UNIVERSITA’ DEGLI STUDI DI PAVIA FACOLTA’ DI SCIENZE MATEMATICHE, FISICHE E NATURALI
Dottorato di Ricerca in Matematica e Statistica
Fourier-Mukai functors and applications to quadric fibrations
Relatori: Prof. Alberto Canonaco Prof. Paolo Stellari Tesi di Dottorato di: Riccardo Moschetti
Anno Accademico 2013/2014
Contents 1 Introduction
3
2 Preliminaries 2.1 Basics on category theory . . . . . . . . . . 2.1.1 Triangulated categories . . . . . . . 2.1.2 Derived categories . . . . . . . . . . 2.1.3 Semiorthogonal decompositions . . . 2.1.4 Indecomposable objects . . . . . . . 2.2 Derived functors and Fourier-Mukai functors 2.3 Dg-categories . . . . . . . . . . . . . . . . . 2.4 Stability conditions . . . . . . . . . . . . . . 2.5 Quadratic forms with values in a line bundle 2.5.1 The Clifford algebra . . . . . . . . . 2.5.2 The stack of square roots . . . . . .
6 . . . . . . . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . . . . . . . . 7 . . . . . . . . . . . . . . . . . . . . . . 9 . . . . . . . . . . . . . . . . . . . . . . 14 . . . . . . . . . . . . . . . . . . . . . . 16 . . . . . . . . . . . . . . . . . . . . . . 16 . . . . . . . . . . . . . . . . . . . . . . 18 . . . . . . . . . . . . . . . . . . . . . . 22 and Azumaya algebras . . . . . . . . 25 . . . . . . . . . . . . . . . . . . . . . . 29 . . . . . . . . . . . . . . . . . . . . . . . 31
3 Derived category of the scheme of dual numbers 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.2 The category Db (A) . . . . . . . . . . . . . . . . . 3.2.1 Indecomposable complexes of Perf (A) . . . 3.2.2 Maps between indecomposable complexes . 3.2.3 Compositions . . . . . . . . . . . . . . . . . 3.3 Fully Faithful Endofunctors of Perf(A) . . . . . . . 3.4 Main Theorem . . . . . . . . . . . . . . . . . . . . 3.5 t-structures and stability conditions . . . . . . . . . 3.5.1 t-structures on Db (A) . . . . . . . . . . . . 3.5.2 Stability conditions on Db (A) . . . . . . . . 3.5.3 Co-t-structure and co-stability conditions . 3.6 Generalizing to k[]/(n ) . . . . . . . . . . . . . . .
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4 Quadric fibrations and Azumaya algebras 4.1 The space of stability conditions on the cubic threefold . . . . . . . . . . . . . . . . 4.2 Sets of generically regular quadric fibrations . . . . . . . . . . . . . . . . . . . . . . 1
. . . .
34 34 36 36 41 46 47 55 57 57 59 60 62 64 65 69
4.3 4.4 4.5
Line bundle valued quadratic forms of rank 4 with simple degeneration . . . . . . . Line bundle valued quadratic forms of rank 3 . . . . . . . . . . . . . . . . . . . . . Azumaya algebras and admissible coverings . . . . . . . . . . . . . . . . . . . . . .
2
72 78 80
Chapter 1
Introduction In this thesis we deal with two main subjects: the former, analyzed in Chapter 3, is the study of the derived category of the double point to provide some results about Fourier-Mukai functors. The latter is the study of the Azumaya algebras related to quadric fibrations, analyzed in Chapter 4. The double point, also known as the spectrum of the ring k[]/(2 ), k being a field, is an interesting object of study. Indeed, although it is quite easy to deal with it, thanks to the fact that it is a singular scheme of dimension zero, the description of the geometry of this scheme and of its derived category is not completely trivial. This makes the double point a good, easy “test” for a lot of conjectures. In particular our starting point is a conjecture about Fourier-Mukai functors, introduced in Section 2.2. Such functors appear in many geometrical contexts and a theorem proved by Orlov in 1997 (Theorem 3.1.1 in this thesis) implies that every fully faithful functor between the derived category of smooth projective varieties is of Fourier-Mukai type. A strong generalization of this theorem was provided by Lunts and Orlov in 2010 using the language of differential graded categories (Theorem 3.1.2 in this thesis). They prove that every exact and fully faithful functor from the category of perfect complexes of a projective variety X to the bounded derived category of coherent sheaves of a noetherian scheme Y is Fourier-Mukai type if X satisfies a technical hypothesis. Such hypothesis seems to be not very natural, indeed, more generally, one would analyze the following question: Given a functor between the derived categories of two varieties, what are the minimal hypothesis either on the varieties or on the functor to guarantee such a functor being of Fourier-Mukai type? In the literature there are lots of works concerning Fourier-Mukai functors. Canonaco and Stellari, in [25] present a survey on this topic and, in [27] manage to weaken the hypothesis fully faithful on the functor, extending the result of Orlov to the case of bounded derived categories of twisted coherent sheaves. Rizzardo and Van den Bergh in [64] provide a counterexample for the case of a non fully faithful functor between the derived categories of quasi-coherent sheaves. The reason to study the derived category of the double point lies in the fact that this variety does not satisfies the technical hypothesis required by the theorem of Lunts and Orlov. Hence,
3
this makes the double point the simplest example to study in order to understand better the hypothesis of the question above. In Theorem 3.1.3, we prove that the result of Lunts and Orlov can be extended to our case. This scores a point on the possibility to extend the validity of the conjecture for a wider class of variety. We give also an explicit description of the derived category of the double point and of the autoequivalences of this category; moreover we provide an example of autoequivalence that is not exact. Eventually in Section 3.6 we try to generalize our theorem considering the spectrum of the ring k[]/(n ) for n greater than two. The explicit description of the derived category turns out to be more complex and our results have not an immediate generalization. Anyway we think that it might be possible to generalize Corollary 3.3.14 without the explicit description of the category but focusing only on the property of exactness that is required by hypothesis. This would immediately imply a generalization of Theorem 3.1.3. Quadric fibrations and quadratic forms with values in a line bundle are the latter main topic of this thesis. They are a generalization of ordinary quadratic forms over a ring. The earliest works on this subjects were done by Mumford, Kanzaki, Bass, Kneser and others starting from the early sixties. Like in this classical case reported by Max-Albert Knus in [43], we can associate an algebra, called the Clifford algebra (See subsection 2.5.1) to such a form. The key ingredient to introduce the problem is the functor that describes how to associate the Clifford algebra to a quadratic form with values in a line bundle. Under certain conditions, this algebra is related to an Azumaya algebra (see Definition 2.5.14 of this thesis) that is especially useful for the calculation. In the case of our interest, this relation is provided by introducing, if the form has even rank, the double cover of S ramified along the so called discriminant locus and, if the form has odd rank, along the so called square root stack of S (See Propositions 2.5.17 and 2.5.21). The problem we are interested in is the relation between quadratic forms of low rank on a scheme S and Azumaya algebras. One of the first results in this direction was provided by Knus, in [43]; he proved that the Clifford map induces a bijection between the space of regular quadratic forms of rank three and the space of Azumaya algebras. There are several generalizations of this result provided by different authors as Chan and Ingalls in [29] and Voight in [69]. We focus on the generalization proposed by Balaji in [9], where he proved that the map induced by the Clifford algebra between the space of quadratic forms of rank three and a suitable space of “limits” of Azumaya algebras is a bijection. We focus on a result concerning the case of quadratic forms of rank four: Auel, Parimala and Suresh in [8] prove the bijection of a map induced by the Clifford algebra between the space of simple degenerate quadratic forms and with fixed discriminant stack and the space of Azumaya algebras of rank four over the same stack. We prove a result similar to the one proved by Auel, Parimala and Suresh for the quadratic forms of rank three. We focussed also on the case of conic bundles on a minimal rational surface. This subject was studied by Bernardara and Bolognesi in [15] with the aim to study whether a standard conic bundle is rational or not. There is an interlude that provides a motivation to our study of the quadratic forms with values in a line bundle. The starting point was a work of Bernardara, Macrì, Mehrotra and Stellari in
4
which they found a stability condition on the derived category of a generic cubic threefold in order to prove a categorical version of Torelli theorem for the cubic threefold. Part of the geometrical construction to achieve this result comes from the remarkable construction introduced by Voisin in [70] and is used also by Kuznetsov in [48], in relation with his well-known conjecture. The notion of stability condition on a triangulated category was introduced by Bridgeland in [22], and this subject had a lot of further development in the context of geometric invariant theory. Here we focus only on the very basic definition to the space of stability conditions on the cubic threefold. By the way, as an application of our techniques we study the space of stability conditions on the derived category of the double point scheme X. It turns out that, as proven in [40] by Jørgensen and Pauksztello, there are no stability conditions on the category Perf (X) and that the stability manifold of the derived category of coherent sheaves on X is isomorphic to C.
5
Chapter 2
Preliminaries 2.1
Basics on category theory
The language of categories has a lot of applications to different aspects of mathematics; in fact they provide a powerful language for abstractions. In this introduction we will assume basic facts about categories and functors. For a basic background we refer to [53]. We will require all the categories and all the functors to be at least additive, according to the following definition Definition 2.1.1. A category A is said to be additive if the following conditions hold true • The space Hom(X, Y ) is an abelian group for all objects X and Y . • The composition of morphisms is bilinear. • There exists a zero object. • For every pair of objects there exist both their direct sum and their direct product and they coincide. A functor between additive categories is said to be additive if the maps induced between the hom-spaces are group homomorphisms. In point of fact we will often deal with abelian categories, that are additive categories satisfying a further condition, which reads: • Every morphism admits kernel and cokernel and the natural map between coimage and image is an isomorphism. A subcategory B of an abelian category A is called thick if B is a full abelian subcateogry and any extension in A of objects in B is again in B. Consider a functor F between two arbitrary categories A and B. A functor H : B → A is said to be right adjoint to F if the isomorphisms Hom(F (A), B) ∼ = Hom(A, H(B)) 6
exist for any two objects A ∈ A and B ∈ B and are functorial in A and B From now on we will write F a H if H is the right adjoint to F . F is said to be left adjoint to H if and only if H is right adjoint to F . Another remarkable property that we will use in the following is the representability of a functor. Let F be a functor between an arbitrary category A and S , the category of sets. Recall that for every object A of A we can define the Hom functor Hom(A, ·), that maps objects X to Hom(A, X) viewed as set. The functor F is said to be representable if it is naturally isomorphic to the functor Hom(A, ·) for some object A of A . There is a representability notion also for contravariant functor, that are functor from the opposite category A op to S . A contravariant functor is representable when it is naturally isomorphic to the functor Hom(·, A) for some object A of A .
2.1.1
Triangulated categories
Triangulated categories were introduced by Grothendieck and Verdier to give an axiomatization of derived categories. After that triangulated categories have been studied a lot on their own, producing a great number of papers, whose main results are gathered together in the monography [58]. Here we start with the presentation of the axioms of triangulated categories and then, in the following section we will describe the realization of these axioms in derived categories framework. Let T : A → A be an automorphism of the category A , a triangle is a set of objects and morphisms like f
g
h
X− →Y → − Z− → T (X) The functor T is called suspension. By analogy with the geometrical case, it is often denoted by [1] and called shift. Definition 2.1.2. The additive category A , endowed with the functor T , is called triangulated if one can describe a family of triangles of A , called distinguished triangles, such that they satisfy the following six axioms. TR1) The family of distinguished triangles is closed by isomorphisms. id
The triangle X −−X → X → 0 → T (X) is distinguished for all object X. f
For every morphism f : X → Y there exists a distinguished triangle X → Y → Z → T (X). f
g
g
h
h
→Y → − Z − → T (X) is distinguished if and only if the triangle Y → − Z − → TR2) The triangle X − −T (f )
T (X) −−−−→ T (Y ) is such. TR3) In the following commutative diagram, X
f
u
X0
f0
/Y
/Z v
/Y0
/ Z0
7
/ T (X)
T (u)
/ T (X 0 )
in which rows are distinguished triangles, there exists a map h : Z → Z 0 that makes all the squares commutative. TR4) Consider the following distinguished triangles f
/Y
/ Z0
/ X[1]
g
/Z
/ X0
/ Y [1]
g◦f
/Z
/Y0
/ X[1].
/ X0
/ Z 0 [1]
X
Y
X
Then there exist a distinguished triangle /Y0
Z0
that makes the following diagram commutative f
X
/Y
X
g◦f
/Z
f
Y
Z0
/ X[1]
g
idX
/ Z0
g
/Y0
/ X[1]
idZ
/Z
/ X0
/Y0
idX[1]
idX 0
/ X0
f [1]
/ Y [1] / Z 0 [1]
The last axiom is also called the octahedral axiom, because the last commutative diagram involved can be rearranged on the form of an octahedron. It is known that these axioms are redundant, see for example [56], but a list of indipendent axioms is not known and neither it is known if TR4 is independent from the others. Classically, a functor F between abelian categories is said to be left exact (right exact, respectively) if any short left (right, respectively) exact sequence is mapped to a left (right, respectively) exact sequence. The functor F is said to be exact if it is both left and right exact. A functor F between two triangulated categories (A , T ), (A 0 , T 0 ) is said to be exact in the sense of triangulated category if F ◦ T ∼ = T 0 ◦ F and if F preserves distinguished triangles. A functor F between a triangulated category (A , T ) and an abelian category D is called cohomological if the sequence F (X) → F (Y ) → F (Z) is exact in the usual sense for all distinguished triangles f g h X− →Y → − Z− → T (X). Proposition 2.1.3 ([37], Section 1). Consider a triangulated category with suspension T . Then, the following properties hold. 8
f
g
h
• If the triangle X − →Y → − Z− → T (X) is distinguished, then g ◦ f is zero. • The functors Hom(X, ·) and Hom(·, X) are cohomological for every object X. • Consider the following morphism of triangles /Y
X φ
/Z ψ
/Y0
X0
/ X[1]
θ
/ Z0
T (φ)
/ X 0 [1]
If φ e ψ are isomorphisms then θ is such. f
• If X − → Y → 0 → T (X) is distinguished, then f is an isomorphim. Proof. It is straightforward to verify that the properties follow directly by the axioms. Let f : X → Y be a morphism in the triangulated category A . The first axiom of Definition 2.1.2 ensures the existence of a distinguished triangle f
X− → Y → Z → T (X) By the properties of the previous proposition, the object Z of this distinguished triangle is unique up to a non unique isomorphism. We call Z the cone of the morphism f .
2.1.2
Derived categories
A remarkable example of triangulated category is given by the homotopy category of complexes of objects of an additive category. The construction of the category comes directly from the geometrical context and it will be useful in order to introduce the notion of derived categories. In fact we will often think of A as the category of sheaves on a given scheme, or the category of modules on a given ring, but the theory also works for an arbitrary additive category. The reasons to study the category of complexes are at least two: study the cohomology of a space and working with resolutions. The objects of the category of complexes of objects in A , named Kom(A ) are families • A := {(An , dn )} indexed by an integer number n, where An is an object of A and dn is a morphism, called differential map, from An to An+1 such that dn+1 ◦ dn is zero for all n. In what follows, we will explicitely write dnA• to specify that the differential dn belongs to the complex A• whenever it will be necessary. A morphism f • in Kom(A ) between two complexes A• and B • is given by a family of maps f n , where f n : An → B n and dnB • ◦ f n = f n+1 ◦ dnA• . The definition can be summarized in the following diagram ···
n−1 / An−1 d / An
···
f n−1
dn
fn
n−1 / B n−1 d / Bn
9
dn
/ An+1
/ ···
f n+1
/ B n+1
/ ···
To simplify the notation we will use the notation • to underline that we are speaking of a complex or of a morphism of complexes only if it is not clear by the context. Notice that the category Kom(A ) is additive, and it is abelian if and only if A is such. An important basic automorphism on the category of complexes Kom(A ) is the shift. For every integer k,the shift [k] sends an object A• of Kom(A ) to a new complex A• [k] defined by (A[k]n , dnA• [k] ) = (An+k , (−1)k dn+k A• ) At the level of the morphisms, the functor [h] sends f • to f • [k] defined by setting f [k]n = f n+k . Dealing with complexes, it is natural to introduce the equivalence relation of homotopy. A morphism of complexes f : A• → B • is homotopy equivalent to zero if there exist morphisms sn : An → B n−1 in A such that for all n the following relation holds true f n = sn+1 ◦ dnA + dn−1 ◦ sn B Moreover, f is homotopy equivalent to g if f − g is homotopy equivalent to zero. Denote by HtA (A, B) ⊂ HomKom(A ) (A, B) the subgroup of the morphisms homotopic to zero. Definition 2.1.4. The homotopy category of complexes of A is the category HKom(A ) obtained by quotienting the group of morphisms by the subgroup HtA , i.e. it is defined by ( Ob(HKom(A )) = Ob(Kom(A )) HomHKom(A ) (X, Y ) = HomKom(A ) (X, Y )/HtA (X, Y ) The category HKom(A ) has a triangulated structure closely related to the geometrical point of view. The shift functor will play the role of the autofunctor T of Definition 2.1.2. We only need a way to choose the family of distinguished triangles. We will use for that purpose the cone construction Definition 2.1.5. Let f : X → Y a morphism in Kom(A ). The mapping cone of f , denoted by Cone(f ) is the following object of Kom(A ) Cone(f )n = X n+1 ⊕ Y n ! n d 0 n X[1] dCone(f ) = f n+1 dnY The family of distinguished triangles that we will consider is given by all triangles isomorphic to the image in HKom(A ) of the triangles of the form f
β
α
{X − →Y − → Cone(f ) − → X[1]}
(2.1)
for all morphisms f : X → Y of Kom(A ). The maps α(f ) : Y → Cone(f ) and β(f ) : Cone(f ) → X[1] are given by ! 0 α(f )n = β(f )n = (idX n+1 , 0) idY n 10
Notice that, despite the fact that the cone is functorial in the category Kom(A ), and then the third axiom of triangulated categories holds with the map h : Z → Z 0 defined in a natural way, there are other axioms that do not hold, for example the second and the first one, and so the set of triangles 2.1 do not induce a structure of triangulated category on Kom(A ). Assume now that A is abelian; one of the reasons to study the language of derived categories is its application to the analysis of the cohomology of the complexes. If A is an object of Kom(A ), we can define the k-th cohomology group of A as B k (A) = Im(dk−1 A )
Z k (A) = Ker(dkA ),
H k (A) = Coker(B k (A) → Z k (A)) We can define the cohomology functor H k from Kom(A ) to A for every integer k. Notice that if a morphism f is homotopic to zero, then the induced morphism in cohomology is the morphism zero, and hence the cohomology functor is well defined on the category HKom(A ). Obviously the image of an isomorphism under the cohomology functor is an isomorphism too. However in the category HKom(A ) there are some maps that are not isomorphisms, whose associated map in cohomology is an isomorphism. We will call these maps quasi-isomorphisms. We want to pretend quasi-isomorphisms to be isomorphisms by a procedure similar to the localization in algebra, where one formally inverts a family of elements with some given characteristics. Definition 2.1.6. Let C be a category and let S be a family of morphisms in C . S is called a multiplicative system if the following properties hold: S1) For all objects X in C , the identity idX belongs to S. S2) S is closed with respect to composition. S3) Given X, Y and Z objects of C and f , g morphisms with g ∈ S, then there exist an object W and morphisms h, t, with h ∈ S, that make the following diagram commutative t
W
h f
X
/Z
g
/Y
And also the same but with the arrows reversed S4) For every f, g ∈ HomC (X, Y ), the two following conditions are equivalent • There exists t : Y → Y 0 , t ∈ S such that t ◦ f = t ◦ g. • There exists s : X 0 → X, s ∈ S such that f ◦ s = g ◦ s. We now want to formally invert the morphisms of the multiplicative system S. Let f : Z → X be in S and consider g : Z → Y ; as for the localization of a ring we will consider equivalence classes of objects like this 11
Z f
g
~
X
/Y
Definition 2.1.7. The category CS , named localization of C by the multiplicative system S, is defined by Ob(CS ) = Ob(C ) HomCS (X, Y ) = {(X 0 , s, f ) s.t. X 0 ∈ Ob(C ), s : X 0 → X, f : X 0 → Y, s ∈ S}/R where R is the following equivalence relation: (X 0 , s, f )R(X 00 , t, g) if and only if there exist a commutative diagram XO 0
f
s
}
X ao
u t
!
X 000 g
=Y
X 00 where u ∈ S. The composition of two morphisms [(X 0 , s, f )] ∈ HomCS (X, Y ) and [(Y 0 , t, g)] ∈ HomCS (Y, Z) is defined by constructing the following diagram using the property (S3) of the definition of multiplicative system X 00 h
X0
}
t0
!
f
X
~
s
Y0
!
g
Y
}
t
Z
with t0 ∈ S. The composition [(Y 0 , t, g)] ◦ [(X 0 , s, f )] will be [(X 00 , s ◦ t0 , g ◦ h)]. Definition 2.1.8. The derived category D(A ) of A is defined to be the category HKom(A ) localized by the multiplicative system given by quasi-isomorphisms. Notice that the objects of D(A ), HKom(A ) and Kom(A ) are the same, only morphisms change giving rise to very different structures. Notice that the multiplicative system of quasi-isomorphisms satisfies the two conditions to be compatible with the triangulated structure of HKom(A ): • For every element s in the multiplicative system S, the shift s[n] belongs again to the multiplicative system for every integer n.
