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To the blessed memory of Andrei Nikolaevich Tyurin
Russian Math.Surveys 58:3 511–591 c
2003 RAS(DoM) and LMS
Uspekhi Mat.Nauk 58:3 89–172 DOI 10.1070/RM2003v058n03ABEH000629
Derived categories of coherent sheaves
and equivalences between them
D. O. Orlov
Abstract. This paper studies the derived categories of coherent sheaves on smooth
complete algebraic varieties and equivalences between them. We prove that every
equivalence of categories is represented by an object on the product of the varieties.
This result is applied to describe the Abelian varieties and K3 surfaces that have
equivalent derived categories of coherent sheaves.
Contents
Introduction 512
Chapter 1. Preliminaries 517
1.1. Triangulated categories and exact functors 517
1.2. Derived categories and derived functors 523
1.3. Derived categories of sheaves on schemes 525
Chapter 2. Categories of coherent sheaves and functors between them 530
2.1. Basic properties of categories of coherent sheaves 530
2.2. Examples of equivalences: flopping birational transformations 538
Chapter 3. Fully faithful functors between derived categories 544
3.1. Postnikov diagrams and their convolutions 544
3.2. Fully faithful functors between the derived categories of coher-
ent sheaves 547
3.3. Construction of the object representing a fully faithful functor 549
3.4. Proof of the main theorem 555
3.5. Appendix: the n-Koszul property of a homogeneous coordi-
nate algebra 564
Chapter 4. Derived categories of coherent sheaves on K3 surfaces 567
4.1. K3 surfaces and the Mukai lattice 567
4.2. The criterion for equivalence of derived categories of coherent
sheaves 570
Chapter 5. Abelian varieties 573
5.1. Equivalences between categories of coherent sheaves on Abe-
lian varieties 573
This work was partially supported by RBRF (under grant no. 02-01-00468), by the Leading
Scientific Schools programme (under grant no. 00-15-96085), and by INTAS (under grant INTAS-
OPEN-2000-269). The investigations described in the paper were partially supported by CRDF
(grant no. RM1-2405-MO-02). The author also expresses his gratitude to the Domestic Science
Support Foundation.
AMS 2000 Mathematics Subject Classification. Primary 14F05, 18E30; Secondary 14J28,
14K05.
512 D. O. Orlov
5.2. Objects representing equivalences, and groups of auto-equi-
valences 581
5.3. Semi-homogeneous vector bundles 583
Bibliography 589
Introduction
The main objects of study in algebraic geometry are algebraic varieties (or
schemes) and morphisms between them. Every algebraic variety Xis a ringed
topological space and thus has a topology (usually the Zariski topology) and a
sheaf of rings of regular functions OX.
To a large extent, the study of an algebraic variety is the study of sheaves on
it. Since the space is ringed, the natural sheaves are sheaves of OX-modules on it,
among which the quasi-coherent and coherent sheaves are distinguished by their
algebraic nature. Recall that a sheaf of OX-modules is quasi-coherent if it is locally
representable as the cokernel of a homomorphism of free sheaves, and coherent if
these free sheaves are of finite rank. (Locally free sheaves on a variety correspond
one-to-one with vector bundles, and we therefore use these terms interchangeably
in what follows.)
Thus, corresponding to every algebraic variety Xwe have the Abelian categories
coh(X) of coherent sheaves and Qcoh(X) of quasi-coherent sheaves. Morphisms
between varieties induce inverse image and direct image functors between these
Abelian categories. However, these functors are not exact, that is, do not take
exact sequences to exact sequences. This causes significant complications when
working with Abelian categories and non-exact functors between them. To preserve
functoriality, Cartan and Eilenberg [11] introduced the notion of derived functors
which give necessary corrections to non-exact functors. This technique was devel-
oped by Grothendieck in [15], which subsequently led to the introduction of the
new concepts: derived category and derived functors between them.
Derived categories, in contrast to Abelian categories, do not have short exact
sequences, and the kernels and cokernels of morphisms are not defined. However,
derived categories admit a certain internal structure, formalized by Verdier as the
notion of triangulated category [44].
Passing from Abelian categories to their derived categories allows us to solve
many problems related to difficulties arising in the study of natural functors.
Among the first examples, we mention the creation of the global intersection
theory and the proof of the Riemann–Roch theorem. These results, achieved by
Grothendieck and his co-authors [41], were made possible by the introduction of
the triangulated category of perfect complexes.
Another example relates to the introduction of perverse sheaves and to the estab-
lishment of the Riemann–Hilbert correspondence between holonomic modules with
regular singularities and constructible sheaves (see [3], [23]); this correspondence
only became possible on applying the notions and techniques of triangulated cate-
gories.
Many problems relating to the study of varieties require the study and descrip-
tion of the derived categories of coherent sheaves on them. In the simplest cases,
Derived categories of coherent sheaves 513
when the variety is a point or a smooth curve, every object in the derived category of
coherent sheaves is isomorphic to a direct sum of some family of coherent sheaves
with suitable shifts; that is, every A∈Db(coh X) is isomorphic to k
i=1 Fi[ni],
where Fiare coherent sheaves. These examples reflect the fact that in these cases
the Abelian category has homological dimension 1. However,forhigherdimen-
sional varieties there are complexes that are not isomorphic in the derived category
to the sum of their cohomology. Thus, describing the derived category for varieties
of dimension greater than 1 is a difficult and interesting problem. The first steps
in this direction were made in [4] and [2], which described the derived category of
coherent sheaves on projective spaces, and subsequently allowed the technique to be
applied to the study the moduli space of vector bundles on P2and P3.Inparticular,
these papers showed that the derived category of coherent sheaves Db(coh Pn)on
projective space is equivalent to the derived category of finite-dimensional modules
over the finite-dimensional algebra A=End
i=n
i=0 O(i). This approach has been
perfected since then, and descriptions of the derived categories of coherent sheaves
on quadrics and on flag varieties have also been obtained ([20]–[22]).
Introducing the notions of exceptional family and semi-orthogonal decomposition
enabled one to formulate new principles for describing the derived categories of
coherent sheaves [5], [6]. It turned out that the existence of a complete exceptional
family always realizes an equivalence of the derived category of coherent sheaves
with the derived category of finite-dimensional modules over the finite-dimensional
algebra of endomorphisms of the given exceptional family [5]. The notion of semi-
orthogonal decomposition allowed us to describe the derived category of a blowup
in terms of the derived category of the variety that is blown up and that of the
subvariety along which the blowup occurs [34].
However, for many types of varieties, no description of the derived category is
possible. Nevertheless, the natural question can be posed roughly as follows: how
much information is preserved on passing from a variety to its derived category of
coherent sheaves? In fact it turns out that ‘almost all’ information is preserved
under this correspondence. In many cases one can even recover the variety itself
from its derived category of coherent sheaves, for example if the canonical (or
anticanonical) sheaf is ample [8].
For certain types of varieties one nevertheless finds examples in which two dis-
tinct varieties have equivalent derived categories of coherent sheaves. The first
example of two different varieties having equivalent derived categories of coherent
sheaves was found by Mukai [29]. He showed that this happens for every Abelian
variety and its dual variety. We generalized this construction in [38]: for any
Abelian variety, we introduced an entire class of Abelian varieties, all of which
have the same derived category of coherent sheaves. On the one hand, these exam-
ples show that there are varieties having equivalent derived category of coherent
sheaves; on the other hand, every class of varieties with equivalent derived cate-
gories of coherent sheaves is ‘small’ (it is finite in all the examples).
To obtain a complete classification of varieties with equivalent derived categories
of coherent sheaves, we need a description of the functors and equivalences between
them. It turns out that equivalences are always geometric in nature, that is, they
are represented by certain complexes of sheaves on the product of the varieties.
514 D. O. Orlov
We explain what we mean. In what follows we write Db(X)todenotethe
bounded derived category of coherent sheaves on X. Any morphism f:X→Y
between smooth complete algebraic varieties induces two exact functors between
their bounded derived categories of coherent sheaves: the direct image functor
Rf∗:Db(X)−→ Db(Y) and the inverse image functor Lf∗:Db(Y)−→ Db(X),
which is left adjoint to Rf∗. Moreover, every object E∈Db(X) defines an exact
tensor product functor ⊗LE:Db(X)−→ Db(X). We can use these standard
derived functors, to introduce a new large class of exact functors between the derived
categories Db(X)andDb(Y).
Let Xand Ybe two smooth complete varieties over a field k. Consider the
Cartesian product X×Y,andwrite
Xp
←− X×Yq
−→ Y
for the projections of X×Yto Xand Yrespectively. Every object E∈Db(X×Y)
defines an exact functor ΦEfrom the derived category Db(X) to the derived cate-
gory Db(Y), given by
ΦE(·):=R·q∗(E⊗Lp∗(·)).(1)
Every functor of this type has left and right adjoint functors.
Thus, to every smooth complete algebraic variety one can assign its derived
category of coherent sheaves, and to every object E∈Db(X×Y) on the product
of two such varieties one can assign an exact functor ΦEfrom the triangulated
category Db(X) to the triangulated category Db(Y). This paper is devoted to the
study of this correspondence.
One of the first questions that arises in the study of derived categories of coherent
sheaves is the following: can every functor between these categories be represented
by an object on the product? that is, is it of the form (1)? In Chapter 3 we give
an affirmative answer to this question if the functor is an equivalence.
Two other central questions here are as follows:
1) When are the derived categories of coherent sheaves on two different smooth
complete varieties equivalent as triangulated categories?
2) What is the group of exact auto-equivalences of the derived category of
coherent sheaves on a given variety X?
Some results in this direction were already known. Exhaustive answers to the
above questions are known when the variety has ample canonical or anticanonical
sheaf: in [8] we proved that a smooth projective variety Xwith ample canonical (or
anticanonical) sheaf can be recovered from its derived category of coherent sheaves
Db(X); moreover, [8] also gives an explicit construction for recovering X.For
varieties of this type, the group of exact auto-equivalences can also be described.
We now describe the contents and structure of this paper. Most of the results
collected here can be found in some form in the papers [7], [8], [34], [35], and [37].
Chapter 1 collects material of a preliminary nature. We first give the definition of
triangulated category and recall the notions of an exact functor between triangu-
lated categories, the localization of a triangulated category with respect to a full
subcategory, and the general definition of derived functor for localized triangulated
categories. After this we define the homotopy category and the derived category
Derived categories of coherent sheaves 515
of an Abelian category, and we also discuss the properties of derived categories of
coherent and quasi-coherent sheaves on schemes and the functors between these
categories.
In Chapter 2 we introduce the class of functors between the bounded derived
categories of coherent sheaves on smooth complete algebraic varieties that are
representable by objects on products, and describe their main properties. Using
results from Chapter 3, we prove that, if two smooth projective varieties Xand Y
have equivalent derived categories, then there exists an isomorphism between the
bigraded algebras HA(X)andHA(Y) defined by the following formula:
HA(X)=
i,k
HAi,k(X)=
i,k
p+q=i
Hp(X,
q
TX⊗ωk
X),
where TXis the tangent bundle and ωXthe canonical bundle of X(Theorem 2.1.8
and Corollary 2.1.10).
In the second section of Chapter 2 we present a whole class of pairs of vari-
eties having equivalent derived categories of coherent sheaves. These examples are
interesting in that the varieties that arise are birationally isomorphic (but not iso-
morphic in general) and are related by a birational transformation called a flop. In
particular, these examples show that we cannot weaken the condition of ampleness
of the canonical (or anticanonical) class in the theorem on recovering Xfrom D(X).
Let Ybe a smoothly embedded closed subvariety in a smooth complete algebraic
variety Xsuch that Y∼
=Pkwith normal bundle NX/Y ∼
=OY(−1)⊕(l+1). We assume
that lkand write
Xto denote the blowup of Xwith centre along Y.Inthiscase
the exceptional divisor
Yis isomorphic to the product of projective spaces Pk×Pl.
There is a blowdown of
Xsuch that
Yprojects to the second factor Pl.Consider
the diagram of projections
Xπ
←−
Xπ+
−→ X+.
The birational map fl: X X+is the simplest example of a flip or flop; it is a
flip for l<kand a flop for l=k.
The main theorem of this section relates the derived categories of coherent
sheaves on the varieties Xand X+. It asserts that for any line bundle Lon
X,the
functor
Rπ∗(Lπ+∗(·)⊗L): Db(X+)−→ Db(X)
is fully faithful, and for k=lthis functor is an equivalence.
Chapter 3 is central. It is concerned with proving that every equivalence between
derived categories of coherent sheaves on smooth projective varieties is represented
by an object on the product. This assertion allows us to describe equivalences
between derived categories of coherent sheaves and to answer the question of when
two different varieties have equivalent derived categories of coherent sheaves.
In fact, in this chapter we prove a more general assertion: namely, that any func-
tor between bounded derived categories of coherent sheaves on smooth projective
varieties that is fully faithful and has an adjoint functor can be represented by an
object Eon the product of these varieties; that is, it is isomorphic to the func-
tor ΦEdefined by the rule (1). Moreover, the object Erepresenting it is uniquely
determined up to isomorphism (Theorem 3.2.1).
516 D. O. Orlov
In Chapter 4 we study the derived categories of coherent sheaves on K3 surfaces.
For any K 3 s urface S, the cohomology lattice H∗(S, Z) has a symmetric bilinear
form defined by the rule
(u, u)=r·s+s·r−α·α∈H4(S, Z)∼
=Z
for any pair u=(r, α, s), u=(r,α
,s
)∈H0(S, Z)⊕H2(S, Z)⊕H4(S, Z). The
cohomology lattice H∗(S, Z) with the bilinear form ( ·,·) is called the Mukai lattice
and denoted by
H(S, Z).
The lattice
H(S, Z) admits a natural Hodge structure. In the present case, by
Hodge structure, we mean that we fix a distinguished one-dimensional subspace
H2,0(S)inthespace
H(S, C). We say that the Mukai lattices of two K3 surfaces
S1and S2are Hodge isometric if there is an isometry between them taking the
one-dimensional subspace H2,0(S1)toH
2,0(S2).
The main theorem of this chapter (Theorem 4.2.1) asserts that the derived cat-
egories Db(S1)andDb(S2) of coherent sheaves on two K3 surfaces over the field
Care equivalent as triangulated categories if and only if there is a Hodge isometry
f:
H(S1,Z)∼
−→
H(S2,Z) between their Mukai lattices. This theorem has another
version in terms of lattices of transcendental cycles (Theorem 4.2.4).
In view of the Torelli theorem for K3 surfaces [39], [27], which says that a K3
surface can be recovered from the Hodge structure on its second cohomology, we
obtain an answer in terms of Hodge structures to the question of when the derived
categories of coherent sheaves on two K3 surfaces are equivalent.
In Chapter 5 we study the derived categories of coherent sheaves on Abelian
varieties and their groups of auto-equivalences. Let Abe an Abelian variety and
Athe dual Abelian variety. As proved in [29], the derived categories of coherent
sheaves Db(A)andDb(
A) are equivalent, and the equivalence, called the Fourier–
Mukai transform, can be given by means of the Poincar´e line bundle PAon the
product A×
Aby the rule (1): F→ R·p2∗(PA⊗p∗
1(F)).
This construction of Mukai was generalized in [38] as follows. Consider two
Abelian varieties Aand Band an isomorphism fbetween the Abelian varieties
A×
Aand B×
B.Writefin the matrix form
f=xy
zw
,
where xstands for a homomorphism from Ato B,yfrom
Ato B,andsoon. We
say that the isomorphism fis isometric if its inverse has the form
f−1=w−y
−zx.
We define U(A×
A, B ×
B) to be the set of isometric isomorphisms f.IfB=A,
then we denote this set by U(A×
A); note that it is a subgroup of Aut(A×
A).
We proved in [38] that if there is an isometric isomorphism between A×
A
and B×
Bfor two Abelian varieties Aand Bover an algebraically closed field,
Derived categories of coherent sheaves 517
then the derived categories of coherent sheaves Db(A)andDb(B)areequivalent.
In Chapter 5 we prove that these conditions are equivalent over an algebraically
closed field of characteristic zero; that is, the derived categories Db(A)andDb(B)
are equivalent if and only if there is an isometric isomorphism from A×
Ato B×
B.
In fact, the “only if” part holds over an arbitrary field (Theorem 5.1.16). As a
corollary, we see that there are only finitely many non-isomorphic Abelian varieties
whose derived categories are equivalent to Db(A) for a given Abelian variety A
(Corollary 5.1.17).
Representing equivalences by objects on the product, we construct a map from
the set of all exact equivalences between Db(A)andDb(B)tothesetofisometric
isomorphisms from A×
Ato B×
B. We then prove that this map is functorial
(Proposition 5.1.12). In particular, we obtain a homomorphism from the group of
exact auto-equivalences of Db(A) to the group U(A×
A) of isometric automorphisms
of A×
A.
In §5.2 we describe the kernel of this homomorphism, which turns out to be
isomorphic to the direct sum of Zand the group of k-valued points of A×
A
(Proposition 5.2.3). Technically, this description is based on the fact that the
object on the product of Abelian varieties that defines the equivalence is in fact a
sheaf, up to a shift in the derived category (Proposition 5.2.2).
In the final §5.3, under the assumption that the ground field is algebraically
closed and char(k) = 0, we give another proof of the assertion in [38]; this proof
uses results in [30] describing semi-homogeneous bundles on Abelian varieties. In
particular, we obtain a description of the group of auto-equivalences as an exact
sequence
0−→ Z⊕(A×
A)k−→ Auteq Db(A)−→ U(A×
A)−→ 1.
CHAPTER 1
Preliminaries
1.1. Triangulated categories and exact functors. A detailed treatment of
the facts collected in this chapter may be found in [14], [24], [25], and [44]. The
notion of triangulated category was first introduced by Verdier in [44]. Let Dbe
some additive category. We define a structure of tri angul ated category on Dby
giving the following data:
a) an additive shift functor [1] : D−→ Dwhich is an auto-equivalence;
b) a class of distinguished (or exact) triangles
Xu
−→ Yv
−→ Zw
−→ X[1]
that must satisfy the following set of axioms T1–T4.
T1. a) For any object Xthe triangle Xid
−→ X−→ 0−→ X[1] is distinguished.
b) If a triangle is distinguished, then any isomorphic triangle is also distin-
guished.
518 D. O. Orlov
c) Any morphism Xu
−→ Yin Dcan be completed to a distinguished triangle
Xu
−→ Yv
−→ Zw
−→ X[1].
T2. A triangle Xu
−→ Yv
−→ Zw
−→ X[1] is distinguished if and only if the
triangle
Yv
−→ Zw
−→ X[1] −u[1]
−→ Y[1]
is distinguished.
T3. Given two distinguished triangles and two morphisms between their first
and second terms that form a commutative square, this diagram can be
completed to a morphism of triangles:
T4. For any pair of morphisms Xu
−→ Yv
−→ Zthere is a commutative diagram
Xu
−−−−→Yx
−−−−→Z−−−−→X[1]
v
w
X−−−−→Zy
−−−−→Yw
−−−−→X[1]
t
u[1]
XXr
−−−−→Y[1]
r
Y[1] x[1]
−−−−→Z[1]
in which the top two rows and the two central columns are distinguished
triangles.
Let Dbe a triangulated category. We say that a full additive subcategory N⊂D
is a tr iangulat ed subcategory if it is closed under the shift functor and under taking
the mapping cone of morphisms; that is, if two objects of some triangle
X−→ Y−→ Z−→ X[1]
belong to N, then so does the third object. We now describe the type of functors
between triangulated categories that it makes sense to consider.
Definition 1.1.1. We say that an additive functor F:D−→ Dbetween two
triangulated categories Dand Dis exact if
a) Fcommutes with the shift functor, that is, there is a given isomorphism of
functors
tF:F◦[1] ∼
−→ [1] ◦F,
b) Ftakes each distinguished triangle in Dto a distinguished triangle in D
(where we use the isomorphism tFto replace F(X[1]) by F(X)[1]).
Derived categories of coherent sheaves 519
It follows at once from the definition that the composite of two exact functors is
again exact. Another property we need concerns adjoint functors.
Lemma 1.1.2 ([6], [8]). If a functor G:D−→ Dis left (or right)adjoint to an
exact functor F:D−→ D,then Gis also exact.
We define and describe the main properties of a Serre functor, the abstract
definition of which was given in [6] (see also [8]).
Definition 1.1.3. Let Dbe a k-linear category with finite-dimensional Hom-spaces
between objects. A covariant functor S:D→Dis a Serre functor if it is an
equivalence of categories, and there exists a bifunctorial isomorphism
ϕA,B :Hom
D(A, B)∼
−→ HomD(B, SA)∗for any objects A, B ∈D.
Lemma 1.1.4 [8]. Any equivalence of categories Φ: D−→ Dcommutes with Serre
functors;that is,there exists a natural isomorphism of functors Φ◦S∼
−→ S◦Φ,
where Sand Sare Serre functors for the categories Dand Drespectively.
If we have two Serre functors for the same category, then they are isomorphic,
and this isomorphism commutes with the bifunctorial isomorphisms ϕA,B in the
definition of Serre functor. Indeed, let Sand Sbe two Serre functors for
the category D. Then for any object Ain Dthere is a natural isomorphism
Hom(A, A)∼
=Hom(A, SA)∗∼
=Hom(SA, SA).
Considering the image of the identity morphism idAunder this identification, we
obtain a morphism SA −→ SA, which gives an isomorphism S∼
−→ S.
Thus, a Serre functor for a category D(if it exists) is uniquely determined
(up to isomorphism). In what follows, we will be interested in Serre functors for
triangulated categories.
Lemma 1.1.5 [6]. A Serre functor for a triangulated category is exact.
