 id Y Y O O 2 To show that η : ηβ S u is equal to ηβ , we can demonstrate that η β induces a = 0 structure of Y -algebra on ◦ and use the uniqueness in the proposition 3.2.3.⊕ O A As v is a isomorphism, we can carry the multiplication m0 of 0 to by A A v v ⊗ 0 0 A ⊗m A A ⊗ A e m0

v 1 0 A − A But, if n is the "multiplication" map induced by η β on , we have ⊕ A u u ⊗ 0 0 E ⊗ E E ⊗ E

S2u n 2 2 m S S 0 E⊗E 0 E0⊗E0 E E η β ηβ β ⊕ 0 ⊕ id u Y ⊕ Y 0 O ⊕ E O ⊕ E The previous diagram implies that n = me , so n induces a structure of Y -algebra on . O A Therefore, by uniqueness, ηβ = η.

Ω and Θ are quasi inverses of each other

For the morphisms, the situation is the same as for the double covers. For the objects :

Y If ( , m) Cov3( ), then (Ω Θ)( , m) = ( Y , n), where = ker Tr ( ,m) and n is • inducedA by∈ the component β ◦: A E of mE ⊕Oaccording to .E We knowA (proposition

E ⊗ E → E M 2 2 3.2.3) that the essential part of n is β η, where, if in a local basis, (z ,zw,w )(β) =  ‹ ⊕ a d c 2 2
− , η is of the form 2(a bd) (ad bc) 2(d ac) . But according b a d − − − − to the− formula 3.4, the local matrix of the essential component of m is the same, so m = n.

33 If ( , β) e3(Y ), then taking Ω( , β) = ( , n) with = Y , we have (Θ • E ∈ D E A A E ⊕ O ◦ Ω)( , β) = (ker Tr ( ,n), β 0). But thanks to the equation 3.5, = ker Tr ( ,n) and it E A E A follows obviously that β = β 0.

3.2.2 Equivalence between e3(Y ) and 3(Y ) D D Let us define two functors F : e3(Y ) 3(Y ) and G : 3(Y ) e3(Y ). D → D D → D F( , β) = ( , δ) where δ is the unique morphism making this diagram commutative • E E β id S2 ⊗ E ⊗µ E E ⊗ν E δ S3 det E E

2 G( , δ) = ( , β) where β is the unique morphism : S such that this diagram • commutesE :E E → E

β id S2 ⊗ E ⊗µ E E ⊗ν E δ 3 S det (3.6) E E Our goal is to show that F and G are quasi inverses of each other ; obviously, only the fact that they are well-defined matters. The theorem 1.4.7 will be the key argument to prove this equivalence of categories.

Action on the objects

As the fact that F( , β) is well-defined is ensured by the definition of e3(Y ), we just have to prove the followingE proposition to show that what we wrote makesD sense. Proposition 3.2.5

Given ( , δ) 3(Y ), there exists a unique morphism β such that the diagram 3.6 commutes.E ∈ D

Lemma 3.2.6

If is locally free of rank 2, there exists a canonical isomorphism γ : ' det ∨. E E → E ⊗ E Proof. Let ν : det be the canonical morphism. Then, thanks to the theorem 1.4.7, E ⊗E → E we get a morphism γ : det ∨. E → E ⊗ E If x Y , let us check that γx is an isomorphism. We can take a basis (v, w) of x . ∈ E By remembering the construction of γ, one sees that if u : Y,x x ( x )∨ is the morphism as in the proof of 1.4.7, then O → E ⊗ E

id u ν id γx : v ⊗ v (v v∗ + w w∗) ⊗ (v v) v∗ + (v w) w∗ = (v w) w∗ 7−→ ⊗ ⊗ ⊗ 7−→ ∧ ⊗ ∧ ⊗ ∧ ⊗

And similarly γx (w) = (v w) v∗, which proves that γx sends a basis on a basis, and thus is an isomorphism. − ∨ ⊗

34 Proof (of the proposition 3.2.5). We want to prove that, if δ : S3 det is a morphism, there exists a unique β : S2 such that E → E E → E β id S2 ⊗ E ⊗µ E E ⊗ν E δ S3 det E E 2 If ϕ = δ µ : S det , then the theorem 1.4.7 gives us a morphism ψ = K(ϕ) : 2 ◦ E ⊗ E → E 2 K 2 S det ∨, where K is the bijection Hom(S E , det ) Hom(S , det ∨) E → E ⊗ E 1 ⊗ E E −→ E E ⊗ E Let us take β = γ− ψ, with γ the map of the lemma 3.2.6. We want ϕ = ν (β id) = λ, 1 ◦ ◦ ⊗ i.e. λ = K− (ψ). 1 But, if ev is the evaluation morphism ζ s ζ(s), K− (λ) is ⊗ 7→ 2 ψ id iddet ev S ⊗ det ∨ E ⊗ det E ⊗ E −→ E ⊗ E ⊗ E −→ E And λ can be decomposed as

ψ id γ 1 id 2 − ν S ⊗ det ∨ ⊗ det E ⊗ E −→ E ⊗ E ⊗1 E −→ E ⊗ E −→ E Therefore it suffices to show that ν (γ− id) = iddet ev. Passing to the stalks x and using the computation made in the proof◦ of⊗ the previousE ⊗ lemma gives us this resultsE by evaluating on a basis.