12
• Given the following morphism of distinguished triangles X
s
X0
/Y
/Z s0
/Y0
t
/ Z0
/ X[1]
s[1]
/ X 0 [1]
where the maps s and s0 are in the multiplicative system, then also the third map t is such. If these conditions hold, then the localized category CS will be triangulated as well and the projection functor from C to CS will be exact. We can define the categories of bounded (bounded above or bounded below, respectively) complexes simply by requiring An = 0 if |n| >> 0 (n >> 0 or n << 0, respectively) for the complex A. By following exactly the same steps mentioned above we can define the bounded derive category Db (A ) (bounded above derived category D− (A ), bounded below derived category D+ (A ), respectively). Since in the geometrical applications we will work over a field k, then the categories that we will study will have some interesting properties related to the base field. The additive category A is called k-linear if the groups of morphisms of A are k-vector spaces and the composition maps are k-linear. Moreover an additive functor F between k-linear categories is a k-linear functor if it acts as a linear map on the space of morphisms. If we consider a scheme X, we will denote by D(X) the derived category of the category of coherent sheaves on X. If we will need to use other categories we will specify it: for example the derived category of the category of quasi-coherent sheaves on X will be denoted by D(Qcoh(X)). Definition 2.1.9. Let A be a k-linear category. A k-linear equivalence S : A → A is called a Serre functor if for every A and B objects of A , there exists a natural isomorphism of k-vector spaces Hom(A, B) ∼ = Hom(B, S(A))∗ In the following we will need the notion of perfect complexes. An object of D(Qcoh(X)) is perfect if it is locally quasi-isomorphic to a bounded complex of locally free sheaves of finite rank. The full, thick subcategory composed by perfect complexes of D(Qcoh(X)) will be denoted by Perf (X). Let X be quasi-projective, then X is regular if and only if Db (X) coincides with Perf (X). Consider a triangulated category T with direct sums. An object X in T is compact if, for every collection Yi {i∈I} of objects in T , the natural morphism: M
Hom(X, Yi ) −→ Hom(X, ⊕Yi )
is an isomorphism. The full, thick subcategory composed by compact objects of T will be denoted by T C . Let X be quasi-projective, and consider an object C in D(Qcoh(X)), then C is perfect if and only if C is compact.
13
2.1.3
Semiorthogonal decompositions
An efficient way to study derived categories is by means of semiorthogonal decompositions. The main idea is that in the definition of triangulated categories there are two ways to produce objects: the shift functor T (X) of an object X and the cone Cone(f ) of a morphism f : X → Y . We can see a triangulated category as generated by some subcategories, by the use of these two operations, according to the following definition. Definition 2.1.10. Let S be a family of objects of a triangulated category T . We say that S generates T if T is the smallest strictly full subcategory of T , containing S, closed with respect to taking shifts and cones. Consider a full triangulated subcategory B ⊂ T . The full subcategory of T right-orthogonal to B, denoted by B ⊥ , is given by all the objects C of T such that the space of morphisms HomT (B, C) is equal to zero for all B in the category B. We have to require a further condition in order to get good properties of the semiorthogonal decompositions. Definition 2.1.11. Let B be a full subcategory of T . B is said to be a right-admissible subcategory if the inclusion B ,→ T has a right adjoint. There is also a specular notion of left-ammissibility. We we call a subcategory admissible if it is both left- and right- admissible. To find a decomposition of the category T the first idea is to find a triangulated subcategory B, taking the right orthogonal B ⊥ and then claim that T is generated by B and B ⊥ . A sequence of full and ammissible subcategories of a triangulated category T T1 , . . . , Tn is called semi-orthogonal if Tj is contained in Ti⊥ for all j < i. If the family T1 , · · · , Tn generates the whole T , we will call this family a semiorthogonal decomposition of T , and we will write T = hT1 , · · · , Tn i
(2.2)
Moreover, if for all j < i we have Ti ⊂ Tj⊥ , we say that 2.2 is an orthogonal decomposition. However it is most common to use semiorthogonal decomposition in the geometric context due to the following theorem, proved by Bridgeland. Theorem 2.1.12 ([21], Example 3.2). Let X be a noetherian scheme, then Db (X) has no orthogonal decomposition if and only if X is connected. If we are dealing with k-linear categories we can find semiorthogonal decompositions starting directly with some particular objects, called exceptional objects. By definition, an object E in a k-linear triangulated category is called exceptional if the morphisms between E and the shifts E[l] are of the type 14
( Hom(E, E[l]) =
k
if l = 0
0
if l 6= 0
If E is an exceptional object of T , then the smallest subcategory TE containing E and closed under taking shifts and cones is an admissible subcategory. A sequence of exceptional objects E1 , . . . , En is called an exceptional sequence if the subcategories TE1 , . . . , TEn form an orthogonal sequence. For example, the derived category of Pn admits the following semiorthogonal decomposition Db (Pn ) = hO(−n), O(−n + 1), . . . , O(−1), Oi where the sheaves O(i) form an exceptional sequence. The proof of this fact uses a result of Beilinson called the resolution of the diagonal, that can be found in [13]. As a consequence we are able to find a semiorthogonal decomposition of an hypersurface X in Pn . The result is a sort of a Lefschetz hyperplane theorem result for the derived categories, more details can be found on [50]. If X is a hypersurface of degree d in Pn then Db (X) = hAX , O, O(1), . . . , O(n − d)i where AX is the right orthogonal to the category hO, O(1), . . . , O(n − d)i, generated by the exceptional sequence. We can think of AX as being the interesting part of the derived category of X. Consider a triangulated category T and let E be an exceptional object. One can consider the following functors from T to T called left and right mutation LE (M ) := Cone(ev : RHom(E, M ) ⊗ E → M ) RE (M ) := Cone(ev ∗ : M → RHom(M, E)∗ ⊗ E)[−1] where ev is the evaluation morphism. Consider the following semiorthogonal decomposition of T T = hT1 , · · · , Tk , E, Tk+1 , · · · , Tn i By these functors we can produce new semiorthogonal decompositions T = hT1 , · · · , Tk , LE (Tk+1 ), E, Tk+2 , · · · , Tn i T = hT1 , · · · , Tk−1 , E, RE (Tk ), Tk+2 , · · · , Tn i An important result that allows us to find semiorthogonal decompositions of the derived category of a blow up is the following theorem due to Orlov Theorem 2.1.13 ([61], Theorem 4.3, [19], Theorem 3.5). Let X be a smooth variety, Y be a smooth σ subvariety of X of codimension n + 1, and let Z − → X be the blow-up of X along Y with exceptional divisor D. Then DbCoh (Z) = hDbCoh (Y )−n , DbCoh (Y )−n+1 , . . . , DbCoh (Y )−1 , σ ∗ DbCoh (X)i 15
where DbCoh (Y )−k is the full subcategory of DbCoh (Z) consisting of objects of the form i∗ (p∗ E ⊗ OD (−k)), where E is an object of DbCoh (Y ), p : D → Y and i : D ,→ Z .
2.1.4
Indecomposable objects
A way to study the objects and morphisms of an additive category is by focusing on the indecomposable complexes. In an additive category, X is called indecomposable if X ∼ = Y ⊕ Z implies Y ∼ = 0 or Z ∼ = 0. A good context to study indecomposable objects is provided by Krull-Schmidt categories, which are explained in details in [57]. A ring R is called semiperfect if every finitely generated right R-module has a projective cover; see [2] for further references. Definition 2.1.14. Let C be an additive category such that EndC (X) is a semiperfect ring for all X ∈ C (in that case C is called a pre-Krull-Schmidt category). C is called a Krull-Schmidt category if every idempotent splits, i.e. for every X in C and for every e ∈ EndC (X) such that e2 = e, there exist Y in C and two morphisms p : X → Y and q : Y → X such that qp = e and pq = 1Y . An additive category in which every idempotent splits is also called Karoubian, and hence a Krull-Schmidt category is a pre-Krull-Schmidt category that is also Karubian. Notice that every abelian category is Karoubian. In [68] one can find another definition of the split property: an idempotent e : X → X splits if and only if there exists a non trivial decomposition X ∼ = Y ⊕Z with e corresponding to the projection on Y . These two definitions are equivalent in a triangulated category, which is the case of the present paper. Thanks to the following result, called Theorem of Krull-Schmidt which is an easy generalization of a result proved by Atiyah in [4], we can confine ourselves to indecomposable elements: Theorem 2.1.15 ([57], Theorem 3.9). In a Krull-Schmidt category every object can be decomposed into a finite direct sum of indecomposable objects. Moreover this decomposition is unique up to isomorphism and permutation of the summands.
2.2
Derived functors and Fourier-Mukai functors
Derived functors arise in a natural way in the process of the extension of a functor F between two abelian categories A and B to a functor between the derived categories D(A ) and D(B). If F is exact there is a natural way to induce a functor at the level of derived category because F preserves quasi-isomorphism and acyclic complexes. A problem arises since a lot of geometrical constructions are not exact. Let X and Y be noetherian schemes and f : X → Y a proper morphism and F a quasi-coherent sheaf on X. Examples of left or right exact functors are: • The global section functor Γ(X, ·) is left exact. • The direct image functor f∗ : Qcoh(X) → Qcoh(Y ) is left exact. • The inverse image functor f ∗ : Coh(Y ) → Coh(X) is right exact. 16
• The functor Hom(F , ·) is left exact. • The tensor product functor F ⊗ · is right exact. Assuming only partial exactness, there is a more complicated construction that allows us to induce a natural functor between the derived categories. This new functor is called derived functor. Without going into details, for which the reader can be referred to [37], the construction requires that the source abelian category contains enough injective objects. An object I is called injective if Hom(·, I) is exact. We say that “A contains enough injectives” if any object of A can be embedded into an injective one, and this implies that the functor ι : HKom+ (I ) → HKom+ (A ) → D+ (A ) is an equivalence. Let F : A → B be a left exact functor; the right-derived functor RF of F is an exact functor from D+ (A ) to D+ (B) that can be obtained by composing the inverse of ι with K(F ), that is the functor induced by F at level of complexes, and then projecting to the derived category of B. This can be summarized in the following formula: ι−1
K(F )
Q
D+ (A ) −−→ HKom+ (I ) ,→ HKom+ (A ) −−−→ HKom+ (B) −−B → D+ (B) In a similar way, given a right exact functor F : A → B one can construct the left derived functor LF : D− (A ) → D− (B) in an analogous way. In some cases, such as when defining the left derived tensor functor, it may be difficult to find enough projective objects. To get around this problem it is possible to use another, more general, class of objects, called F -adapted, in the place of the injectives and then proceed in the same way as above to obtain the derived functor of F . Fourier-Mukai functors are a significant class of objects to study. First of all, they have a lot of nice properties; for example they are exact, are closed with respect to the composition and arise in a clear way from geometrical transformation. A functor F : Perf (X) −→ Db (Y ) is of Fourier-Mukai type if there exists an object E ∈ Db (X × Y ), called kernel of the functor, such that F ∼ = ΦE , with L
ΦE (−) := R(p)∗ (E ⊗ q ∗ (−))
ΦE : Perf (X) −→ Db (Y ),
where p : X × Y → Y and q : X × Y → X are the projections. Fourier-Mukai functors play an important role in many geometric contexts. For example, if S is a projective K3 surface, then any other K3 surface Y for which there exists a Fourier-Mukai equivalence with kernel E, ΦE : Db (S) −→ Db (Y ) is isomorphic to a moduli space of stable sheaves on S (see, e.g., [60]). In fact all the functors mentioned until now are naturally isomorphic to Fourier-Mukai functors. In order to prove it we need to specify the kernels of these functors. As before, let X and Y be noetherian schemes and f : X → Y be a proper morphism and F be a coherent sheaf on X. Below there is a list of the kernels of the above-mentioned functors. • The kernel of the identity functor is the structure sheaf O∆ of the diagonal ∆ ⊂ X × X. • The kernel of the push forward functor f∗ : Db (X) → Db (Y ) is the structure sheaf OΓf of the graph Γf ⊂ X × Y . 17
• The shift functor [1] has O∆ [1] as kernel. • The kernel of the Serre functor SX [−n] is ι∗ ωX , where ι is the diagonal embedding and n is the dimension of X. The composition of two Fourier-Mukai functors is again of Fourier-Mukai type, and this simple observation makes it possible to produce a lot of examples of Fourier-Mukai functors. For example, to obtain the Serre functor SX one could compose the functor of the last example above with the Fourier-Mukai functor given by the shift [n]. One of the main results about Fourier-Mukai functors states that, if X and Y are smooth projective varieties and F : Db (X) −→ Db (Y ) is an exact fully faithful functor, then F is of Fourier-Mukai type. In addition, the kernel is unique up to isomorphisms (see [60]). Actually, it is expected the same for every exact functor. More precisely, the question is: are all exact functors between the bounded derived categories of smooth projective varieties of Fourier-Mukai type? There is not a positive answer to this question yet, although an effective counterexample is still missing. Anyway, even if the answer to this question were positive, the Fourier-Mukai kernel would certainly not be unique, as it was shown in [26]. Fourier-Mukai functors are current subject of study, and a lot of well known geometric problems can be reformulated by using this kind of functors. A survey about recent development of the theory can be found in [25].
2.3
Dg-categories
Triangulated categories and derived categories are not completely natural in their construction. If we look at the third axiom of a triangulated category (axiom TR3 in Definition 2.1.2, that is the one that allows us to find a map between triangles by knowing only the first two terms) we see that this axiom guarantees only the existence of such a map and not its unicity. In fact the map is not unique in general. From the geometric viewpoint, this problem can be faced using the mapping cone; this construction is functorial if we work in the category of complexes Kom(A ), but if we work in the triangulated category HKom(A ), this property is lost. The idea is to look at other categories in which this problem can be reformulated in a better way. In the differential graded category, here abbreviated DG categories, the hom-set are cochain complexes. Definition 2.3.1. A DG category is a k-linear category A consisting of the following data: • Hom(X, Y ) is a Z-graded k-module for every X, Y objects of A. • There is a differential d : Hom(X, Y ) → Hom(X, Y ) of degree one, such that for every X, Y, Z the composition Hom(X, Y ) ⊗ Hom(Y, Z) → Hom(X, Z) is a morphism of complexes of k vector spaces. A DG functor F : A → B between two DG categories is given by a map on the objects as the classical functors and maps on the spaces of morphisms: F(X, Y ) : HomA (X, Y ) → HomB (F(X), F(Y )) 18
which are morphisms of DG k-modules and are compatible with the compositions and the units. Now we provide some examples of DG categories. The DG category of complexes of k modules has all complexes of modules on k as objects. Let M and N be two complexes and n an integer. The graduation on the morphisms between M and N is given by setting as the component of Hom(M, N ) of degree n, named Hom(M, N )n , the k module morphisms f from M to N of the underlying graded modules of degree n. The differential d : Hom(M, N )i → Hom(M, N )i+1 maps the family {f i } into the family d(f )i := dM f i + (−1)n f i+1 dN where dM and dN are the differentials of the complexes M and N . Given a DG category A, we can construct an usual category, denoted by H 0 (A) and called the homotopy category associated to A. The objects are the same as the ones of the DG category A and the morphisms are defined by taking the zeroth cohomology H 0 (HomA (X, Y )). The homotopy category of the DG category of complexes of k modules is exactly the homotopy category of complexes of k modules defined in the previous sections. Conversely, we can see an ordinary k-linear category A as a DG category by taking again the same objects and by setting the complex of morphisms from X to Y as a complex concentrated in degree zero A (X, Y ). Notice that a DG category A with only one object can be viewed as a DG algebra. Indeed, the composition induces a unital and associative DG algebra structure on A(X, X). Conversely, from a unital and associative DG algebra we can construct a DG category with only one object by using the algebra multiplication to define the composition. Another example of DG category is obtained by considering the opposite Aop of a DG category A. The set of objects of Aop is the same as the one of A, and we set HomAop (X, Y ) := HomA (Y, X) together with the composition maps, given up to a sign by those of A. Definition 2.3.2. A DG functor F : A → B is called a quasi-equivalence if F(X, Y ) is a quasiisomorphism for all objects X, Y ∈ A and the induced functor H 0 (F) : H 0 (A) −→ H 0 (B) is an equivalence. We say that two objects a, b ∈ A are homotopy equivalent if they are isomorphic in H 0 (A). A notion similar to the triangulated categories in the theory of DG categories is given by pretriangulated DG categories: Let A be a DG category and define the n translation of an object X by the object X[n] representing the functor Hom(·, X)[n] Define the cone of a morphism f : X → Y of degree zero as an object Cone(f ) representing the functor 19
f∗
Cone(Hom(·, X) −→ Hom(·, Y )). A is a strong pretriangulated DG category if it admits a zero object, all translations of all objects, and all cones of all morphisms. A pretriangulated DG category is a DG category quasi equivalent to a strong pretriangulated DG category. One can prove that for every pretriangulated DG category A, the homotopy category H 0 (A) is naturally a triangulated category. Definition 2.3.3. Let Mod-k be the DG category of DG k-modules. Given a small DG category A, every DG functor M : Aop −→ dgMod-k is called a right DG A-module. Notice that Mod-k is an example of pretriangulated DG category. Definition 2.3.4. The derived category D(A) is defined by localizing the homotopy category associated with Mod-A by quasi-isomorphisms. Every object X ∈ A defines a representable DG module hX (−) := Hom(−, X). The functor h• is called the Yoneda functor, and it is fully faithful. Definition 2.3.5. A DG A-module M is called free if it is isomorphic to a direct sum of shifts of representable DG modules of the form hX [n], where X ∈ A, n ∈ Z. Definition 2.3.6. DG A-module P is called semi-free if it has a filtration 0 = φ0 ⊂ φ1 ⊂ φ2 ⊂ . . . = P such that each quotient φi /φi−1 is free. If φm = P for some m and φi /φi−1 is a finite direct sum of DG modules of the form hY [n], then we call P a finitely generated semi-free DG module. Denote by SF(A) the full DG subcategory of semi-free DG modules. Definition 2.3.7. Given a small DG category A we denote by P erf (A) the DG category of perfect DG modules, that is the full DG subcategory of SF(A) consisting of all DG modules which are homotopy equivalent to a direct summand of a finitely generated semi-free DG module. Recall that, given two DG categories A and B, their tensor product A ⊗ B is again a DG category. See [18] for references. Let A and B be two DG categories, a A-B-bimodule is a DG Aop ⊗B-module. A quasi-functor from A to B is a A-B-bimodule X ∈ D(Aop ⊗ B) such that the tensor functor (−) ⊗A X : D(A) −→ D(B) 20
takes every representable A-module to an object which is isomorphic to a representable B-module. Some further references on the tensor functor can be found in [41]. Given a triangulated category T , a way to find a pretriangulated DG category A, such that the homotopy category H 0 (A) corresponds to the initial category T , is called enhancement. Definition 2.3.8. Let T be a triangulated category. An enhancement of T is a pair (A, ), where A is a pretriangulated DG category and : H 0 (A) −→ T is an equivalence of triangulated categories. It is a natural question to wonder about the existence or the unicity of enhancements. Two enhancements (A, ) and (A0 , 0 ) of a triangulated category T are said to be equivalent if there exists a quasi-equivalence F between A and A0 . The same enhancements are said to be strong equivalent if they are equivalent and 0 ◦ H 0 (F ) ∼ = . These relations allows us to speak of uniqueness and strong uniqueness on an enhancement. The following result, proven by Lunts and Orlov, implies the uniqueness of the enhancement of the derived category of a smooth projective variety. Theorem 2.3.9 ([52], Theorem 9.9). Let X be a projective scheme over k such that the maximal torsion subsheaf of dimension zero T0 (OX ) is trivial. Then the categories Perf (X) and Db (X) have strongly unique enhancements. Such an enhancement for the category Db (X) was described by Töen in Section 8.3 of [67]. He defines the DG-category LQcoh (X) as the category of quasi coherent fibrant complexes and its full sub-DG-category Lparf (X) composed by all perfect complexes. Then Lparf (X) is an enhancement of Db (CohX) for X smooth and proper over k. The property of an object of being fibrant belongs to the theory of model categories, not analyzed here, for which the reader can refer to [30]. Other enhancements that will be useful in the following can be found on the work of Lunts and Orlov [52], where the authors find enhancements of the derived categories D(Qcoh(X)), Perf (X) and Db (X). Example 2.3.10 ([52], Corollary 7.6). The derived category D(Qcoh(X)) of a quasi-compact and separated scheme X that has enough locally free sheaves has a unique enhancement. Recall that “having enough locally free sheaves” means that for any finitely presented sheaf F there exist an epimorphism E F for a locally free sheaf E of finite type. By Proposition 1.17 of the same article of Lunts and Orlov, we have an explicit description of the enhancement of D(Qcoh(X)) by using the notion of semi-free DG modules of Definition 2.3.7: Corollary 2.3.11. There exist a DG-equivalence between the DG-category enhancement of D(Qcoh(X)), called Ddg (Qcoh(Y )) and the DG-category SF(P erf (Y )). 21
Example 2.3.12 ([52], Theorem 7.9). The category of perfect complexes Perf (X) of a quasi projective scheme X has a unique enhancement This is a consequence of a more general statement concerning the existence of an enhancement of the triangulated category of compact objects. In particular consider a category A as DG-category and L a localizing subcategory of D(A) generated by compact objects L ∩ D(A)C . Theorem 2.8 of [52] ensures that, if the following condition holds for every object Y and Z of A Hom(π(hY ), π(hZ )[i]) = 0 when i < 0 where π : D(A) → D(A )/L is the quotient functor, then the subcategory of compact objects (D(A)/L)C has a unique enhancement. Example 2.3.13 ([52], Theorem 8.13). The category Db (X) of a quasi-projective scheme X has a unique enhancement.