We recall the definition of localization of a category and, in particular, the local-
ization of a triangulated category with respect to a full triangulated subcategory
(see [13]). Let Cbe a category and Σ a class of morphisms in C; the localization of
Cwith respect to Σ has a good description if Σ admits a calculus of left fractions;
that is, if the following properties hold:
L1. All the identity morphisms of the category belong to Σ.
L2. The composite of any two morphisms in Σ again belongs to Σ.
L3. Each diagram of the form Xs
←− Xu
−→ Ywith s∈Σcanbecompletedto
a commutative square
with t∈Σ.
L4. If fand gare two morphisms, and there exists a morphism s∈Σ satisfying
fs =gs, then there also exists t∈Σ such that tf =tg.
520 D. O. Orlov
If Σ admits a calculus of left fractions, then the category C[Σ−1] can be described
as follows. The objects of C[Σ−1]arejustthoseofC. The morphisms from Xto Y
are equivalence classes of diagrams (s, f)inCof the form
Xf
−→ Ys
←− Ywith s∈Σ,
where two diagrams (f,s)and(g, t) are equivalent if they fit into a commutative
diagram
with r∈Σ.
The composite of two morphisms (f, s)and(g, t) is the morphism (gf, st)con-
structed using the square of axiom L3:
One sees readily that C[Σ−1] constructed in this way is indeed a category (with
morphisms between objects forming a set), and that the canonical functor
Q:C−→ C[Σ−1] defined by X→ X, f → (f, 1)
inverts all morphisms in Σ, and is universal in this sense (see [13]).
Consider a full subcategory B⊂Cand write Σ ∩Bfor the class of morphisms
in Balso belonging to Σ. We say that Bis right cofinal in Cwith respect to Σ if
for any s:X−→ Xin Σ with X∈Bthere is a morphism f:X−→ Ysuch that
fs ∈Σ∩B.
Lemma 1.1.6 ([17], [25]). The class Σ∩Balso admits a calculus of left fractions
and,if Bis right cofinal in Cwith respect to Σ, the canonical functor
B[(Σ ∩B)−1]−→ C[Σ−1]
is fully faithful.
We recall the definition of fully faithful functor.
Derived categories of coherent sheaves 521
Definition 1.3. We say that a functor F:C−→ Dis fully faithful if the natural
map
Hom(X, Y )−→ Hom(FX,FY)
is a bijection for any two objects X, Y ∈C.
Now let Dbe a triangulated category and Na full triangulated subcategory. We
write Σ for the class of morphisms sin Dthat fit in an exact triangle
N−→ Xs
−→ X−→ N[1],
with N∈N, and call Σ the multiplicative system associated with the subcategory
N. It follows from the general theory of localization that there exists an additive
category D[Σ−1] and an additive localization functor Q:D−→ D[Σ−1]. We can
give the category D[Σ−1] the shift functor induced by [1] : D−→ D.Moreover,
we define distinguished triangles in D[Σ−1] to be the triangles isomorphic to the
images of distinguished triangles in Dunder the localization. We set
D/N:= D[Σ−1].
Proposition 1.1.8. Giving D/Nthe structure described above makes it into a
tr iangu lat ed category,and makes Q:D−→ D/Ninto an exact functor.
Note that in our situation the system Σ admits a calculus of left (and right)
fractions, so that the category D/Nadmits a good description as given above.
Following Deligne [12] (see also [25]), we now describe the general construction of
derived functors for the localizations of triangulated categories. Let Cand Dbe
triangulated categories and F:C−→ Dan exact functor. Let M⊂Cand N⊂D
be full triangulated categories. Since we do not assume that FM⊂N, the functor
Fdoes not induce any functor from C/Mto D/N. However, there may exist a
certain canonical approximation to an induced functor, namely, an exact functor
RF:C/M−→ D/N, and a morphism of exact functors can : QF −→ (RF)Q.The
construction proceeds as follows. Write Σ for the multiplicative system associated
with the subcategory M.LetYbe an object of C/M. We define a contravariant
functor rFY from D/Nto the category of Abelian groups by the following rule:
the value of rFY(X) at an object X∈D/Nis the equivalence classes of pairs (s, f )
Ys
−→ Y,X
f
−→ FY,
with s∈Σandfa morphism in D/N.Twosuchpairs(s, f )and(t, g)areequivalent
if there exist commutative diagrams in Cand D/Nof the form
522 D. O. Orlov
with r∈Σ. If the functor rFY is representable, we define RFY as the object that
represents it, and say that the right derived functor RFis defined on Y.Inthis
case we have an isomorphism
Hom(X, RFY)∼
=rFY(X).
One sees readily that a morphism of functors rFα:rFY −→ rFZ is defined for
any morphism α:Y−→ Zin C/M. Now if the derived functor RFis defined
on both Yand Z, the morphism RFα is also defined. This makes RFa functor
W−→ D/Non some full subcategory W⊂C/M, consisting of the objects on which
RFis defined.
Proposition 1.1.9 [12]. Suppose that
X−→ Y−→ Z−→ X[1]
is a distinguished triangle in C/Mand RFis defined on Xand Z.Then it is also
defined on Y,and takes the given triangle into a distinguished triangle of D/N.
Thus,WisatriangulatedsubcategoryinC/Mand RF:W−→ D/Nis an exact
functor.
It follows at once from the construction of the derived functor that there is a
canonical morphism can: QF Y −→ (RF)QY for any object Y∈C(provided, of
course, that RFis defined on QY ∈C/M). All these morphisms define a natural
transformation of triangulated functors can: QF |W−→ (RF)Q|W.
The left derived functor LFis defined in the dual way: for Y∈C/M, we define
a covariant functor lFY whose value at X∈D/Nis the equivalence classes of pairs
(s, f),
Ys
−→ Y, FXf
−→ Y,
with s∈Σandfa morphism in D/N.ThenLFY (if it exists) is the object rep-
resenting the functor lFY;thatis,Hom(LFY,X)∼
=lFY(X). There is a canonical
morphism can : LFQY −→ QFY .
Suppose that the functor F:C−→ Dtakes the subcategory Minto N.Inthis
case the derived functors RFand LFare both isomorphic to the canonical functor
C/M−→ D/Ninduced by F.
Let j:V→Cbe the inclusion of a full triangulated subcategory which is right
cofinal with respect to Σ. By Lemma 1.1.6 the induced functor V/(V∩M)−→ C/M
is fully faithful. We denote it by Rj.
Lemma 1.1.10. Fo r a n y o b j ect V∈Vthe functor RFis defined on Vif and only
if R(Fj)is defined on V,and there is an isomorphism of functors R(Fj)V∼
−→
RFRjV .
We now describe conditions under which the right derived functor is defined on
the entire category C.
Definition 1.1.11. An object Y∈Cis said to be (right) F-split with respect to
Mand Nif RFis defined on Yand the canonical morphism QF Y −→ (RF)QY
is an isomorphism.
The following lemma gives a characterization of F-split objects.
Derived categories of coherent sheaves 523
Lemma 1.1.12. An object Y∈Cis F-split if and only if for any morphism
s:Y−→ Yin Σthe morphism QF s admits a retraction,that is,there exists a
p:QF Y −→ QF Y such that p◦QF s =id.
We say t hat Cadmits enough F-split objects (with respect to Mand N)iffor
any Y∈Cthere exists a morphism s:Y−→ Y0in Σ such that Y0is F-split. In
this case RFis defined on the entire category C/M, and there are isomorphisms
RFY ∼
−→ RFY
0∼
←− FY
0.
To conclude this section we say a few words on adjoint functors. Suppose that a
functor Fhas a left adjoint G:D−→ Cand assume that the derived functors RF
and LGexist (that is, that they are everywhere defined). Then LGis again a left
adjoint to RF, and hence there are functorial isomorphisms
Hom(LGX, Y )∼
=Hom(X, RFY)forX∈D/Nand Y∈C/M.(2)
1.2. Derived categories and derived functors. Let Abe an additive category.
We wr i te C(A) to denote the category of differential complexes. Its objects are the
complexes
M·=(···−→Mpdp
−→ Mp+1 −→···)withMp∈Afor p∈Z,andd2=0,
and the morphisms f:M·−→ N·are families of morphisms fp:Mp−→ Npin A
that commute with the differentials; that is,
dNfp−fp+1 dM=0 forany p.
We wr i te C+(A), C−(A)andCb(A) for the full subcategories of C(A)formed
by complexes M·for which Mp=0forallp0, respectively for all p0,
respectively for all p0andallp0.
We say that a morphism of complexes f:M·−→ N·is nul l-homotopic if fp=
dNhp+hp+1dMfor all p∈Zfor some family of morphisms hp:Mp+1 −→ Np.We
define the homotopy category H(A) to be the category having the same objects as
C(A) and the morphisms in H(A) are classes fof morphisms fbetween complexes
modulo null-homotopic morphisms.
We define the shift functor [1] : H(A)−→ H(A)bytherule
(M[1])p=Mp+1,d
M[1] =−dM.
We define a standard triangle in H(A) to be a sequence
Lf
−→ Mg
−→ Cf h
−→ L[1],
where f:L−→ Mis some morphism of complexes, Cf =M⊕L[1] is a graded
object of C(A), with the differential
dCf =dMf
0−dL,
gis the canonical embedding M−→ Cf,and−hthe canonical projection. As
usual, Cf is called the mapping cone of f.
524 D. O. Orlov
Lemma 1.2.1. The category H(A)with [1] as shift functor and the class of tri-
angles isomorphic to standard triangles as distinguished triangles is a triangulated
category.
We wr i te H+(A), H−(A)andHb(A) for the images in H(A) of the categories
C+(A), C−(A)andCb(A) respectively. These categories are also triangulated.
Suppose now that Ais an Abelian category. To define the derived category of
an Abelian category, we must recall the notions of acyclic complex and of quasi-
isomorphism. For any complex N·and each p∈Z, the cohomology Hp(N·)∈A
is defined as Ker dp/Im dp−1. Thus, for any integer pwe have an additive functor
Hp:C(A)−→ AtakingacomplexN·to its pth cohomology Hp(N·)∈A.
We say that a com p l ex N·∈C(A)isacyclic at the nth term if Hn(N·) = 0, and
simply acyclic if all its cohomology vanishes, Hn(N·)=0forn∈Z.Wedenoteby
Nthe full subcategory of H(A) consisting of all acyclic complexes. The following
lemma is practically obvious.
Lemma 1.2.2. T he subcategor y Nis a full triangulated subcategory of H(A).
We say that a morphism f:X−→ Yin H(A)isaquasi-isomorphism if its
mapping cone is an acyclic complex. In other words, fis a quasi-isomorphism if
the map it induces on cohomology is an isomorphism. Let Quis be the multiplicative
system associated with N, that is, the system consisting of all quasi-isomorphisms.
Definition 1.2.3. The derived category D(A) of an Abelian category Ais defined
as the localization of the homotopy category H(A) with respect to the subcategory
of all acyclic complexes, that is,
D(A):=H(A)/N=H(A)[Quis−1].
For ∗∈{+,−,b}, we define the corresponding derived category D∗(A)inthesame
way as the localization H∗(A)/(H∗(A)∩N).
Lemma 1.2.4. Fo r ∗∈{+,−,b},the canonical functors
D∗(A)−→ D(A)
define equivalences with the full subcategories of D(A)formed by complexes that are
acyclic respectively for n0, for n0, and for n0and n0. T he subcate-
gory H+(A)is right cofinal in H(A)with respect to the class of quasi-isomorphisms,
and H−(A)is left cofinal.
Suppose that the Abelian category Ahas enough injective objects; that is, every
object embeds in an injective. We denote by Ithe full subcategory of Aconsisting
of the injective objects. In this case, one can prove that the composite functor
H+(I)−→ H+(A)Q
−→ D+(A)
is an equivalence of categories (see [17]). Similarly, if an Abelian category Ahas
enough pro jectives, then the composite functor
H−(P)−→ H−(A)Q
−→ D−(A)
is an equivalence, where Pin Ais the full subcategory of pro jectives.
Derived categories of coherent sheaves 525
Let F:A−→ Bbe an additive functor (not necessarily exact) between Abelian
categories. Then Finduces in an obvious way an exact functor H(A)−→ H(B),
which we denote by the same symbol F. The general construction of (right) derived
functor described in the previous section gives a functor RF,definedonacertain
full triangulated subcategory of D(A), and taking values in D(B). The same applies
to the left derived functor. We define the nth right (respectively left) derived functor
of Fas the cohomology
RnFX =Hn(RFX) (respectively LnFX =H−n(LFX)) for n∈Z.
In applications, the right adjoint functor usually turns out to be well defined on the
subcategory D+(A). Using Lemmas 1.1.10 and 1.2.4, we can say that the restric-
tion of the functor RFto D+(A) coincides with the derived functor of the
restriction of Fto H+(A)⊂H(A).
We now describe the conditions under which the right derived functor RFis
defined on the entire category D+(A). We say that a full additive subcategory
R⊂Ais right adapted to a functor Fif
a) Ftakes acyclic complexes in C+(R) to acyclic ones;
b) every object of Aembeds in some object of R.
We say that the objects of Rare right F-acyclic. If there exists a subcategory R
right adapted to F, one often says that Ahas enough (right) F-acyclic ob jects.
Suppose that F:A−→ Bis a functor for which a right adapted subcategory
R⊂Aexists. Applying Lemma 1.1.12, one checks readily that every right bounded
complex of objects in Ris right F-split. From condition b) one deduces that for
each object X∈H+(A) there is a quasi-isomorphism X−→ Xwith X∈H+(R).
As a corollary, we see that the canonical functor
H+(R)[Quis−1]−→ D+(A)
is an equivalence of triangulated categories.
Lemma 1.2.5. Suppose that Fis a functor for which a right adapted subcategory
R⊂Aexists.Then the functor RFis defined on D+(A), and for any left bounded
compl ex Xthere is an isomorphism RFX ∼
−→ X,where X−→ Xis a quasi-
isomorphism with X∈H+(A).
If Ahas enough injectives, then the full subcategory I⊂Aconsisting of all
injectives is right adapted to every additive functor. In this case we can compute
the right derived functor RFX by applying Fto an injective resolution Xof the
complex X.
Dually,one can introduce the notion of subcategory left adapted to a functor F.
If such a subcategory exists, the left derived functor LF:D−(A)−→ D(B)is
defined.
1.3. Derived categories of sheaves on schemes. Several Abelian categories
of sheaves can be assigned to any scheme. Let Xbe a scheme over a field k,with
structure sheaf OX.WedenotebyOX-Mod the Abelian category of all sheaves
of OX-modules in the Zariski topology. The category OX-Mod has all limits and
526 D. O. Orlov
colimits, and has a set of generators. Direct colimits are exact. For this reason,
the category OX-Mod is a Grothendieck Abelian category, and has enough injectives
(see [15], [42], Exp. IV).
From now on, we consider only Noetherian schemes (although many of the facts
treated below also hold in the more general situation). We denote by Qcoh(X)
the full Abelian subcategory of OX-Mod consisting of quasi-coherent sheaves. On
a Noetherian scheme Xevery quasi-coherent sheaf is the direct colimit of its sub-
sheaves of finite type (see [16], EGA1, 9.4). In this case the category Qcoh(X)has
a set of generators and is a Grothendieck Abelian category, and thus has enough
injectives.
The third category that we can assign to a scheme Xis the category of coher-
ent sheaves coh(X); it is a full Abelian subcategory of Qcoh(X). Although the
definition of (quasi-)coherent sheaves is local, in fact they do not depend on
the topology. We could, for example, consider not just the Zariski topology but
also, say, the etale or flat topology. In this case, although the notion of sheaf of
OX-modules depends on the choice of topology, (quasi-)coherent sheaves do not (see
[36]). In particular, for an affine scheme X, the category Qcoh(X)isequivalentto
the category of modules over the algebra corresponding to X.
In what follows we will focus on the category of coherent sheaves, and, more
precisely, on the derived category of coherent sheaves. However, since coh(X)does
not have enough injectives, in constructing derived functors we make use of the
categories Qcoh(X)andOX-Mod.
For a Noetherian scheme X, the full embedding of Abelian categories Qcoh(X)→
OX-Mod takes injectives to injectives. From this, we can deduce by a simple
procedure (see [17], I.4.6, [43], Appendix B) that the triangulated subcategory
H+(Qcoh) is right cofinal in the triangulated category H+(OX-Mod). Thus, apply-
ing Lemma 1.1.6, we obtain the following assertion.
Proposition 1.3.1 ([17], [41], Exp. II). If Xis a Noetherian scheme,the canonical
functor
D+(Qcoh(X)) −→ D+(OX-Mod)
is fully faithful and defines an equivalence with the full subcategory
D+(OX-Mod)Qcoh ⊂D+(OX-Mod)
consisting of complexes with quasi-coherent cohomology.
Under additional conditions on the scheme we can also prove the analogous
assertion for unbounded derived categories.
Proposition 1.3.2 ([41],Exp.II). If Xis a finite-dimensional Noetherian scheme,
then the canonical functor
D(Qcoh(X)) −→ D(OX-Mod)
is fully faithful and defines an equivalence with the full subcategory
D(OX-Mod)Qcoh ⊂D(OX-Mod),
which consists of complexes with quasi-coherent cohomology.
Derived categories of coherent sheaves 527
The proof makes use of the fact that the embedding functor has a right adjoint
Q:OX-Mod −→ Qcoh(X), and for finite-dimensional schemes this functor has
finite cohomological dimension (see [41], II.3.7).
We now consider the embedding of Abelian categories coh(X)⊂Qcoh(X).
Assertions similar to those just described are also known for the canonical functor
between derived categories; however, these assertions relate only to right bounded
derived categories.
Proposition 1.3.3 ([41],Exp.II).For a Noetherian scheme X,the canonical
functor
D−(coh(X)) −→ D−(Qcoh(X))
is fully faithful and gives an equivalence with the full subcategory D−(Qcoh(X))coh .
Combining this proposition with Propositions 1.3.1 and 1.3.2, we obtain the
following corollary.
Corollary 1.3.4 ([41],Exp.II).Let Xbe a Noetherian scheme (respectively,a
finite-dimensional Noetherian scheme ). Then the canonical functor
Db(coh(X)) −→ Db(OX-Mod) (respectively D−(coh(X)) −→ D−(OX-Mod))
is fully faithful and defines an equivalence with the full subcategory
Db(OX-Mod)coh (respectively D−(OX-Mod)coh).
We now describe the main derived functors between the derived categories of
sheaves on schemes. Let f:X−→ Ybe a morphism of Noetherian schemes. There
exists an inverse image functor
f∗:OY-Mod −→ OX-Mod,
which is right exact. Since OY-Mod has enough flat OY-modules and they are
f∗-acyclic, it follows that the left derived functor
Lf∗:D−(OY-Mod) −→ D−(OX-Mod)
is defined. One proves readily that Lf∗takes the categories D−(OY-Mod)Qcoh and
D−(OY-Mod)coh to D−(OX-Mod)Qcoh and D−(OX-Mod)coh respectively. Thus, for
finite-dimensional Noetherian schemes we obtain a derived inverse image functor
Lf∗on right bounded derived categories of quasi-coherent and coherent sheaves.
If f∗has finite cohomological dimension (in which case we say that fhas finite
Tor-dimension), we can extend the derived functor Lf∗to the unbounded derived
categories. Moreover, if fhas finite Tor-dimension, the derived inverse image func-
tor takes the bounded derived category to the bounded derived category. In par-
ticular, we have the functor
Lf∗:Db(OY-Mod)coh −→ Db(OX-Mod)coh.
528 D. O. Orlov
Let E,F∈C(OX-Mod) be two complexes of OX-modules. We define the tensor
product E⊗Fas the complex associated to the double complex Ep⊗Fq,thatis,
(E⊗F)n=
p+q=n
Ep⊗Fq,
with the differential d=dE+(−1)ndF. A homotopy between morphisms of com-
plexes extends to the tensor product, and we obtain a functor
E⊗:H(OX-Mod) −→ H(OX-Mod).
Suppose now that E∈C−(OX-Mod). The category H−(OX-Mod) has enough
objects that are left split with respect to the functor E⊗; indeed, right bounded
complexes of flat OX-modules have this property. Therefore, there exists a left
derived functor
E⊗L:D−(OX-Mod) −→ D−(OX-Mod).
If E1and E2are quasi-isomorphic, then E1⊗Land E2⊗Lare isomorphic. In fact,
we obtain a functor in two variables
⊗L:D−(OX-Mod) ×D−(OX-Mod) −→ D−(OX-Mod),
which is exact with respect to both arguments. The derived functor of the tensor
product is obviously associative and symmetric.
Suppose that an object Ehas finite Tor-dimension, that is, Eis quasi-isomorphic
to a bounded complex of flat OX-modules. Then, on the one hand, E⊗Lextends to
the unbounded derived category, and on the other, by restriction we obtain a functor
from the bounded derived category to itself. We obtain the functors
E⊗L:D(OX-Mod) −→ D(OX-Mod),E⊗L:Db(OX-Mod) −→ Db(OX-Mod).
Note that if Eis in D−(OX-Mod)coh (respectively D−(OX-Mod)Qcoh ), then E⊗L
takes objects with (quasi-)coherent cohomology to objects with (quasi-)coherent
cohomology.
Let f:X−→ Ybe a morphism of Noetherian schemes. The direct image functor
f∗:OX-Mod −→ OY-Mod
is left exact. Since the category of OX-modules has enough injectives, it follows
that the right derived functor
Rf∗:D+(OX-Mod) −→ D+(OX-Mod)
exists. Moreover, in this case, Rf∗takes the subcategory D+(OX-Mod)Qcoh to the
subcategory D+(OY-Mod)Qcoh .