Action on the morphisms

Proposition 3.2.7 If u Hom , β , , β , and δ, δ are the images of β, β by F, then the following e3(Y )(( ) ( 0 0)) 0 0 diagram∈ commutesD E : E

S3u 3 3 S S 0 E E δ δ0 det det det u 0 E E

Proof. As the morphisms ν : det and µ : S2 S3 are canonical, we obviously have the followingE diagramsE ⊗ E → : E E E ⊗ E → E

u u S2u u 2 2 ⊗ ⊗ 0 0 S S 0 0 E ⊗ν E E ⊗ν E E ⊗µ E E ⊗µ E 0 E 0 E det u E S3u E 3 3 det det 0 and S S 0 E E E E But we also have, by taking the tensor product of the compatibility diagram of u with

β, β 0 and u : S2u u 2 2 ⊗ S S 0 0 E ⊗ E E ⊗ E β id β0 id ⊗ u u ⊗ ⊗ 0 0 E ⊗ E E ⊗ E Therefore :

35 S2u u 2 2 ⊗ S S 0 0 E ⊗ E E ⊗ E β id β0 id ⊗ u u ⊗ ψ ⊗ ψ 0 0 0 E ⊗ν E E ⊗ν E 0 E det u E det det 0 E E where the composed arrow ψ (resp. ψ0) is, by property of δ and β equal to ϕ = δ µ (resp. to ϕ δ µ ). This implies that in the following diagram, the whole rectangle◦ E 0 = 0 0 commutes. ◦ E

S2u u 2 ⊗ SE S 0 0 ⊗µ E E ⊗µ E E S3u E ϕ 3 3 ϕ S S 0 0 Eδ Eδ det u det det 0 E E As we already know that the upper diagram commutes and the vertical arrows of this diagram are surjective (this principle is easy to understand set-theoretically, and we get this result here by looking at the stalks), then the lower diagram commutes.

Proposition 3.2.8 If u Hom , δ , , δ , and β, β are the images of δ, δ by G, then the following 3(Y )(( ) ( 0 0)) 0 0 diagram∈ commutesD E : E

S2u 2 2 S S 0 E E β β0 u 0 E E

Proof. We keep the notations of the previous proof. If ϕ = δ µ , ϕ0 = δ0 µ , then glueing ◦ E ◦ E 0 the compatibility diagram of u with δ, δ0 and the canonical diagram of µ , µ , we get : E E 0

S2u u 2 2 ⊗ S S 0 0 E ⊗ϕ E E ⊗ E ϕ0 det u det det 0 E E 1 1 Tensorising this diagram with ∨, ( 0)∨, where (u− )∨ : f f u− is the transposition 1 E E 7→ ◦ of u− , we have :

2 1 S u u (u− )∨ 2 ⊗ ⊗ S2 S ∨ 0 0 ( 0)∨ E ⊗ E ⊗ E E ⊗ E ⊗ E ϕ id ϕ0 id ⊗ 1 ⊗ det u (u− )∨ ⊗ det det ∨ 0 ( 0)∨ E ⊗ E E ⊗ E

Finally, using the canonical morphisms τ : Y ∨ and γ : det ∨ (con- structed respectively in 1.4.7 and 3.2.6), we haveO :→ E ⊗ E E → E ⊗ E

36 2 2 S u 2 S Y S 0 Y E ⊗ O E ⊗ O id τ id τ0 ⊗ 2 1 ⊗ S u u (u− )∨ 2 ⊗ ⊗ S2 S ∨ 0 0 ( 0)∨ E ⊗ Eϕ ⊗id E E ⊗ E ⊗ E ϕ0 id ⊗ 1 ⊗ det u (u− )∨ ⊗ det det ∨ 0 ( 0)∨ E ⊗ E1 E ⊗ E1 γ− γ0− u 0 E E By construction of the β, β 0 corresponding to δ, δ0 (proposition 3.2.5), the vertical ar- rows are actually β and β 0.

These two propositions show that both functors are well-defined for the morphisms, which completes the proof of the equivalence of categories.

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