2.4
Stability conditions
The notion of stability condition was introduced by Bridgeland, that gave an analogue of the Hilbert-Mumford stability in the context of triangulated categories. The notion of stability condition on a triangulated category has some important features: it can be defined in a purely categorical way and, under reasonable hypotheses, we can put a natural topology on the space of the stability condition of a triangulated category A . The space Stab(A ) endowed by this topology becomes a complex manifold. Moreover there is a wall and chamber structure on this manifold, but we will focus only on how to construct Stab(A ) starting by the triangulated category A . Denote by K(T ) the Grothendieck group of the triangulated category T , that is the abelian group generated by isomorphism classes [T ] of objects of T modulo the equivalence relation [T2 ] = [T1 ] + [T3 ] for all the distinguished triangles T1 → T2 → T3 → T1 [1] Definition 2.4.1. A stability condition σ = (Z, P) on a triangulated category T is provided by a group homomorphism Z : K(T ) → C, called central charge, and full additive subcategories P(φ) of T , indexed by φ ∈ R, such that: 1. For every object 0 6= E ∈ P(φ), Z(E) has phase φ(E) := φ, i.e the complex number Z(E) is ρeiφ for a real non-negative number ρ. 2. P(φ + 1) = P(φ)[1] for all φ ∈ R. 3. If φ1 > φ2 , Ei ∈ P(φi ), then HomT (E1 , E2 ) = 0. 4. Any non-zero object E admits a Harder-Narasimhan filtration, that is a finite number of morphisms 0 = E0 → E1 → · · · → En−1 → En = E
22
such that Fj = Cone(Ej−1 → Ej ) are semistable objects with phase φ(F1 ) > · · · > φ(Fn−1 ) > φ(Fn ). The non-zero objects of P(φ) are called semistables and the objects of P(φ) that do not have proper subobjects are called stables. The family of subcategories P(φ) is also called slicing of the triangulated category T . Notice − that, for every non-zero object E of T , one has two real numbers φ+ P (E) and φP (E) corresponding to the maximum and minimum φ in point 4 of Definition 2.4.1. We then denote by P((a, b)), the + set of the zero objects of T together with those objects E which satisfy a < φ− P (E) ≤ φP (E) < b. The study of stability conditions on a triangulated category T based on the search of slicings can be a big deal. A good instrument to simplify the problem is given by the notion of t-structure. A stability condition is called locally finite if there exists greater than zero such that, for all reals φ, each subcategory P(φ − , φ + )) is of finite length, that is, it is artinian and noetherian. We require a stability condition σ to satisfy a further condition. At first we have to fix a finite dimensional lattice Γ with a map λ from K(T ) to Γ. One example of such a lattice is the numerical grothendieck group Knum (T ). Now we restrict ourselves only to stability conditions for which the morphism Z factors through Γ. We have the following Definition 2.4.2. The stability condition σ is said to have the support property if there exists a constant C > 0 such that, for every semistable object E and for every fixed norm || · || on Γ ⊗ C one has ||E|| ≤ C|Z(E)| This notion is essential to have a good behaviour of the space of stability conditions. In particular if the support property holds then all the stability conditions are locally finite. Notice that if K(T ) is discrete then the support property holds, hence all the stability conditions are locally finite. This will be the case of all the categories we will deal with in this thesis. Let Stab(T ) denote the set of stability conditions which are locally finite. Bridgeland proved that this space has a natural topology defined by a generalized metric. Stab(T ) endowed with this topology, turns out to be a complex manifold. If the Grothendieck group is finitely generated, as in our case, this manifold is of finite dimension. Definition 2.4.3. Let T be a triangulated category. A t-structure on T is given by a full additive subcategory F such that: • F [1] ⊂ F • For every object E in T , there exists a distinguished triangle F →E→G with F ∈ F and G ∈ F ⊥ . 23
The heart of a t-structure is the subcategory A := F ∩ F ⊥ [1]. Beilinson, Bernstein and Deligne introduced the notion of t-structure in [14]. Here it was proven that the heart of a t-structure is an abelian category. One can think of a t-structure as a way to see the embedding of different abelian categories into a triangulated category. A t-structure is said to be bounded if every object E ∈ T belongs to F [i] ∩ F ⊥ [j] for some i and j. The trivial t-structures on T are given by F = 0 or F = T . It seems natural to wonder if a specific heart identifies a unique t-structure. An answer to this question has been given by Bridgeland in the following Proposition 2.4.4 ([22], Lemma 3.2). Let A be a full additive subcategory of a triangulated category T . Then, A is the heart of a bounded t-structure if and only if the following properties hold: 1. For every object A and B of A and for every integers h1 > h2 , one has Hom(A[h1 ], B[h2 ]) = 0. 2. For every object E of T , there exist a finite sequence of integers h1 > h2 > . . . > hn and a collection of distinguished triangles: / E1 [
0Z
A1
/ E2
A2
/ ···
/ En−1 ^
/ En = E
An
~
with Aj ∈ A [hj ] for all j. The subcategory F is then generated by extension of the subcategories A [h], h ≥ 0. The following result gives some well known properties about t-structures and hearts. Lemma 2.4.5. Given a heart of a bounded t-structure, the filtration provided by Proposition 2.4.4 has the following properties: 1. The filtration is essentially unique up to isomorphism. In particular the shifts hj are fixed. 2. The filtration of the object X[h] can be deduced from the filtration of X. 3. The filtration of the object X ⊕ Y can be deduced from the filtrations of X and of Y . Proof. A deep description of the subject and of the first property can be found in [14]. The second property follows by shifting all the elements of the filtration by h. Finally, given a filtration with integers k1 > k2 > . . . > kn , one can always complete the filtration such that the integers are ∼ consecutive by adding the distinguished triangles Ei − → Ei → 0, notice that the object 0 belong to the category A [kj ] for every kj . The third property follows by completing the two filtration and then adding them term to term, in view of the fact that the direct sum of distinguished triangle is again distinguished.
24
Definition 2.4.6. Let A be an abelian category. A stability function on A is a group homomorphism Z : K(A ) → C such that for every non zero object E of A , the number Z(E) belongs to: H = {z ∈ C s.t. z = ρ exp(iπφ), ρ > 0, 0 < φ ≤ 1} . The notion of t-structure is deeply related to the stability conditions on a triangulated category T . Bridgeland proved in [22] that if P is the slicing of a triangulated category T , then the subcategories P(> φ) and P(≥ φ) defines t-structures on T . The hearts of such t-structures are the abelian categories P((φ, φ + 1]) and P([φ, φ + 1)), respectively. In particular one can associate the t-structure with heart P((0, 1]) to the slicing P. A stability function is said to satisfy the Harder-Narashimhan property if every non-zero object of T admits a Harder-Narasimhan filtration. Proposition 2.4.7 ([22], Proposition 5.3). Giving a stability condition on a triangulated category T is equivalent to providing a bounded t-structure on T and a stability function on its heart satisfying the Harder-Narasimhan property. Two non trivial examples of spaces of stability conditions were given by Bridgeland in [23]. He studied the space of stability conditions of K3 surfaces and of abelian curves. He proved that, in the space of stability condition of an algebraic K3 surface X over C, there is a connected component Stab† (X) that can be described by the map π : Stab(X) → N (X) ⊗ C. The image of Stab† (X) under π defines a covering and can be described in a precise way by a particular open subset of N (X) ⊗ C, where N (X) can be defined in terms of the cohomology of X. Another example was found by Macrì that studied, in [55], the space of stability conditions associated to a complete exceptional collection on a triangulated category. He manages to find the description of the stability manifold of P1 , that is a connected and simply connected 2 dimensional manifold. Moreover, in every stability conditions of P1 there exists an integer k such that the line bundles O(k) and O(k + 1) are stable. In the same article Macrì found a complete description of the stability manifold of smooth projective curves of positive genus. Theorem 2.4.8 ([55], Theorem 2.7). If C is a smooth projective curve with genus g(C) ≥ 1, then + ] the action of GL (2, R) on Stab(C) is free and transitive, and so one has + ] Stab(C) ∼ (2, R) ∼ = GL =C×H
where H denotes the complex upper half plane
2.5
Quadratic forms with values in a line bundle and Azumaya algebras
Here is a collection of results concerning quadric fibrations of low rank, in particular we will focus on tools to deal with the quadric fibrations arising from cubic threefolds and from cubic fourfolds containing a plane, topic of the last chapter. A good tool to study quadric fibrations are quadratic forms with values in a line bundle (Definition 2.5.1). We can associate to a line bundle valued 25
quadratic form a particular quadric fibration in such a way that some geometric properties of the fibration reflect on the quadratic form and vice versa (see, for example, Proposition 2.5.9 and Proposition 2.5.10). Definition 2.5.1. Let S be a scheme. A line bundle valued quadratic form on S is a triple (E , q, L ) where E is a vector bundle on S, L is a line bundle on S and q : E → L is a morphism of vector bundles such that • q(av) = a2 q(v) where a is a section of OS and v is a section of E • the morphism bq : E × E → L , defined for every v and w sections of E by bq (v, w) = q(v + w) − q(v) − q(w) is OS -bilinear . The rank of the quadratic form (E , σ, L ) is the rank of E . From now on, for the sake of simplicity, we will refer to line bundle valued quadratic form as quadratic form. This definition is the one used in [7], where Auel, Bernardara and Bolognesi provide the following useful lemma to compare different definitions of quadratic form. Lemma 2.5.2 ([7], Lemma 1.1.1). The following sets are in natural bijection: (i) Line bundle valued quadratic forms (E , q, L ). (ii) Global sections of Hom(S 2 E , L ) on S, that corresponds to morphisms of OS modules S 2 E → L. (iii) Global sections of Hom(L ∗ , S 2 E ∗ ) on S, that corresponds to morphisms of OS modules L ∗ → S2E ∗. (iv) Global sections of S 2 E ∗ ⊗ L on S. (v) Global sections of OPS (E )/S (2) ⊗ p∗ L on PS (E ), where p : PS (E ) → S is the projectivization of E on S. i
A subform of a quadratic form (E , q, L ) is given by (E 0 , q 0 , L 0 ) where E 0 ,→ E and the following diagram is commutative i /E . E 0
q0
L0
∼ =
q
/L
Notice that, in [49], the line bundle L is replaced by its dual L ∗ . The quadratic form (E , q, L ) is said to be primitive if the morphism given in part (ii) of Lemma 2.5.2 is an epimorphism. We can associate to bq a morphism ψq : E → Hom(E , L ) called polar morphism. 26
Definition 2.5.3. A quadratic form (E , q, L ) is called regular if ψq is an isomorphism. A quadratic form (E , q, L ) is called generically regular if the form is regular over the generic point of S. Problems can arise if the characteristic of the base field is equal to two. For example, in the case of a quadratic form of odd rank, the determinant of ψq turns out to be twice a certain polynomial, called half-discriminant of q in [9]. One can introduce the notion of semiregular quadratic form if its half-discriminant is a unit. The notions of regularity and semiregularity coincide if the characteristic is different from two and hence we will restrict our attention to this case. By following [17], we observe that given a regular quadratic form (E , q, L ), the polar morphism induces a k-linear involution τ of the algebra End(E ) defined by τ (f ) := h−1 f ∗ h, where h : E → E ∗ ⊗ L . Conversly, we can associate to a k-linear involution τ of End(E ), for a projective k-module finitely generated E , a quadratic form (E , q, L ) with adjoint involution being exactly τ . Definition 2.5.4. Let (E , q, L ) be a generically regular quadratic form of rank n on S. By part (iii) of Lemma 2.5.2, there exists an morphism of vector bundles σ : L ∗ → S 2 E ∗ . The d-th degeneration locus of the quadratic form, denoted by Sd ⊂ S, is a closed subscheme defined by the following sheaf of ideals Λn+1−d σ
Id = Im(Λn+1−d E ⊗ Λn+1−d E ⊗ (L ∗ )n+1−d −−−−−−→ OS ). Notice that Si+1 is contained in Si for all positive integers i, and that S1 is a divisor on S that is called discriminant divisor. In [8] we find another definition that involves the regularity of a quadratic form Definition 2.5.5. A quadratic form (E , q, L ) has simple degeneration if for every point s of S the form admits a regular subform of rank at least n − 1. By Proposition 1.5 of [8] one can immediatly see a relation between simple degeneration and the d-th degeneration loci. Proposition 2.5.6. Let (E , q, L ) be a generically regular quadratic form. Then S1 is a divisor on S and the form is regular on S r S1 . Moreover (E , q, L ) has simple degeneration if and only if S2 is empty. Proof. The proof can be done simply by recalling the definitions: the degeneration locus S1 is defined as the zero locus of the section det σ of (det E )2 ⊗ (L ∗ )n , a form is regular if the polar morphism ψq : E → Hom(E , L ) is an isomorphism. Let n be the rank of the form (E , q, L ), then det ψq : det E → det E ∗ ⊗ L n gives rise to a global section of (det E )2 ⊗ (L ∗ )n corresponding to det σ by Lemma 2.5.2. If a form is regular the section det σ does not vanish. Hence if the form is generically regular det σ defines a divisor. Now turn to the simple degeneration. S2 is defined by the vanishing locus of (Λn−1 E )2 ⊗(L ∗ )n−1 ; hence the rank of a regular subform on a point of S2 is at most n − 2. 27
We now want to describe some morphisms between quadratic forms. Definition 2.5.7. Two quadratic forms (E , q, L ) and (E 0 , q 0 , L 0 ) are similar if there exist two isomorphisms of OS -modules ψ : E → E 0 and λ : L → L 0 such that the following diagram is commutative q
E
ψ
E0
q0
/L
λ
/ L0
Two quadratic forms (E , q, L ) and (E 0 , q 0 , L 0 ) are projective similar if (E 0 , q 0 , L 0 ) is similar to the form (N ⊗ E , qN , N ⊗2 ⊗ L), where N is a line bundle on S and qN is the squaring form. The name projective similarity is used in [7] and [8], lax-similarity in [11]. In [9] and [69] the same relation is described by the theory of twisted vector bundles. We will call quadric fibration a morphism p : X → S such that the fiber Xs over a point s is a quadric. Starting with a quadratic form (E , q, L ), one can consider the projection π : PS (E ) → S, where PS (E ) is the projectivization of E on S. By denoting X ⊂ PS (E) as the zero locus of σ, it is straightforward to prove that the restriction of the projection π to X is a quadric fibration. In particular, X is called the quadric fibration associated to (E , q, L ). From now on we require all quadric fibrations to be flat, in the same spirit we will use only quadratic form with a flat associated quadric fibration. If we start with a conic fibration we have the following result. Proposition 2.5.8 ([29], Proposition 2.2). Let X be a Gorenstein conic fibration over a smooth variety S. Then π : X → S is the conic fibration associated to a quadratic form. Properties of quadratic forms reflect on the corresponding quadric fibrations. For example a quadratic form is regular if and only if the associated quadric fibration has only nondegenerate fibers, and a quadratic form is generically regular if and only if the locus on S that parametrizes degenerate fiber is a divisor of S. In general we have Proposition 2.5.9. Let (E , q, L ) be a generically regular quadratic form and let X be the associated quadric fibration. Then the d-th degeneration locus Sd ⊂ S has codimension at least d. We can relate the morphisms described by Definition 2.5.7 to the corresponding quadric fibrations. By the functoriality of the construction, two quadric fibrations associated to similar quadratic forms are isomorphic. For the converse to be true we have to use the notion of projective similarity. Proposition 2.5.10 ([7], Proposition 1.2.1). Two generically regular quadratic forms (E , q, L ) and (E 0 , q 0 , L 0 ) on a locally factorial scheme S are in the same projective similarity class if and only if the two corresponding quadric fibrations X and X 0 are isomorphic on S. In most cases we will work on a nice base S, and so we will use directly the characterization of projective similarity given by the proposition instead of the definition.
28
2.5.1
The Clifford algebra
For this subsection, we will follow the work of Knus in [43]. Let M be a module over the commutative ring R, and q : M → R be a quadratic form on R, that is a map satisfying the following properties • q(rx) = r2 q(x) for all x in M and r in R. • The form bq defined for all x, y in M by bq (x, y) = q(x + y) − q(x) − q(y) is R-bilinear. Definition 2.5.11. A Clifford algebra for (M, q) is an R-algebra C with a R-linear map i : M → C such that (i(x))2 = q(x)1C (2.3) for all x in M , and such that for any R-algebra B and any R-linear map f : M → B satisfying a property similar to (2.3), there is a unique homomorphism of R-algebras h : C → B with hi = f . Theorem 2.5.12 ([43], chapter 4, Theorem 1.1.2). For any quadratic form (M, q) there is, up to isomorphism, a unique Clifford algebra C (M, q). In the same spirit, we can associate a Z-graded algebra to each line bundle valued quadratic form, following a construction proposed by Bichsel and Knus in [17]. The submodule of even degrees, denoted by C0 , has a structure of algebra, and is called the even (part of the) Clifford algebra. The submodule of odd degrees, denoted by C1 , has a structure of C0 -bimodule, and is called the odd part of the Clifford algebra. These spaces play an important geometric role, that can be found for example in [48]. The construction proposed in [7] allows us to construct directly the even Clifford algebra: starting from a quadratic form of rank n on a scheme S, not necessarely regular, consider the tensor algebra T (E ⊗ E ⊗ L ∗ ), and define the ideals J1 = (v ⊗ v ⊗ f − f (q(v))) J2 = (u ⊗ v ⊗ f ⊗ v ⊗ w ⊗ g − f (q(v))u ⊗ w ⊗ g) where u, v, w are section of E and f, g are section of L ∗ . The even Clifford algebra of the quadratic form (E , q, L ) is defined by the quotient C0 (E , q, L ) = T (E ⊗ E ⊗ L ∗ )/(J1 + J2 ) This is not the only way to construct the even Clifford algebra of a quadratic form; we want to recall the construction of Kuznetsov in [49], where he defines a sheaf of Clifford algebras C , graded with an even and an odd parts, respectively C0 and C1 , and the description in terms of the initial quadratic forms turns out to be C0 ∼ = OS ⊕ (Λ2 E ⊗ L ∗ ) ⊕ (Λ4 E ⊗ (L ∗ )2 ) ⊕ · · · C1 ∼ = E ⊕ (Λ3 E ⊗ L ∗ ) ⊕ (Λ5 E ⊗ (L ∗ )2 ) ⊕ · · · 29
as OS -modules. Other ways to define the even Clifford algebra are the so-called splitting construction and the gluing construction see [7], Appendix A for further references. They turn out to be all equivalent, since the Clifford algebra of a quadratic form over a ring is unique. A good description can be found in Section 1.8 of [6]. The splitting construction can be used to descibe the “even Clifford algebra” C0 as a functor. We denote by C0 (q) := C0 (E , q, L ) the Clifford algebra associated to the quadratic form (E , q, L ). Proposition 2.5.13. A projective similarity between (E , q, L ) and (E 0 , q 0 , L 0 ) corresponds to an isomorphism between C0 (E , q, L ) and C0 (E 0 , q 0 , L 0 ). Recall that if S is a commutative R-algebra, we say that S is finite étale if for any maximal ideal m of R, the finite dimensional R/m-algebra S/mS is separable. Definition 2.5.14. A is an Azumaya algebra over a local commutative ring R if A is a free R-algebra of finite rank and such that the map A ⊗R Aop → EndR (A) a ⊗ b 7→ (x 7→ axb) is an isomorphism. If X is a scheme, an OX -algebra A is Azumaya if it is coherent as an OX -module and, for every closed point x of X, Ax is Azumaya over OX,x . On the set of Azumaya algebras over R we have an equivalence relation ∼: A ∼ A0 if and only if there exists two natural numbers n and n0 such that A ⊗R Mn (R) ∼ = A0 ⊗R Mn0 (R), where Mn (R) is the Azumaya algebra of the n × n matrices with coefficients in R. Definition 2.5.15. The Brauer group of a ring R, denoted by Br(R) is the set of ∼-equivalence classes of Azumaya algebras over R with the operation induced by the tensor product. The Brauer group of a scheme S, denoted by Br(S) is defined in a similar way by using the notion of Azumaya algebras over a scheme in Definition 2.5.14 and the following equivalence relation ≈: A ≈ A 0 if and only if there exist two locally free OX -modules E and E 0 of finite rank such that A ⊗OX EndOX (E) ∼ = A 0 ⊗OX EndOX (E 0 ) Sometimes, the even Clifford algebra of a quadric fibration can be described in terms of an Azumaya algebra. We have to consider two separate cases, according to the rank of the quadric fibration n being even or odd. If we consider regular quadratic form we have the following result. Proposition 2.5.16 ([5], Proposition 1.32). Let (E , q, L ) be a regular quadratic form of rank n on S. • If n is odd, then C0 (E , q, L ) is an Azumaya OS algebra of rank 2n−1 . • If n is even, then the center Z of C0 (E , q, L ) is an ètale quadratic OS algebra, and C0 (E , q, L ) is an Azumaya algebra over Z of rank 2n−2 . 30
If we consider a generically regular quadratic form (E , q, L ) with even rank, we will call discriminant cover the double cover of S ramified at the discriminant locus S1 . Proposition 2.5.17 ([49], Proposition 3.13). Let (E , q, L ) be a generically regular quadric fibration on S of even rank n, and let f : T˜ → S be the discriminant cover. Then there exists a sheaf of algebras A0 on T˜ such that f∗ (A0 ) ∼ = C0 := C0 (E , q, L ) and the functor ∼ f∗ : Coh(T˜, A0 ) − → Coh(S, C0 )
is an equivalence of categories. Moreover, the restriction of A0 to the complement of f −1 (S2 ) ⊂ T˜ is a sheaf of Azumaya algebras.
2.5.2
The stack of square roots
The stack of square roots will be useful in the following to deal with the case of quadratic form with odd rank. The first step is the definition of group functors, that are contravariant functors from the category of schemes on S to the category of groups. Then, for each scheme X on S we have a group F (X) and for each morphism X → Y , we have a morphism of groups F (Y ) → F (X). Some basic examples are the following • The additive group functor Ga , that associates to each scheme X on S the additive group of Γ(X, OX ). • The general linear group functor GL(n), that associates to each scheme X on S the group of n × n invertible matrices with entries in Γ(X, OX ) with the operation of matrix multiplication. • The particular case GL(1), called the multiplicative group functor and denoted by Gm . The group Gm (X) is the group of units of Γ(X, OX ) • The special linear group functor SL(n), that associates to every scheme X on S the group A ∈ GL(n)(X) s.t. det(A) = 1 • The r-th roots of unity group functor µr , that associates to every scheme X on S the group A ∈ Gm (X) s.t. Ar = 1 If a group functor F is represented by a scheme G, G is called a group scheme. By the Yoneda lemma one can translate the group axioms for a group scheme G. • Associativity axiom. There exists a morphism m : G ×S G → G on S such that the following diagram commutes G ×S G ×S G
1×m
m×1
G ×S G 31
m
/ G ×S G
m
/G
• Identity axiom. There exists a section : S → G for the structure morphism πG → S such that the following diagrams commute m
G ×O S G ×1
∼ =
S ×S G
/G O
G ×O S G
1
1×
/G
m
/G O 1
S ×S G
∼ =
/G
• Inverse axiom. There exists a morphism inv : G → G on S such that the following diagrams commute, where ∆ denotes the diagonal.