If in addition, f∗has finite cohomological dimension, then Rf∗can be extended
to the category of unbounded complexes. This holds, for example, if Xis a finite-
dimensional Noetherian scheme. On the other hand, in this case (that is, when f∗
has finite cohomological dimension), the right derived functor between the bounded
derived categories
Rf∗:Db(OX-Mod) −→ Db(OX-Mod)
exists.
For the right derived functor to be defined between derived categories of coherent
sheaves, we need additional conditions on the morphism.
Derived categories of coherent sheaves 529
Proposition 1.3.5 ([16], III, 3.2.1, [17]). Suppose that f:X−→ Yis a proper
morphism of Noetherian schemes.Then the functor Rf∗takes the subcategory
D+(OX-Mod)coh to the subcategory D+(OY-Mod)coh .If in addition,Xis finite-
dimensional,then the analogous assertion holds for the bounded and unbounded
derived categories.
Let E,F∈C(OX-Mod) be two complexes of OX-modules. We define a complex
Hom ·(E,F)bytherule
Hom n(E,F)=
p
Hom (Ep,Fp+n)
with the differential d=dE+(−1)n+1 dF. A homotopy between morphisms of
complexes extends to the local Hom , and we obtain a bifunctor
Hom :H(OX-Mod)op ×H(OX-Mod) −→ H(OX-Mod).
Since every left bounded complex has an injective resolution, we obtain a derived
bifunctor
RHom :D(OX-Mod)op ×D+(OX-Mod) −→ D(OX-Mod).
In this situation we define the local hyper-Ext
Ext i(E,F):=Hi(RHom (E,F)).
For a Noetherian scheme X,ifEand Fare (quasi-)coherent OX-modules, then the
sheaves Ext i(E,F) are also (quasi-)coherent for any i0.
Now if E∈D−(OX-Mod)coh and F∈D+(OX-Mod)coh,thenRHom(E,F)
belongs to D(OX-Mod)coh.
We describe the main properties and relations between the derived functors
introduced in this section. Consider two morphisms f:X→Yand g:Y→Z.
In this situation we have two functors L(gf)∗and Lf∗Lg∗from D−(OZ-Mod) to
D−(OX-Mod). Then the natural transformation
L(gf)∗∼
−→ Lf∗Lg∗
is an isomorphism. The proof of this assertion follows from the fact that the functor
g∗takes flat OZ-modules to flat OY-modules (see, for example, [17]).
Inthesameway,wehaveanisomorphism
R(gf)∗−→ Rg∗Rf∗
of functors from D+(OX-Mod) to D+(OZ-Mod). This assertion follows from the
fact that f∗takes injective sheaves to flabby sheaves on Y,whichinturnareg∗-
acyclic (see [17]).
The other relations that we use fairly frequently are called the projection formula
and flat base change.
530 D. O. Orlov
Proposition 1.3.6 ([17], II.5.6). Let f:X→Ybe a morphi sm between finite -
dimensional Noetherian schemes.Then for any objects E∈D−(OZ-Mod) and
F∈D−(OX-Mod)Qcoh there is a natural isomorphism of functors
Rf∗E⊗LF∼
−→ Rf∗(E⊗LLf∗F).(3)
Proposition 1.3.7 ([17], II.5.12). Let f:X→Ybe a morphism of finite type
between fini te-dimensional Noetherian schemes and g:Y→Ya flat morphism.
We consider the Cartesian square
X×YYg
−−−−→X
f
f
Yg
−−−−→Y
.
In this situation there is a natural isomorphism of functors
Lg∗Rf∗E∼
−→ Rf
∗Lg∗Efor any E∈D(OX-Mod)Qcoh.(4)
We state another relation that we need.
Proposition 1.3.8 ([17], II.5.16). Let Ebe a bounded compl ex of loca l ly free sheaves of
finite rank on a Noetherian scheme X.Then the following natural isomorphisms
of functors
RHom (F,G)⊗LE∼
−→ RHom (F,G⊗LE)∼
−→ RHom (F⊗LE∨,G)(5)
hold for any F∈D−(OX-Mod), G∈D+(OX-Mod), where E∨:= RHom(E,OX).
CHAPTER 2
Categories of coherent sheaves and functors between them
2.1. Basic properties of categories of coherent sheaves. From now on,
we consider only bounded derived categories of coherent sheaves on smooth com-
plete algebraic varieties. For brevity, we always write simply Db(X) instead of
Db(coh(X)). Moreover, we omit the symbol of derived functor if the functor is
exact, for example, for inverse image under a flat morphism or for tensor product
by a locally free sheaf.
For a smooth complete variety Xof dimension nthe bounded derived category of
coherent sheaves admits a Serre functor (see Definition 1.1.3), given by ( ·)⊗ωX[n],
where ωXis the canonical sheaf (see [6]). Thus, we have an isomorphism
Hom(E,F)=Hom(F,E⊗ωX[n])∗(6)
for any pair of objects E,F∈Db(X).
As shown in the previous section, every morphism f:X→Ybetween smooth
complete algebraic varieties induces two exact functors, the direct image functor
Derived categories of coherent sheaves 531
Rf∗:Db(X)−→ Db(Y) and inverse image functor Lf∗:Db(Y)−→ Db(X), and
these functors are mutually adjoint. Moreover, each object E∈Db(X) defines the
exact tensor product functor ⊗LE:Db(X)−→ Db(X).
We can use these standard derived functors to introduce a large new class of
exact functors between the derived categories Db(X)andDb(Y).
Let Xand Ybe two smooth complete varieties over a field k,ofdimensionn
and mrespectively. Consider the Cartesian product X×Yand write pand qfor
the projections of X×Yto Xand Yrespectively:
Xp
←− X×Yq
−→ Y.
Every object E∈Db(X×Y) determines an exact functor ΦEfrom the derived
category Db(X) to the derived category Db(Y), defined by the formula
ΦE(·):=R·q∗(E⊗Lp∗(·)).(7)
Moreover, to the same object E∈Db(X×Y) one can assign another functor ΨE
from the derived category Db(Y) to the derived category Db(X), defined by a rule
similar to (7):
ΨE(·):=Rp∗(E⊗Lq∗(·)).
One checks readily that the functor ΦEhas both left and right adjoint functors.
Lemma 2.1.1. The functor ΦEhas left and right adjoint functors Φ∗
Eand Φ!
E
respectively,defined by the formulae
Φ∗
E∼
=ΨE∨⊗q∗ωY[m]and Φ!
E∼
=ΨE∨⊗p∗ωX[n].(8)
Here ωXand ωYare the canonical sheaves on Xand Yrespectively,and E∨is a
convenient notation for RHom(E,OX×Y).
Proof. We give the proof for the left adjoint functor. It comes from the following
sequence of isomorphisms:
Hom(A, Rq∗(E⊗Lp∗B)) ∼
=Hom(q∗A, E⊗Lp∗B)
∼
=Hom(p∗B, E∨⊗Lq∗A⊗ωX×Y[n+m])∗
∼
=Hom(B, Rp∗(E∨⊗Lq∗(A⊗ωY[m])) ⊗ωX[n])∗
∼
=Hom(Rp∗(E∨⊗Lq∗(A⊗ωY[m])),B).
Here we have used the adjunction between direct and inverse image functors, Serre
duality (6) (twice), and also formula (5).
We note that, of course, any diagram of the form
Xp
←− Zq
−→ Y
and any object E∈Db(Z) can be assigned a functor from the derived category of
coherent sheaves on Xto the derived category of coherent sheaves on Y,givenbya
formula similar to (7). However, any functor of this kind is isomorphic to a functor
532 D. O. Orlov
of the form (7), with the object R(p, q)∗Eon X×Y,where(p, q) is the canonical
morphism from Zto the direct product X×Y.
Now let X,Yand Zbe three smooth complete varieties and E,Fand Gobjects
of the derived categories Db(X×Y), Db(Y×Z)andDb(X×Z) respectively.
Consider the following diagram of projections:
The objects E,Fand Gdefine three functors,
ΦE:Db(X)−→ Db(Y),ΦF:Db(Y)−→ Db(Z),ΦG:Db(X)−→ Db(Z),
given by formula (7), that is,
ΦE:= Rπ2
12∗(E⊗Lπ1
12∗(·)),ΦF:= Rπ3
23∗(F⊗Lπ2
23∗(·))
and ΦG:= Rπ3
13∗(G⊗Lπ1
13∗(·)).
We consider the object p∗
12E⊗Lp∗
23F∈Db(X×Y×Z), which we always denote
by E
YFin what follows. The following assertion gives the composition rule for the
exact functors between derived categories represented by objects on the product.
Proposition 2.1.2. The composite of functors ΦF◦ΦEis isomorphic to the functor
ΦGrepresented by
G=Rp13∗E
YF.(9)
The proof is a direct verification.
Thus, to each smooth complete algebraic variety we assign its derived category
of coherent sheaves, and to every object E∈Db(X×Y) on the product of two
varieties we assign an exact functor ΦEfrom the triangulated category Db(X)to
the triangulated category Db(Y), with the composition law give just described.
The following problems are fundamental to understanding this correspondence:
1) When are the derived categories of coherent sheaves on two different smooth
complete algebraic varieties equivalent as triangulated categories?
2) What is the group of exact auto-equivalences of the derived category of
coherent sheaves for a given variety X? (By this we mean the group
of isomorphism classes of exact auto-equivalences.)
3) Is every exact functor between derived categories of coherent sheaves rep-
resented by an object on the product, that is, of the form (7)?
Some results in this direction are already known. For example, one can give
definitive answers to the first two questions when the variety has ample canonical
or anticanonical sheaf.
Derived categories of coherent sheaves 533
Theorem 2.1.3 [8]. Let Xbe a smooth projective variety whose canonical (or
anticanonical)sheaf is ample.Suppose that the category Db(X)is equivalent as
a triangulated category to the derived category Db(X)for some smooth algebraic
variety X.Then Xis isomorphic to X.
The proof of this theorem given in [8] is constructive, and gives a method
for recovering a variety from its derived category of coherent sheaves. Moreover,
in the assumptions of the theorem one can assume that the derived categories
are equivalent only as graded categories rather than as triangulated categories
(see [8]).
In this situation one can also describe the group of exact auto-equivalences.
Theorem 2.1.4 [8]. Let Xbe a smooth projective variety whose canonical (or
anticanonical)sheaf is ample.Then the group of isomorphism classes of exact auto-
equivalences of the category Db(X)is generated by automorphisms of the variety,
twists by line bundles,and shifts in the derived category.
For any variety Xthe group Auteq Db(X) of exact auto-equivalences always
contains the subgroup G(X) which is the semidirect product of the normal subgroup
G1=Pic(X)⊕Zand the subgroup G2=AutXacting naturally on G1. Under this
inclusion G(X)⊂Auteq Db(X), the generator of Zgoes to the shift functor [1], a
line bundle L∈Pic(X) goes to the functor ⊗L, and an automorphism f:X→X
induces the auto-equivalence Rf∗. We proved in [8] that, under the assumption
of Theorem 2.1.4, the group Auteq Db(X) of exact auto-equivalences equals G(X);
that is, in this case
Auteq Db(X)∼
=Aut X(Pic(X)⊕Z).
To study the problem of when two varieties have equivalent derived categories of
coherent sheaves and to describe their groups of auto-equivalences, it is desirable
to have explicit formulae for all exact functors. There is a conjecture that they
are always representable by objects on the product, that is, are of the form (7).
In the next chapter we give the proof of this conjecture for fully faithful functors
and, in particular, for equivalences. The whole of the next chapter is taken up with
the proof of this result. This will thus allow us to consider only functors of the
form (7) in studying equivalences between derived categories of coherent sheaves
on smooth projective varieties. Another problem that arises in connection with the
solution of these questions is the need for a criterion to determine whether a given
functor is an equivalence. To prove that a functor Fis an equivalence, it is enough
to show that both Fand its right (or left) adjoint are fully faithful functors (see
Definition 1.1.7).
There is a method to decide whether a functor ΦE:Db(X)−→ Db(Y) is fully
faithful.
Theorem 2.1.5 [7]. Let Mand Xbe smooth projective varieties over an alge-
braically closed field of characteristic 0and E∈Db(M×X). In this case the
functor ΦEis fully faithful if and only if the following orthogonality conditions
hold:
1) Homi
X(ΦE(Ot1),ΦE(Ot2)) = 0 for all iand all t1=t2;
534 D. O. Orlov
2) Hom0
X(ΦE(Ot),ΦE(Ot)) = kand Homi
X(ΦE(Ot),ΦE(Ot)) = 0 for any i/∈
{0,dim M}.
Here t,t1and t2are points of M,and Otithe corresponding skyscraper sheaves.
The assumptions of this theorem are in general rather difficult to verify; however,
the criterion works rather well when the object Eon the product is a vector bundle.
Consider four smooth complete algebraic varieties X1,X2,Y1and Y2.Wetake
two objects E1and E2belonging to the categories Db(X1×Y1)andDb(X2×Y2)
respectively, and consider the object
E1E2∈Db((X1×X2)×(Y1×Y2)),
which is p∗
13(E1)⊗Lp∗
24(E2) by definition. As above (see (7)), the objects E1,E2,
and E1E2define functors
ΦE1:Db(X1)−→ Db(Y1),ΦE2:Db(X2)−→ Db(Y2),
and ΦE1E2:Db(X1×X2)−→ Db(Y1×Y2).
We consider an object G∈Db(X1×X2)andwriteHto denote the object
ΦE1E2(G)∈Db(Y1×Y2). To each of these two objects one can assign functors by
the rule (7):
ΦG:Db(X1)−→ Db(X2)andΦ
H:Db(Y1)−→ Db(Y2).
Proposition 2.1.6. In the above notation there is an isomorphism of functors
ΦH∼
=ΦE2◦ΦG◦ΨE1.
The proof follows at once from Proposition 2.1.2.
Now if Z1and Z2are two other smooth complete varieties and F1and F2objects
of Db(Y1×Z1)andDb(Y2×Z2) respectively, then there are also functors ΦF1,
ΦF2,andΦ
F1F2. By the rule (9) we can find objects G1and G2belonging to
Db(X1×Z1)andDb(X2×Z2) such that
ΦG1∼
=ΦF1◦ΦE1and ΦG2∼
=ΦF2◦ΦE2.
A direct check shows that there is a natural relation
ΦF1F2◦ΦE1E2∼
=ΦG1G2.(10)
Using this, one readily proves the following assertion.
Proposition 2.1.7. Under the above conditions,assume that ΦE1and ΦE2are
fully faithful (respectively,are equivalences). Then the functor
ΦE1E2:Db(X1×X2)−→ Db(Y1×Y2)
is also fully faithful (respectively,an equivalence of categories ).
Derived categories of coherent sheaves 535
Proof.IfFhas a left adjoint F∗(say), then it is fully faithful if and only if the
composite F∗Fis isomorphic to the identity functor. The functors ΦEihave left
adjoints Φ∗
Eidefined by (8). Since they are fully faithful, it follows that the com-
posites Φ∗
Ei◦ΦEiare isomorphic to the identity functors, which are representable
by the structure sheaves of the diagonals ∆i∈Xi×Xi. One sees readily that the
sheaf O∆1O∆2is isomorphic to the structure sheaf of the diagonal O∆,where
∆ is the diagonal in (X1×X2)×(X1×X2). Using formula (10), we see that the
composite Φ∗
E1E2◦ΦE1E2is represented by the structure sheaf of the diagonal ∆,
and is thus isomorphic to the identity functor. Thus, ΦE1E2is fully faithful. The
assertion concerning equivalences can be proved in a similar way.
Now assume that the functor ΦE:Db(X)−→ Db(Y) is an equivalence and that
F∈Db(X×Y) is an object such that ΨF∼
=Φ−1
E.By(8),wehaveisomorphisms
F∼
=E∨⊗p∗ωX[n]∼
=E∨⊗q∗ωY[m],
which imply at once that the dimensions nand mof the varieties Xand Yare
equal.
Consider the functor
ΦFE:Db(X×X)−→ Db(Y×Y) (11)
and denote it by Ad E. By Proposition 2.1.7 it is also an equivalence. Moreover,
by Proposition 2.1.6, for any object G∈Db(X×X) there is an isomorphism of
functors
ΦAd E(G)∼
=ΦE◦ΦG◦Φ−1
E.(12)
Consider the special case when Gis the structure sheaf of the diagonal O∆X,
representing the identity functor. Thus, applying (12), we see that the functor Ad E
takes the structure sheaf of the diagonal O∆Xto the structure sheaf of the diagonal
O∆Y.
Consider the more general situation. We denote by iXand iYthe embeddings
of the diagonals in X×Xand Y×Yrespectively. We apply the functor Ad Eto the
object iX∗ωk
X,whereωXis the canonical sheaf of X(as above). The object iX∗ωk
X
represents the functor Sk[−nk], where Sis the Serre functor of Db(X). Since every
equivalence commutes with Serre functors by Lemma 1.1.4, we see that
Ad E(iX∗ωk
X)∼
=iY∗ωk
Y.(13)
Now for every variety Xwe define the bigraded algebra
HA(X)=
i,k
HAi,k(X):=
i,k
Exti
X×X(O∆X,i
X∗ωk
X).
The algebra structure is defined here by composition of Ext’s, bearing in mind the
canonical identification
Exti
X×X(O∆X,i
X∗ωk
X)∼
=Exti
X×X(iX∗ωm,i
X∗ωm+k
X).
To prove the next theorem, we need to apply the main result of Chapter 3, which
states that every equivalence is represented by an object on the product.
536 D. O. Orlov
Theorem 2.1.8. Let Xand Ybe smooth projective varieties whose derived cat-
egories of coherent sheaves are equivalent as triangulated categories.Then the
bigraded algebras HA(X)and HA(Y)are isomorphic.
Proof. By Theorem 3.2.2, every equivalence F:Db(X)→Db(Y) is represented by
some object on the product, and is thus isomorphic to a functor of the form ΦE
for some E∈Db(X×Y). Each equivalence of this kind defines an equivalence
Ad E:Db(X×X)−→ Db(Y×Y),
taking iX∗ωk
Xto iY∗ωk
Y. The equivalence AdEinduces isomorphisms
Exti
X×X(O∆X,i
X∗ωk
X)∼
=Exti
Y×Y(O∆Y,i
Y∗ωk
Y),
and hence an isomorphism of the bigraded algebras HA(X)andHA(Y).
We note that one can obtain both the canonical and anticanonical algebras of
Xfrom the bigraded algebra HA(X). Indeed,
k0
H0(X, ωk
X)=
k0
HA0,k(X)and
k0
H0(X, ωk
X)=
k0
HA0,k(X).
Thus, Theorem 2.1.8 implies the following corollary.
Corollary 2.1.9. If the derived categories of coherent sheaves on two smooth pro-
jective varieties Xand Yare equivalent,then the canonical (and anticanonical)
algebras of Xand Yare isomorphic.
The statement of this corollary is very close to Theorem 2.1.3. However, we
should note that the proof of Theorem 2.1.3 given in [8] does not depend on the
main result of the next chapter and, moreover, is constructive. Also note that in
Theorem 2.1.3, we do not assume that the canonical (or anticanonical) sheaf of the
second variety Xis ample; this follows from the proof of the theorem.
We can also describe all the other spaces HAi,k (X). In [40] it is proved that the
spectral sequence that computes
HAi,k(X)=Ext
i(O∆X,i
X∗ωX)
in terms of the cohomology of O∆Xrestricted to the diagonal degenerates at the
term E2. In particular, there are isomorphisms
HAi,k(X)∼
=
p+q=i
Hp(X,
q
TX⊗ωk
X),(14)
where TXis the tangent bundle to X. Moreover, this isomorphism turns into an
algebra isomorphism, that is,
HA(X)∼
=
i,k
p+q=i
Hp(X,
q
TX⊗ωk
X)
as bigraded algebras. This relation and Theorem 2.1.8 imply the following corollary.
Derived categories of coherent sheaves 537
Corollary 2.1.10. If the derived categories of coherent sheaves on two smooth
projective varieties Xand Yare equivalent,then there are vector space isomorphisms
p+q=i
Hp(X,
q
TX⊗ωk)∼
=
p+q=i
Hp(Y,
q
TY⊗ωk).(15)
In particular,we obtain isomorphisms between the verticals of the Hodge diamond:
p−q=i
Hp(X, Ωq
X)∼
=
p−q=i
Hp(Y, Ωq
Y).(16)
Proof. The isomorphisms (15) follow at once from Theorem 2.1.8 and the equality
(14). The isomorphisms (16) are the special case k= 1 of (15).
The isomorphisms between the verticals of the Hodge diamond can also be
obtained in another way. Suppose that the ground field kis C.
For any element ξ∈H∗(X×Y, Q) we can define linear maps
vξ:H
∗(X, Q)−→ H∗(Y, Q)andwξ:H
∗(Y, Q)−→ H∗(X, Q)
by the formulae
vξ(−)=q∗(ξ·p∗(−)) and wξ(−)=p∗(ξ·q∗(−)).(17)
For these maps one can write out a composition formula similar to formula (9) for
the composition of functors. Let X,Y,andZbe three smooth complete varieties
and ξand ηelements of H∗(X×Y, Q)andH
∗(Y×Z, Q) respectively. Then the
composite vη◦vξcoincides with the map vζ,whereζ∈H∗(X×Z, Q)isgivenby
the formula
ζ=pXZ∗p∗
YZ(η)∪p∗
XY (ξ).
To any functor of the form ΦE:Db(X)−→ Db(Y) we can assign a linear map
ϕE:H
∗(X, Q)→H∗(Y, Q). For this, define an element ε∈H(X×Y, Q)bytherule
ε=p∗tdX·ch(E)·q∗tdY,(18)
where tdXand tdYare the Todd classes of Xand Yrespectively, and ch(E)isthe
Chern character of E.Wedefinethemaps
ϕE(−):=vε(−)=q∗(ε·p∗(−)),
ψE(−):=wε(−)=p∗(ε·q∗(−)).(19)
The next proposition follows immediately from the Grothendieck form of the
Riemann–Roch theorem.
Proposition 2.1.11. Suppose that ΦE:Db(X)−→ Db(Z)is a composite ΦG◦ΦF
for some
ΦF:Db(X)−→ Db(Y),ΦG:Db(Y)−→ Db(Z).
Then ϕE=ϕG◦ϕF.
This implies at once the following corollary.
538 D. O. Orlov
Corollary 2.1.12. If the functor ΦE:Db(X)−→ Db(Z)is an equivalence,then
the map ϕE:H
∗(X, Q)→H∗(Y, Q)is an isomorphism,and its complexification
induces the isomorphisms (16) between the verticals of the Hodge diamond.
Proof. It follows from Proposition 2.1.11 that the quasi-inverse functor to ΦE
induces the inverse map of ϕE. Moreover, since the element ε∈H∗(X×Y, Q)
corresponds to an algebraic cycle by (18), one checks readily that the complexifi-
cation of ϕEpreserves the verticals of the Hodge diamond.
In conclusion we observe also that every functor ΦE:Db(X)−→ Db(Y) induces
amapΦE:K(X)−→ K(Y) between the Grothendieck groups K(X)andK(Y)of
the categories Db(X)andDb(Y). Consider the map
ch ·tdX:K(X)−→ H∗(X, Q)
that takes an element of K(X) to its Chern character times the square root of the
Todd class. Using the Riemann–Roch theorem, one can show that the diagram
K(X)ΦE
−−−−→K(Y)
ch ·√tdX
ch ·√tdY
H∗(X, Q)ϕE
−−−−→H∗(Y, Q)
is commutative.
2.2. Examples of equivalences: flopping birational transformations. In
this section we present an entire class of examples of pairs of smooth varieties for
which the derived categories of coherent sheaves are equivalent. Examples of such
varieties were of course already known (the first example is an Abelian variety and
its dual, considered by Mukai [29]). The principal difference with the examples
treated in this section is that here we obtain pairs of (in general non-isomorphic)
varieties related by a birational transformation which is a flop. It also follows from
our examples that the conditions on the (anti-)canonical sheaf in Theorem 2.1.3
cannot be weakened.
To start this section we recall the definitions of admissible subcategories and
semi-orthogonal decompositions (see [5], [6]).
Definition 2.2.1. Let Bbe a full additive subcategory of an additive category A.
By the right orthogonal to Bin Awe mean the full subcategory B⊥⊂Aconsisting
of all objects Csuch that Hom(B, C ) = 0 for any B∈B. The left orthogonal ⊥B
is defined dually.
Note that, if Bis a triangulated subcategory in a triangulated category A,then
⊥Band B⊥are also triangulated subcategories.
Definition 2.2.2. Let I:N−→ Dbe an embedding of a full triangulated sub-
category in a triangulated category D.WesaythatNis right admissible (or left
admissible) if the embedding functor Ihas a right adjoint P:D−→ N(respectively
left adjoint).
Derived categories of coherent sheaves 539
For a subcategory N, the property of being right admissible (or left admissible) is
equivalent to the following property, stated in terms of orthogonals: for any object
X∈D, there is a distinguished triangle N→X→Mwith N∈Nand M∈N⊥
(respectively, M→X→Nwith M∈⊥Nand N∈N). We say simply that a
subcategory is admissible if it is both right and left admissible.
If N⊂Dis an admissible subcategory, we say that Dadmits a semi-orthogonal
decomposition of the form N⊥,Nor N,⊥N. This process of decomposition can
sometimes be extended further, decomposing the subcategory Nor its orthogonals.
We give the general definition of semi-orthogonal decomposition.
Definition 2.2.3. A sequence (N0,...,Nn) of admissible subcategories of a trian-
gulated category Dis said to be semi-orthogonal if Nj⊂N⊥
ifor all 0 j<in.
We say that a semi-orthogonal sequence is complete if it generates the category D,
that is, the minimal triangulated subcategory in Dcontaining all the Nicoincides
with D. In this case this sequence is called a semi-orthogonal decomposition of the
category Dand is represented as follows:
D=N0,...,Nn.
The simplest example of a semi-orthogonal decomposition is when Dhas a com-
plete exceptional family.
Definition 2.2.4. We say that an object Ein a triangulated category Dis excep-
tional if Homi(E,E)=0fori= 0 and Hom(E, E)=k. An ordered family
(E0,...,E
n) of exceptional objects is called a comp let e except ion al fa mil y if it gen-
erates Dand Hom·(Ei,E
j)=0fori>j.
The best-known example of a complete exceptional family is provided by pro-
jective space.
Example 2.2.5 [2]. On projective space PN,givenanyi∈Z, the family
(O(i),...,O(i+N))
is exceptional and complete. In particular, we obtain that the derived category of
coherent sheaves Db(PN) is equivalent to the derived category of finite-dimensional
modules over the finite-dimensional algebra EndN
j=0 O(j)of endomorphisms of
the exceptional family.
Similar decompositions exist for some other varieties, for example, for quadrics
and flag varieties [20]–[22].
We now present some facts we need on blowups and the behaviour of the derived
categories of coherent sheaves under blowups. All these results are contained in
the paper [34] (see also [7]). Let Xbe a smooth complete algebraic variety and
Y⊂Xa smoothly embedded closed subvariety of codimension r.Wedenoteby
X
the blowup of Xwith centre along Y.Thevariety
Xis also smooth, and there is
a commutative diagram:
Yj
−−−−→
X
p
π
Yi
−−−−→X
540 D. O. Orlov
with iand jclosed embeddings and p:
Y→Ythe projective bundle of the excep-
tional divisor of
Yover the centre Yof the blowup; in particular, pis a flat mor-
phism. We recall that
Y∼
=P(NX/Y ), where NX/Y is the normal bundle to Yin X.
We denote by O
Y(1) the canonical relatively ample line bundle on
Y=P(NX/Y ).
It is well known that this bundle is isomorphic to the restriction of the line bundle
O(−
Y)to
Y.
Proposition 2.2.6 [34]. The derived inverse image functors
Lπ∗:Db(X)−→ Db(
X)and p∗:Db(Y)−→ Db(
Y)
are fully faithful.
Proof. The projection formula (3) gives an isomorphism
Hom(Lπ∗F, Lπ∗G)∼
=Hom(F, Rπ∗Lπ∗G)∼
=Hom(F, Rπ∗O
Y⊗LG)
for F, G ∈Db(X). Similarly for p∗. Combining these with Rπ∗O
X∼
=OXand
Rp∗O
Y=OYgives the proof.
Proposition 2.2.7 ([34], [7]). For any invertible sheaf Lon
Y,the functor
Rj∗(L⊗p∗(·)): Db(Y)−→ Db(
X)
is fully faithful.
Proof. To prove that the functor is fully faithful, it is enough to show that conditions
1)–2) of Theorem 2.1.5 hold. For any closed point y∈Ythe image Φ(Oy)isthe
structure sheaf of the corresponding fibre of the map p,viewedasasheafon
X.
Since the fibres over distinct points are disjoint, the orthogonality condition 1) of
Theorem 2.1.5 is satisfied.
Consider the structure sheaf OFof some p-fibre F⊂
Y. Wehaveanisomorphism
Homi(j∗OF,j
∗OF)∼
=Homi(Lj∗j∗OF,OF).
In the derived category Db(
Y) we have a distinguished triangle
OF⊗O
Y(1)[1] −→ Lj∗j∗OF−→ OF,
where O
Y(1) is the relatively ample line bundle on
Y, isomorphic to O(−
Y)|
Y.The
fibre of Fis a projective space, and the restriction of O
Y(1) to Fis isomorphic to
O(1). Thus,
Homi(OF⊗O
Y(1),OF)=0
for all i. Hence,
Homi(j∗OF,j
∗OF)∼
=Homi(OF,OF).
Therefore, condition 2) of Theorem 2.1.5 also holds.
We wr i te D(X) for the full triangulated subcategory of Db(
X) which is the image
of Db(X) under Lπ∗,andD(Y)kfor the full subcategory in Db(
X)whichisthe
image of Db(Y) under Rj∗(O
Y(k)⊗p∗(·)), where O
Y(k)=O
Y(1)⊗kand O
Y(1)
is the canonical relatively ample line bundle on
Y=P(NX/Y ). It follows from
Propositions 2.2.6 and 2.2.7 that D(X)∼
=Db(X)andD(Y)k∼
=Db(Y).
Derived categories of coherent sheaves 541
Theorem 2.2.8 [34]. The sequence of admissible subcategories
D(Y)−r+1,...,D(Y)−1,D(X)
is semi-orthogonal,and it gives a semi-orthogonal decomposition of the category
Db(
X).
This theorem provides a description of the derived category of the blowup
X
in terms of the blown up variety Xand the centre of the blowup Y.Usingthis
description of the derived category of a blowup, we now study the behaviour of the
derived category under the simplest flipping and flopping transformations. Consider
the following example.
Let Ybe a smoothly embedded closed subvariety in a smooth complete algebraic
variety Xsuch that Y∼
=Pkwith the normal bundle NX/Y ∼
=OY(−1)⊕(l+1).We
suppose that lk.
Wri te
Xfor the blowup of Xalong the centre Y. In this case the exceptional
divisor
Yis isomorphic to the product of projective spaces Pk×Pl.Moreover,in
this situation we have the following description of the normal sheaf to
Yin
X:
N
X/
Y=O
X(
Y)|
Y∼
=O(−1; −1),
where O(−1; −1) := p∗
1OPk(−1) ⊗p∗
2OPl(−1). These facts allow us to assert that
there is a blowdown of
Xunder which
Yprojects to the second factor Pl.This
blowdown exists in the analytic category, and its result is a smooth variety X+
which in general may not be algebraic. We assume that X+is algebraic. All the
geometry described above is contained in the following diagram:
(20)
The birational map fl: X−→ X+is the simplest example of a flip or flop. It is
aflipforl<kand a flop for l=k. In what follows, we need a formula for the
restriction of the canonical sheaf ω
Xto the divisor
Y. For the blowup of a smooth
subvariety we obtain
ω
X∼
=π∗ωX⊗O
X(l
Y).
The adjunction formula gives
ωX|Y∼
=ωY⊗
l+1
N∗
X/Y ∼
=OY(l−k).
Combining these facts together, we obtain the isomorphism
ω
X|
Y∼
=(π∗ωX⊗O
X(l
Y))|
Y∼
=p∗(ωX|Y)⊗O
X(l
Y)|
Y∼
=O(−k;−l).(21)
The main theorem of this section relates the derived categories of coherent
sheaves on Xand X+.
542 D. O. Orlov
Theorem 2.2.9. Let Lbe a line bund le o n
X.In the above notation,the functor
Rπ∗(Lπ+∗(·)⊗L): Db(X+)−→ Db(X)
is fully faithful.
Proof. We first consider the restriction of Lto
Y.Since
Y=Pk×Pl, it follows
that L|
Y∼
=O(a;b) for some integers aand b.
We must show that for any pair A, B ⊂Db(X+) the composite map
Hom(A, B)∼
−→ Hom(Lπ+∗A, Lπ+∗B)
−→ Hom(Rπ∗(Lπ+∗A⊗L),Rπ∗(Lπ+∗B⊗L)) (22)
is an isomorphism. Using adjunction of the functors, we obtain an isomorphism
Hom(Rπ∗(Lπ+∗A⊗L),Rπ∗(Lπ+∗B⊗L))
∼
=Hom(Lπ∗Rπ∗(Lπ+∗A⊗L),Lπ+∗B⊗L).
Consider the distinguished triangle
Lπ∗Rπ∗(Lπ+∗A⊗L)−→ Lπ+∗A⊗L−→ A. (23)
Thus, to prove that the composite (22) is an isomorphism, it is necessary and
sufficient to show that
Hom(A, Lπ+∗B⊗L)=0.(24)
Since by Proposition 2.2.6 the composite Rπ∗Lπ∗is isomorphic to the identity
functor, by applying the functor Rπ∗to the distinguished triangle (23) we see that
Rπ∗A=0. Thus,Hom(Lπ∗C, A) = 0 for any object C∈Db(X+). Hence, A
belongs to the subcategory D(X)⊥.
Theorem 2.2.8 implies the semi-orthogonal decomposition
D(X)⊥=D(Y)−l,...,D(Y)−1.
Since Yis a projective space, it follows from Example 2.2.5 that each D(Y)−i
admits a complete exceptional family. Collecting these families, we obtain a com-
plete exceptional family in D(X)⊥. The following family will be convenient for our
purposes:
D(X)⊥=Rj∗O(a−k;−l), ... ... Rj∗O(a;−l),
Rj∗O(a−k+1;−l+1), ... ... Rj∗O(a+1;−l+1),
.
.
..
.
.
Rj∗O(a−k+l−1; −1), ... ... Rj∗O(a+l−1; −1).
Derived categories of coherent sheaves 543
We can now regroup this exceptional sequence to obtain a semi-orthogonal
decomposition of D(X)⊥of the form
D(X)⊥=B,A,
where Aand Bare the subcategories generated by Rj∗O(i;s)withiaand
with i<arespectively. For 1 ikand 1 sl, the objects Rj∗O(a−i;−s)
belong simultaneously to the subcategories D(X)⊥and D(X+)⊥⊗L.Inparticular,
B⊂D(X)⊥∩(D(X+)⊥⊗L). Applying Hom to the distinguished triangle (23),
we obtain
Hom(A, Rj∗O(a−i;−s)) = 0 for 1 ikand 1 sl.
Since A∈D(X)⊥and Ais orthogonal to the subcategory B, it follows at once
that A∈A. Now note that if the object Rj∗O(a+i;s) belongs to the subcategory
A,thenisatisfies the inequalities 0 i<l. Taking account of the formula (21)
for the canonical class ω
X|
Y∼
=O(−k;−l) and of the condition lk,weseethat
A⊗ω
X⊂D(X+)⊥⊗L. Hence, for any ob ject B∈Db(X+)wehave
Hom(Lπ+∗B⊗L, A ⊗ω
X)=0.
Applying Serre duality (6) gives the desired equality Hom(A, Lπ+∗B⊗L)=0
immediately.
Theorem 2.2.10. In the above notation,if l=k(and thus flis a flop), the functor
Rπ∗(Lπ+∗(·)⊗L)is an equivalence of triangulated categories.
Proof. By the previous theorem, the functor in question is fully faithful. Its left
adjoint is of the form Rπ+
∗(Lπ∗(·)⊗L), where L=L−1⊗ω
X⊗π+∗ω−1
X+.Thus,
it is also fully faithful by the previous theorem. This proves that both functors are
equivalences.
We note that the proof of the above assertions remains valid if a flop is carried out
simultaneously in some finite set Y1,...,Y
sof disjoint subvarieties, each satisfying
the condition of the theorems. This simple remark is essential in connection with
our assumption that the variety X+we obtain is algebraic. The point is that there
are many examples in which birational transformations of the above kind carried
out in just one of the subvarieties Yilead to non-algebraic varieties, whereas a
flip (or flop) carried out simultaneously in the whole set gives a variety which is
algebraic.
A second remark is that the flopped varieties Xand X+of Theorem 2.2.10
are of course not isomorphic in general; flops often occur in birational geometry,
for example, in the construction used to describe Fano threefolds by the method
known as double projection from a line (see [19], §8). Suppose that we have a
Fano threefold Vof index 1 with Pic V=Zembedded in projective space by its
anticanonical system. Then the blowup of this variety in a line gives a variety
Xwhose anticanonical class is ‘almost’ ample; that is, the map defined by its
anticanonical system contracts a certain set of curves on this variety, namely the
proper transforms of the lines on Vmeeting the blown up line. In many examples
544 D. O. Orlov
these curves have the normal sheaf O(−1)⊕O(−1), which puts us in the situation of
our theorem. Making simultaneous flops in these curves gives a variety X+that is
not isomorphic to X; however, by Theorem 2.2.10, it has the same derived category
of coherent sheaves. In particular, this example shows that the ampleness condition
on the anticanonical class in Theorem 2.1.3 on recovering Xfrom Db(X) cannot
be weakened. There are similar examples for varieties of general type arising in the
minimal model programme.
These results have another natural generalization. Suppose that a smooth sub-
variety Yin a smooth complete algebraic variety Xis the projectivization of a
vector bundle Eof rank k+ 1 over a smooth variety Z,thatis,Y∼
=P(E)→Z.We
also assume that the normal bundle NX/ Y when restricted to the fibre of the map
Y→Zis isomorphic to OPk(−1)⊕(l+1) . We again assume that lk.Denotingby
Xthe blowup of Xwith centre along Y, we again obtain a diagram of the form
(20), where Y+is the projectivization of a bundle of rank l+1 over Z.Inthis
situation we can assert that the analogues of Theorems 2.2.9 and 2.2.10 remain
valid.
Other similar examples arise when Xis a threefold and Yis a rational curve
satisfying Y·KX= 0. In this case the normal bundle on Ycan be of the form
O(−1) ⊕O(−1), O⊕O(−2) or O(1) ⊕O(−3). In each of these cases there exists a
flopping birational transformation, fl : X X+. Moreover, in each of these cases
the derived categories of coherent sheaves of Xand X+are equivalent. The first
case is a special case of Theorem 2.2.10. The second case was treated in [7]. More
recently, the equivalence of categories was proved in all these cases together in [10].
CHAPTER 3
Fully faithful functors between derived categories
3.1. Postnikov diagrams and their convolutions. In this section we con-
sider Postnikov diagrams in triangulated categories and find conditions under which
a Postnikov diagram admits a convolution and this convolution is uniquely deter-
mined.
Let X·={Xcdc
−→ Xc+1 dc+1
−−−→···−→X0}with c<0 be a bounded complex of
objects in a triangulated category D. This means that all the composites di+1 ◦di
vanish.
By definition, a left Postnikov system associated with X·is a diagram of the
form
Derived categories of coherent sheaves 545
in which the triangles marked with are all distinguished, and those marked with
are all commutative (that is, jk◦ik=dk). An object E∈Ob Dis called a left
convolution of the complex X·if there is a left Postnikov system associated with
X·such that E=Y0. WedenotebyTot(X·) the class of all convolutions of the
complex X·. Postnikov systems and their convolutions are obviously stable under
exact functors between triangulated categories.
Note that the class Tot(X·) may contain many non-isomorphic objects, or may
also be empty. In what follows we shall describe a sufficient condition for the class
Tot(X·) to consist of a single object up to isomorphism. The following lemma was
proved in [3].
Lemma 3.1.1. Let gbe a mor phi sm betw een object s Yand Ythat are in turn
included into distinguished triangles:
If vgu =0,then there exist morphisms f:X→Xand h:Z→Zsuch that the
triple (f, g, h)is a morphism of triangles.
Suppose in addition that Hom(X[1],Z)=0. Then the morphisms fand h
(respectively making the first and second squares of the diagram commute)are
uniquely determined by these conditions.
We now prove two lemmas that generalize the previous lemma to Postnikov
diagrams.
Lemma 3.1.2. Let X·={Xcdc
−→ Xc+1 dc+1
−−−→···−→X0}be a bounded com plex
of objects in a triangulated category D.Suppose that it satisfies
Homi(Xa,Xb)=0 for i<0and for all a<b. (25)
Then a convolution of X·exists,and all convolutions are (non-canonically)iso-
morphic.
Suppose in addition that
Homi(Xa,Y0)=0 for i<0and for al l a(26)
holds for some convolution Y0(and therefore for any convolution). Then all con-
volutions of X·are canonically isomorphic.
Lemma 3.1.3. Let X·
1and X·
2be bounded complexes satisfying condition (25) and
(fc,...,f
0)a morphism between these complexes:
Xc
1
dc
1
−−−−→Xc+1
1−−−−→ ··· −−−−→X0
1
fc
fc+1
f0
Xc
2
dc
2
−−−−→Xc+1
2−−−−→ ··· −−−−→X0
2
.
546 D. O. Orlov
Suppose that
Homi(Xa
1,Xb
2)=0 for i<0and for a<b. (27)
Then for each convolution Y0
1of X·
1and for each convolution Y0
2of X·
2there is
a morphism f:Y0
1→Y0
2that commutes with the morphism f0.If in addition we
have
Homi(Xa
1,Y0
2)=0 for i<0and for any a, (28)
then this morphism is uniquely determined.
Proof. We prove both lemmas at the same time by induction based on Lemma 3.1.1.
Let Yc+1 be the mapping cone of the morphism dc,
Xcdc
−→ Xc+1 α
−→ Yc+1 −→ Xc[1].(29)
By assumption, dc+1 ◦dc= 0 and Hom(Xc[1],Xc+2) = 0. Thus, there is a unique
morphism dc+1 :Yc+1 →Xc+2 such that dc+1 ◦α=dc+1 . Consider the composite
dc+2 ◦dc+1 :Yc+1 −→ Xc+3 .
It is known that dc+2 ◦dc+1 ◦α=dc+2 ◦dc+1 = 0; moreover, we have the equality
Hom(Xc[1],Xc+3 ) = 0. This immediately implies that the composite dc+2 ◦dc+1
also vanishes.