G
∆
/ G ×S G inv×1 / G ×S G
π
S
m
/G
G
∆
/ G ×S G 1×inv / G ×S G
π
S
m
/G
A group scheme can be defined by a scheme G on S endowed with a triple of morphisms m : G ×S G → G, : S → G and inv : G → G satisfying the axioms of associativity, identity and inverse. Let now G be an affine smooth group S-scheme, X be a noetherian S-scheme and consider right action of G on X defined by ρ : X × G → X. By following [59] we define the quotient stack [X/G] as follows: p
f
Ob([X/G](U )) = {(U ← −P − → X), P → U a G − bundle, f equivariant} where U is an S-scheme and the maps are given by G-equivariant isomorphisms of G-bundles. Recall that a map is G-equivariant if it commutes with the action of G. A G-bundle is a locally trivial bundle E on S with a right action of G on E that preserve fibers. Now turn to definition of square root stack. Further reference, in particular for the definition of r-th root stack can be found in [24]. Lemma 2.5.18 ([24], Lemma 2.1.1). Let S be a scheme, and denote by [A1 /Gm ] the quotient stack where Gm acts on A1 by multiplication. There is an equivalence between the category of morphisms from S to [A1 /Gm ] and the category whose objects are couples (L, s) where L is an invertible sheaf on S and s is a section of L. Morphisms between (L, s) and (M, t) are given by an isomorphism φ : L → M such that φ(s) = t. We can define a morphism ·2 : [A1 /Gm ] → [A1 /Gm ] induced by the square power map on A1 and Gm . By the equivalence of Lemma 2.5.18, ·2 sends the element (L, s) to its square tensor power (L2 , s2 ).
32
Definition 2.5.19. Let S be a scheme. The square root stack S(L,s,2) defined by an invertible sheaf L on S and a section s of L is the fiber product S ×·2 [A1 /Gm ] taken over the map described above. A detailed description of the n-th root stack can be found in [24]; we will focus only on the following two properties 1. Let S be a scheme and g : D → S be a morphism. There is a natural isomorphism D(g∗ L,g∗ s,2) ∼ = S(L,s,2) ×S D 2. The objects of S(L,s,2) over a scheme T can be described by four data f
(T − → S, M, t, φ) where f is a morphism, M is an invertible sheaf on T , t is a global section of M and φ : M 2 → f ∗ L is an isomorphism such that φ(t2 ) = f ∗ s. ψ
In characteristic different from two, the square root stack S(L,s,2) − → S is a Deligne-Mumford ˜ stack, and there exists an ètale morphism η : S → S(L,s,2) such that φ = ψ ◦ η is a double covering φ S˜ − → S. ˆ is Definition 2.5.20. Let Aˆ be a sheaf of algebras on S(L,s,2) . Aˆ is said to be Azumaya if η ∗ (A) φ → S as described in definition 2.5.14. Azumaya on the double covering S˜ − If the rank of (E , q, L ) is odd we will call discriminant stack the square root stack given by the line bundle det(E ∗ )⊗2 ⊗ L ⊗n and the section det σ, where σ : L ∗ → S 2 E is the map associated to q. Proposition 2.5.21 ([49], Proposition 3.15). Let (E , q, L ) be a generically regular quadric fibration on S of odd rank n, and let f : Tˆ → S be the discriminant stack. Then there exists a sheaf of algebras A0 on T˜ such that f∗ (A0 ) ∼ = C0 and the functor ∼ f∗ : Coh(Tˆ, A0 ) − → Coh(S, C0 )
is an equivalence of categories. Moreover, the restriction of A0 to the complement of f −1 (S2 ) ⊂ Tˆ is a sheaf of Azumaya algebras.
33
Chapter 3
Derived category of the scheme of dual numbers 3.1
Introduction
This chapter is devoted to the study of the derived category of the scheme of dual numbers. The starting point was to analyze a particular case not covered by the theorem of Orlov, reported in Section 2.2 of the preliminaries. This chapter is the extended version of a work joint with Francesco Amodeo ([1]), in particular here are reported some explicit calculations not mentioned in the article, such as our explicit proof of the classification of indecomposable objects of Perf (A), analyzed in Section 3.2, or some remarks about the generalization of the work to other rings, performed in Section 3.6. We will work over an arbitrary field k. A historical motivation to study Fourier-Mukai functor is the following theorem due to Orlov. Theorem 3.1.1 (Orlov, [60], Theorem 2.2). Let X and Y be smooth projective varieties and let F : Db (X) → Db (Y ) be an exact fully faithful functor. Then F is of Fourier-Mukai type. One could try to weaken the hypothesis either on the functor, for example as has been done by Rizzardo in [63], or on the varieties. The latter is the case of our interest, and in particular we drop the smoothness hypothesis. Let X be a projective scheme, and let Perf (X) be the subcategory of D(Qcoh(X)) consisting of the objects which are quasi-isomorphic to bounded complexes of locally free sheaves of finite type on X. The category Perf (X) is always included in Db (X), in view of the natural equivalence Db (X) ∼ = Dbcoh (Qcoh(X)), where Dbcoh (Qcoh(X)) is the full subcategory of Db (Qcoh(X)) consisting of objects with coherent cohomology; the equality holds if X is smooth. Let now Y be a noetherian separated scheme. The following theorem was proved, by the use of DG categories, by Lunts and Orlov in [52]. Theorem 3.1.2 (Lunts, Orlov, [52], Corollary 9.13). Let Y be a quasi-compact and separated scheme and X be a projective scheme such that the maximal torsion subsheaf of dimension zero T0 (OX ) ⊂ OX is trivial. Let: / D(QcohY )
F : Perf (X) 34
be a fully faithful functor. Then there is an object E ∈ D(Qcoh(X × Y )) such that: ΦE |Perf (X) ∼ = F. Furthermore, if Y is noetherian and F sends Perf (X) to Db (Y ), then E ∈ Db (X × Y ). The hypothesis T0 (OX ) = 0 is related to the use of ample sequences and it seems not to be a very natural assumption. What happens if we consider a projective scheme X such that T0 (OX ) 6= 0? The simplest example of such a scheme is given by Spec k. In this case the result is trivial (see [26], Remark 2.2) in view of the simple description of Db (Spec k). Thus, we could consider a zero dimensional non-smooth scheme. In such a way, the maximal torsion subsheaf of dimension zero is certainly not trivial. A basic model of such type of objects is given by the "double point scheme", which is the spectrum of the ring of dual numbers A := k[]/(2 ). We are interested in studying the subcategory Perf (Spec A) ⊂ Db (Spec A). The category Perf (Spec A) coincides with the full subcategory of compact objects. We are working on the ring A and on the scheme Spec(A), to fix the notations let Perf (A) be the full subcategory of Db (A) := Db (A − modfg ) consisting of bounded complexes of finitely generated projective modules. Since Coh(Spec A) is equivalent to A − modfg , then Perf (Spec A) and Db (Spec A) are equivalent to Perf (A) and Db (A). Our main result is the following: Theorem 3.1.3. Let Y be a quasi-compact and separated scheme. Let: / D(Qcoh(Y ))
F : Perf (A)
be a fully faithful functor. Then there is an object E ∈ D(Qcoh(Spec A × Y )) such that: ΦE |Perf (A) ∼ = F. Furthermore, if Y is noetherian and F sends Perf (A) to Db (Y ), then E ∈ Db (Spec A × Y ). Thus we show that Theorem 3.1.2 still holds in a case in which the maximal torsion subsheaf of dimension zero is not trivial, hence we do expect it is possible to avoid this hypothesis and prove the same result in a more general case. Eventually we deal with the problem of classifying all the stability conditions on the category b D (A). The main result is the following: Theorem 3.1.4. Stab(Db (A)) is isomorphic to C, the universal covering of C∗ . In order to prove the results concerning such a classification, we will exploit the study on the category Db (A) following an argument originally used by Jørgensen and Pauksztello in [40], Holm, 35
Jørgensen and Yang in [36] for the category Perf (A). In Section 3.2 we will describe the categories Perf (A) and Db (A) by using indecomposable objects. The first part is devoted to the description of indecomposable objects and the second part describes the classification of the morphisms between such objects; as we are in a k-linear category, we describe the generators of the spaces of morphisms and how the compositions between those morphisms works. In Section 3.3 we focus our attention on fully faithful endofunctors of Perf (A). We prove that every exact fully faithful functor F : Perf (A) −→ Perf (A) is an equivalence. More precisely, it is isomorphic to the composition of a shift and a push forward along an automorphism of Spec A. On the other hand, we give a concrete example of an autoequivalence of Perf (A) that is not exact. In Section 3.4 we prove the main theorem using the language of DG categories. Section 3.5 is devoted to the study of the t-structures on the category Db (A) and the space Stab(Db (A)) of the stability conditions. Eventually in Section 3.6 there are some further notes about the possible generalization of the work for a ring like k[]/(n ).
3.2 3.2.1
The category Db (A) Indecomposable complexes of Perf (A)
In order to study the category Perf (A) we want to focus on its indecomposable objects. Theorem 2.1.15, which ensures the unicity of the decomposition of the objects, can be applied in our case thanks to the following Proposition 3.2.1. Let X be a projective variety. Then Perf (X) and Db (X) are Krull-Schmidt categories. Proof. Since X is projective, the endomorphism ring of every object of Perf (X) and of Db (X) is a finitely generated k-algebra of finite dimension, and then it is semiperfect. Moreover, D(Qcoh(X)) is Karoubian, because it is a triangulated category with countably many direct sums. The subcategories Perf (X) and of Db (X) are thick and, hence, Karoubian. Each finitely generated module N over A, by the well-known structure theorem, is of the form: N = An ⊕ k m , with n and m unique natural numbers. We will see that the element k, seen as a complex concentrated in degree zero, does not belong to Perf (A). The aim of this section is to show that indecomposable complexes of Db (A) are, up to shifts, of the form given by the following definition. Definition 3.2.2. For every i ∈ N, i > 0 let: Xi := { 0
/ A(−i)
/ ···
36
/ A(−1)
/ 0 }.
X∞ := { · · ·
/A
/ ···
/ A(−1)
/ 0 }.
Where A(l) stands for the module A in the position l ∈ Z and the map corresponds to the multiplication by . This work has been already done by Keller, Yang and Zhou in [42] and by Künzer in [46]. In this section we propose an explicit way to prove the claim similar to the one followed by Künzer. We will proceed in the following way: • We reduce the problem to minimal complexes, that are complexes in which all the maps are multiples of , up to homotopy. • We show that each complex of the Definition (3.2.2) is indecomposable. • We prove that every other complex is either decomposable or isomorphic to a complex of the Definition (3.2.2). Notice that there is an equivalence between the categories HKom− (P) and D− (A), where P denotes the category of projective and finitely generated A-modules. The objects of P are isomorphic to An for some positive integer n. Hence every object of the category D− (A) can be written as
···
ft
/ Ant
/ Ant−1 ft−1 / . . .
/ An2
f2
/ An1
/ 0.
Maps are given by matrices fi with coefficients in the ring A. We will show in the following proposition that this complex is homotopy equivalent to a minimal complex. Proposition 3.2.3. Every complex of HKom− (P) is homotopy equivalent to a minimal complex. This complex is unique up to isomorphism of complexes. Moreover let C • be the complex defined by C • := (· · · → An2 → An1 → 0) if C • is homotopy equivalent to the minimal complex D• D• := (· · · → Am2 → Am1 → 0) Then mi ≤ ni for every integer i. Proof. Consider a complex C • and start by focusing on a map fi . ...
/ Ani+2
fi+2
/ Ani+1
fi+1
/ An i
fi
/ Ani−1
fi−1
/ Ani−2
fi−2
/ Ani−3
where
fi : Ani
/ Ani−1
a1,1 ··· . . fi = . ani−1 ,1 · · · 37
a1,ni .. . ani−1 ,ni
/ ...
If fi is not minimal, then there exists a complex that is homotopy equivalent to C • in which the map corresponding to fi is minimal. There exists an element ah,k of the form x + y with x = 6 0. Hence ah,k is invertible, with inverse −1 −1 −2 given by (x + y) = x − yx . Up to change the basis of Ani and Ani−1 , we can assume that such an element is a1,1 . a−1 1,1
···
a1,1
. . . ani−1 ,1
. . . ···
a1,ni
ani−1 ,ni
1
0
−a2,1 a−1 1,1
1
0
.
.
0
.
.
.
=
1 . 1
−ani−1 ,1 a−1 1,1
−a1,ni a−1 1,1
···
. . . 0
···
. . .
−a1,2 a−1 1,1
1
0
1
···
0
a2,1 a−1 1,1 . . .
M
ani−1 ,1 a−1 1,1
···
0
1 a2,1 a−1 1,1 . . .
M
ani−1 ,1 a−1 1,1
1 0 = . . . 0
0
···
0
0
0 := fi
M
The calculation above shows that there exist two matrices H and K with Kf H = fi0 . We can use them to construct an isomorphism between the complex C • and another one, that will be called C 0• in which fi0 appears. ...
...
/ Ani+2
fi+2
/ Ani+1
id
/ Ani+2
fi+2
fi+1
/ An i
id H −1 fi+1
/ Ani+1
/ Ani−1
fi
H −1
/ An i
fi−1
/ Ani−2
K
id
−1 / Ani−1fi−1 K / Ani−2
fi0
fi−2
/ Ani−3
/ ...
id
fi−2
/ Ani−3
/ ...
We observe that matrices H −1 fi+1 and fi−1 K −1 must have, respectively, a zero row and a zero column at the position of 1 in the matrix f 0 , since the compositions have always to be 0. H −1 fi+1 =
0
! fi−1 K −1 =
N
0 N0
We observe that C 0• is homotopy equivalent to the following one, that we will call D• . ...
/ Ani+2
fi+2
/ Ani+1
N
/ An i
/ Ani−1
M
N0
/ Ani−2
/A
/0
fi−2
/ Ani−3
/ ...
Therefore we have: C ∼ = D• ⊕ 0•
0
/A
id
and the last complex is homotopy equivalent to zero. We have solved the problem of non-minimality given by a1,1 . If there were others non-minimal elements one could repeat the process above until fi is minimal. To prove the claim for all the complex C • it is sufficient to apply this procedure for every non-minimal map of the complex. Now turn to the unicity part. Suppose that two minimal complexes C • and D• are equivalent in HKom− (P). It means that there exist two maps f • : C • → D• and g • : D• → C • such that g • ◦ f • ∼ idC • and f • ◦ g • ∼ idD• . That is, by considering the single elements of the complexes i i+1 g i ◦ f i = idiC • +di−1 ◦ diC • C• ◦ h + h
38
it follows that g i ◦ f i it is equal to the identity plus a nilpotent morphism, since the differential of a minimal complex is nilpotent. Hence the composition g i ◦ f i is an isomorphism. The same holds true for the composition f i ◦ g i too and so the complexes are isomorphic.
Corollary 3.2.4. The object defined by the field k concentrated in degree zero does not belong to Perf (A). Proof. It is easy to see that this object is quasi-isomorphic to the minimal complex X∞ . Suppose that X∞ is a perfect object, then it would be isomorphic in HKom(A) to a (not necessairly minimal) bounded complex C • := (0 → An1 → · · · → Ank → 0) but for the previous proposition the minimal complex associated to C • will be bounded too, and this is not possible. As a consequence of Proposition 3.2.3 we can, up to homotopy, look for the indecomposable element among the minimal complexes. The following proposition clarifies that all the objects in the family introduced in Definition 3.2.2 are indecomposables. Proposition 3.2.5. All the complexes Xn , defined in 3.2.2 are indecomposables. Proof. Let us suppose that Xn is the sum of two complexes A• and B • . By Proposition 3.2.3 we can consider the minimal complexes A0• and B 0• associated to A• and B • . The direct sum A0• ⊕ B 0 • = (· · · → Ank → · · · ) is again minimal. Since Xn is minimal too, then every exponent nk has to be smaller or equal than the corresponding exponent in Xn , that will be zero or one. That implies immediately that A0• or B 0• is zero.
The next step is to show that indecomposable elements can be found only if the exponents ni of the elements Ani are equal to 1. Lemma 3.2.6. Consider the minimal complex ···
/ Ant
ft
/ Ant−1 ft−1 / · · ·
/ An2
fi : Ani
/ Ani−1
f2
a1,1 ··· . . fi = . ani−1 ,1 · · ·
/ An1
/0
a1,ni .. . ani−1 ,ni
and suppose that the element a1,1 is non zero. Then there exists Hi (invertible, upper triangular matrix) and Ki (invertible, lower triangular matrix) such that: Ki · fi · Hi = Di 39
where: Di =
!
0
1
0 M
The proof is similar to the one of proposition 3.2.3. This lemma does not imply directly that the whole complex admit a decomposition. The proof of this last fact follows by induction. Proposition 3.2.7. Consider the minimal complex A• ft
/ Ant
···
/ Ant−1 ft−1 / · · ·
f2
/ An2
/ An1
/0
If at least one ni is not equal to 1, then the complex can be decomposed. Proof. We will proceed by induction from the right. One can always assume, by base change, that ft has a non-zero element in the place (1, 1). Using Lemma 3.2.6 one can find an isomorphism between A• and a complex in which the map corresponding to ft is of the form Dt : ! 10 Dt := 0· · · Assume by induction that all the maps after fi to be!in the form D... described above, where we 10 will use the notation Dj to indicate a matrix like . There are three cases depending on the 0· · · map fi . 1] If the element at place (1, 1) is non-zero, Lemma 3.2.6 gives two matrices Hi and Ki that realize an isomorphism between fi and a map Di . fi
/ Ani
···
Hi−1
/ Ani
···
Di−1
/ Ani−1
Di
Ki 0 Di−1
/ Ani−1
/ ...
/ Ani−2
Xi−2
/ An
/ ...
i−2
0 , . . . D 0 such that every X is invertible. This We want to show that there exists Xi−2 , . . . , X2 , Di−1 ... 2 is done in Lemma 3.2.8. 2] If the element at place (1, 1) is zero and there is a non-zero element in the first row, this element can be moved in the (1, 1) place by multiplying fi on the left by the matrix
10
!
1j id where 1j is the (j − 1)-vector with 1 in the place j and 0 elsewhere. This is an invertible, lowertriangular matrix, so, like in the case (1), we can extend the map given by this matrix to an isomorphism of A• . 3] If the first row contains only zeroes, the complex looks like
···
/
Ani
0 M
/
n A i−1
1 0 0· · ·
/
n A i−2
/
···
/
An2
and this can be decomposed in the sum of the following complexes 40
1 0 0· · ·
/
An1
/0
··· ···
/0
0
/A
/A
/ ···
/A
/ Ani M / Ani−1 −1 ··· / Ani−2 −1 ··· / · · ·
/A
/ An2 −1 ··· / An1 −1
/0 /0
To conclude the proof, it is sufficient to notice that either all the steps reduce to case 1 - 2, and so the complex has X∞ as a direct summand, or one step, say the k-th, falls into case 3, and so the complex has Xk as a direct summand. Lemma 3.2.8. Consider the following diagram, where Ki is an invertible, lower-triangular matrix. 0 , which make the Then there exist a lower triangular invertible matrix Xi−2 and a matrix Di−1 diagram commute. Ani−1
Proof. Suppose Xi−2 =
0
Z 1
· Ki =
0 Di−1
Xi−2
/ Ani−2
! 0 K , so We want to have Xi−2 Di−1 = Di−1 i
Xi−2 Di−1 = 0 Di−1
/ Ani−2
Ki
Ani−1
1
Di−1
1
1
0
Z 1 ! 0
0 M0
!
1
0
0 M ! 1 0 A N
! =
=
1
0
!
Z M 1
0
!
M 0A M 0N
Since N is invertible, then we must have M 0 = M · N −1 e Z = M · N −1 A. Hence: ! 1 0 Xi−2 = M N −1 A 1 This is still a lower triangular matrix, then we can use again the same result to get Xi−3 , . . . , X2 .
3.2.2
Maps between indecomposable complexes
In order to describe the category Perf (A) we have to study the morphisms. We can focus only on the morphisms between indecomposable objects in view of Theorem 2.1.15. If we consider A• and / B • , then we will have f = g B • two indecomposable complexes and two morphisms f, g : A• in HomPerf (A) (A• , B • ) if and only if f is homotopy equivalent to g in HomKom(A) (A• , B • ). We preliminarly study how the single squares can appear. Since complexes can be made up only by 0 or A, we can study all the cases that could happen:
41
1. In the following 9 cases we can only have zero maps between the complexes involved. 0
0
/0
0
0
0
0
/0
0
0
/0
0
A
/A
A
0
/0
0
0 0
0
0
/A
A
0
0
/A
0
0
0
0
0
/0
A
0
/0
0
/0
0
0
0
0
0
/0
0
0
0
0
0
0
0
/0 /A 0
A
0
/0
A
/A
/A
0
0
0
0
/0
0
0
0
0
/0
2. In the following two cases we could have only a non-zero map, which is a + b for every a, b ∈ k; there are no further reduction by homotopy to do. 0
0
0
0 0
/A a+b
0
A
/A
a+b
0
A
/0
0
/0
3. The diagram X is always commutative, for every a and b; notice that the map a + b is locally homotopy equivalent to the map a, the homotopy is specified in the diagram Y , and so the map can be described by the diagram Z. 0 / 0 / 0 / 0 A A A 0 0 b X := Y := Z := a a+b a a+b 0 0 0
A
/A
A
/A
A
/A
4. The diagram X is always commutative, for every a and b; as above, the map a + b is locally homotopy equivalent to the map a, the homotopy is specified in the diagram Y , and so the map can be described by the diagram Z. / / / A A A A A A a+b a a X := Y := Z := a+b 0 0 b 0 / 0 / 0 / A 0 A 0 A 0 5. The following diagram must be commutative, so we have to choose the right vertical map to be 0. There are no more reductions. A
b
A 42
/0
0
/A
6. The following diagram must be commutative, so we have to choose the left vertical map to be 0. There are no more reductions. / A A
0
0
b
/A
0
7. The following diagram is commutative if the maps have the same term in a. Moreover we can move all the terms in by homotopy. A a+b
/A
a+c
A
/A
b
a
a+b
/A
A x
a+c
A
A a+(c−b)
/A
/A
a
A
a+(c−b)
/A
Now we pass to the study of maps between indecomposable objects. Notice that for every complexes Xi , Xj and for every integers α, β: Hom(Xi [α], Xj [β]) ∼ = Hom(Xi , Xj [β − α]) and so it is sufficient to study Hom(Xi , Xj [α]) for a certain integer α. We start with the morphisms in Perf (A) by considering the space V := Hom(Xi , Xj [α]) with i, j 6= ∞. If i > j, there are five cases, from α ≤ −j to α ≥ i. 1. α ≤ −j. 0
/A
/ ···
/A
/0
/ ···
/ ···
/ ···
/ ···
/0
/ ···
/ ···
/ ···
/ ···
/0
/A
/ ···
/A
/0
0
It is clear that in this case all the vertical arrows are zero and thus V = 0.
2. −j < α ≤ 0. By the commutativity of the squares we have: 0
/A
/ ···
/ ···
/0
0
/A
/ ···
b1 /A / ···
/A
/0
bk /A / ···
/ ···
/0
/A
/0
with k = j + α. Define: B :=
k X
(−1)l+1 bk−l+1 .
l=1
Up to homotopy we can reduce the diagram to be the following one: 0
/A
/ ···
/A
/ ···
/A
/ ···
/0
0 /A
/ ···
B /A / ···
0
43
/0
/ ···
/0
/A
/0
This shows that in this case the space of morphisms V is isomorphic to k.
3. 0 < α < i − j. By the commutativity of the squares we have: 0
/A
/ ···
/ ···
/0
0
/A
/ ···
/A bk /A
b1 /A / ···
/ ···
/A
/0
/0
/ ···
/0
Up to homotopy the morphism is zero, hence V = 0.
4. i − j ≤ α < i and α 6= 0. This case is similar to (2). By the commutativity of the squares we have: 0
/ ···
/0
/A
/ ···
0
/A
/ ···
/A
/ ···
a+bk /A /0
a+b1 /A / ···
/A
/0
/ ···
/0
Up to homotopy we can reduce the diagram to be the following one: 0
/ ···
/0
/A
/ ···
/A
/ ···
/A
/0
/A
/ ···
a /A
/ ···
a /A
/0
/ ···
/0
0
Thus V is still isomorphic to k. 5. i ≤ α. This case is similar to (1). Thus, V is equal to zero. If i = j, the calculations are similar to the previous case. Notice that (3) can not hold in this case. However, if α = 0, by the commutativity of the squares we obtain: 0
/A
a+b1 /A / ···
0 Define:
/ ···
/A
/0
a+bh /A /0
i X C := (−1)l+1 bi−l+1 . l=1
Up to homotopy we can reduce the diagram to be the following one: 0
/A
/ ···
/A
a /A
/ ···
a+C /A /0
0
/0
This shows that in this case the space of morphisms V is equal to k ⊕ k.
44
If i < j, the calculations are similar to the case i > j. We can sum it all up in the following: Proposition 3.2.9. Consider the space V = Hom(Xi , Xj [α]): • If −j < α ≤ min{0, i − j} and (i − j, α) 6= (0, 0) then V has dimension 1 and it is generated by ij[α] . These morphisms are named of k -type. • If max{0, i − j} ≤ α < i and (i − j, α) 6= (0, 0) then V has dimension 1 and it is generated by 1ij[α] . These morphisms are named of k1 -type. • If i = j and α = 0 then V has dimension 2 and it is generated by both ii[0] and 1ii[0] . These morphisms are named of k 2 -type • V = {0} for all the remaining cases. A morphism between two indecomposable objects can be described by the couple (a, b) of elements of k, where a is the coefficient of the generator 1 and b is the coefficient of the generator . Remark 3.2.10. The results of this proposition can be extended to Db (A); one can easly prove that • Hom(X∞ , X∞ [h]) is generated by 1 if h ≥ 0 and is 0 otherwise. • Hom(X∞ , Xi [h]) is generated by if −i < h ≤ 0 and is 0 otherwise. • Hom(Xi , X∞ [h]) is generated by 1 if 0 ≤ h < i and is 0 otherwise. Corollary 3.2.11. For all A• , B • ∈ Perf (A), one has HomPerf (A) (A• , B • ) ∼ = HomPerf (A) (B • , A• ) Proof. Every object A• is homotopy equivalent to direct sum of indecomposable object, so A• ∼ = L L • ∼ j (Ij ); here we denote by Ii one indecomposable object as, for example, Xn [α]. i (Ii ) and B = The statement is true for indecomposable object, so one has M M HomPerf (A) (A• , B • ) ∼ (Ii ), (Ij )) ∼ = HomPerf (A) ( = i
∼ =
M
j
HomPerf (A) (Ii , Ij ) ∼ =
i,j
M
HomPerf (A) (Ij , Ii ) ∼ =
i,j
∼ = ... ∼ = HomPerf (A) (B • , A• )
Since the Hom-spaces are vector spaces of finite dimension, it can be proved that the isomorphism HomPerf (A) (A• , B • ) ∼ = HomPerf (A) (B • , A• )∗ is natural. (See Residue and Duality by R. Hartshorne, section 6. for further details). 45
3.2.3
Compositions
We are now wondering how the composition of morphisms in Perf (A) works; that is, given two morphisms between indecomposable objects f : Xi → Xj [α] and g : Xj [α] → Xk [β], we are asking for what type the morphism g ◦ f is. We are going to study the compositions of the generators of the morphism described in Proposition 3.2.9. The situation is summed up in the following table. Clearly, if either f or g is the zero morphism, then also the composition g ◦ f is zero. ◦
0
1ij[α]
ij[α]
0
0
0
0
j[α] 1k[β] j[α] k[β]
0
(i)
(ii)
0
(iii)
(iv)
Proposition 3.2.9 gives the bounds for the generators to be well defined. Case (i) holds when max{0, i − j} ≤ α < i and max{0, j − k} ≤ β − α < j. Case (ii) holds when −j < α ≤ min{0, i − j} and max{0, j − k} ≤ β − α < j. Case (iii) holds when max{0, i − j} ≤ α < i and −k < β − α ≤ min{0, j − k}. Case (iv) holds when −j < α ≤ min{0, i − j} and −k < β − α ≤ min{0, j − k}. j[α]
(i) The composition of 1k[β] ◦ 1ij[α] is a morphism from Xi to Xk [β]. If max{0, i − k} ≤ β < i holds, that is the condition of having a morphism of k1 -type between Xi and Xk [β], then j[α] 1k[β] ◦ 1ij[α] = 1ik[β] , as shown in the following diagram:
A 1
A
/ ···
/ ··· 1
/ ···
A
/ ···
/ ···
/ ···
1 /A
/ ···
1
/ ···
1
1 /A
1 /A
/ ···
1
1 / A.
/A
j[α]
Otherwise 1k[β] ◦ 1ij[α] = 0 j[α]
(ii) The composition 1k[β] ◦ ij[α] is a morphism from Xi to Xk [β]. If −k < β ≤ min{0, i − k} holds, that is the condition of having a morphism of k -type between Xi and Xk [β], then j[α] 1k[β] ◦ ij[α] = ik[β] , as shown in the following diagram: A
/ ···
/ ··· 0
A
1
A
/ ···
/ ···
/A
0
/ ··· 1
/ ···
46
/A
/ ···
/ ···
1 /A
/ ···
1
1 / A.
/A
j[α]
Otherwise 1k[β] ◦ ij[α] = 0. j[α]
(iii) The composition of k[β] ◦ 1ij[α] is a morphism from Xi to Xk [β]. If −k < β ≤ min{0, i − k} holds, that is the condition of having a morphism of k -type between Xi and Xk [β], then j[α] k[β] ◦ 1ij[α] = ik[β] , as shown in the following diagram:
A
/ ···
A
/ ···
/ ···
1 /A
/ ···
1
1 /A
/ ···
0
/ ···
A
/ ···
/A
/ ···
/ ···
0
/ A.
j[α]
Otherwise k[β] ◦ 1ij[α] = 0. (iv) The composition of two morphisms of k -type is always zero. As in Remark 3.2.10, the above results hold, with the same inequalities, also in Db (A).
3.3
Fully Faithful Endofunctors of Perf(A)
In this section we will deal with k−linear functors that commute with the shifts. Two simple examples of such type of functors are given by the shift [n] and the push forward Rf∗ along a proper morphism f of projective varieties. Also, these two functors are exact and of Fourier-Mukai type; see [37] for a deeper discussion. For a more general analysis, in this section we will suppose F to be fully faithful but we will not require the functor to be exact. Recall that the category we are working on are Krull-Schmidt, as proved in Proposition 3.2.1, hence are Karoubian and then the following lemma can be applied. Lemma 3.3.1. Let C be a Karoubian triangulated category, D an additive category and F : / D a fully faithful additive functor. Then F sends indecomposable objects of C to indecomC posable objects of D. Proof. Let X • be an indecomposable object of C . Since C is Karoubian, Hom(X • , X • ) does not contain any idempotent except the identity and zero. Suppose F (X • ) ∼ = Y • ⊕ Z • , with Y • and Z • non zero. Since F is fully faithful and additive we have an isomorphism of rings: Hom(X • , X • ) ∼ = Hom(F (X • ), F (X • )) ∼ = Hom(Y • ⊕ Z • , Y • ⊕ Z • ). The last space contains the projection Y • ⊕ Z • the identity and zero, giving a contradiction. 47
/ Y • , which is an idempotent different from
First of all we can describe the action of the functor F on the object of the category Perf (A). / Perf (A) be a fully faithful functor. On the objects, F Proposition 3.3.2. Let F : Perf (A) is isomorphic to the shift functor [n] for some integer n.
Proof. F commutes with the shifts, so we can focus on the image of an indecomposable object Xi for any integer i > 0. By Proposition 3.3.1, F sends indecomposable objects to indecomposable objects, so F (Xi ) ∼ = Xj [α] for some integer j > 0 and some α. F is also fully faithful, thus:
Hom(Xi , Xi [β]) ∼ = Hom(F (Xi ), F (Xi )[β]) ∼ = Hom(Xj [α], Xj [α + β]) ∼ = ∼ = Hom(Xj , Xj [β]). It follows from Proposition 3.2.9 that i = j, and this proves that F (Xi ) ∼ = Xi [hi ] for some integer hi . Actually hi does not depend on i:
Hom(Xi , Xj ) ∼ = Hom(F (Xi ), F (Xj )) ∼ = Hom(Xi [hi ], Xj [hj ]) ∼ = ∼ = Hom(Xi , Xj [hj − hi ]). Again, by Proposition 3.2.9, hj − hi = 0. Corollary 3.3.3. Every fully faithful functor F : Perf (A)
/ Perf (A) is an equivalence.
Proof. It is clear from Proposition 3.3.2 that every fully faithful functor Perf (A) −→ Perf (A) is also essentially surjective, hence it is an equivalence. With similar arguments, and by including the indecomposable objects X∞ , Proposition 3.3.2 and Corollary 3.3.3 can be extended to a fully faithful functor F : Db (A) −→ Db (A). Remark 3.3.4. Due to Proposition 3.3.2, F (Xi ) is isomorphic to Xi [h] for a fixed h ∈ Z. Up to composition with a shift [−h], we can assume that F is isomorphic to the identity functor on the objects. We now want to study the action of F on the morphisms between indecomposable elements. Proposition 3.3.5. Consider a morphism (a, b) as described in Remark 3.2.10 from an indecomposable object Xi to itself, that is 0
/A
0
a
/A
/A
/ ···
/A
a
/A
/ ···
a
/A
/A
a+·b
/A
/0 /0
with a, b ∈ k. The action of the functor on the morphism (a, b) is given by an invertible matrix: ! 1 0 . 0 δi with δi ∈ k. 48
Proof. Since F is a functor, it must preserve compositions and the identity. The action of the functor on the morphism (a, b) is given by an invertible matrix: ! λ1 λ2 . λ3 λ4 The identity morphism (1, 0) is sent to (λ1 , λ3 ), then λ1 = 1 and λ3 = 0. Furthermore, if we compose (a, b) with itself, we get the morphism (a2 , 2ab) which is sent to (a2 + 2abλ2 , 2abλ4 ). On the other hand, if we compose the image of (a, b) with itself we obtain ((a + bλ2 )2 , 2(bλ4 ) · (a + bλ2 )); then (a2 + 2abλ2 , 2abλ4 ) = ((a + bλ2 )2 , 2(bλ4 ) · (a + bλ2 )) and this proves λ2 = 0.
In section 3.2.3 we have seen that one morphism between indecomposable objects can range among four different types: 0, k1 , k and k 2 . By proposition 3.3.5 and since F is a k-linear functor, its action can be described as a multiplication by specified coefficients of the field k; in particular, we can think the functor F acts on the morphisms between Xi and Xj [α] by multiplying the generator i ; in the particular case of Hom(X , X ) the previous proposition 1ij[α] or ij[α] by the coefficient kj[α] i i i i states that ki[0] refers to the generator i[0] . This shows that if (a, b) is a morphism of k 2 -type from Xi to itself, then F acts only on its second component, which is the one generated by k . Hence the following definition makes sense. i Definition 3.3.6. For all i, j ∈ N and α ∈ Z we define kj[α] ∈ k such that:
• if (a, b) is a morphism of k 2 -type from Xi to Xi , then i F (a, b) = (a, ki[0] b).
• if (a, 0) is a morphism of k1 -type from Xi to Xj [α], then i F (a, 0) = (kj[α] a, 0).
• if (0, b) is a morphism of k -type from Xi to Xj [α], then i F (0, b) = (0, kj[α] b). i Notice that the element ki[0] corresponds to δi in Proposition 3.3.5. The functor F is fully i faithful, hence all the coefficients kj[α] are non zero. i Proposition 3.3.7. ki[0] does not depend on i ∈ N r {0}.
49
i 1 Proof. We prove that ki[0] = k1[0] for i > 1. Consider the following morphisms from Xi to X1 and from X1 to Xi :
Xi :
/A
0
/ ···
/A
i1[0]
/0
0
X1 :
/0
11i[0]
Xi :
/A
0
/ ···
/0
/A
/0
1
/A
/ 0.
i i 1 1 1 1 i The functor F sends i1[0] to k1[0] 1[0] and 1i[0] to ki[0] 1i[0] ; moreover, the composition 1i[0] ◦ 1[0] = i 1i . As F preserves 1ii[0] is a morphism between Xi and Xi and then it is sent by F to ki[0] i[0] compositions: F (11i[0] ◦ i1[0] ) = F (11i[0] ) ◦ F (i1[0] ), i i 1i 1 i i 1 i which means ki[0] i[0] = ki[0] k1[0] 1i[0] . It follows ki[0] = ki[0] k1[0] . By composing these morphisms in the inverse order we get i1[0] ◦ 11i[0] = 111[0] , a morphism between X1 and X1 . It is sent by F to 1 11 ). Again, F preserves compositions, hence k 1 1 i i 1 (0, k1[0] 1[0] 1[0] = ki[0] k1[0] , that is ki[0] = k1[0] .
Proposition 3.3.8. Up to composing with a shift and a push forward along an automorphism of Spec(A), the functor F is isomorphic to a functor which is the identity on the objects and has i coefficients ki[0] equal to 1. Proof. Assume, as in Remark 3.3.4, that F is isomorphic to the identity on the objects. Moreover, i it acts as the multiplication by µ := ki[0] on the morphisms of k 2 -type, which is constant over i ∈ N / A defined as follow: by Proposition 3.3.7. Now consider the map φµ : A a + b 7−→ a + µb. The induced push forward functor (φµ )∗ on Perf (A) is isomorphic to the identity on the objects and it acts as multiplication by µ−1 on morphism of k 2 -type. Up to isomorphisms of functors, described explicitly in the Remark 3.3.9, the composition (φµ )∗ ◦ F is the identity on the objects and acts as the identity on morphisms of k 2 -type. Remark 3.3.9. Let F be an automorphism of Perf (A), and let F be isomorphic to the identity on the objects. Then, for all X ∈ Perf (A), there exists an isomorphism φX : F (X) → X. One can use φX to define a new functor naturally isomorphic to F . On the objects we define: G(X) := X Now, for all f : X → Y we define G(f ) to be the morphism that makes the following diagram commute: F (X)
φX
F (f )
F (Y )
φY
/ G(X)
G(f ):=φY ◦F (f )◦(φX )−1
/ G(Y )
50
By proposition 3.3.7, the action of G on the morphism of k 2 is the multiplication on the i / Xi -component by ki[0] = λ 6= 0. Let’s consider the following isomorphism: ψi : Xi /A
0
(1,0)
/A
0
/A
λ
/A
(λ,0)
/A
λ
/ ···
/A
(λ2 ,0)
/A
λ
/0
(λi−1 ,0)
/ · · · λ / A
/0
Now, one can define a functor H from functor G using ψi instead of φXi , and the identity morphism in the other cases. ψi
G(Xi )
/ H(Xi )
G(f )
G(Xi )
H(f ):=ψi ◦G(f )◦(ψi )−1
ψi
/ H(Xi )
H sends the indecomposable object 0
/A
/A
0
/A
λ
/A
λ
/ ···
/A
/0
/ · · · λ / A
/0
to the object
On the morphisms of k 2 type, H(f ) = G(f ), then H multiplies by λ on the -component of every morphism between objects. From now on, in view of Proposition 3.3.8, we can assume that the functor F satisfies the following condition: i (C1) F is the identity on the objects of Perf (A) and the coefficients ki[0] of F are equal to 1. i Lemma 3.3.10. Let kj[α] be the coefficient of a functor F satisfying (C1). The following relations hold: j i (R1) ki[α] kj[−α] =1 if −i < α ≤ min{0, j − i}
(R2) (R3) (R4) (R5)
j j i−1 ki[α] = ki−1[α] ki[0] j j i−1 ki[α] = ki−1[0] ki[α] i−1 i−1 i−1 ki[α] = ki−1[α] ki[0] i−1 i−1 i−1 ki[2−i] = ki−1[1] ki[1−i]
or max{0, j − i} ≤ α < j. 0 ≤ α < j ≤ i, (i − j, α) 6= (0, 0), (1, 0). j < i − 1 and −i < α ≤ j − i. 1 − i < α < 0. i > 2.
Proof. (R1) For i = j and α = 0 the statement is trivial. In the other cases notice that, when the j i first inequality holds, ki[α] is related to a morphism of k -type, kj[−α] is related to a morphism of k1 -type and the composition is a non zero morphism of k -type between Xj and Xj . When the second inequality holds, the types are swapped and the composition is still non zero. So we have: j j i ki[α] kj[−α] = kj[0] = 1.
51
(R2) The morphisms from Xj to Xi [α], from Xj to Xi−1 [α] and from Xi−1 to Xi [0] are of k1 -type, hence case (i) of Section 4 implies that the composition: j 1ji[α] = 1i−1 i[0] ◦ 1i−1[α]
is non zero. (R3) The morphism from Xj to Xi−1 [0] is of k1 -type, the morphisms from Xj to Xi [α] and the morphism from Xi−1 to Xi [α] are both of k -type, hence case C of Section 4 implies that the composition: j ji[α] = i−1 i[α] ◦ 1i−1[0] is non zero. (R4) The morphism from Xi−1 to Xi [0] is of k1 -type, the morphisms from Xi−1 to Xi [α] and the morphism from Xi−1 to Xi−1 [α] are both of k -type, hence case (ii) of Section 4 implies that the composition: i−1 i−1 i−1 i[α] = 1i[0] ◦ i−1[α] is non zero. (R5) The morphism from Xi−1 to Xi−1 [1] is of k1 -type, the morphisms from Xi−1 to Xi [1 − i] and the morphism from Xi−1 to Xi [2 − i] are both of k -type, hence case (iii) of Section 4 implies that the composition: i−1 i−1 i−1 i[1−i] ◦ 1i−1[1] = i[2−i] is non zero.
Given a set of objects E ⊂ Ob(Perf (A)) we denote by Add {E } the smallest full subcategory of Perf (A) containing E and closed under shifts and finite direct sums. Lemma 3.3.11. Let F be a functor satisfying (C1). The functor F is isomorphic to a functor F 0 0i−1 satisfying (C1) and such that the coefficients ki[0] of F 0 , are equal to 1 for all i > 1. Proof. The isomorphism of functors between F and F 0 is given by the coefficients: φ1 = 1 φi =
i−1 Y
h−1 −1 (kh[0] ) : Xi
/ Xi
h=1 0i−1 The following diagram is commutative, and shows that ki[0] = 1 concluding the proof of the lemma.
52
Qi−2
Xi−1
h=1
h−1 −1 (kh[0] )
/ Xi−1 0i−1 f ·ki[0]
i−1 f ki[0]
Qi−1
h=1
Xi
/ Xi .
h−1 −1 (kh[0] )
Now we can choose a functor F satisfying (C1) and, by Lemma 3.3.11, the condition i−1 (C2) The coefficients ki[0] of F are equal to 1 for all i > 1.
Theorem 3.3.12. Let F be a functor satisfying (C1) and (C2). The action of F on the morphisms 2 is completely determined by its coefficient k1[1] = λ. In particular: i kj[α] = λα
(3.1)
for −j < α ≤ min{0, i − j} or max{0, i − j} ≤ α < i. Proof. We proceed by induction on the number of the indecomposable objects generating the subcategory Add {X1 , . . . , Xi }. On the subcategory Add {X1 , X2 } we have: (R1)
(R1)
1 2 2 1 k2[0] k1[0] = 1 and k1[1] k2[−1] = 1, (R2)
(C2)
2 2 1 k2[1] = k1[1] k2[0] = λ, (R1)
2 2 k2[−1] = (k2[1] )−1 = λ−1 .