Considering the distinguished triangle (29), we see that
Homi(Yc+1 ,Xb)=0
for i<0andb>c+1. Thus, the complex Yc+1 −→ Xc+2 −→ · · · −→ X0also
satisfies (25). This complex has a convolution by induction. Thus, X·also has a
convolution, and hence the class Tot(X·)isnotempty.
We now show that, under condition (27), every morphism of complexes extends
to a morphism of Postnikov systems. Consider the mapping cones Yc+1
1and Yc+1
2
of the morphisms dc
1and dc
2. There is a morphism gc+1 :Yc+1
1→Yc+1
2completing
the pair (fc,f
c+1) to a morphism of triangles,
Xc
1
dc
1
−−−−→Xc+1
1
α
−−−−→Yc+1
1−−−−→Xc
1[1]
fc
fc+1
gc+1
fc[1]
Xc
2
dc
2
−−−−→Xc+1
2
β
−−−−→Yc+1
2−−−−→Xc
2[1]
.
As already shown above, there exist morphisms dc+1
i:Yc+1
i→Xc+2
ifor i=1,2,
which are uniquely determined. Consider the diagram
Yc+1
1
dc+1
1
−−−−→Xc+2
1
gc+1
fc+2
Yc+1
2
dc+1
2
−−−−→Xc+2
2
.
Derived categories of coherent sheaves 547
We prove that the square is commutative. Indeed, write h=fc+2 ◦dc+1
1−dc+1
2◦gc+1
for the difference. We have the equality h◦α=fc+2 ◦dc+1
1−dc+1
2◦fc+1 =0. And
by the assumption of the lemma, Hom(Xc
1[1],Xc+2
2) = 0. This implies immediately
that h=0.
Thus, we obtain a morphism of complexes
Yc+1
1
dc+1
1
−−−−→Xc+2
1−−−−→ ··· −−−−→X0
1
gc+1
fc+2
f0
Yc+1
2
dc+1
2
−−−−→Xc+2
2−−−−→ ··· −−−−→X0
2
.
These complexes satisfy conditions (25) and (27). By the induction assumption, a
morphism between these complexes extends to a morphism between the Postnikov
systems. We thus obtain a morphism between the Postnikov systems associated
with X·
1and X·
2.
Moreover, one sees that, if all morphisms fiare isomorphisms, then the morphism
between the Postnikov systems is also an isomorphism. Hence, if condition (25)
holds, all objects in Tot(X·) are isomorphic.
In conclusion, consider a morphism between the distinguished triangles taking
part in the Postnikov diagrams,
Y−1
1
j1,−1
−−−−→X0
1
i1,0
−−−−→Y0
1−−−−→Y−1
1[1]
g−1
f0
g0
g−1[1]
Y−1
2
j2,−1
−−−−→X0
2
i2,0
−−−−→Y0
2−−−−→Y−1
2[1]
.
If the complexes X·
isatisfy condition (28) (that is, Homi(Xa
1,Y0
2)=0fori<0and
for all a), then Hom(Y−1
1[1],Y0
2) = 0. By Lemma 3.1.1, the morphism g0is defined
uniquely. This completes the proof of the lemmas.
3.2. Fully faithful functors between derived categories of coherent
sheaves. Let Xand Mbe two smooth complete varieties over some field k.As
before, we denote by Db(X)andDb(M) the bounded derived categories of coherent
sheaves on Xand Mrespectively. We proved above that these categories have the
structure of triangulated categories.
Consider the product M×Xand write pand πfor the projections of M×Xto
Mand Xrespectively:
Mp
←− M×Xπ
−→ X.
For every object E∈Db(M×X) we defined an exact functor ΦEfrom Db(M)to
Db(X)by(7):
ΦE(·):=Rπ∗(E⊗Lp∗(·)).(30)
The functor ΦEhas left and right adjoint functors Φ∗
Eand Φ!
Erespectively, given
by the formulae (8),
Φ∗
E(·)=Rp∗(E∨⊗Lπ∗(ωX[dim X]⊗(·))),
Φ!
E(·)=ωM[dimM]⊗Rp∗(E∨⊗L(·)),
where ωX,ωMare the canonical sheaves of Xand Mand E∨:= R·Hom (E,OM×X).
548 D. O. Orlov
To study the problem of when two varieties have equivalent derived categories of
coherent sheaves, and to describe their groups of auto-equivalence, it is desirable to
have explicit formulae for all exact functors. There is a conjecture that they can all
be represented by objects on the product, that is, are of the form (30). However,
at present it is not known whether or not this assertion is true. Nevertheless, it
turns out that a special case of this conjecture is valid. Namely, if a functor is
fully faithful and has an adjoint functor, it can be represented by an object on the
product. The present chapter is devoted to the proof of this fact. More exactly,
the main theorem of this chapter is as follows.
Theorem 3.2.1. Let Fbe an exact functor from the category Db(M)to the cat-
egory Db(X), where Mand Xare smooth projective varieties.Suppose that F
is fully faithful and has a right (or left )adjoint functor.Then there is an object
E∈Db(M×X)such that Fis isomorphic to the functor ΦEdefined by (30), and
the object Eis determined uniquely up to isomorphism.
It follows at once that every equivalence is representable by an object on the
product, because every equivalence has an adjoint, which coincides with a quasi-
inverse functor.
Theorem 3.2.2. Let Mand Xbe two smooth projective varieties.Suppose that
an exact functor F:Db(M)∼
−→ Db(X)is an equivalence of triangulated categories.
Then there exists an object E∈Db(M×X), unique up to isomorphism,such that
Fis isomorphic to the functor ΦE
These results allow us to describe all equivalences between derived categories
of coherent sheaves, and answer the question of when two distinct varieties have
equivalent derived categories of coherent sheaves.
Before starting on the proof of these theorems, we make a remark. Let Fbe
an exact functor from Db(M)toDb(X). We write F∗and F!respectively for the
left and right adjoint functors of F, assuming that they exist. If a left adjoint F∗
exists, the right adjoint F!also exists, and is defined by the formula
F!=SM◦F∗◦S−1
X,
where SXand SMare Serre functors of the categories Db(X)andDb(M). These
functors exist and are equal to ( ·)⊗ωX[dim X]and(·)⊗ωM[dim M] respectively
(see (6)).
Let Fbe an exact functor from a derived category Db(A) to a derived category
Db(B). We say that Fis bounded if there exist z∈Zand n∈Nsuch that the
cohomology Hi(F(A)) vanishes for i/∈[z, z +n] and for any object A∈A.
Lemma 3.2.3. Let Mand Xbe projective varieties and Masmoothvariety.If
an exact functor F:Db(M)−→ Db(X)has a left adjoint,then Fis bounded.
Proof.WedenotebyG:Db(X)−→ Db(M) the left adjoint of Fand choose a
very ample line bundle Lon X. It defines an embedding i:X→PN. For any
k<0, the sheaf O(k)onPNhas a right resolution in terms of the sheaves O(j)for
j=0,1,...,N,oftheform
O(k)∼
−→ V0⊗O−→ V1⊗O(1) −→ · · · −→ VN⊗O(N)−→ 0,
Derived categories of coherent sheaves 549
where all the Vjare vector spaces [2]. Restricting this resolution to Xgives a
resolution of the sheaf Lkin terms of the sheaves Ljfor j=0,1,...,N.Since
for any j=0,1,...,N the non-zero cohomology of the objects G(Lj)belongto
some interval, one can find an integer zand a positive integer nsuch that the
cohomology Hl(G(Lk)) vanishes for all k0andforl/∈[z,z
+n]. This follows
at once from the existence of a spectral sequence
Ep,q
1=Vp⊗Hq(G(Lp)) =⇒Hp+q(G(Li)).
Let A∈Db(M) be some object. Since Lis ample, it follows that, if for a chosen
jwe have Homj(Li,F(A)) = 0 for any i0, then the cohomology Hj(F(A))
vanishes. By assumption, Gis left adjoint to F. Hence,
Homj(Li,F(A)) ∼
=Homj(G(Li),A).
Now consider a sheaf Fon M. Since for all i<0 the cohomology of the objects
G(Li) is concentrated in the interval [z,z
+n], it follows that Homj(G(Li),F)=0
for any i<0andj/∈[−z−n,−z+dimM]. (Here we use the fact that the
homological dimension of the category coh(M)isequaltodimM.) Hence, for
thesamevaluesofjwe have Hj(F(F)) = 0 for any sheaf F. Therefore, the functor
Fis bounded.
Remark 3.2.4. After shifting Fin the derived category if necessary, we assume from
now on and throughout this chapter that for any sheaf Fon Mthe cohomology
Hi(F(F)) is non-zero only for i∈[−a, 0], where ais a fixed positive integer.
3.3. Construction of the object representing a fully faithful functor. In
this section, starting from an exact fully faithful functor F, we construct a certain
object E∈Db(M×X); in the next section, we prove that the functors Fand ΦEare
isomorphic. The construction of Eproceeds in a number of steps. We first consider
a closed embedding j:M→PNand construct a certain object E∈Db(PN×X).
We then prove that Ein fact comes from the subvariety M×X,thatis,there
exists an object E∈Db(M×X) such that E=RJ∗E,whereJ=(j×id) is the
closed embedding M×Xin PN×X.
We choose a very ample line bundle Lon Msuch that Hi(Lk) = 0 for all k>0
and all i=0,andwritejfor the closed embedding of Min PNdefined by L.
The product PN×PNhas a so-called resolution of the diagonal (see [2]). This
is a complex of sheaves of the form:
0−→ O(−N)ΩN(N)d−N
−−−→O(−N+1)ΩN−1(N−1)
−→ ...−→ O(−1) Ω1(1) d−1
−−−→OO.(31)
This complex is a resolution of the structure sheaf O∆, where ∆ is the diagonal of
the product PN×PN.
Wri te Ffor the functor from Db(PN)toDb(X) obtained as the composite
F◦Lj∗, and consider the diagram of projections
PN×Xπ
−−−−→X
q
PN
.
550 D. O. Orlov
Wri te
d
−i∈HomPN×XO(−i)F(Ωi(i)),O(−i+1)F(Ωi−1(i−1))
for the image of the morphism d−iunder the following composite map:
HomO(−i)Ωi(i),O(−i+1)Ωi−1(i−1)
∼
−→ HomOΩi(i),O(1) Ωi−1(i−1)
∼
−→ HomΩi(i),H0(O(1)) ⊗Ωi−1(i−1)
−→ HomF(Ωi(i)),H0(O(1)) ⊗F(Ωi−1(i−1))
∼
−→ HomOF(Ωi(i)),O(1) F(Ωi−1(i−1))
∼
−→ HomO(−i)F(Ωi(i)),O(−i+1)F(Ωi−1(i−1)).
One sees readily that the composite d
−i+1 ◦d
−ivanishes. Hence, we can consider
the following bounded complex of objects of the derived category Db(PN×X):
C·:= O(−N)F(ΩN(N)) d
−N
−−−→···
−→ O(−1) F(Ω1(1)) d
−1
−−→ OF(O).(32)
For l<0wehave
HomlO(−i)F(Ωi(i)),O(−k)F(Ωk(k))
∼
=HomlOF(Ωi(i)),H0(O(i−k)) ⊗F(Ωk(k))
∼
=Homlj∗(Ωi(i)),H0(O(i−k)) ⊗j∗(Ωk(k))=0.
Thus, by Lemma 3.1.2, C·has a convolution, and all convolutions are isomorphic.
Wri te Efor a convolution of C·and γ0for the morphism OF(O)γ0
−→ E.(In
fact, we see below that all convolutions of C·are canonically isomorphic.) Now let
ΦEbe the functor from Db(PN)toDb(X) defined by (7).
Lemma 3.3.1. Fo r al l k∈Zthere are canonical isomorphisms
fk:F(O(k)) ∼
−→ ΦE(O(k)),
and these isomorphisms are functorial ;that is,for any α:O(k)→O(l)the diagram
F(O(k)) F(α)
−−−−→F(O(l))
fk
fl
ΦE(O(k)) ΦE(α)
−−−−→ ΦE(O(l))
is commutative.
Proof. Assume first that k0 and consider the resolution (31) of the diagonal
∆⊂PN×PN.TensoritbyO(k)O, then take its direct image under the
Derived categories of coherent sheaves 551
projection to the second factor. As a result we obtain the following resolution of
the sheaf O(k) on projective space PN:
H0(O(k−N))⊗ΩN(N)−→ · · ·−→ H0(O(k−1))⊗Ω1(1) −→ H0(O(k))⊗Oδk
−→O(k).
since Fis exact by assumption, it follows that F(O(k)) is a convolution of the
complex
H0(O(k−N))⊗F(ΩN(N))−→ · · · −→ H0(O(k−1))⊗F(Ω1(1))−→ H0(O(k))⊗F(O)
of objects of the category Db(X). We denote this complex by D·
k.
Recall now that, by construction, Eis a convolution of the complex C·(32).
Consider the complex C·
k:= q∗O(k)⊗C·on PN×X.Thenq∗O(k)⊗Eis a
convolution of C·
k. And there is a morphism γk:O(k)F(O)−→ q∗O(k)⊗E
canonically obtained from γ0.Thecomplexπ
∗(C·
k), the direct image of (C·
k) under
the projection to the second factor, is canonically isomorphic to D·
k. Thus, we see
that the objects F(O(k)) and ΦE(O(k)) := Rπ
∗(q∗O(k)⊗E) are both convolutions
of the same complex D·
k.
By assumption, the functor Fis full and faithful. Hence, for locally free sheaves
Gand Hon PNwe have the equality
Homi(F(G),F(H)) = Homi(j∗(G),j
∗(H)) = 0 for i<0.
This implies in particular that the complex D·
ksatisfies conditions (25) and (26) of
Lemma 3.1.2. Hence, by the lemma, there exists a uniquely defined isomorphism
fk:F(O(k)) ∼
−→ ΦE(O(k)) that makes the following diagram commutative:
H0(O(k)) ⊗F(O)F(δk)
−−−−→F(O(k))
id
fk
H0(O(k)) ⊗F(O)Rπ
∗(γk)
−−−−−→ ΦE(O(k))
.
We now prove that these isomorphisms are functorial. For any α:O(k)→O(l)
there are commutative squares of the form
H0(O(k)) ⊗F(O)F(δk)
−−−−→F(O(k))
H0(α)⊗id
F(α)
H0(O(l)) ⊗F(O)F(δl)
−−−−→F(O(l))
and
H0(O(k)) ⊗F(O)Rπ
∗(γk)
−−−−−→ ΦE(O(k))
H0(α)⊗id
ΦE(α)
H0(O(l)) ⊗F(O)Rπ
∗(γl)
−−−−−→ΦE(O(l))
.
552 D. O. Orlov
These three commutative squares imply the following equalities:
fl◦F(α)◦F(δk)=fl◦F(δl)◦(H0(α)⊗id) = Rπ
∗(γl)◦(H0(α)⊗id),
ΦE(α)◦fk◦F(δk)=Φ
E(α)◦Rπ
∗(γk)=Rπ
∗(γl)◦(H0(α)⊗id).
The complexes D·
kand D·
lsatisfy the conditions of Lemma 3.1.3, and hence, there
is a unique morphism h:F(O(k)) →ΦE(O(l)) for which
h◦F(δk)=Rπ
∗(γl)◦(H0(α)⊗id).
Thus, the morphism hcoincides simultaneously with fl◦F(α)andwithΦ
E(α)◦fk,
which implies that these two are equal.
Now consider the case k<0. Take the right resolution
O(k)∼
−→ Vk
0⊗O−→···−→Vk
N⊗O(N)
of the sheaf O(k)onPN. Applying Lemma3.1.3 again, we see that the morphism of
complexes
Vk
0⊗F(O)−−−−→ ··· −−−−→Vk
N⊗F(O(N))
id ⊗f0
id ⊗fN
Vk
0⊗ΦE(O)−−−−→ ··· −−−−→Vk
N⊗ΦE(O(N))
gives a uniquely defined morphism fk:F(O(k)) −→ ΦE(O(k)). A direct check
(which we omit) shows that these morphisms are functorial.
Remark 3.3.2. We note that the object E∈Db(PN×X) constructed from the
functor Fis uniquely determined.
We now prove the existence of an object in the category E∈Db(M×X)such
that RJ∗E∼
=E,where,asabove,Jis the embedding of M×Xin PN×X.
Let Lbe a very ample line bundle on Mand j:M→PNthe embedding into
projective space it defines. We denote by Athe graded algebra ∞
i=0 H0(M, Li).
Set B0=kand B1=A1.Form2, we define Bmby the rule
Bm=Ker
Bm−1⊗A1
um−1
−−−→ Bm−2⊗A2,(33)
where um−1is the natural map defined by induction.
Definition 3.3.3. We say that an algebra Ais an n-Koszul algebra if the sequence
of right A-modules
Bn⊗kA−→ Bn−1⊗kA−→ · · · −→ B1⊗kA−→ A−→ k−→ 0
is exact. An algebra is called a Koszul algebra if it is an n-Koszul algebra for any n.
Suppose that Ais an n-Koszul algebra. We set R0=OMand for m1we
write Rmfor the kernel of the canonical morphism
Bm⊗OM−→ Bm−1⊗L
Derived categories of coherent sheaves 553
defined by the natural embedding Bm−→ Bm−1⊗A1. Using (33), we obtain
a canonical morphism Rm−→ A1⊗Rm−1(in fact, one checks that there is an
isomorphism Hom(Rm,R
m−1)∼
=A∗
1).
Moreover, if Ais an n-Koszul algebra, then the following complex of sheaves is
exact for mn:
0−→ Rm−→ Bm⊗OM−→ Bm−1⊗L−→ · · · −→ B1⊗Lm−1−→ Lm−→ 0.(34)
On the projective space PNthere is an exact complex of the form
0−→ Ωm(m)−→
m
A1⊗O−→
m−1
A1⊗O(1) −→···−→O(m)−→ 0.(35)
There is a canonical map fm:j∗Ωm(m)−→ Rm. Indeed, since Ais commutative,
there are natural embeddings iA1⊂Bi. Therefore, there exists a morphism from
the complex (35) restricted to Mto the complex (34), and hence a canonical map
fm:j∗Ωm(m)−→ Rm.
It is known that for any nthere exists an lsuch that the Veronese algebra
Al=∞
i=0 H0(M, Lil)isn-Koszul; moreover, it was proved in [1] that the alge-
bra Alis in fact a Koszul algebra for l0.
However, in what follows, along with the n-Koszul property of the Veronese
algebra, we need some additional properties. Namely, using the technique of [18]
and replacing the sheaf Lby a sufficiently high power Lj, one can prove the following
assertion.
Proposition 3.3.4. For any integer nthere is a very ample line bundle Lsuch
that
1) the algebra Ais an n-Koszul algebra,that is,the sequence
Bn⊗kA−→ Bn−1⊗kA−→ · · · −→ B1⊗kA−→ A−→ k−→ 0
is exact;
2) the complex of sheaves on Mgiven by
Ak−n⊗Rn−→ Ak−n+1 ⊗Rn−1−→
···−→Ak−1⊗R1−→ Ak⊗R0−→ Lk−→ 0
is exact for any k0(if k−i<0, then Ak−i=0by definition);
3) the complex of sheaves on M×Mof the form
L−nRn−→···−→L−1R1−→ OMR0−→ O∆
is exact,that is,it gives an n-resolution of the diagonal on M×M.
The proof of this proposition is given in §3.5.
Wri te Tkfor the kernel of the canonical morphism Ak−n⊗Rn−→ Ak−n+1⊗Rn−1.
In view of property 2) of Proposition 3.3.4 and the fact that Extn+1(Lk,T
k)=0
for n0, we see that every convolution of the complex
Ak−n⊗Rn−→ Ak−n+1 ⊗Rn−1−→···−→Ak⊗R0
is canonically isomorphic to Tk[n]⊕Lk.
554 D. O. Orlov
The canonical morphisms Rk−→ A1⊗Rk−1induce morphisms
L−kF(Rk)−→ L−k+1 F(Rk−1).
This follows from the existence of isomorphisms
Hom(L−kF(Rk),L−k+1 F(Rk−1)) ∼
=Hom(F(Rk),H0(L)⊗F(Rk−1))
∼
=Hom(Rk,A
1⊗Rk−1).
Moreover, we have the following complex of objects in the category Db(M×X):
L−nF(Rn)−→ · · · −→ L−1F(R1)−→ OMF(R0).(36)
By Lemma 3.1.2, the complex (36) has a convolution, and all its convolutions
are isomorphic. We denote this convolution by G∈Db(M×X).
For any k0 the object Rπ∗(G⊗p∗(Lk)) is a convolution of the complex
Ak−n⊗F(Rn)−→ Ak−n+1 ⊗F(Rn−1)−→···−→Ak⊗F(R0).(37)
On the other hand, the object F(Tk[n]⊕Lk) is also a convolution of this complex,
obviously satisfying the condition of Lemma 3.1.2. Hence, there is an isomorphism
Rπ∗(G⊗p∗(Lk)) ∼
=F(Tk[n]⊕Lk).
It follows from Lemma 3.2.3 and Remark 3.2.4 that for all k>0 the non-
trivial cohomology sheaves Hi(Rπ∗(G⊗p∗(Lk))) = Hi(F(Tk)[n]) ⊕Hi(F(Lk)) are
concentrated in the union [−n−a, −n]∪[−a, 0] (where ais the number defined in
Remark 3.2.4). Since Lis ample, it follows that the cohomology sheaves Hi(G)
are also concentrated in [−n−a, −n]∪[−a, 0]. We can assume that n>dim M+
dim X+a. Since the category of coherent sheaves on M×Xhas homological
dimension dim M+dimX, we see in this case that G∼
=C⊕E,whereE,C are
objects of Db(M×X)forwhichHi(E)=0fori/∈[−a, 0] and Hi(C)=0for
i/∈[−n−a, −n]. Hence, in particular, Rπ∗(E⊗p∗(Lk)) ∼
=F(Lk). Note that since
the object Gis uniquely determined as the convolution of the complex (36), the
object Eis also uniquely determined up to isomorphism.