Notice that, by Proposition 3.2.9, these equalities determine the behaviour of F on all the coefficients and prove (3.1) for the subcategory Add {X1 , X2 }. Now assume that (3.1) holds true for the subcategory Add {X1 , . . . , Xi−1 } and prove it for the i subcategory Add {X1 , . . . , Xi }. By assumption ki[0] = 1. By the description of the morphism of Proposition 3.2.9, it is clear that the following steps cover all the remaining coefficients of the functor on Add {X1 , . . . , Xi }. j i (i) ki[0] for all j < i (deducing the case of kj[0] by (R1)). j i (ii) ki[α] for all 0 < α < j, j < i (deducing the case of kj[−α] by (R1)). j i (iii) ki[α] for all −i < α ≤ j − i, j < i (deducing the case kj[−α] by (R1)). i i (iv) ki[α] for all 0 < α < i (deducing the case ki[−α] by (R1)).
As for the proof: 53
i−1 (i) If j = i − 1 one obtains ki[0] = 1 by (C2). j For j < i − 1, by induction ki−1[0] = 1. We have: (R2)
(C1)
j j i−1 ki[0] = ki−1[0] ki[0] = 1. j (ii) By induction ki−1[α] = λα . The claim is true because (R2)
j j i−1 ki[α] = ki−1[α] ki[0] = λα . j (iii) If j 6= i − 1, by induction ki−1[0] = 1, then: (R3)
j j i−1 i−1 ki[α] = ki−1[0] ki[α] = ki[α] . i−1 Therefore we have to prove the claim only for ki[α] . In this case 1 − i ≤ α < 0. i−1 • If 1 − i < α < 0, by induction ki−1[α] = λα , then: (C2)
(R4)
i−1 i−1 i−1 = λα ki[0] ki[α] = ki−1[α] i−1 i−1 • If α = 1 − i, it is sufficient to notice that ki[α+1] = ki[2−i] belongs to the previous case i−1 and by induction ki−1[1] = λ. Hence: (R5)
i−1 i−1 i−1 ki[1−i] = ki[2−i] (ki−1[1] )−1 = λα+1 λ−1 = λα .
(iv) We have: (R2)
(iii)
i−1 i i i ki[α] = ki−1[α] ki[0] = ki−1[α] = λα .
So the claim is true. Corollary 3.3.13. Let F be a functor satisfying (C1) and (C2). If F is exact, then it is isomorphic to the identity functor. 2 Proof. It suffices to show that, if F is exact, then λ = k1[1] = 1. Consider the following distinguished triangle:
X1
11[0]
/ X1
i
/ C(1 ) 1[0]
p
/ X1 [1],
since the cone C(11[0] ) on the morphism 11[0] is isomorphic to X2 , the triangle becomes: X1
11[0]
/ X1
112[0]
/ X2
121[1]
/ X1 [1].
(3.2)
Now F sends the previous triangle in to the following one: X1
11[0]
/ X1
112[0]
/ X2
54
121[1] λ
/ X1 [1].
(3.3)
Since F is exact, the triangle (3.3) is distinguished, hence it is isomorphic to the distinguished triangle (3.2). So we have X1
11[0]
id
X1
11[0]
/ X1
112[0]
/ X2
id
/ X1
112[0]
/
λ121[1]
/ X1 [1]
a+b 121[1] / X2
id
X1 [1].
The diagram is commutative up to homotopy, hence: ( a=λ a=1 Thus λ = 1 and, by Theorem 3.3.12, the functor F is the identity. Corollary 3.3.14. Every exact autoequivalence of Perf (A) is of Fourier-Mukai type. Proof. See [37], Proposition 5.10, for the proof that the composition of Fourier-Mukai functor is again of Fourier-Mukai type. F is the identity up to shifts and push forwards functors, which are both of Fourier-Mukai type. Hence F itself is a Fourier-Mukai functor. Corollary 3.3.15. If k 6= Z2 , then there exist an autoequivalence Perf (A) that is not exact. 1 6= 0, 1, set all the coefficients as described in the Theorem 3.3.12. Proof. Choose the coefficient k2[1] The functor F is well defined since all the compositions are well posed: j[α]
j i i i kj[α] kl[β] = kj[α] kl[β−α] = λα+β−α = λβ = kl[β] .
By Corollary 3.3.13, F is not exact.
3.4
Main Theorem
In the following we give the proof of Theorem 3.1.3 Proof. We know by ([52]) that there exist enhancements of the derived categories D(Qcoh(Spec A)) and D(Qcoh(Y )), we call them Ddg (Qcoh(Spec A)) and Ddg (Qcoh(Y )) respectively. Also, by [52] Proposition 1.17, these enhancements are quasi equivalent to the DG categories SF(P erf (A)) and SF(P erf (Y )). Denote by: φA : Ddg (QcohA)
/ SF(P erf (A))
φY : Ddg (QcohY )
/ SF(P erf (Y ))
the corresponding quasi-functors. The functor F induces an equivalence: F˜ : Perf (A)
55
∼
/ H 0 (C)
where C is the full DG subcategory in SF(P erf (Y )) consisting of all objects in the essential image of H 0 (φY ) ◦ F . By [52], Theorem 6.4, there is a quasi-equivalence: /C
F : P erf (A) which induces a quasi-equivalence:
/ SF(C) .
F ∗ : SF(P erf (A))
Let D ⊂ SF(P erf (Y )) be a DG subcategory that contains P erf (Y ) and C. Denote by J : / D and I : P erf (Y ) / D the respective embeddings. Let H := φ−1 ◦ I∗ ◦ J ∗ ◦ F ∗ ◦ φA : C Y Ddg (QcohA)
/ Ddg (QcohY ) be the functor that makes the following diagram be commutative:
/ Ddg (QcohY )
H
Ddg (QcohA) φA
SF(P erf (A))
F∗
J∗
/ SF(C)
/ SF(D)
I∗
φY
/ SF(P erf (Y ))
Notice that H 0 (H) commutes with direct sums, hence ([52], Theorem 9.10) the functor H 0 (H) is isomorphic to ΦE with E ∈ D(Qcoh(Spec A × Y )). As observed in the proof of [52], the restriction of I∗ ◦ J ∗ on C is isomorphic to the inclusion / SF(P erf (Y )) , hence the restriction ΦE |Perf (A) is fully faithful. C Let A be the full subcategory of Perf (A) whose object is only A, and let j : A → Perf (A) be the natural embedding. Define: / Perf (A) G := H 0 (F)−1 ◦ F˜ : Perf (A) By [52], Theorem 6.4, there is an isomorphism of functors: j
/G◦j
∼
on the category A. Hence, by Corollary 3.3.13, the functor G is the identity on the whole Perf (A). Therefore, the functors H 0 (F) and F˜ are isomorphic, that is: (H 0 (φY ) ◦ H 0 (H))|Perf (A) ∼ = (H 0 (φY ) ◦ F ) ⇒ ΦE |Perf (A) ∼ = F. Finally if Y is noetherian and F sends Perf (A) to Db (Y ), then [52], Corollary 9.13, implies: E ∈ Db (Coh(Spec A × Y )).
Corollary 3.4.1. Let Y be a quasi-compact and separated scheme. Let: F : Db (Spec A)
/ D(QcohY )
be a fully faithful functor that commutes with homotopy colimits. Then there is an object E ∈ D(Qcoh(X × Y )) such that: ΦE |Db (A) ∼ = F. 56
Proof. Corollary 9.14 in [52] shows a similar result: if X is a projective scheme such that T0 (OX ) = 0 and Y is a quasi-compact separated scheme, then for every fully faithful functor that commutes with homotopy colimits: F : Db (X) −→ D(Qcoh(Y )) there is an object E ∈ D(Qcoh(X × Y )) such that: ΦE |Db (X) ∼ = F. The authors assume T0 (OX ) = 0 in order to prove that the restriction of the functor F to the subcategory of perfect complexes Perf (X) is of Fourier-Mukai type. In our case we have actually T0 (OA ) 6= 0, but we have already shown in Theorem 3.1.3 that the restriction of F to Perf (A) is a Fourier-Mukai functor. Hence we do not need this hypothesis and the proof follows as in Corollary 9.14, [52].
3.5 3.5.1
t-structures and stability conditions t-structures on Db (A)
As a preliminary to the case of Db (A), we are interested in the case of Perf (A). Holm, Jørgensen and Yang proved, in the context of spherical objects ([36]), that all the t-structures on Perf (A) are trivial. The proof of this fact follows easily by a direct calculation: a similar argument to Proposition 3.5.4 allows us to consider only the subcategory Add {X1 [h]} as heart, and it is easy to verify that such a subcategory can not satisfy the property (2) of Proposition 2.4.4. Our proof of this fact will follow from the description of the category Perf (A) given in the previous sections. Consider an abelian subcategory A of Perf (A), that will be the heart of a t-structure. Lemma 3.5.1. Every abelian subcategory C ⊂ Perf (A) is thick. Proof. If C contains the direct sum A ⊕ B, then C contains the kernel of the morphism (0,id)
A ⊕ B −−−→ A ⊕ B that is A. The next proposition immediately follows by looking at the indecomposable objects in A , and gives a strong restriction on the possible hearts of bounded t-structures on Perf (A). Proposition 3.5.2. Up to a shift, A is the subcategory Add {X1 }. Proof. By Lemma 3.5.1 it is sufficent to check which indecomposable object does A contain. By the classification of the morphism between indecomposable objects given in Proposition 3.2.9, we know that Hom(Xi [h], Xi [h − 1]) = Hom(Xi , Xi [−1]) 6= 0 57
and this is in contradiction with the property (1) of 2.4.4. It follows that the indecomposable object Xi [h] does not belong to A if i > 1. Since the subcategory A is not trivial, up to a shift we can assume that the indecomposable X1 belongs to A . To conclude the proof, A can not contain other indecomposable X1 [h], h 6= 0, because there would be again a contradiction with the property (1) of 2.4.4. Since A is additive, Add {X1 } = A . The previous proposition is not sufficient to prove that A is the heart of a bounded t-structure, because of the property (2) of Proposition 2.4.4. Proposition 3.5.3. If an object E satisfies the property (2) of Proposition 2.4.4 then it is of the form ⊕i X1 [ki ] for a finite different number of ki . Proof. By the distinguished triangle ∼
0 → ⊕X1 [k1 ] − → ⊕X1 [k1 ] → 0 it is clear that the object E1 of the filtration can be only of the form ⊕X1 [k1 ] for a certain integer k1 . Now, by looking at the next distinguished triangle one has E1 → E2 → A2 → E1 [1] where E1 = ⊕X1 [k1 ] and A2 = ⊕X1 [k2 ]. The integer k2 must be less than k1 , then the map A2 → E1 [1] is zero, and so E2 = ⊕X1 [k1 ] ⊕ X2 [k2 ]. The claim follows by repeating the argument a finite number of times. The previous proposition makes clear that the subcategory Add {X1 } does not satisfy the property (2) of Proposition 2.4.4. Then there are no possible hearts of bounded t-structures in Perf (A), and then no possible t-structures. Let us turn to analyze the case of the category Db (A). In fact, it is generated, as triangulated category, by the indecomposable object X∞ . This case is not covered by [36], since Db (A) is not generated by any spherical object. First of all we have to consider that, since we are dealing with the derived category of a projective variety, we are sure that at least the standard t-structure on this category exists: F = {X ∈ Db (A) s.t. H i (X) = 0 for every i ≥ 0} is the standard t-structure on Db (A). Its heart is the subcategory Add {X1 , X∞ } = A − modfg [1]. Proposition 3.5.4. Up to shift, the unique t-structure on Db (A) is the standard one. Proof. We look for all possible hearts satisfying the two properties of Proposition 2.4.4. Since the heart A is abelian, it is sufficient to check which indecomposable objects does A contain. Thanks to the first part of Proposition 2.4.4 it is easy to verify that, up to shifts, the only admissible candidates for hearts are A = Add {X1 }, A = Add {X∞ } and A = Add {X1 , X∞ }. The first case is not possible, since X1 does not generate the whole category Db (A). The distinguished triangle:
1
X∞ → − X1 → − X∞ is an extension of X1 by elements of Add {X∞ }, and so if X∞ is an element of A , then X1 is such. It follows that the unique possibility is A = Add {X1 , X∞ }. 58
It could be interesting to look at the explicit construction of the filtration for the objects of D (A). The first step is to write the filtration of the indecomposable objects of Db (A). X1 , X∞ and all the other elements Z of the heart have the filtration provided by the distinguished triangle 0 → Z → Z. As for other indecomposables, by taking the cone one has the following exact triangle, for 1 < i < ∞: X∞ → − Xi [−i + 1] → X∞ [−i + 1]. b
The filtration of the indecomposable object Xi [−i + 1] is the following: / X∞ c
0Z
/ Xi [−i + 1]
(3.4)
x
X∞
X∞ [−i + 1]
By Lemma 2.4.5, the filtration of other indecomposable objects can be obtained by shifting these ones above. Moreover, the filtration of every object X of Db (A) can be constructed by taking direct sums of the filtration of indecomposable objects that generate X.
3.5.2
Stability conditions on Db (A)
In this section we will describe the space Stab(Db (A)) of stability conditions on Db (A). The proofs of this section are inspired by [40], where Jørgensen and Pauksztello describe the space of co-stability conditions on Perf (A). Thanks to the results of previous section, we know that all the t-structures on Db (A) are given by shifts of the standard one. In particular all the possible heart are Ah = Add {X1 [h], X∞ [h]}. The exact sequence:
1
0 → X∞ → − X1 → − X∞ → 0 gives a relation in the Grothendieck group [X1 [h]] = 2 [X∞ [h]]. It follows then that the Grothendieck group is the free abelian group generated by X∞ [h]. In order to give the stability function, it suffices to choose a vector v in H as the image of X∞ [h]. All objects of the hearts Ah are semi-stable. Proposition 3.5.5. A stability condition on Db (A) can be given by an integer h and a vector v ∈ H. Proof. By Proposition 5.3 of [22], it is sufficient to provide a heart of a bounded t-structure and a stability function. The Harder-Narasimhan property required in the theorem is assured since the heart is artinian and noetherian. The integer h specifies the heart Ah as described above, and v describes the stability function Z(X∞ [h]) = v. The Grothendieck group K(Ah ) is, for every h, isomorphic to the Grothendieck group of the whole category Db (A) required in 2.4.1 (see [65] for details). The data (h, v) correspond to the stability condition σ = (Z, P) where the group homeomorphism Z is given by the stability function as observed above. Let φ be the phase of v; P is given by: P(φ) = Add {X1 [h], X∞ [h]} 59
and it is zero for all the other φ ∈ (0, 1], these data extend to all φ ∈ R by the property (2) of Definition 2.4.1. + ] Definition 3.5.6. The universal covering GL (2, R) of GL+ (2, R) is defined by pairs (G, f ) where G ∈ GL+ (2, R) and f : R → R such that:
• f is an increasing map with f (x + 1) = f (x) + 1 for all x ∈ R. G exp iπφ |G exp iπφ|
•
= exp iπf (φ).
There are two group actions on the space Stab(T ) (See [22], Lemma 8.2): a right action of + ] GL (2, R) and a left action by isometries of the group of auto-equivalences of the category D. Let + ] S be the subgroup of GL (2, R) generated by rotations (exp iπθ, f (x) = x + θ), θ ∈ R and scalings + (k, f (x) = x), k ∈ R where by k ∈ GL+ (2, R) we mean the diagonal matrix k · id. We have: S = {(k exp iπθ,
f (x) = x + θ) ∀θ ∈ R,
k ∈ R+ }.
The action on Stab(T ) is given by: (G, f ) (Z, P(φ)) = (G−1 ◦ Z, P(f (φ)). Lemma 3.5.7. The action of S on Stab(Db (A)) is free and transitive, hence Stab(Db (A)) ∼ = S. Proof. Let (Z, P) be the given stability condition, as in Proposition 3.5.5, by the couple (h, v). Notice that v = |v| exp iπφv . P(φv ) = Add {X1 [h], X∞ [h]} thus P(φv − h) = Add {X1 , X∞ }. For + ] every stability condition (Z, P), there is an element of the group GL (2, R) that sends (Z, P) to −1 (0, −1). If we set θ = −h − φv and h = (|v|) this element has the form (h exp iπθ, f (x) = x + θ). Then the action is transitive. Moreover it is straightforward to verify that the action is also free. Theorem 3.5.8. Stab(Db (A)) is isomorphic to C, the universal covering of C∗ . Proof. C is the universal covering of C∗ by standard arguments. By Lemma 3.5.7, it is sufficient to verify the claim for the subgroup S. S is the universal covering of the subgroup of GL+ (2, R) given by {k exp(iπθ), k ∈ R+ , θ ∈ R}, which is isomorphic to C∗ .
3.5.3
Co-t-structure and co-stability conditions
In [36], Holm, Jørgensen and Yang study the existence of co-t-structures on the category Perf (A). Such structures were introduced in [62] and [20]. Definition 3.5.9. Let T be a triangulated category. A co-t-structure on T is given by a full additive subcategory F such that: • F [−1] ⊂ F • For all objects E in T , there exists a distinguished triangle: F →E→G where F ∈ F and G ∈ F ⊥ . 60
The co-heart of a co-t-structure is the subcategory A := F ∩ F ⊥ [−1]. Co-t-structures satisfy similar properties of t-structures, in particular the existence of a bounded co-t-structure is linked to the existence of a co-heart; the property (2) of Proposition 2.4.4 holds but with the shifts kj in the opposite order; see for example section 2 of [40] for a reference. Moreover, properties like the one in the Lemma 2.4.5 can be proved. On the category Perf (A) there exist a bounded co-t-structure, the idea is to look at the co-heart. We know that the indecomposable X1 generates by shift and cones the whole category Perf (A). Proposition 3.5.10. Every indecomposable object Xi of Perf (A) sits in a collection of triangles like / X1 / X2 / · · · / Xn−1 / Xn = E 0Z ^ c
X1
z
X1 [N − 1]
X1 [1]
Proof. One can show that Xn = Cone(X1 [n − 2] → − Xn−1 ), and so the following triangle is distinguished Xn−1 → Xn → X1 [n − 1] → − Xn−1 [1]
By this proposition one can see that, for all the indecomposable objects of Perf (A), a co-version of the property (2) of 2.4.4 holds. The proof of the Proposition 3.5.2 still works in the context of co-t-structures, so it turns out (see [36]) that the subcategory Add {X1 } of Perf (A) is, up to a shift, the unique possible heart of a co-t-structure on Perf (A). A similar Lemma to 2.4.5 can be stated in the context of co-t-structures (see [20] and [40]). Notice that we can prove directly, as we made for the case of the t-structure, that up to a shift, the unique candidate to be the co-heart of a bounded co-t-structure on Db (A) is Add {X1 , X∞ }. To prove the co-version of the fibration (2) of 2.4.4, the co-filtration for every object of the category Db (A) can be build explicitely in a similar way of the filtration in (3.4). There are important differences between t-structures and co-t-structures: there are examples of co-hearts of a co-t-structures that are not abelian and in general the filtration (2) of Proposition 2.4.4 is not unique. Proposition 1.3.3 of [20] makes clear that the proof of Proposition 3.5.4 still work in the context of co-t-structures. The notion of co-stability conditions is also rather similar to the one of stability condition given in Definition 2.4.1. Notice that the inequality of part (3) of definition is reversed (co-3) If φ1 < φ2 , Ei ∈ P(φi ), then HomT (E1 , E2 ) = 0. as are reversed the inequalities involving the shifts in the Harder-Narasimhan filtration (co-4) Any non zero object E admits a Harder-Narasimhan filtration, that is a finite number of inclusions 0 = E0 ,→ E1 ,→ · · · ,→ En−1 ,→ En = E 61
such that Fj = Cone(Ej−1 ,→ Ej ) are semistable objects of with phase φ(F1 ) < · · · < φ(Fn−1 ) < φ(Fn ) The space co-Stab(Db (A)) consisting of all the co-stability condition on a triangulated category T is a topological manifold. By following [40] and mimicking the proof of Theorem 3.5.8 one gets the following proposition. Proposition 3.5.11. co-Stab(Db (A)) is isomorphic to C.
3.6
Generalizing to k[]/(n )
With the aim of generalizing what has been proven in the previous sections, one can wonder what happens for the spectrum of the ring k[]/(n ). The first step is to analyze the case n = 3. Call B = k[]/(3 ) and, as before, A = k[]/(2 ); then the finitly generated modules on B are of the form k n ⊕ Am ⊕ B t . The following proposition is useful to reduce the possible cases, considering only the complexes in which only the term B t appears. Proposition 3.6.1. In the category Db (B) the following complexes are quasi isomorphic 2
2
2
2
(0 → k → 0) q.i. (· · · → B −→ B → − B −→ B → − B → 0) (0 → A → 0) q.i. (· · · → B → − B −→ B → − B −→ B → 0) And the following one allows us to consider only complexes without the sum in the maps. +2
− Proposition 3.6.2. The complex (0 → B −−−→ B → 0) is isomorphic to the complex (0 → B → B → 0) By mimicking the case of Db (A) one can try to prove that all the indecomposable objects are again made up of terms in B t with t equal to one. Unfortunately that is not true due to the following example Example 3.6.3. The following complex is an indecomposable element of Perf (B). 2
•
X := 0 → B(−3)
( ) 2 (,0) −−→ B(−2) −−−→ B(−1) → 0
here the index (i) denotes the position as in Definition 3.2.2. Proof. The complex X • has only three non-trivial cohomology groups: H −3 (X • ) ∼ =k
H −2 (X • ) ∼ =A
62
H −1 (X • ) ∼ =k
Since we want to have cohomology k at places (−3) and (−1), the only possibility is to decompose X • in two complexes of the following form
0→B→ − B→0 This complex has two non-trivial cohomologies, both isomorphic to k, and so it is not possible to obtain A at place (−2) then the complex X • is indecomposable.
63
Chapter 4
Quadric fibrations and Azumaya algebras This chapter is devoted to the study of quadric fibrations. Our motivation is very geometrical and it is explained in Section 4.1. Quadric fibrations arise in many geometrical contexts, in our case we studied the theory of line bundle valued quadratic form related to the study of the derived category of the cubic threefold or of the cubic fourfold containing a plane. Kuznetsov, in [49] described a semiorthogonal decomposition of the derived category Db (X) of a quadric fibration X → S. A non trivial developments of the theory is the Kuznetsov conjecture stated in [48]. In [16], Bernardara, Macrì, Mehrotra and Stellari described a stability condition on the derived category of the cubic threefold in order to prove a categorical version of the Torelli theorem for such a variety. As a consequence the space Stab(X) of the cubic threefold X is non empty and a natural question is whether the only stability condition, up to isomorphism, is the one found in [16] or not. It is a natural problem to classify line bundle valued quadratic forms in terms of Azumaya algebra. In case of rank 4, Auel Parimala and Suresh classified the space of quadric fibration with simple degeneration. Theorem 4.0.4 ([8], Theorem 1). There is a bijection between the set of rank four quadratic fibration with simple degeneration and with fixed discriminant cover and the space of Azumaya algebras of rank four on the same discriminant cover. A theorem of Balaji allows us to classify all the form of rank 3 (Theorem 4.4.3). By using the bijection described by Balaji we find a bijection similar to the one of Theorem 4.3.1 (Theorem 4.4.6). Theorem 4.0.5. There is a bijection between the set of rank three quadratic fibration with simple degeneration and with fixed discriminant stack and the space of Azumaya algebra of rank four on the same discriminant stack. If the base S is a rational surface, we can use, as in the work of Bernardara and Bolognesi in [15] the exact sequence of Artin Mumford to study the Clifford Algebra.
64
Section 4.1 is devoted to describe the motivation to study the theory of line bundle valued quadratic forms. Moreover we investigate a preliminary step of the problem of describing the whole space of stability conditions of the cubic threefold. In Section 4.2 we fix the notation that we will use to describe the spaces of the forms and Azumaya algebras. We keep a notation similar to the work of Auel, Parimala and Suresh in [8], that we will recall in section 4.3. Eventually, in section 4.4 we prove our specialization of the classification of Balaji about the ternary line bundle valued quadratic form and we focus on the case, studied by Bernardara and Bolognesi, of the conic fibrations with a regular surface as base. We will work on a base field k of characteristic different from 2.
4.1
The space of stability conditions on the cubic threefold
We will follow the notation of [51], where Lahoz, Macrì and Stellari described the arithmetically Cohen Macaulay bundles on the cubic threefold developing the same techniques of [49]. Let X be a smooth cubic threefold contained in P4 . As pointed out in Theorem 2.1.13, its derived category can be decomposed in three parts: Db (X) = hTX , OX , OX (1)i
(4.1)
where OX (1) is the restriction of the sheaf OP4 (1) to X and TX is the right orthogonal of hOX , OX (1)i. The scheme of lines on the cubic threefold was studied by Fano in [32]; he proves that its dimension is two. Let us consider a generic line l0 on the cubic threefold, and a plane P2 disjoint from l0 . The projection map from l0 to P2 is a rational map and becomes a morphism by blowing up P4 along the line l0 . / Bll X / Bll P4 D 0 0 σ
l0
/X
π
τ
$ 2 /P
Notice that the exceptional divisor that one obtains by blowing up the line on the cubic threefold is isomorphic to the normal bundle of the line in the cubic threefold. Such normal bundles were described by Clemens and Griffiths in [31]. We denote by D the exceptional divisor of the blow ˜ := Bll X the strict transform of X under the blow up τ , and by π the projection up σ, by X 0 morphism. By Theorem 2.1.13 and by mutating the terms to put on the right the exceptional objects concerning the blowup, the decomposition in (4.1) gives rise to the following decomposition ˜ of the derived category of the blow up X. ˜ = hσ ∗ TX , O ˜ , O ˜ (H), OD , OD (H)i Db (X) X X
(4.2)
Where H denotes the hyperplane class on P4 and its pullbacks. The fibers of the rational morphism X 99K P2 are obtained by intersecting the cubic threefold X with a plane that contains l0 , and so ˜ → P2 are conics. By they contain the line l0 and a conic. Then, the fiber of the morphism π : X 65
following [16], π is a quadric fibration related to the line bundle valued quadratic form with base ∼ S = P2 , E = OP⊕2 2 ⊕ OP2 (−h) and L = OP2 (h), here we are denoting by h the class of a line on the plane P2 and its pullbacks. The following theorem of Kuznetsov gives another semiorthogonal decomposition of the derived ˜ category of X. Theorem 4.1.1 ([49], Theorem 4.2). Let (E , q, L ) be a quadric fibration of rank n on S, and π : X → S the corresponding regular morphism. There exists a semiorthogonal decomposition Db (X) = hDb (S, C0 ), π ∗ (Db (S)) ⊗ OX/S (1), π ∗ (Db (S)) ⊗ OX/S (2), · · · , π ∗ (Db (S)) ⊗ OX/S (n − 2)i Where C0 denotes the even part of the Clifford algebra associated to the quadric fibration, and Db (S, C0 ) is the derived category of coherent sheaves of C0 -modules on S. ˜ such that We can define a fully faithful functor Φ : Db (P2 , C0 ) → Db (X) ˜ = hΦ(Db (P2 , C0 )), O ˜ (−h), O ˜ , O ˜ (h)i Db (X) X X X
(4.3)
Some mutations of (4.2) and (4.3) allow us to find new semiorthogonal decompositions
˜ = hΦ0 (Db (P2 , C0 )), O ˜ , O ˜ (h), O ˜ (H)i Db (X) X X X ˜ = hσ ∗ TX , O ˜ (h − H), O ˜ , O ˜ (h), O ˜ (H)i Db (X) X X X X By comparing the two equations, one can find a fully faithful functor Ξ from the subcategory TX ⊂ Db (X) to Db (P2 , C0 ). The functor Ξ can be used, as in [16], to find a stability condition on TX . Remark 4.1.2 ([16], Section 2.3). The category Coh(P2 , C0 ) admits a notion of µ-stability, that is a slope function µ from the objects of this category with non-zero rank to R that satisfies the following properties • The Harder-Narasimhan filtration exists for every µ-(semi)stable object. • If F and F 0 are µ-semistable torsion-free sheaves with µ(F ) > µ(F 0 ) then Hom(F, F 0 ) = 0. • The Serre functor preserves µ-stability. • The exceptional object C1 is µ-stable. The starting point is to provide a stability condition on Db (P2 , C0 ); as described in Proposition 2.4.7, for which we need a heart of a bounded t-structure and a stability function. Start by defining the following T0 := {K ∈ Coh(P2 , C0 ) s.t. K of torsion or µ− (Ktf ) > −5/4} F0 := {K ∈ Coh(P2 , C0 ) s.t. K is torsion free and µ+ (K) ≤ −5/4} 66
Where Ktf stands for the torsion free part and µ+ and µ− are the maximum and the minimum values of the Harder-Narasimhan filtration relative to the µ-stability of Remark 4.1.2 as defined in Section 2.4. Then one gets a bounded t-structure on Db (P2 , C0 ) with heart given by H i (C) = 0 for all i 6= 0, −1 b 2 0 A0 = C ∈ D (P , C0 ) s.t. H (C) ∈ T0 H −1 (C) ∈ F0 where the cohomology H • denotes the cohomology with respect to the t-structure with heart Coh(P2 , C0 ). Eventually one sets B := (σ∗ ◦ Φ0 )(A0 ) ∩ TX . Lemma 4.1.3 ([16], Lemma 3.4). The category B is the heart of a bounded t-structure on the category TX . The other ingredient to define the stability condition is the stability function on the heart B. Start by the following group homomorphism Z from the Grothendieck group K(P2 , C0 ) to C Z([C]) = rk(C) + i(deg(C) − µ(C0 )rk(C)) Now to obtain the stability function on TX it is sufficient to exploit the previous calculation done in the category Db (P2 , C0 ) and define Z([A]) := Z([Ξ(A)]) for every object A of TX . Proposition 4.1.4 ([16], Proposition 3.7). The pair (B, Z) defines a locally finite stability condition on TX . For the purpose of determining how many stability conditions on the derived category of the cubic threefold can be found, we wonder if an argument based on a construction similar to the one presented in [16], but with a different fibration in conics, would be useful in order to provide new stability conditions. First of all, we may analyse the situation in which we consider another plane P2 . As before we denote by H the hyperplane class on P4 , and by h0 and h1 the hyperplane classes on the two planes, ˜ depends only on the choice of the line, and so P20 and P21 , that we are considering. The blow up X we will denote again by D the exceptional divisor and by H the pullbacks of the hyperplane class ˜ and to Bll P4 . to X 0 ˜ X π0
P20 o
σ
X 67
π1
/ P2 1
Using the same calculation as above, we obtain two different semiorthogonal decomposition of the derived category of the blow up ˜ = hΦ0 (Db (P2 , C 0 )), O ˜ , O ˜ (π ∗ h0 ), O ˜ (H)i Db (X) 0 0 0 X X 0 X
(4.4)
˜ = hΦ0 (Db (P2 , C 1 )), O ˜ , O ˜ (π ∗ h1 ), O ˜ (H)i Db (X) 1 1 0 X X 1 X Now they depend on the choice of the plane because of the exceptional object OX˜ (πi∗ hi ) for i = 0, 1. ˜ does not depend on the choice of the Proposition 4.1.5. The subcategory Φ0i Db (P2i , C0i ) of Db (X) 2 plane Pi . Proof. By the semiorthogonal decomposition described in (4.5) it follows that Φ0i (Db (P2i , C0i )) is the right orthogonal of the category hOX˜ , OX˜ (πi∗ hi ), OX˜ (H)i for i = 0, 1. We now perform some mutation to make the latter category not depending on the choice of the plane. In other words we want to substitute OX˜ (πi∗ hi ) with something not depending on the plane. Note that the left mutation of the pair (OX˜ (H), OD (H)) is (OX˜ (πi∗ hi ), OX˜ (H)) for both i = 0, 1. So by mutation one obtains the following semiorthogonal decomposition in which Φ0i (Db (P2i , C0i )) is the right orthogonal of a category that does not depend on the plane. ˜ = hΦ0i (Db (P2i , C0i )), O ˜ , O ˜ (H), OD (H)i Db (X) X X
(4.5)
Now we want to study the relations between the categories Φ00 (Db (P2 , C00 )) and Φ01 (Db (P2 , C01 )), obtained by using the projection lines l0 and l1 . By the previous result the projection plane is not relevant and, by generality we can choose the same plane P2 for both the lines. Now we have to ˜ 0 and X ˜ 1 , each with a different structure of conic bundle. deal with two different blow ups X We would like to know if, as in the case of changing the plane, the two categories Φ0 (Db (P2 , C00 )) and Φ0 (Db (P2 , C01 )) are equivalent. First we try to use a similar reasoning of the previous proof, that is starting with a decomposition and then mutating it. π0
X˜0
/ P2 o π1 O
σ0
l0
X˜1
σ1
~
/Xo
? _ l1
˜i , for i = We will denote by H the hyperplane class on P4 and by Hi its pullback to X 2 ∗ ˜i for i = 0, 1. Moreover h will denote the hyperplane class on P and πi h its pullback on X 0, 1. Eventually, the two exceptional divisors will be denoted by Di for i = 0, 1. Notice that the pair OX˜i (Hi ), O(Di ) is completely orthogonal and that the left mutation of the pair OX˜i (mHi ), ODi (mHi ) is OX˜i (πi∗ h + (m − 1)Hi ), OX˜i (mHi ) . Since these mutations involve in both sides something related to the projection lines or to the blowups we were not able to use them to prove our claim, and this method does not work unless we 68
are able to find other mutations. A second way to prove the claim is to blow up twice, as described in the following diagram
σ1
/ Bll X˜1 0
∼
Bll1 X˜0
X˜0
σ0
X˜1 σ0
#
X
{
σ1
X˜0 denotes the strict transform of X in the blow up of P4 along the line l0 and Bll1 X˜0 denotes the strict transform of X˜0 in the blow up of Bll0 P4 along the line l1 . To simplify the problem suppose the lines l0 and l1 to be disjoint. Then Bll1 X˜0 and Bll0 X˜1 are isomorphic; we will call this ˜ spaces X. ˜ obtained by the By applying Theorem 2.1.13 we find two semiorthogonal decompositions of X two different chains of blow ups.
˜ = hσ1∗ (Φ0 (Db (P2 , C00 ))), O ˜ , O ˜ (σ1∗ π0∗ h), O ˜ (σ1∗ H0 ), OD , OD (H1 )i Db (X) 1 1 X X X ˜ = hσ ∗ (Φ0 (Db (P2 , C 1 ))), O ˜ , O ˜ (σ ∗ π ∗ h), O ˜ (σ ∗ H1 ), OD , OD (H0 )i Db (X) 0 0 0 0 X X 0 1 X 0 However, we could not manage to perform a comparison between these two semiorthogonal decompositions. Since by the previous ways of reasoning we were not able to provide an answer to our problem, we focused on the problem of finding the relations between conic fibrations and Azumaya algebras. Is it true that the existence of an isomorphism between the conic fibrations π0 and π1 would entail an isomorphism between the Azumaya algebras provided by the work of Kuznetsov? Notice that, even if one could prove the existence of such an isomorphism, the problem of finding all the stability conditions on a cubic threefold would be far away from being solved. In fact even if all the stability conditions arising from the changing of the projection line were the same, there could be another way to produce a new, non isomorphic, stability condition. Conversely, it might be possible to find non isomorphic Azumaya algebras associated to different lines that give rise to the same stability condition, since the group that acts on the space Stab(X) is composed by autoequivalences and not only by isomorphisms.
4.2
Sets of generically regular quadric fibrations
The even Clifford algebra functor associates to a quadric fibration with values in a line bundle its corresponding Clifford algebra. In the regular case, this algebra turns out to be Azumaya. It is a natural problem to study the correspondences between these two sets of objects. In particular is it possible to associate an Azumaya algebra to a quadric fibration in the non-regular case? Is this correspondence injective or surjective? Let us state more clearly the settings. 69
Definition 4.2.1. Let Qn (S) be the set of quadratic forms of rank n on a smooth scheme S, and consider the following subsets: • Qrn (S) the subset of Qn (S) composed by regular forms. • Qgr n (S) the subset of Qn (S) composed by generically regular forms. gr • Qsd n (S) the subset of Qn (S) composed by generically regular forms with simple degenerations.
We can consider on Qn (S) the equivalence relation given by projective similarity. Notice that this relation induces other relations on the subsets of Qn (S). Two elements (E , q, L ) and (E 0 , q 0 , L 0 ) belonging to one of the subsets in the previous definition are in the same projective similarity class if and only if the same elements, consider as elements of Qn (S) are such. The following corollary is an easy consequence of Proposition 2.5.10, and proves that the subsets introduced by Definition 4.2.1 are closed by projective similarity. Corollary 4.2.2. Let (E , q, L ) and (E 0 , q 0 , L 0 ) be two quadratic forms on S in the same projective similarity class. • (E , q, L ) is regular if and only if (E 0 , q 0 , L 0 ) is such. • (E , q, L ) is generically regular if and only if (E 0 , q 0 , L 0 ) is such. • (E , q, L ) and (E 0 , q 0 , L 0 ) have the same n-th discriminant locus, in particular (E , q, L ) is simple degenerate if and only if (E 0 , q 0 , L 0 ) is such. We will use the convention to name Quad the sets of forms quotiented by this relation as in the following Definition 4.2.3. Let Quadn (S) be the quotient of Qn (S) by projective similarity Quadrn (S) is defined to be the quotient of Qrn (S) by projective similarity. gr Quadgr n (S) is defined to be the quotient of Qn (S) by projective similarity. sd Quadsd n (S) is defined to be the quotient of Qn (S) by projective similarity.
The aim now is to relate quadratic forms with algebras to study the property of the correspondences induced by the Clifford algebra. Definition 4.2.4. Let Azn (S) be the set of isomorphism classes of rank n Azumaya algebras over S. We now want to define some subset of Quadgr n (S) paying attention to the locus where the form is non regular. Recall that, if the form has even rank, we speak about the discriminant double cover T˜/S, and if the form has odd rank, we speak of discriminant stack Tˆ/S. If we do not care about the rank of a quadratic form, stating a result that holds in both even and odd cases, we will use the notation T /S.
70
Definition 4.2.5. Qn (T /S, S2 ) will denote the set of generically regular quadratic form with fixed discriminant stack or discriminant double cover T /S, and with 2-th degeneration locus S2 . The notation Qn (T /S, ∅) specifies that the 2-th discrimination locus is empty, and then we are considering forms with simple degeneration. sd The spaces Qn (T /S, S2 ) and Qn (T /S, ∅) can be included in Qgr n (S) and Qn (S), respectively. Notice that the information given by S2 is redundant because the discriminant cover or the discriminant stack T /S of a quadratic form determines all the degeneration loci Sd ; then knowing T /S fixes the property of (E , q, L ) of having simple degeneration. Consider (E , q, L ) a form in Qn (T /S), and its projective similarity class P = [(E , q, L )] in the set Qn (S). Projective similarity is induced by considering Qn (T /S) as a subset of Qn (S), and so the equivalence relation is well defined; this allow us to speak of the space Quadn (T /S). By Corollary 4.2.2 it is clear that P is contained in Qgr n (S) and moreover, that all the elements of P have the same discriminant loci.
Lemma 4.2.6. The discriminant stacks associated to two generically regular quadric fibrations that are similar are isomorphic. Proof. Recall the definition of similarity given in 2.5.7. It is straightforward to verify that the isomorphisms ψ and λ induce an isomorphism φ between the line bundle G := det(E ∗ )⊗2 ⊗ L ⊗n and G0 := det(E 0∗ )⊗2 ⊗ L 0⊗n , that maps the section t := det σ in t0 := det σ 0 . By Lemma 2.5.18 the maps g, g 0 : S → [A1 /G1 ] corresponding to (G, t) and (G0 , t0 ) are isomorphic. The square root stacks SG,t,2 and SG0 ,t0 ,2 are given by pull-back: [A1 /G1 ] g0
S
g
//
·2
[A1 /G1 ]
And these pullback are isomorphic being g and g 0 isomorphic. From the algebraic side, we need to fix the notation of the space of Azumaya algebras. The following proposition is a specialization of Propositions 2.5.17 and 2.5.21 and will be the key ingredient to associate an Azumaya algebra to a quadratic form with simple degeneration. In fact in this case it turns out that the Clifford algebra is Azumaya itself. Proposition 4.2.7 ([7], Propositions 1.6.2 and 1.7.2). Let (E , q, L ) be a quadratic form on S with simple degeneration, then the Clifford algebra C0 (E , q, L ) is Azumaya on the discriminant cover T˜ or on the discriminant stack Tˆ depending on the rank being even or odd respectively. Like in Definition 4.2.4, we now want to define the space of Azumaya algebras related to a generically regular quadratic form. It is defined as follows f
Definition 4.2.8. Let (E , q, L ) be a quadratic form and let T − → S be, as above, the discriminant double cover or the discriminant stack. Azn (T /S, S2 ) denotes the set of isomorphism classes of algebra over T /S that are Azumaya of rank n over the complement of f −1 (S2 ). 71
Notice that, if we are considering a form with simple degeneraton, the set S2 is empty.