We now show that there is an isomorphism RJ∗E∼
=E. For this, we consider
the map of complexes over Db(PN×X),
O(−n)F(Ωn(n)) −−−−→ ··· −−−−→OF(O)
canF(fn)
canF(f0)
Rj∗L−nF(Rn)−−−−→ ··· −−−−→Rj∗OMF(R0)
.
Applying Lemma 3.1.3, we obtain the existence of a morphism ϕ:K−→ RJ∗G
between the convolutions.
If N>n, then the object Kis not isomorphic to E, but there is a distinguished
triangle
S−→ K−→ E−→ S[1].
Derived categories of coherent sheaves 555
As above, we can show that the cohomology sheaves Hi(S) are non-zero only for
i∈[−n−a, −n]. This implies that Hom(S, RJ∗E)=0andHom(S[1],RJ∗E)=0,
because the cohomology RJ∗Eis concentrated in the closed interval [−a, 0]. This
implies the existence of a unique morphism ψ:E−→ RJ∗Esuch that the following
diagram is commutative:
Kϕ
−−−−→RJ∗G
Eψ
−−−−→RJ∗E
.
As we know,
Rπ
∗(E⊗q∗(O(k))) ∼
=F(Lk)∼
=Rπ∗(E⊗p∗(Lk)).
Wri te ψkfor the morphisms Rπ
∗(E⊗q∗(O(k))) −→ Rπ∗(E⊗p∗(Lk)) induced
by ψ.Theψkfit into a commutative diagram
SkA1⊗F(O)can
−−−−→F(Lk)∼
−−−−→Rπ
∗(E⊗q∗(O(k)))
can
ψk
Ak⊗F(O)can
−−−−→F(Lk)∼
−−−−→Rπ∗(E⊗p∗(Lk))
.
This implies that the morphisms ψkare isomorphisms for any k0. Hence, ψis
also an isomorphism. Thus, we have proved the following assertion.
Proposition 3.3.5. There is an object E∈Db(M×X)such that RJ∗E∼
=E,
where Eis the object of Db(PN×X)constructed in §3.3; and this Eis unique up
to isomorphism
3.4. Proof of the main theorem. In the previous section, starting from a fully
faithful functor Fbetween the derived categories of coherent sheaves on varieties
Mand X, we constructed an object Eon the product M×X, and thus obtained
a new functor ΦE. The main objective of the present section is to show that these
two functors Fand ΦEare isomorphic. For this, we must construct a natural
transformation between these functors which is an isomorphism. By construction,
the transformation is already given on an ample sequence of line bundles on M.
Our task is to extend this transformation to the entire derived category.
We start by proving some assertions on Abelian categories that we need below.
Let Abe a k-linear Abelian category (in what follows we always consider Abelian
categories that are k-linear).
Definition 3.4.1. We say that a sequence of ob jects {Pi|i∈Z0}(with negative
indices)inanAbeliancategoryAis ample if for every object X∈Athere exists
an integer Nsuch that the following conditions hold for any index i<N:
a) the canonical morphism Hom(Pi,X)⊗Pi−→ Xis surjective,
b) Extj(Pi,X) = 0 for all j=0,
c) Hom(X, Pi)=0.
556 D. O. Orlov
Example 3.4.2. Fo r Lan ample line bundle on a projective variety, the sequence
{Li|i∈Z0}is ample in the Abelian category of coherent sheaves.
Lemma 3.4.3. Le t {Pi}be an ample sequence in an Abelian category A.If an
object Xin the category Db(A)satisfies the equality
Hom·(Pi,X)=0 for all i0,
then Xis the zero object.
Proof. It follows from the definition of ampleness that
Hom(Pi,Hk(X)) ∼
=Homk(Pi,X)=0 fori0.
However, the morphism Hom(Pi,Hk(X)) ⊗Pi−→ Hk(X) must be surjective for
i0. Hence, Hk(X) = 0 for all k. This means that Xis the zero object.
Lemma 3.4.4. Let Abe an Abelian category of finite homological dimension and
{Pi}an ample sequence in A.If an object X∈Db(A)is such that Hom·(X, Pi)=0
for any i0, then Xis the zero object.
Proof. Suppose that the object Xis non-trivial. After shifting Xin the derived
category if necessary, we can assume that the rightmost non-zero cohomology of
Xis H0(X). Consider the canonical morphism X−→ H0(X). For some i1,there
exists a surjective map P⊕k1
i1−→ H0(X); write Y1for its kernel. By assumption,
Hom·(X, Pi1) = 0, and hence also Hom1(X, Y1)= 0. Next, take a surjective map
P⊕k2
i2−→ Y1,whichexistsforsomei20, and write Y2for its kernel. The con-
dition Hom·(X, Pi2) = 0 again gives Hom2(X, Y2)= 0. Continuing this procedure,
we obtain a contradiction to the finite homological dimension of A.
Lemma 3.4.5. Let Aand Bbe Abelian categories and suppose that Ahas finite
homological dimension.Let {Pi}be an ample sequence in A.Suppose that Fis an
exact functor from Db(A)to Db(B)that has right and left adjoint functors F!and
F∗respectively.If the maps
Homk(Pj,P
i)∼
−→ Homk(F(Pj),F(Pi))
are isomorphisms for j<iand for al l k,then Fis fully faithful.
Proof. Consider the canonical morphism fi:Pi−→ F!F(Pi) and the distinguished
triangle
Pi
fi
−→ F!F(Pi)−→ Ci−→ Pi[1].
By assumption, for j<iwe have isomorphisms
Homk(Pj,P
i)∼
−→ Homk(F(Pj),F(Pi)) ∼
=Homk(Pj,F!F(Pi)).
Hence, Hom·(Pj,C
i)=0forj<i. By Lemma 3.4.3, Ci=0. Thus,fiis an
isomorphism.
Derived categories of coherent sheaves 557
For an arbitrary object Xwe now consider the canonical map gX:F∗F(X)−→ X
and the distinguished triangle
F∗F(X)gX
−−→ X−→ CX−→ F∗F(X)[1].
There is a sequence of isomorphisms
Homk(X, Pi)∼
−→ Homk(X, F !F(Pi)) ∼
=Homk(F∗F(X),P
i).
This implies that Hom·(CX,P
i) = 0 for all i. By Lemma 3.4.4 we get that CX=0.
Thus, gXis an isomorphism. Therefore, Fis fully faithful.
We now state and prove the main proposition of this section, which we need in
the proof of Theorem 3.2.1, the main result of this chapter. The proposition is also
of independent interest.
Let Abe an Abelian category with an ample sequence {Pi|i∈Z0}.Writej
for the embedding of the full subcategory Cwith objects Ob C:= {Pi|i∈Z0}
into Db(A). In this situation, given a functor F:Db(A)−→ Db(A), one proves
that, if there exists an isomorphism of F|Cwith the identity functor on C, then this
transformation extends to an isomorphism on the entire category Db(A).
Proposition 3.4.6. Let Abe an Abelian category and {Pi|i∈Z0}an ample
sequence in A.Write jfor the embedding of the full subcategory Cwith objects
Ob C:= {Pi|i∈Z0}into Db(A). Let F:Db(A)−→ Db(A)be some a uto-
equiv alence.Suppose that there exists an isomorphism of functors f:j∼
−→ F|C.
Then fextends to an isomorphism id ∼
−→ Fon the entire category Db(A).
Proof.First,sinceFcommutes with direct sums, the transformation fextends
componentwise to direct sums of objects in the category C. We note that an object
X∈Db(A) is isomorphic to an object of Aif and only if Homj(Pi,X)=0forj=0
and all i0. It follows that in this case the object F(X) is also isomorphic to an
object of A, because
Homj(Pi,F(X)) ∼
=Homj(F(Pi),F(X)) ∼
=Homj(Pi,X)=0
for j= 0 and for all i0.
Step 1. Let Xbe an object of the category A. We fix a surjective morphism
v:P⊕k
i−→ X. There exists an isomorphism fi:P⊕k
i∼
−→ F(P⊕k
i) together with
two distinguished triangles
Yu
−−−−→P⊕k
i
v
−−−−→X−−−−→Y[1]
fi
F(Y)F(u)
−−−−→F(P⊕k
i)F(v)
−−−−→F(X)−−−−→F(Y)[1]
.
Let us prove that F(v)◦fi◦u= 0. For this, consider a surjective morphism
w:P⊕l
j−→ Y; it is enough to show that F(v)◦fi◦u◦w=0. Let fj:P⊕l
j∼
−→ F(P⊕l
j)
558 D. O. Orlov
be the canonical isomorphism. Using the commutation relations for fiand fj,we
obtain the equalities
F(v)◦fi◦u◦w=F(v)◦F(u◦w)◦fj=F(v◦u◦w)◦fj=0.
Since Hom(Y[1],F(X)) = 0, there is a unique morphism fX:X−→ F(X)com-
muting with fiby Lemma 3.1.1.
Now consider the mapping cone CXof fX. Using the isomorphisms
Hom(Pi,X)∼
=Hom(F(Pi),F(X)) ∼
=Hom(Pi,F(X)),
we see that Homj(Pi,C
X) = 0 for all jand i0. Hence, CX= 0 by Lemma 3.4.3,
and fXis an isomorphism.
Step 2. We now show that fXdoes not depend on the choice of the covering
v:P⊕k
i−→ X. Consider two such surjective morphisms v1:P⊕k1
i1−→ Xand
v2:P⊕k2
i2−→ X. We can always fix up two surjective morphisms w1:P⊕l
j−→ P⊕k1
i1
and w2:P⊕l
j−→ P⊕k2
i2such that the following diagram is commutative:
P⊕l
j
w2
−−−−→P⊕k2
i2
w1
v2
P⊕k1
i1
v1
−−−−→X
.
It is obviously enough to check that the transformations fXconstructed from v1
and v1◦w1coincide. For this, consider the commutative diagram
P⊕l
j
w1
−−−−→P⊕k1
i1
v1
−−−−→X
fj
v2
fX
F(P⊕l
j)F(w1)
−−−−→F(P⊕k1
i1)F(v1)
−−−−→F(X)
.
Here the isomorphism fXis constructed from v1. Both squares of the diagram
commute. Since there only exists one morphism from Xto F(X)thatcommutes
with fj, it follows that the morphism fXconstructed from v1coincides with that
constructed from v1◦w1.
Step 3. Now we have to check that the morphisms fXdefine a natural transfor-
mation of functors on A. That is, for any morphism Xϕ
−→ Y,wemustprove
that
fY◦ϕ=F(ϕ)◦fX.
Consider a surjective morphism P⊕l
j
v
−→ Y. We choose an index i0anda
surjective morphism P⊕k
i
u
−→ Xsuch that the composite ϕ◦ulifts to a morphism
ψ:P⊕k
i−→ P⊕l
j. This is possible because for i0themapHom(P⊕k
i,P⊕l
j)→
Hom(P⊕k
i,Y) is surjective. We obtain a commutative square
P⊕k
i
u
−−−−→X
ψ
ϕ
P⊕l
j
v
−−−−→Y
.
Derived categories of coherent sheaves 559
We wr i te h1and h2for the composites fY◦ϕand F(ϕ)◦fXrespectively. We have
the equalities
h1◦u=fY◦ϕ◦u=fY◦v◦ψ=F(v)◦fj◦ψ=F(v)◦F(ψ)◦fi
and
h2◦u=F(ϕ)◦fX◦u=F(ϕ)◦F(u)◦fi=F(ϕ◦u)◦fi=F(v◦ψ)◦fi=F(v)◦F(ψ)◦fi.
Thus, for t=1,2 the morphisms htmake the following diagram commute:
Z−−−−→P⊕k
i
u
−−−−→X−−−−→Z[1]
F(ψ)◦fi
ht
F(W)−−−−→F(P⊕l
j)F(v)
−−−−→F(Y)−−−−→F(W)[1]
.
Since Hom(Z[1],F(Y)) = 0, it follows from Lemma 3.1.1 that h1=h2.Thus,
fY◦ϕ=F(ϕ)◦fX.
Step 4. We define a transformation fX[n]:X[n]−→ F(X[n]) ∼
=F(X)[n] for any
X∈Aby the formula
fX[n]=fX[n].
One proves readily that the transformations defined in this way commute with
any morphism u∈Extk(X, Y ). Indeed, every element u∈Extk(X, Y )canbe
represented as a composite u=u0u1···ukof certain elements ui∈Ext1(Zi,Z
i+1),
where Z0=Xand Zk=Y. Thus, it is enough to verify that fX[n]commutes with
elements u∈Ext1(X, Y ). For this, consider the diagram
Y−−−−→Z−−−−→Xu
−−−−→Y[1]
fY
fZ
fY[1]
F(Y)−−−−→F(Z)−−−−→F(X)F(u)
−−−−→F(Y)[1]
.
By one of the axioms of triangulated category, there is a morphism h:X→
F(X) such that (fY,f
Z,h) is a morphism of triangles. On the other hand, since
Hom(Y[1],F(X)) = 0, it follows from Lemma 3.1.1 that the morphism his uniquely
determined by the condition that it commutes with fZ. However, fXalso commutes
with fZ. Hence, h=fX, and thus
fY[1] ◦u=F(u)◦fX.
Step 5. We carry out the final part of the proof by induction on the length of
the interval to which the non-trivial cohomology of the object belongs. For this,
consider the full subcategory jn:Dn→Db(A)ofDb(A) consisting of objects with
non-trivial cohomology in some interval of length n(the interval is not fixed). We
now prove that there is a unique extension of the natural transform fto a natural
560 D. O. Orlov
functorial isomorphism fn:jn−→ F|Dn. We have already proved this above for
n= 1, as the basis of the induction.
Now to prove the inductive step, suppose that the assertion is already proved
for some n=a1. Let Xbe an object of Da+1 , and suppose for definiteness
that the cohomology Hp(X) is non-trivial for p∈[−a, 0]. We take Piin the ample
sequence, where iis a sufficiently negative index such that
a) Homj(Pi,Hp(X)) = 0 for all pand j=0,
b) there exists a surjective morphism u:P⊕k
i−→ H0(X),
c) Hom(H0(X),P
i)=0.
(38)
It follows from a) and the standard spectral sequence that there is an isomorphism
Hom(Pi,X)∼
−→ Hom(Pi,H0(X)). Thus, there is a morphism v:P⊕k
i−→ Xwhose
composite with the canonical morphism X−→ H0(X)coincideswithu.Consider
the distinguished triangle
Y[−1] −→ P⊕k
i
v
−→ X−→ Y.
Since the object Ybelongs to Da, it follows from the induction assumption that
the isomorphism fYalready exists and commutes with fi. We have the diagram
(39)
Next, the sequence of isomorphisms
Hom(X, F (P⊕k
i)) ∼
=Hom(X, P ⊕k
i)∼
=Hom(H0(X),P⊕k
i)=0
allows us to apply Lemma 3.1.1 with g=fY, and it follows from this that there
is a unique morphism fX:X−→ F(X) completing the diagram to a morphism of
triangles. It is obvious that fXis in fact an isomorphism, because fiand fYare.
Step 6. We now have to prove that the isomorphism fXdoes not depend on the
choice of iand u. Suppose that we are given two surjective morphisms u1:P⊕k1
i1−→
H0(X)andu2:P⊕k2
i2−→ H0(X) satisfying a), b) and c). Then we can choose a
sufficiently negative index jand surjective morphisms w1and w2that make the
diagram
P⊕l
j
w2
−−−−→P⊕k2
i2
w1
u2
P⊕k1
i1
u1
−−−−→H0(X)
commute. Write v1:P⊕k1
i1−→ Xand v2:P⊕k2
i2−→ Xfor the morphisms cor-
responding to u1and u2.SinceHom(Pj,X)∼
−→ Hom(Pj,H0(X)), we see that
v2w2=v1w1.
Derived categories of coherent sheaves 561
There is a morphism ϕ:Yj−→ Yi1such that the triple (w1,id,ϕ) is a morphism
of triangles
P⊕l
j
v1◦w1
−−−−→Xy
−−−−→Yj−−−−→P⊕l
j[1]
w1
id
ϕ
w1[1]
P⊕k1
i1
v1
−−−−→Xy1
−−−−→Yi1−−−−→P⊕k1
i1[1]
,
that is, ϕy =y1.
Since Yjand Yi1only have non-trivial cohomology in the interval [−a, −1], by
induction we have the following commutative square:
Yj
ϕ
−−−−→Yi1
fYj
fYi1
F(Yj)F(ϕ)
−−−−→F(Yi1)
.
Wri te fj
X,fi1
Xand fi2
Xfor the morphisms constructed by the above rule; these
can be completed to a commutative diagram (39) for v=v1w1,v=v1and v=v2
respectively. We have already proved in Lemma 3.1.1 above that the morphism fi1
X
is uniquely determined by the condition
F(y1)fi1
X=fYi1y1.
On the other hand, we have the relations
F(y1)fj
X=F(ϕy)fj
X=F(ϕ)F(y)fj
X=F(ϕ)FYjy=fYi1ϕy =fYi1y1,
which imply at once that fj
X=fi1
X.Inthesamewaywegetfj
X=fi2
X. Hence,
the morphism fXdoes not depend on the choices of the index iand the morphism
u:P⊕k
i−→ H0(X), and is thus absolutely uniquely defined.
Step 7. We have thus obtained an extension of fato Da+1 . It remains to show that
this extension is again a natural transformation from ja+1 to F|Da+1;thatis,that
for any morphism ϕ:X−→ Ywith Xand Yin Da+1, we obtain a commutative
diagram
Xϕ
−−−−→Y
fX
fY
F(X)F(ϕ)
−−−−→F(Y)
.(40)
We will reduce this problem to the case in which both objects Xand Ybelong
to Da. There are two cases.
Case 1. We consider the case when the highest non-trivial cohomology of the
object X(which we can assume to be H0(X) without loss of generality) has
index strictly greater than that for Y. Asabove,wetakeasurjectivemorphism
u:P⊕k
i−→ H0(X) satisfying a), b) and c) and construct a lift of uto a mor-
phism v:P⊕k
i−→ X. We have a distinguished triangle
P⊕k
i
v1
−−−−→Xα
−−−−→Z−−−−→P⊕k
i[1].
562 D. O. Orlov
If iis sufficiently negative, then Hom(P⊕k
i,Y) = 0. Applying Hom(−,Y)tothis
triangle, we see that there exists a morphism ψ:Z−→ Yfor which ϕ=ψα.Itis
known that the isomorphism fXconstructed above satisfies the relation
F(α)fX=fZα.
If we assume that
F(ψ)fZ=fYψ,
then we obtain
F(ϕ)fX=F(ψ)F(α)fX=F(ψ)fZα=fYψα =fYϕ.
This means that, to check that the square (40) is commutative, we can replace
Xby Z. But the upper bound for the non-trivial cohomology of Zis one less than
for X. Moreover, one can see that, if Xbelongs to Dkwith k>1, then Zbelongs
to Dk−1, and, if Xbelongs to D1,thenZalso belongs to D1, but the index for its
non-trivial cohomology is one less than for X.
Case 2. We now consider the other case: the highest non-trivial cohomology of Y
(which we can again assume to be H0(Y)) has index greater than or equal to that
of X. Take a surjective morphism u:P⊕k
i−→ H0(Y) satisfying conditions a), b)
and c) and construct a morphism v:P⊕k
i−→ Y, which is uniquely determined
by u. Consider the distinguished triangle
P⊕k
i
v
−−−−→Yβ
−−−−→W−−−−→P⊕k
i.(41)
Wri te ψfor the composite β◦ϕ.
If we now assume that
F(ψ)fX=fWψ,
then, since F(β)fY=fWβ,weobtain
F(β)(fYϕ−F(ϕ)fX)=fWβϕ −f(βϕ)fX=fWψ−F(ψ)fX=0.(42)
We again choose ito be sufficiently negative, so that the vanishing condition
Hom(X, P ⊕k
i) = 0 is satisfied. Since F(P⊕k
i) is isomorphic to P⊕k
i,wehavethe
equality Hom(X, F (P⊕k
i)) = 0. Now applying Hom(X, F (−)) to the triangle (41),
we see that F(β) defines an embedding of Hom(X, F (Y)) into Hom(X, F (W)). It
now follows at once from (42) that fYϕ=F(ϕ)fX.
Thus, to check that the square (40) is commutative, we can replace Yby an
object Wthat has upper bound of the non-trivial cohomology one less than Y.If
Ybelongs to Dkwith k>1, then Wbelongs to Dk−1.IfYbelongs to D1,then
Walso belongs to D1, but has non-trivial cohomology of index one less than Y.
Suppose now that Xand Ybelong to the category Da+1 with a>1. Depending
on which of the cases 1) or 2) is applicable, we can replace either Xor Yby
an object that already belongs to Da. Repeating this procedure if necessary, we
can reduce the upper bound of the cohomology of this object until the other case
becomes applicable. Then we will be able to reduce the length of the non-trivial
Derived categories of coherent sheaves 563
cohomology of the other object, and arrive at the situation in which both objects
already belong to Da. This is our induction step.
In conclusion we note that during our construction, the isomorphisms fXwere
uniquely determined at each point. Hence, the natural transformation from id to
Fthat we have constructed is unique. This completes the proof of the proposition.