4.3
Line bundle valued quadratic forms of rank 4 with simple degeneration
Proposition 2.5.16 describes how to associate an Azumaya algebra to a regular quadratic form. The properties of the Clifford algebra can be used to prove that this association is well defined on the space of quadratic forms defined in the previous section. An interesting problem could be to find some properties of this map induced by the Clifford algebra. In particular is this map bijective? The Clifford algebra induces a map between the space of simple degenerate quadratic forms to a space of Azumaya algebras. Auel, Parimala and Suresh prove that, if we consider forms of rank 4 and simple degeneration over a regular integral scheme of dimension less or equal than 2, the map is a bijection. Here below there is a sketch of the proof of this fact, as given in [8], with a particular attention to the injectivity part. Recall that the notation used in [8] is slightly different, since the authors use the codimension of the fibration instead of the dimension of the vector bundle. The map defined by the Clifford algebra C0 goes from the space Quad4 (T˜/S, ∅) to Az4 (T˜/S, ∅) by associating to a quadric fibration the Clifford algebra, that is Azumaya on T˜ by Proposition 4.2.7. The theorem proved by Auel, Parimala and Suresh is the following Theorem 4.3.1 ([8], Theorem 1). The map induced by the Clifford algebra between the sets Quad4 (T˜/S, ∅) and the subspace composed by Azumaya algebras with generically trivial corestriction to S, is a bijection. Notice that this result holds only if we consider Azumaya algebras with generically trivial corestriction to S. For sake of simplicity in all this section we will refer to this subspace as Az4 (T˜/S, ∅). The proof of surjectivity is carried on by providing an inverse to the map induced by the Clifford algebra, that is called the norm functor Quad4 (T˜/S, ∅) o
C0 NT˜ /S
/
Az4 (T˜/S, ∅)
We now want to report the part of the proof concerning the injectivity of the map. The proof is based on the relation between the group of similarity class of the quadratic form and the group of isomorphisms of the Azumaya algebras. We will start with a brief description of such groups. GO(E , q, L ) and O(E , q, L ) denote the group schemes on S of the similitudes and isometries of the quadratic form (E , q, L ), respectively. If the form we are considering has even rank n = 2m, then there is a homomorphism GO(E , q, L ) → µ2 that sends a similitude to its determinant rescaled by the similarity factor. The kernel of this map is denoted by GO+ (E , q, L ). Notice that we are assuming 2 invertible. Eventually PGO(E , q, L )
72
is the cokernel sheaf of the subgroupschemes Gm → GO(E , q, L ) of homotheties; and, in the same way, PGO+ (E , q, L ) is the cokernel sheaf of Gm → GO+ (E , q, L ). The key point to prove the injectivity is the following theorem Theorem 4.3.2 ([8], Theorem 2.1). Let S be a regular scheme with 2 invertible and (E , q, L ) a quadratic form of rank 4 on S in the space Quad4 (T˜/S, ∅), and C0 (E , q, L ) is the even Clifford algebra. The canonical homomorphism induced from the Clifford algebra PGO+ (E , q, L ) → RT˜/S PGL(C0 (E , q, L )) is an isomorphism of S-group schemes. Here RT˜/S denote the restriction functor. It is a generalization of an analogous theorem for regular forms, due to Knus Theorem 4.3.3 ([44], Proposition 10.5). Let S be a scheme and (E , q, L ) a regular quadratic form of rank 4 on S. Let T˜ → S be its discriminant cover, that in the regular case corresponds to an ètale double covering of S, and C0 (E , q, L ) its even Clifford algebra. The canonical homomorphism induced from the Clifford algebra PGO+ (E , q, L ) → RT˜/S PGL(C0 (E , q, L )) is an isomorphism of S-group schemes. Before turning to the proof of Theorem 4.3.2 we will need some local results to study simple degeneration. Notice that if we localize a line bundle valued quadratic form, we can trivialize the line bundle, and consider only the form (E , q) := (E , q, OS ) instead of (E , q, L ). Lemma 4.3.4 ([8], Lemma 1.4). Let (E , q) be a quadratic form with simple degeneration over the spectrum of a local ring R with 2 invertible. Then (E , q) ∼ = (E 0 , q 0 ) ⊥ (R, hri) where (E 0 , q 0 ) is a regular quadratic form and r is an element of R. We need the group schemes involved in the proof to be smooth. The validity of this asserction is assured by the following lemmas. Lemma 4.3.5 ([8], Proposition 2.2). Let (E , q) be a rank n quadratic form with simple degeneration defined on a regular local ring R with 2 invertible. Then O(q) and O+ (q) are smooth R-group schemes. A similar result holds if we consider quadratic forms on a regular scheme S with 2 invertible. Lemma 4.3.6 ([8], Proposition 2.3). Let (E , q, L ) a simple degenerate quadratic form of even rank on S. Then the group schemes O(q) and O+ (q), GO(q) and GO+ (q), PGO(q) and PGO+ (q) are smooth on S. Moreover, if T˜ → S is the discriminant cover of (E , q, L ) and C0 (q) the Clifford algebra, then the schemes RT˜/S GL1 (C0 (q)), RT˜/S SL1 (C0 (q)) and RT˜/S PGL1 (C0 (q)) are smooth on S 73
Proof of Theorem 4.3.2. In order to prove that the map γ : PGO+ (E , q, L ) → RT˜/S PGL(C0 (E , q, L )) is an isomorphism of S-group schemes we will use the following proposition. Proposition 4.3.7 ([45], Proposition 22.5). Let f : G → H be a homomorphism of algebraic group schemes with H smooth. Then f is an isomorphism if and only if the map is an isomorphism at the level of schemes and the Lie algebra map df : Lie(G) → Lie(H) is injective. Hence we need to prove that the map γ is an isomorphism of schemes on S, and that the induced Lie algebra map dγ is an isomorphism. To prove the first claim we will use the following result of Grothendieck. Proposition 4.3.8 ([34], Corollary 17.9.5). Consider a morphism of S-schemes g : X → Y locally of finite presentation and let X be flat on S. Then g is an isomorphism if and only if its fiber gs : Xs → Ys is an isomorphism over each geometric point s of S. If we consider a point s not in the discriminant divisor, the form is regular, and so we can use directly Theorem 4.3.3 to prove that γs : PGO+ (qs ) → RT /S PGL(C0 (qs )) is an isomorphism. Now we have to consider the geometric points s belonging to the discriminant divisor D. By the hypothesis of simple degeneration, the radical of q has rank 1, generated by an element v ∈ S. A simple calculation shows that the fiber T˜s is isomorphic to A := k[]/(2 ). We will show that the map cs is an isomorphism also in this case by proving that the following sequence is exact 1 → µ2 → O+ (qs ) → PGL1 (C0 (qs )) → 1 Then the claim follows from the fact that that PGO+ is the quotient of O+ (q) by µ2 . Recall that, by Lemma 4.3.4, the local structure of the form (E , q, L ) near the simple degeneration is q = q 0 ⊥ hri. Moreover, over the points where the form q is regular we have the decomposition q = q 0 ⊥ h1i, then the following proof works instead of Theorem 4.3.3 for the points outside the discriminant divisor D. Consider the following diagram
74
1
1
µ2
µ2
1
/U + O (q)
/ O+ (q)
/ O(q 0 )
/1
1
/ I + c0 (q)
/ PGL1 (C0 (q))
/ PGL1 (C0 (q 0 ))
/1
1
1
Where UO+ (q) denotes the unipotent radical of O+ (q) and c0 (q) is the affine scheme of reduced trace zero elements of C0 (q). We want to prove that the first column on the left is exact; by a diagram chasing argument, it suffices to prove that all the squares of the diagram commute and that the two rows and the column on the right are exact. Let start with the proof of the exactness of the sequences. Start from the first row of the diagram. f
i
1 → UO+ (q) → − O+ (q) − → O(q 0 ) → 1 is defined by the inclusion i and by the map f defined by ! M ··· f( )=M ··· ··· In particular the image of the space UO+ (q) , restricted to the generic point k, by the inclusion i is the group of block matrices of the form ! I 0 ··· 1 The ring O(q 0 ) is semisimple, and so the kernel contains the unipotent radical UO+ (q) . By a dimension count the kernel coincides exactly with the unipotent radical, and so the sequence is exact. Now consider the second row of the diagram 1 → I + c0 (q) → PGL1 (C0 (q)) → PGL1 (C0 (q 0 )) → 1 It is proven in [8], Lemma 1.11, that the Clifford algebra C0 (q) is isomorphic to C0 (q 0 ) ⊗k Z (q), that correspond to C0 (q 0 ) ⊗k A if we work over a geometric point of the discriminant divisor. Then the map from PGL1 (C0 (q)) to PGL1 (C0 (q 0 )) is defined by the reduction A → k. By the definition of c0 (q) the kernel of the maps turns out to be exactly I + c0 (q). To conclude this part, consider the right column of the diagram 1 → µ2 → O(q 0 ) ∼ = µ2 × O+ (q 0 ) → PGL1 (C0 (q 0 )) → 1 75
the proof of it’s exactness follows by applying a result concerning the split isogeny of type A1 = B1 , that can be found in [45], IV.15.A. All the squares of the diagram are trivially commutative with the exception of the following one UO+ (q)
/ O+ (q)
/ PGL1 (C0 (q))
I + c0 (q)
To conclude the proof, it suffices to prove that the map UO+ (q) (k) → I + c0 (q)(k) is an isomorphism. This is done in Lemma 4.3.9. To conclude this part it is sufficient to apply the five lemma. It remains to prove that the Lie algebra map dγ is an isomorphism. Start by considering the following commutative diagram 1
/ I + xso(q)(k)
1
/ O+ (q)(k[x]/(x2 ))
1+xdγ
/ I + xg(k)
γ(k[x]/(x2 ))
/ PGL1 (C0 (q))(k[, x]/(2 , x2 ))
/ O+ (q)(k)
/1
/ PGL1 (C0 (q))(k[]/(2 ))
/1
where so(q) denotes the Lie algebra of O+ (q) and g denotes the Lie algebras of RA/k PGL1 (C0 (q)). The Lie algebra so(q 0 ) of O(q 0 ) corresponds to the scheme of 3 by 3 matrices A such that the product A · diag(1, −1, 1) is skew-symmetric. Then I + xso(q)(k) consists of block matrices of the form ! I + xA 0 M := xw 1 for w ∈ A3 (k) and A ∈ so(q 0 )(k). The matrix M can be obtained by the following (commutative) product ! ! I 0 I + xA 0 xw 1 0 1 This proves that I + xso(q) has a direct product decomposition UO+ (q) × (I + xso(q 0 )). As above, we can give an explicit form of the map 1 + xdγ to prove directly that it is an isomorphism. The image under this map of the matrix M is given by (I − β(xw))(I − α(xA)) = I − x(α(A) + β(w)) where α : so(q 0 ) → sl2 is the Lie algebra isomorphism induced from the isomorphism between PSO(q 0 ) and PGL2 and defined by ! 0 x −y x −y + z 1 α(x 0 z ) = 2 y+z −x y z 0 and β : A3 → sl2 is the Lie algebra isomorphism 1 β( x y z ) = 2
x −y + z y+z −x
!
Then the map dγ is an isomorphism and this concludes the proof of the theorem. 76
f Proof of injectivity in Theorem 4.3.1. Start by fixing the double cover T˜ − → S and by considering two elements in Quad4 (T˜/S, ∅) with the same image in Az4 (T˜/S, ∅). That means considering (E1 , q1 , L1 ) and (E2 , q2 , L2 ) such that there exist an isomorphism φ : C0 (E1 , q1 , L1 ) → C0 (E2 , q2 , L2 ). ∼ q2 ⊗ N , for some line bundle By Theorem 4.3.2, there exsist a similarity transformation ψ : q1 = N on S defined on S. If we replace (E2 , q2 , L2 ) by (E2 ⊗ N , q2 ⊗ h1i, L2 ⊗ N ⊗2 ) we have that ψ : q1 ∼ = q2 , that proves the injectivity.
Lemma 4.3.9 ([8], Theorem 2.1). The map UO+ (q) (k) → I + sl2 (k) defined in the proof of Theorem 4.3.2 is an isomorphism. Proof. Since k is algebraically closed of characteristic different from two, we can diagonalize q as h1, −1, 1, 0i. If we call the corresponding orthogonal basis {v1 , v2 , v3 , v4 } we have that C0 (q 0 )(k) is generated by {1, v1 v2 , v2 v3 , v1 v3 } and we have an isomorphism φ between C0 (q 0 )(k) and the space of two by two matrices with coefficients in k ! ! ! ! 1 0 0 1 1 0 0 1 1 7→ , v1 v2 7→ , v2 v3 7→ , v1 v3 7→ 0 1 1 0 0 −1 −1 0 In a similar way we can give explicitely the generators of C0 (q) over A, that are 1, v1 v2 , v1 v3 and v2 v3 since v1 v4 = v2 v3 , v2 v4 = v1 v3 , v3 v4 = v1 v2 , v1 v2 v3 v4 = Hence we have an extension of the isomorphism φ to an isomorphism ψ between C0 (q) and the space of the matrices of coefficient in A. Moreover, the isomorphism φ induces the following group isomorphism PGL1 (C0 (q))(k) ∼ = PGL2 (A) and the following isomorphism of Lie algebras c0 (q)(k) ∼ = sl2 (k) where sl2 is the scheme of two by two matrices with trace equal to zero. In order to prove this, we can consider the linear transformation given by the matrix 1 0 0 0 0 1 0 0 0 0 1 0 x y z 1 and its image σx,y,z in I + sl2 (k). It can be thought of as an automorphism of C0 (q)(A). Then we have
σx,y,z (v1 v2 ) = v1 v2 + yv2 v3 − xv1 v3 σx,y,z (v2 v3 ) = v2 v3 + zv1 v3 − yv1 v2 σx,y,z (v1 v3 ) = v1 v3 + zv2 v3 − xv1 v2 σx,y,z () = 77
A simple calculation shows that the map σ coincides with the conjugation in the Clifford algebra of the element 1 1 − (zv1 v2 + xv2 v3 − yv1 v3 ) 2 ! x −y + z ψ(zv1 v2 + xv2 v3 − yv1 v3 ) = y+z −x and so the map UO+ (q) (k) → I 1 0 0 x
+ sl2 (k) is given explicitly by 0 0 0 ! 1 0 0 x −y + z 7→ I − /2 0 1 0 y+z −x y z 1
(4.6)
and is an isomorphism.
Let us give a brief description on how to prove the surjectivity part of Theorem 4.3.2. The key point is the definition of the norm functor, here given for regular quadratic forms. Let V˜ → U be the étale double covering of a regular quadratic form and let L/K be the corresponding quadratic extension of the fields of functions. The corestriction map from the Brauer group of L to the Brauer group of K, induces a well-defined map NT /S : Br(V˜ ) → Br(U ) Consider a sheaf of Azumaya algebras A on V˜ /U . If there is an involution τ on A , and NT /S (A ) ∼ = End(EU ) for some finitely generated projective k-module EU , then NT /S (τ ) is also an involution and it is adjoint to some quadratic form (E , q, L ) on U . The norm functor associates to the Azumaya algebra A , that in the case we are considering, has an involution, the associated quadratic form (E , q, L ). Auel, Parimala and Suresh manage to prove that this functor can also be defined for a quadratic form with simple degeneration, and then prove that this functor is the inverse of the map induced by Clifford algebra.
4.4
Line bundle valued quadratic forms of rank 3
The problem studied by Auel, Parimala and Suresh can be reformulated in the context of quadratic forms of rank 3, that correspond to conic fibrations. In fact that is the case of interest to us for the problem related to cubic threefolds. Max-Albert Knus proves the following result Theorem 4.4.1 ([43], Corollary 3.2.5). The set Quadr3 (S) and the set Az4 (S) are in bijection. V. Balaji generalized this result to the whole set Quad3 (S), dealing with the fact that the Clifford algebra of a generic conic fibration is not Azumaya everywhere. For this purpose Balaji introduced the following notion of limit of Azumaya algebras 78
Definition 4.4.2 ([10], Theorem 3.4, Theorem 3.8). SpAzu4 (X) denotes the set of isomorphism classes of associative unital algebra structures of rank 4 over X that are locally isomorphic to even Clifford algebras of rank 3 quadratic bundles. The result obtained by Balaji is the following Theorem 4.4.3 ([9], Theorem 3.1). The set Quad3 (S) is in bijection with the set SpAzu4 (S) via the map induced by the Clifford algebra C0 . This bijection can be restricted to Quadr3 (S), obtaining the same bijection of Theorem 4.4.1. By an abuse of notation we will call this bijection C0 . We are interested in describing the image in SpAzu4 (S) of the subspace Quad3 (Tˆ/S) under the map C0 of Theorem 4.4.3. In order to prove a result similar to the one proved by Auel, Parimala and Suresh described in the previous section, we will exploit the theorem of Balaji, specializing it to the case of conic fibrations with simple degeneration and describing the spaces with respect to the discriminant stack. We will focus on the subspace of Quad3 (S) given by generically regular quadratic forms, and study its image in SpAzu4 (S) under the map C0 . Notice that given an element of Az4 (Tˆ/S, S2 ) we can consider the pushforward by the morphism f : Tˆ → S, and obtain an algebra over S. If we start with a conic fibration of rank three, by Proposition 2.5.10 we obtain an element of Az4 (Tˆ/S, S2 ). Its pushforward to S is locally the Clifford algebra associated to the conic fibration. By looking at the definition of the set SpAzu4 (S) we want first to study the relation between Az4 (Tˆ/S, S2 ) and the map described in Theorem 4.4.3. Proposition 4.4.4. The bijection C0 described in Theorem 4.4.3, restricted to the subset Quad3 (Tˆ/S, S2 ) factors throught Az4 (Tˆ/S, S2 ). Proof. Let us denote by U the subset of SpAzu4 (S) image of Quad3 (Tˆ/S, S2 ), and consider the following diagram. Az4 (Tˆ/S, S2 ) 6
π∗
K
Quad3 (Tˆ /S, S2 )
C0
_
Quad3 (S)
C0
/' U _ / SpAzu (S) 4
π The maps involved are K, the map that associate a conic fibration with given stack Tˆ − → S to the algebra described in Proposition 2.5.21 and π∗ the pushforward along π. The commutativity is an easy consequence of the definition of SpAzu4 (S) and of the description of the map K.
Corollary 4.4.5. The map K : Quad3 (Tˆ/S, S2 ) → Az4 (Tˆ/S, S2 ) is injective. The same holds for the space of simply degenerate forms Proof. By Proposition 4.4.4, since the map C0 is injective it follows that the map K is injective too. 79
Theorem 4.4.6. The bijection described by Balaji in Theorem 4.4.3 induces a bijection between the ˆ ˆ sets Quadsd 3 (T /S) and Az4 (T /S, ∅). ˆ Proof. Let U sd be the subset of SpAzu4 (S) image of Quadsd 3 (T /S) by the map C0 . Proposition 4.2.7 guarantees that locally the Clifford algebra of a quadratic form with simple degeneration is Azumaya, then the identity map identifies U sd with Az4 (Tˆ/S, ∅). The situation is summed up in the following diagram. Az4 (Tˆ/S, ∅) 7
g
K
ˆ Quadsd 3 (T /S)
id C0
_
Quad3 (S)
C0
/ U sd _ / SpAzu (S) 4
K is the composition of two surjective functions, hence K is surjective.
4.5
Azumaya algebras and admissible coverings
We now want to give a brief discussion about the case of a generically regular conic fibration where the base S is a surface. In such a way, the discriminant divisor will be of dimension at most 1. Our reference for this section is the work of Bernardara and Bolognesi in [15], in which they study the rationality of conic bundles. As described in Section 2.5, we can associate a quadric fibration to a quadratic form with values in a line bundle. In the case we are interested in, we will require this conic bundle π : X → S to have the following properties i) The fiber Vs := π −1 (s) is isomorphic, as a scheme, to a (possibly degenerate) conic in P2k(s) where k(s) is the residue field. ii) For any irreducible curve C ⊂ S, the surface VC := π −1 (C) is irreducible We can always assume that we are working with a conic bundle with this properties, often called standard conic bundle, thanks to the following result. Proposition 4.5.1 ([66]). Every conic bundle is birationally equivalent to a standard one. In this section we will always assume the base S to be a smooth, rational and simply connected surface and the conic bundles to be standard. Lemma 4.5.2 ([39], Lemma 1). Let (E , q, L ) be a quadratic form and suppose the associated conic bundle π : X → S to be standard. Let Sd be the d-th degeneration loci of (E , q, L ). Then S1 is a reduced normal crossing divisor, and so Sn = ∅ for n > 2, and the fiber Vs can be • P1 , a line, if s ∈ / S1 80
• P1 ∨ P1 , two intersecting lines if s ∈ S1 r S2 • 2P1 , a double line, if s ∈ S2 In order to parametrize the component of fibers, we will use the definition of admissible covering, called a pseudo-revêtement in [12]. Definition 4.5.3. Let D be a connected curve, with only ordinary double points. A morphism ˜ → D is an admissible covering if π:D • For s ∈ / Sing(D), there exists an open neighbourhood U of s such that π|π−1 (U ) is étale of degree two • For s a double point of D, such that OD,s ∼ = k[u, v]/(uv), we have that π −1 (s) is a point s˜ ∗ 2 ∗ 2 ∼ such that OD,˜ ˜ s = k[x, y]/(xy) and π u = x , π v = y . Given a standard conic bundle on S we can consider the curve S1 parametrizing the components of fibers in the surface VS1 . In such a way, VS1 can be thought as a P1 bundle over a certainly curve S˜1 . S˜1 turns out to satisfy the definition of admissible covering of S1 . We will refer to this curve as the associated admissible cover of a conic fibration (E , q, L ). Some reference can be found in [39]. The following lemma, proved by Bernardara and Bolognesi, shows that the Clifford algebras C0 are completely determined by the admissible covers of the discriminant locus. Lemma 4.5.4 ([15], Lemma 3.2). Let S be a smooth rational simply connected surface and (E , q, L ), (E 0 , q 0 , L 0 ) be two quadratic forms with associated sheaves of even parts of Clifford algebras C0 and ˜ → D, then C0 C00 respectively. If the two conic bundles have the same associated admissible cover D 0 is isomorphic to C0 . The proof of the lemma involves the exact sequence of Artin and Mumford. In [3] Artin and Mumford prove that if S is a complete non singular algebraic surface over an algebraic closed field k and H 1 (S, Q/Z) = 0 then the following sequence is exact M M r a r s 1 0 → Br(S) → − Br(k(S)) − → Het (k(C), Q/Z) → − µ−1 → − µ−1 → 0 P ∈S
C⊂S
S where µn denotes the group of n-th roots of unity and µ−1 = n µ−1 n . All the elements of order 1 (k(C), Q/Z) two in the Brauer group of k(S) are Azumaya algebras of rank 4. Moreover the space Het parametrizes the admissible covering of the curve C. If S is rational, then Br(S) = 0 and so we L 1 (k(C), Q/Z). Then the proof of the lemma follows easily. obtain an injection Br(k(S)) ,→ C Het A consequence of this lemma is the following Corollary 4.5.5. Two quadratic forms with the same associated admissible cover belong to the same projective similarity class on Quad3 (S) Proof. A posteriori this is a consequence of Lemma 4.5.4 and of the bijection C0 of Theorem 4.4.3.
81
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Acknowledgments My two advisors Alberto Canonaco and Paolo Stellari. The other people I met during my PhD Francesco Amodeo, Anna Barbieri, Marcello Bernardara, Michele Bolognesi, Matteo Bonfanti, Andrea Cattaneo, Francesco Genovese, Valeria Marcucci, Carlo Orrieri, Gian Pietro Pirola, Daniele Rosmondi, Andrea Seppi My girlfriend Eleonora. My family Arianna, Ivano, Ornella, Pina and Oreste. My office mates Davide Calza, Francesco Matticchio, Enrica Nicolino, Giuseppe Vacca. All the others Federico Benedetti, Davide Bettini, Alessandro Cattaneo, Cristina Cattoni, Domingo Combi, Adriana Cosentino, Francesco Cristofaro, Maria Dedó, Filippo Favale, Stefano dell’Agostino, Andrea Gaburri, Alessandra Mazza, Giovanni Tocalli, Valentina Quadrio, Marco Riccardi.