Proof of Theorem 3.2.1. 1) Existence. Starting from the functor F,wecanuse
Proposition 3.3.5 and Lemma 3.3.1 to construct an object E∈Db(M×X)forwhich
there exists an isomorphism of functors f:F|C∼
−→ ΦE|Con the full subcategory
C⊂Db(M)withObC={Li|i∈Z},whereLis a very ample bundle on Mfor
which Hi(M, Lk)=0fork>0andi=0.
By Lemma 3.4.5, ΦEis fully faithful. Moreover, since there are isomorphisms
F!(f): F!◦F|C∼
=idC∼
−→ F!◦ΦE|C,
Φ∗
E(f): Φ∗
E◦F|C∼
−→ Φ∗
E◦ΦE|C∼
=idC,
it follows again from Lemma 3.4.5 that the functors F!◦ΦEand Φ∗
E◦Fare also
fully faithful. Since they are adjoint to one another, it follows that they are in fact
equivalences.
Consider again the isomorphism F!(f): F!◦F|C∼
=idC∼
−→ F!◦ΦE|Con the
subcategory C. By Proposition 3.4.6, it extends to an isomorphism on the entire
category Db(M), that is, id ∼
−→ F!◦ΦE.
Since F!is right adjoint to F, we obtain a morphism of functors f:F−→ ΦE
for which f|C=f. It remains to show that fis an isomorphism. Indeed, take the
mapping cone CZof the canonical morphism fZ:F(Z)−→ ΦE(Z). Since F!(fZ)
is an isomorphism, we see that F!(Z) = 0. Hence, Hom(F(Y),C
Z) = 0 for any
object Y.Moreover,sinceF(Lk)∼
=ΦE(Lk) for all k, we obtain a sequence of
isomorphisms
Homi(Lk,Φ!
E(CZ)) = Homi(ΦE(Lk),C
Z)=Hom
i(F(Lk),C
Z)=0
for all kand i.
Applying Lemma 3.6 gives at once the equality Φ!
E(CZ) = 0. It follows that
Hom(ΦE(Z),C
Z) = 0. Therefore, the triangle for the morphism fZmust be split,
that is, F(Z)=CZ[−1] ⊕ΦE(Z). However, we have already proved above that
Hom(F(Y),C
Z) = 0 for any Y, and hence also for Z[1]. However, this can only
happen if CZ=0,andfZis an isomorphism.
2) Uniqueness. The uniqueness of the object representing Fin fact follows from
our construction, because each time we construct some object it is unique. However,
let us go through this once more. Suppose that there exist two objects E1and E2in
Db(M×X)forwhichΦ
E1∼
=F∼
=ΦE2. Consider the functor F=Lj∗◦F,where,
as above, j:M−→ PNis an embedding by a suitable very ample line bundle. The
objects RJ∗Eifor i=1,2 must both be convolutions of the complex (32)
C·:= O(−N)F(ΩN(N)) d
−N
−−−→···−→O(−1) F(Ω1(1)) d
−1
−−→ OF(O).
However, as we proved above, all convolutions of this complex are isomorphic by
Lemma 3.1.2. Thus, RJ∗E1∼
=RJ∗E2. Applying Proposition 3.3.5, we now see that
the objects E1and E2are themselves also isomorphic.
564 D. O. Orlov
3.5. Appendix: the n
n
n-Koszul property of a homogeneous coordinate
algebra. The facts collected in this appendix are not original, and are well known
in one form or another. However, in the absence of a good reference, we are obliged
to present our own proof of the assertion used in the main text in the form we need
it. Here we mainly use the technique of [18].
Let Xbe a smooth projective variety and La very ample line bundle on X
satisfying the additional condition Hi(X, Lk) = 0 for all k>0andi=0. We
write Afor the homogeneous coordinate algebra of Xwith respect to L,thatis,
A=∞
k=0 H0(X, Lk).
Consider the variety Xnfor some n∈N. In what follows we write π(n)
ifor the
projection of Xnto the ith factor and π(n)
ij for its projection to the product of
the ith and jth factors. Define a subvariety ∆(n)
(i1,...,ik)(ik+1,...,im)⊂Xnas follows:
∆(n)
(i1,...,ik)(ik+1 ,...,im):= (x1,...,x
n)xi1=···=xik;xik+1 =···=xm.
For brevity, we write S(n)
iinstead of ∆(n)
(n,n−1,...,i); obviously, S(n)
i∼
=Xi.
Now set
T(n)
i:=
i−1
k=1
∆(n)
(n,n−1,...,i)(k,k−1),Σ(n):=
n
k=1
∆(n)
(k,k−1).
(By definition, T(n)
1and T(n)
2are the empty subset.) It is clear that T(n)
i⊂S(n)
i.
We wr i te JΣ(n)for the ideal sheaf of the subscheme Σ(n)⊂Xnand I(n)
ifor the
sheaf on Xnwhich is the kernel of the natural map OS(n)
i−→ OT(n)
i−→ 0.
Let us temporarily fix mand km.Letsbe the embedding of the subvari-
ety S(m)
k∼
=Xk−1×Xinto Xn, which by the definition of S(m)
kis the identity on the
first k−1 factors and the diagonal on the final kth factor. We write pfor
the projection of S(m)
kto Xk−1, which is the product of the first k−1factors.
Lemma 3.5.1. The sheaf I(m)
1is isomorphic to O∆(n)
(n,...,1)
;and I(m)
kfor k>1is
isomorphic to s∗p∗(JΣ(k−1) ). In particular,for k>1there are isomorphisms
a) Hj(Xm,I(m)
k⊗(L···Li))
=H
j(Xk−1,J
Σ(k−1) ⊗(L···L)) ⊗Am−k+ifor all i>0;
b) Rjπ(m)
1∗(I(m)
k⊗(OL···Li))
∼
=Rjπ(k−1)
1∗(JΣ(k−1) ⊗(OL···L)) ⊗Am−k+ifor all i>0;
c) Rjπ(m)
1m∗(I(m)
k⊗(OL···Li))
∼
=Rjπ(k−1)
1∗(JΣ(k−1) ⊗(OL···L)) Lm−k+ifor all i.
Proof. The assertion that I(m)
kis isomorphic to s∗p∗(JΣ(k−1) )fork>1 follows at
once from the definition of I(m)
kand of the subschemes T(m)
kand S(m)
k.Therest
follows at once from this.
By induction on n, one sees readily that the complex
P·
n:0−→ JΣ(n)−→ I(n)
n−→ I(n)
n−1−→···−→I(n)
2−→ I(n)
1−→ 0
Derived categories of coherent sheaves 565
on Xnis exact. For example, for n=2 this complex is the short exact sequence on
X×X,
P·
2:0−→ J∆−→ OX×X−→ O∆−→ 0.
Lemma 3.5.2. Let Xbe a smooth projective variety with ample line bundle M.
Then for any positive integer kthere exists an isuch that the bundle L=Mihas
the following properties for al l 1<mk:
a) Hj(Xm,J
Σ(m)⊗(L···L)) = 0 for j=0;
b) Rjπ(m)
1∗(JΣ(m)⊗(OL···L)) = 0 for j=0;
c) Rjπ(m)
1m∗(JΣ(m)⊗(OL···LO)) = 0 for j=0.
(43)
Proof. For any m, the line bundles M··· M,OM···M,andO
M···MOon Xmare ample, π(m)
1-ample, and π(m)
1m-ample, respectively.
Therefore, for each of them there is an integer such that properties a), b), and c)
hold for all powers of these bundles larger than this integer. Take the maximum of
these numbers over all mk, and denote it by i. Then properties a), b), and c)
also hold for L=Mi.
We introduce the following notation:
Bn:= H0(Xn,J
Σ(n)⊗(L···L))
and Rn−1:= R0π(n)
1∗(JΣ(n)⊗(OL···L)).
Proposition 3.5.3. Let Lbe a very ample bundle on a smooth projective variety
Xsatisfying condition (43) for all mwith 1<mn+dimX+2. Then
1) Ais an n-Koszul algebra,that is,the sequence
Bn⊗kA−→ Bn−1⊗kA−→ · · · −→ B1⊗kA−→ A−→ k−→ 0
is exact;
2) the complex of sheaves on Xof the form
Ak−n⊗Rn−→ Ak−n+1 ⊗Rn−1−→
···−→Ak−1⊗R1−→ Ak⊗R0−→ Lk−→ 0
is exact for any k0(if k−i<0, then Ak−i=0by definition);
3) the complex of sheaves
L−nRn−→···−→L−1R1−→ OMR0−→ O∆
on X×Xis exact;that is,it gives an n-resolution of the diagonal of X×X.
566 D. O. Orlov
Proof. 1) First, combining Lemmas 3.5.1 and 3.5.2, for any 1 <mn+dimX+2
we see that
1) H0(Xm,I(m)
k⊗(L···Li))
=H
0(Xk−1,J
Σ(k−1) ⊗(L···L)) ⊗Am−k+i=Bk−1⊗Am−k+i;
2) Hj(Xm,I(m)
k⊗(L···Li)) = 0 for j=0.
(44)
Consider the complexes P·
m⊗(L···L)formn+dimX+ 1. Applying H0
to them and using condition (44), we obtain the exact sequences
0−→ Bm−→ Bm−1⊗kA1−→ · · · −→ B1⊗kAm−1−→ Am−→ 0
for any mn+dimX+1.
We now set m=n+dimX+1and write W·
mfor the complex
I(m)
m−→ I(m)
m−1−→ · · · −→ I(m)
2−→ I(m)
1−→ 0,
which is a right resolution of JΣ(m).Wetakethecomplex
W·
m⊗(L···LLi)
and apply the functor H0to it. We obtain the sequence
Bm−1⊗kAi−→ Bm−2⊗kAi+1 −→···−→B1⊗kAm+i−2−→ Am+i−1−→ 0.
It follows from (44), 2) that the cohomology of this complex equals
Hj(Xm,J
Σ(m)⊗(L···LLi)).
And (43), b) gives us that
Hj(Xm,J
Σ(m)⊗(L···LLi))
=H
j(X, R0π(m)
m∗(JΣ(m)⊗(L···LO)) ⊗Li).
Hence, this cohomology is trivial for j>dim X, and thus there is an exact sequence
of the form
Bn⊗kAm−n+i−1−→ Bn−1⊗kAm−n+i−→ · · · −→ B1⊗kAm+i−2−→ Am+i−1−→ 0
for i1. However, the exactness for i1 has been proved above. Thus, Ais an
n-Koszul algebra.
2) The proof is similar to that of 1). We have isomorphisms
1) R0π(m)
1∗(I(m)
k⊗(OL···L))
∼
=R0π(k−1)
1∗(JΣ(k−1) ⊗(OL···L)) ⊗Am−k+1 ;
2) Rjπ(m)
1∗(I(m)
k⊗(OL···L)) = 0 for all j=0.
(45)
Derived categories of coherent sheaves 567
Applying the functor R0π(m)
1∗to the complexes P·
m⊗(OL··· L)for
mn+dimX+ 2, we obtain an exact complex of sheaves on X,
0−→ Rm−1−→ A1⊗Rm−2−→···−→Am−2⊗R1−→ Am−1⊗R0−→ Lm−1−→ 0
for mn+dimX+2.
We consider the case m=n+dimX+ 2. Applying the functor R0π(m)
1∗to
W·
m⊗(OL···LLi),
gives the complex
Ai⊗Rm−2−→···−→Am+i−3⊗R1−→ Am+i−2⊗R0−→ Lm+i−2−→ 0.
By property (45), its cohomology is
Rjπ(m)
1∗(JΣ(m)⊗(OL···LLi))
∼
=Rjp1∗(R0π(m)
1m∗(JΣ(m)⊗(OL···LO)) ⊗(OLi)),
which is trivial for j>dim X. Thus, the sequence of sheaves
Ak−n⊗Rn−→ Ak−n+1 ⊗Rn−1−→ · · · −→ Ak−1⊗R1−→ Ak⊗R0−→ Lk−→ 0
on Xis exact for all k0.
3) Consider the complex W·
n+2 ⊗(OL···LL−n)onXn+2 . Applying
R0π(n+2)
1(n+2)∗to it, we obtain the following complex on X×X:
L−nRn−→ · · · −→ L−1R1−→ OMR0−→ O∆.(46)
Recalling condition c) of Lemma 3.5.1 and condition (43), b), we see that its coho-
mology sheaves are isomorphic to
Rjπ(n+2)
1(n+2)∗(JΣ(n+2) ⊗(OL···LO)) ⊗(OL−n),
which vanish for j>0 by (43), c). That is, the complex (46) is exact.
CHAPTER 4
Derived categories of coherent sheaves on K3 surfaces
4.1. K3 surfaces and the Mukai lattice. This chapter is entirely taken up
with derived categories of coherent sheaves on K3 surfaces over the field of complex
numbers. The main question we are interested in, and answer in this chapter, is
as follows: when do two distinct K3 surfaces have equivalent categories of coherent
sheaves? As before, we view derived categories as triangulated categories, and
equivalences are understood as equivalences between triangulated categories.
We recall that for smooth projective varieties with ample canonical or anti-
canonical class there is a procedure (see Theorem 2.1.3) for recovering the variety
568 D. O. Orlov
from its derived category of coherent sheaves. However, for varieties of other types
this is a non-trivial question, and is especially interesting for varieties with trivial
canonical class.
We start with the main facts we need concerning K3 surfaces. Recall that a K3
surface is a smooth compact algebraic surface Swith KS=0andH
1(S, Z)=0.
These surfaces are actually simply connected. One can show that the second coho-
mology H2(S, Z) is torsion-free and is an even lattice of rank 22 with respect to the
intersection form. Moreover, it follows from the Noether formula that pg(S)=1
and h1,1(S) = 20.
One of the main invariants of a K3 surface is its N´eron–Severi group NS(S)⊂
H2(S, Z), which coincides in this case with the Picard group Pic(S). The rank of
NS(S)ish1,1= 20. We write TSfor the lattice of transcendental cycles which,
by definition, is the orthogonal complement to the N´eron–Severi lattice NS(S)in
the second cohomology H2(S, Z).
We denote by tdSthe Todd class of the surface S; this class is an element of
the form 1 + 2win H∗(S, Q), where 1 ∈H0(S, Z) is the identity of the cohomology
ring H∗(S, Z)andw∈H4(S, Z) is the fundamental cocycle of S. We consider the
positive square root √tdS=1+w; for any K3 surface it belongs to the integral
cohomology ring H∗(S, Z).
One introduces the Chern character for any coherent sheaf on S, and extends it
by additivity to the entire derived category of coherent sheaves. If Fis an object
of Db(S), its Chern character
ch(F)=r(F)+c1(F)+ 1
2(c2
1−2c2)
belongs to the integral cohomology H∗(S, Z). For any object Fwe define the element
v(F)=ch(F)·tdS∈H∗(S, Z)
and call it the vector associated with F(or the Mukai vector of F).
We define a symmetric bilinear form on the cohomology lattice H∗(S, Z)bythe
rule
(u, u)=r·s+s·r−α·α∈H4(S, Z)∼
=Z
for any pair u=(r, α, s), u=(r,α
,s
)∈H0(S, Z)⊕H2(S, Z)⊕H4(S, Z). The
cohomology lattice H∗(S, Z) together with this bilinear form ( ·,·) is called the
Mukai lattice and denoted by
H(S, Z). Note that on H2the bilinear form ( ·,·)
differs from the usual intersection form by the minus sign. Thus, the Mukai lattice
H(S, Z) is isomorphic to the lattice U ⊥−H2(S, Z), where U is the hyperbolic
lattice 01
10
and ⊥means orthogonal direct sum.
For any two objects Fand G, the pairing (v(F),v(G)) is by definition the com-
ponent in H4of the element ch(F)∨·ch(G)·tdS. Hence, by the Grothendieck
Riemann–Roch theorem we have the equality
(v(F),v(G)) = χ(F, G):=
i
(−1)idim Exti(F, G).
The lattices
H(S, Z)andTSadmit natural Hodge structures. Here by a Hodge
structure we mean that the spaces
H(S, C)andTS⊗Chave a fixed one-dimensional
subspace H2,0(S).
Derived categories of coherent sheaves 569
Definition 4.1.1. Let S1and S2be two K3 surfaces. We say that the Mukai
lattices of S1and S2(or their lattices of transcendental cycles) are Hodge isometric
if there is an isometry between the lattices that takes the one-dimensional subspace
H2,0(S1)toH
2,0(S2).
Let Ein Db(S1×S2) be an arbitrary object of the derived category of the product.
Consider the algebraic cycle
ZE:= p∗tdS1·ch(E)·π∗tdS2(47)
on the product S1×S2,wherepand πare the projections in the diagram
S1×S2π
−−−−→S2
p
S1
.
InthecaseofK3surfacesthecycleZE,whichisapriorirational, is in fact integral:
Lemma 4.1.2 [31]. Fo r a ny objec t E∈Db(S1×S2), both the Chern character
ch(E)and the cycle ZEare integral,that is,they belong to H∗(S1×S2,Z).
Thus, the cycle ZEdefines a map from the integral cohomology lattice of S1to
that of S2,
fZE:H
∗(S1,Z)−→ H∗(S2,Z)
∪∪
α−→ π∗(ZE·p∗(α))
.(48)
The following proposition is analogous to Theorem 4.9 in [31].
Proposition 4.1.3. Fo r a n o b j ect E,if the functor ΦE:Db(S1)−→ Db(S2)is
fully faithful,then
1) fZEis an isometry between the lattices
H(S1,Z)and
H(S2,Z),
2) the inverse map of fcoincides with the homomorphism
f:H
∗(S2,Z)−→ H∗(S1,Z)
∪∪
β−→ p∗(Z∨
E·π∗(β))
defined by the cycle
Z∨
E=p∗tdS1·ch(E∨)·π∗tdS2,
where E∨:= R·Hom (E,OS1×S2).
Proof. The left and right adjoint functors to ΦEare isomorphic; they are given by
the formula
Φ∗
E=Φ
!
E=Rp∗(E∨⊗Lπ∗(·))[2].
Since ΦEis fully faithful, it follows that the composite Φ∗
E◦ΦEis isomorphic to the
identity functor idDb(S1). The identity functor idDb(S1)is defined by the structure
sheaf O∆of the diagonal ∆ ⊂S1×S1.
570 D. O. Orlov
By the projection formula and the Grothendieck Riemann–Roch theorem, one
finds that the composite f◦fis represented by the cycle p∗
1tdS1·ch(O∆)·p∗
2tdS1,
where p1and p2are the projections of S1×S1to its factors. Using the Grothendieck
Riemann–Roch theorem again, we see that this cycle is equal to ∆. Hence, the
composite f◦fis the identity map, and thus fis an isomorphism from H∗(S1,Z)
to H∗(S2,Z), because both groups are free Abelian groups of the same rank.
Wri te νS:S−→ Spec Cfor the structure morphism of S. Then we can express
the pairing (α, α)on
H(S, Z)asν∗(α∨·α). It follows from the projection formula
that
(α, f(β)) = νS2,∗(α∨·π∗(π∗tdS2·ch(E)·p∗tdS1·p∗(β)))
=νS2,∗π∗(π∗(α∨)·p∗(β)·ch(E)·tdS1×S2)
=νS1×S2,∗(π∗(α∨)·p∗(β)·ch(E)·tdS1×S2)
for arbitrary α∈H∗(S2,Z)andβ∈H∗(S1,Z). In the same way we see that
(β, f(α)) = νS1×S2,∗(p∗(β∨)·π∗(α)·ch(E)∨·tdS1×S2).
Hence, (α, f (β)) = (f(α),β). Since f◦fis the identity map, it follows that
(f(α),f(α)) = (ff(α),α
)=(α, α).
Thus, fis an isometry.
4.2. The criterion for equivalence of derived categories of coherent
sheaves. In this section we give a criterion for the derived categories of coher-
ent sheaves on two K3 surfaces to be equivalent as triangulated categories. The
form of this criterion is very reminiscent of the Torelli theorem for K3 surfaces,
which says that two K3 surfaces S1and S2are isomorphic if and only if their
lattices of second cohomology are Hodge isometric, that is, there is an isometry
H2(S1,Z)∼
−→ H2(S2,Z)
whose extension to complex cohomology takes H2,0(S1)toH
2,0(S2) (see [39], [27]).
The main result of this chapter is as follows.
Theorem 4.2.1. Let S1and S2be two smooth projective K3surfaces over the field
of complex numbers C.Then the derived categories of coherent sheaves Db(S1)and
Db(S2)are equivalent as triangulated categories if and only if there exists a Hodge
isometry f:
H(S1,Z)∼
−→
H(S2,Z)betwee n the Mukai latt ices of S1and S2.
There is another version of this theorem (Theorem 4.2.4) which may also be of
interest.
We break up the proof of Theorem 4.2.1 into two propositions. The proof of the
first proposition depends essentially on the main Theorem 3.2.1 of the preceding
chapter, since it uses the fact that every equivalence of derived categories can be
represented by an object on the product.
Derived categories of coherent sheaves 571
Proposition 4.2.2. Let S1and S2be t wo K 3surfaces whose derived categories of
coherent sheaves are equivalent.Then there is a Hodge isometry between the lattices
of transcendental cycles TS1and TS2.
Proof. By Theorem 3.2.2, there is an object Eon the product S1×S2that defines
the equivalence. It follows from Proposition 4.1.3 that fZEdefines a Hodge isometry
between the Mukai lattices
H(S1,Z)and
H(S2,Z). Since the cycle Zis algebraic,
we obtain two isometries
falg :−NS(S1)⊥U∼
−→ − NS(S2)⊥Uandfτ:TS1∼
−→ TS2,
where NS(S1)andNS(S2)aretheN´eron–Severi lattices and TS1and TS2the lattices
of transcendental cycles. It is obvious that fτis a Hodge isometry.
The proof of the converse uses in an essential way the results of [31], which
studied moduli spaces of bundles on K3 surfaces, and it also uses Theorem 2.1.5,
which gave a criterion for a functor to be fully faithful (see [7]).
Proposition 4.2.3. Let S1and S2be two projective K3surfaces.Suppose that
there exists a Hodge isometry
f:
H(S2,Z)∼
−→
H(S1,Z).
Then the bounded derived categories of coherent sheaves Db(S1)and Db(S2)are
equiv alent.
Proof.Wesetv=f(0,0,1) = (r, l, s)andu=f(1,0,0) = (p, k, q). Without loss
of generality we can assume that r>1. Indeed, a Mukai lattice has two types of
Hodge isometries. The first type is multiplication by the Chern character exp(m)
of a line bundle:
ϕm(r, l, s)=r, l +rm, s +(m, l)+ r
2m2.
Thesecondtypeisthetranspositionofrand s. Using these two types of permu-
tations, one can replace fin such a way that rbecomes greater than 1.
The vector v∈U⊥−NS(S1) is obviously isotropic, that is, (v, v)=0. In
his brilliant paper [31] Mukai proved that, in this case, there is a polarization A
on the K3 surface S1such that the moduli space MA(v) of vector bundles whose
Mukai vector coincides with vand that are stable with respect to Ais a smooth
projective K3 surface. Moreover, since there is a vector u∈U⊥−NS(S1)such
that (v, u) = 1, it follows that MA(v) is a fine moduli space. Hence, there exists a
universal bundle Eon the product S1×MA(v).
The universal bundle Edefines a functor ΦE:Db(MA(v)) −→ Db(S1). One sees
readily that this functor satisfies the conditions of Theorem 2.1.5. Indeed, we have
ΦE(Ot)=Et,whereEtis a stable bundle on S1for which v(Et)=v.Allthe
sheaves Etare simple, and we of course have Exti(Et,Et)=0fori/∈{0,2}.This
gives condition 2) of Theorem 2.1.5.
Since the Etare stable, it follows that Hom(Et1,Et2) = 0. By Serre duality, also
Ext2(Et1,Et2) = 0. Since the vector vis isotropic, we also have Ext1(Et1,Et2)=0.
572 D. O. Orlov
Thus, the sheaves Et1and Et2are orthogonal for any two distinct points t1and t2.
Theorem 2.1.5 gives us that the functor ΦEis fully faithful.
In fact, ΦEis not just fully faithful, but an equivalence of categories. This can
be shown by the following argument, which is based on the proof of Theorem 3.3
of [9]. Write Dfor the image of Db(MA(v)) in Db(S1). Since it is an admissible
subcategory (see Definition 2.2.2), it admits right and left orthogonals; since the
canonical class of a K3 surface is trivial, it follows that these orthogonals coin-
cide. Thus, the semi-orthogonal decomposition of the form D⊥,Dis completely
orthogonal. Consider a very ample line bundle Lon MA(v). All the powers Liare
indecomposable objects, and therefore belong to one or other of the subcategories
Dor D⊥, and they all belong to the same one, because no pair of these objects
is completely orthogonal. However, the powers {Li}form an ample sequence (see
Definition 3.4.1). By Lemma 3.4.3, the orthogonal to a subcategory generated by
an ample sequence is 0. Thus, since Dis non-trivial, it follows that D⊥= 0. Hence,
ΦEis an equivalence.
Next, the cycle ZEdefined by (47) induces a Hodge isometry
g:
H(MA(v),Z)→
H(S1,Z)
for which g(0,0,1) = v=(r, l, s). Hence, f−1◦gis also a Hodge isometry, and
takes (0,0,1) to (0,0,1). Thus, f−1·ginduces a Hodge isometry between the
second cohomology lattices of S2and MA(v). Therefore, by the Torelli theorem
([39], [27]), S2and MA(v) are isomorphic.
This proposition together with Proposition 4.1.3 prove Theorem 4.2.1. There is
another version of Theorem 4.2.1, which gives a criterion for equivalence of derived
categories in terms of the lattices of transcendental cycles.
Theorem 4.2.4. Let S1and S2be two smooth projective K3surfaces over C.
Then the derived categories of coherent sheaves Db(S1)and Db(S2)are equivalent as
triangulated categories if and only if there exists a Hodge isometry fτ:T
S1∼
−→ TS2
between t he lat tice s of transcendental cycles of S1and S2.
This assertion is a corollary of Theorem 4.2.1 and the following proposition.
Proposition 4.2.5 [33]. Let ϕ1,ϕ
2:T−→ Hbe two primitive embeddings of the
lattice Tinto an even unimodular lattice H. Suppose that the orthogonal complement
N:=ϕ1(T)⊥in Hcontains t he hyper bolic l att ice U=01
10
as a sublattice.Then
ϕ1and ϕ2are equivalent,that is,there is an isometry γof Hsuch that ϕ1=γϕ2.
Indeed, suppose that there is a Hodge isometry
fτ:T
S2∼
−→ TS1.
As we know, the orthogonal complement to the lattice of transcendental cycles TS
in the Mukai lattice
H(S, Z) is isomorphic to the lattice −NS(S)⊥U. Thus, by
the previous proposition (Proposition 4.2.5), there is an isometry
f:
H(S2,Z)∼
−→
H(S1,Z)
such that f|TS2=fτ. Thus, the isometry fis also a Hodge isometry. Therefore,
by Theorem 4.2.1, the derived categories of coherent sheaves on S1and S2are
equivalent.
Derived categories of coherent sheaves 573
CHAPTER 5
Abelian varieties
5.1. Equivalences between categories of coherent sheaves on Abelian
var ie t i e s . In this chapter we study derived categories of coherent sheaves on
Abelian varieties and their groups of auto-equivalences. Let Abe an Abelian vari-
ety of dimension nover a field k.Wewritem:A×A→Afor the composition
morphism, which is assumed to be defined over k,andefor the k-point which is
the identity of the group structure. For any k-point a∈Athere is a translation
automorphism m(·,a): A→A, which we denote by Ta.
We wr i te
Afor the dual Abelian variety, which is the moduli space of line bundles
on Abelonging to Pic0(A). Moreover,
Ais a fine moduli space. Therefore, there
exists a universal line bundle Pon the product A×
A, called the Poincar´ebundle.It
is uniquely determined by the condition that for any k-point α∈
Athe restriction
of Pto A×{α}is isomorphic to the line bundle in Pic0(A) corresponding to α,
and, in addition, the restriction P|{e}×
Ashould be trivial.
Definition 5.1. In what follows we denote the line bundle on Acorresponding to
ak-point α∈
Aby Pα. Moreover, given a number of Abelian varieties A1,...,A
m
and a k-point (α1,...,α
m)∈
A1×···×
Am,wedenotebyP(α1,...,αk)the line bundle
Pα1···Pαkon the product A1×···×Ak.
For any homomorphism f:A→Bof Abelian varieties one defines a dual homo-
morphism
f:
B→
A. Pointwise, it acts by taking a point β∈
Bto α∈
Aif and
only if the line bundle f∗Pβon Acoincides with the bundle Pα.
The double dual or bidual Abelian variety
Ais naturally identified with Ausing
the Poincar´e bundles on A×
Aand on
A×
A. In other words, there is a unique
isomorphism κA:A∼
−→
Asuch that the pull-back of the Poincar´e bundle P
Aunder
the isomorphism 1 ×κA:
A×A∼
−→
A×
Acoincides with the Poincar´e bundle PA,
that is, (1 ×κA)∗P
A∼
=PA.Thus,is an involution on the category of Abelian
varieties; that is, it is a contravariant functor whose square is isomorphic to the
identity functor: κ:id ∼
−→
.
The Poincar´e bundle Pgives an example of an exact equivalence between the
derived categories of coherent sheaves on two varieties Aand
Athat are not in
general isomorphic. Consider the projections
Ap
←− A×
Aq
−→
A
and the functor ΦP:Db(A)−→ Db(
A) defined by (7):
ΦP(·)=Rq∗(P⊗p∗(·)).
The following proposition was proved in [29].
574 D. O. Orlov
Proposition 5.1.2 [29]. Let Pbe the Poincar´ebundleonA×
A.Then the functor
ΦP:Db(A)−→ Db(
A)is an exact equivalence,and there exists an isomorphism of
functors
ΨP◦ΦP∼
=(−1A)∗[n],
where (−1A)is the group inverse of A.
Remark 5.1.3. In [29] this assertion was proved for Abelian varieties over an alge-
braically closed field. However, it also holds over an arbitrary field because the
dual variety and the Poincar´e bundle are always defined over the same field (see,
for example, [32]). And the assertion concerning the equivalence of categories will
follow from Lemma 5.1.9.
Consider a k-point (a, α)∈A×
A. To any such point one can assign a functor
from Db(A)toitselfbytherule
Φ(a,α)(·):=Ta∗(·)⊗Pα=T∗
−a(·)⊗Pα.(49)
The functor Φ(a,α)is represented by the sheaf
S(a,α)=OΓa⊗p∗
2(Pα) (50)
on the product A×A,whereΓ
astands for the graph of the translation auto-
morphism Ta. The functor Φ(a,α)is obviously an auto-equivalence.
The set of all functors Φ(a,α)parametrized by A×
Acan be collected into a single
functor from Db(A×
A)toDb(A×A) that takes the structure sheaf O(a,α)of a
point to S(a,α). (We note that this condition does not define the functor uniquely,
but only uniquely up to multiplication by a line bundle lifted from A×
A.)
We define the required functor ΦSA:Db(A×
A)−→ Db(A×A)asthecomposite
of two other functors.
Consider the object PA=p∗
14O∆⊗p∗
23P∈Db((A×
A)×(A×A)), and write
µA:A×A−→ A×Afor the morphism taking (a1,a
2)to(a1,m(a1,a
2)). We obtain
two functors,
ΦPA:Db(A×
A)−→ Db(A×A),RµA∗:Db(A×A)−→ Db(A×A).
Definition 5.1.4. The functor ΦSAis the composite RµA∗◦ΦPA.
Proposition 5.1.5. The functor ΦSAis an equivalence of categories.For any
k-point (a, α)∈A×
Ait takes
a) the structure sheaf O(a,α)of (a, α)to the sheaf S(a,α)defined by (50),
b) thelinebundleP(α,a)on A×
Ato the object O{−a}×A⊗p∗
2Pα[n].
Proof. By definition, ΦSAis the composite of the functors RµA∗and ΦPA,which
are equivalences; this is obvious for the first functor and for the second it follows
from Propositions 2.1.7 and 5.1.2.
The functor ΦPAtakes the structure sheaf O(a,α)of a point to OA×{a}⊗p∗
1Pα.
Moreover, RµA∗takes OA×{a}⊗p∗
1Pαto OΓa⊗p∗
1(Pα).
Derived categories of coherent sheaves 575
In the same way, applying Proposition 5.1.2, we see that ΦPAtakes the line
bundle P(α,a)to the object O{−a}×A⊗p∗
2Pα[n], and RµA∗takes O{−a}×A⊗p∗
2Pα[n]
to itself.
Suppose that Aand Bare two Abelian varieties whose derived categories of
coherent sheaves are equivalent. Let us fix some equivalence. By Theorem 3.2.2,
it can be represented by an object on the product. Thus, there is an object E∈
Db(A×B) and an equivalence ΦE:Db(A)∼
−→ Db(B).
Consider the functor
Ad E:Db(A×A)∼
−→ Db(B×B),
defined by (11), which is an equivalence. And consider the composite of functors
Φ−1
SB◦Ad E◦ΦSA.
Definition 5.1.6. We denote by J(E) the object representing the functor
Φ−1
SB◦Ad E◦ΦSA.
Thus, we have the commutative diagram
Db(A×
A)ΦSA
−−−−→Db(A×A)
ΦJ(E)
Ad E
Db(B×
B)ΦSB
−−−−→Db(B×B)
.(51)
The following theorem allows us to compute the object J(E); it is the main tool
for describing Abelian varieties having equivalent derived categories of coherent
sheaves.
Theorem 5.1.7. There exists a homomorphism of Abelian varieties fE:A×
A−→
B×
Bwhich is an isomorphism,andalinebundleLEon A×
Asuch that the object
J(E)is isomorphic to i∗(LE), where iis the embedding of A×
Ain (A×
A)×(B×
B)
as the graph of the isomorphism fE.
Before proceeding to the proof of the theorem, we state two lemmas that allow
us to assume that the field kis algebraically closed. We write kfor the algebraic
closure of k,setX:= X×Spec(k)Spec(k), and write Ffor the inverse image of F
under the morphism X−→ X.
Lemma 5.1.8 [37]. Let Fbe a cohe rent sheaf on a smooth va riety X.Suppose that
there exist a closed subvariety j:Z→Xand an invertible sheaf Lon Zsuch
that F∼
=j∗L.Then there exist a closed subvariety i:Y→Xand an invertible
sheaf Mon Ysuch that F∼
=i∗Mand j=⊂.
The next lemma tells us that the property that a functor is fully faithful (or an
equivalence) is stable under field extensions.
576 D. O. Orlov
Lemma 5.1.9 [37]. Let Xand Ybe smooth projective varieties over a field kand
Ean object of the derived category Db(X×Y). Consider a field extension F/k
and the varieties
X=X×Spec(k)Spec(F),Y
=Y×Spec(k)Spec(F).
Let Ebe the lift of Eto the category Db(X×Y). Then the functor ΦE:Db(X)−→
Db(Y)is fully faithful (or an equivalence)if and only if ΦE:Db(X)−→ Db(Y)
is fully faithful (respectively,an equivalence).
ProofofTheorem5.1.7. Using Lemmas 5.1.8 and 5.1.9, we can pass to the algebraic
closure of the field k.
Step 1. Write e∈A×
Aand e∈B×
Bfor the closed points which are the identity
elements of the group structures. We consider the skyscraper sheaf Oeand evaluate
its image under the functor ΦJ(E). By definition,
ΦJ(E)=Φ
−1
SB◦Ad E◦ΦSA.
By Proposition 5.1.5, the functor ΦSAtakes Oeto the structure sheaf O∆(A)of the
diagonal in A×A. Since the structure sheaf of the diagonal represents the identity
functor, it follows from (12) that Ad E(O∆(A)) is the structure sheaf O∆(B)of the
diagonal in B×B. In turn, this sheaf goes to the structure sheaf Oeunder
the action of the functor Φ−1
SB, by Proposition 5.1.5 again.
Step 2. Thus, we see that
J(E)⊗LO{e}×(B×
B)∼
=O{e}×{e}.
It follows from this that there is an affine neighbourhood U=Spec(R)ofein the
Zariski topology such that the object J:= J(E)|U×(B×
B)is a coherent sheaf whose
support intersects the fibre {e}×(B×
B)atthepoint{e}×{e}. We recall that
the support of any coherent sheaf is a closed subset.
Consider now some affine neighbourhood V=Spec(S)ofthepointein B×
B.
The intersection of the support of Jwith the complement B×
B\Vis a closed
subset whose projection to A×
Ais a closed subset not containing the point e.
Thus, reducing Uif necessary, we can assume that it is still affine and the support
of Jis contained in U×V. This means that there is a coherent sheaf Fon U×V
such that j∗(F)=J,wherejis the embedding of U×Vinto U×(B×
B). We
denote by Mthe finitely generated R⊗S-module corresponding to the sheaf F,that
is, F=!
M. Moreover, we note that Mis a finitely generated R-module, because
the direct image under projection of a coherent sheaf J=j∗Fis a coherent sheaf.
Let mbe the maximal ideal of Rcorresponding to the point e.Asweknow,
M⊗RR/m ∼
=R/m.
Hence, there exists a homomorphism of R-modules ϕ:R−→ Mwhich becomes
an isomorphism after tensoring with R/m. Thus, the supports of the coherent
sheaves Ker ϕand Coker ϕdo not contain the point e. Therefore, replacing U
Derived categories of coherent sheaves 577
by a smaller affine neighbourhood of edisjoint from the supports of the sheaves
Ker ϕand Coker ϕ,weseethatϕis an isomorphism. Hence, there is a subscheme
X(U)⊂U×(B×
B) such that the projection X(U)−→ Uis an isomorphism and
J=J(E)|U×(B×
B)∼
=OX(U).
Step 3. We have thus proved that for any closed point (a, α)∈U,
ΦJ(E)(O(a,α))∼
=O(b,β)
for some closed point (b, β)∈B×
B. If we now consider an arbitrary closed point
(a, α)∈A×
A, we can always express it as a sum (a, a)=(a1,α
1)+(a2,α
2),
where the points (a1,α
1)and(a2,α
2)belongtoU.Write(b1,β
1)and(b2,β
2)for
the images of these points under the functor ΦJ(E). As we know, the functor ΦSA
takes the structure sheaf O(a,α)to the sheaf S(a,α).WedenotebyGthe object
Ad E(S(a,α)). We compute it using (12). We have an isomorphism
ΦG∼
=ΦE◦Φ(a,α)◦Φ−1
E.
However, the functor Φ(a,α), equal by definition (49) to T∗
a(·)⊗Pα, can be expressed
as the composite Φ(a1,α1)Φ(a2,α2). We thus obtain a chain of isomorphisms
ΦG∼
=ΦE◦Φ(a,α)◦Φ−1
E∼
=ΦE◦Φ(a1,α1)◦Φ−1
E
∼
=ΦE◦Φ(a2,α2)◦Φ−1
E∼
=Φ(b1,β1)◦Φ(b2,β2)∼
=Φ(b,β),
where (b, β)=(b1,β
1)+(b2,β
2). Therefore, the object Gis isomorphic to S(b,β).
We finally obtain
ΦJ(E)(O(a,α))∼
=O(b,β)for any closed point (a, α)∈A×
A.
Now repeating the procedure of Step 2, for any closed point (a, a) we can find a
neighbourhood Wand a subscheme X(W)⊂W×(B×
B) such that the projection
X(W)−→ Wis an isomorphism, and J|W×(B×
B)∼
=OX(W). Gluing all these
neighbourhoods together, we find a subvariety i:X→(A×
A)×(B×
B) such that
the projection X−→ A×
Ais an isomorphism, and the sheaf J(E) is isomorphic
to i∗L,whereLis a line bundle on X.ThesubvarietyXdefines a homomorphism
from A×
Ato B×
Bwhich induces an equivalence of derived categories. Hence,
this homomorphism is an isomorphism.
In particular, it follows at once from the theorem that, if two Abelian varieties A
and Bhave equivalent derived categories of coherent sheaves, then the varieties
A×
Aand B×
Bare isomorphic. We show below that this isomorphism must
satisfy a certain additional condition (see Proposition 5.1.15).
578 D. O. Orlov
Corollary 5.1.10. The isomorphism fEtakes a k-po int (a, α)∈A×
Ato a point
(b, β)∈B×
Bif and only if the equivalences
Φ(a,α):Db(A)∼
−→ Db(A),Φ(b,β):Db(B)∼
−→ Db(B),
defined by the formula (49) are related as follows:
Φ(b,β)◦ΦE∼
=ΦE◦Φ(a,α),
or,in terms of the objects,
Tb∗E⊗Pβ∼
=T−a∗E⊗Pα=T∗
aE⊗Pα.
Proof. By Theorem 5.1.7, ΦJ(E)takes the structure sheaf O(a,α)of (a, α)tothe
structure sheaf O(b,β)of (b, β)=fE(a, α). It follows from Proposition 5.1.5 that
ΦSAtakes O(a,α)to S(a,α). In turn, the sheaf S(a,α)represents the functor
Φ(a,α)=Ta∗(·)⊗Pα.
Now using diagram (51), we see that fEtakes (a, α)to(b, β) if and only if S(b,β)∼
=
Ad E(S(a,α)). Applying formula (12), we see that Φ(b,β)∼
=ΦE◦Φ(a,α)◦Φ−1
E.
In what follows we need an explicit formula for the object J(E) in the special
case when A=Band the equivalence ΦEis equal to Φ(a,α)defined by the formula
(49).
Proposition 5.1.11. Le t A=B.Consider the object S(a,α)on A×Arepresenting
the equivalence Φ(a,α)given by (49). Then J(S(a,α))is equal to ∆∗P(α,−a),where
∆is the diagonal embedding of A×
Ainto (A×
A)×(A×
A)and P(α,a)is the line
bundle on A×
Adefined in 5.1.1.
Proof. It follows from Proposition 5.1.5 that ΦSAtakes O(a,α)to the sheaf S(a,α)
on A×A(50). Moreover, Ad S(a,α)takes S(a,α)to itself because, by formula (12),
the object Ad S(a,α)(S(a,α)) represents the functor
Φ(a,α)◦Φ(a,α)◦Φ−1
(a,α),
which is in turn isomorphic to Φ(a,α)because all such functors commute with one
another. Thus, we see that the functor defined by J(S(a,α)) takes the structure
sheaf of every point to itself, and thus the sheaf J(S(a,α)) is some line bundle L
concentrated on the diagonal.
Now to find the line bundle L, we ask where the functor sends the bundle P(α,a).
Applying Proposition 5.1.5 again, we see that the functor ΦSAtakes P(α,a)to the
object O{−a}×A⊗p∗
2(Pα)[n]. Next, one sees readily that this goes to the object
O{−a+a}×A⊗p∗
2(Pα+α)[n] under the functor Ad S(a,α). Hence, under the action
of the functor given by the sheaf J(S(a,α)), the bundle P(α,a)goes to the bundle
P(α+α,a−a).Thatis,Lis isomorphic to P(α,−a).
Derived categories of coherent sheaves 579
For Abelian varieties Aand B,writeEq(A, B) for the set of all exact equivalences
from Db(A)toDb(B) up to isomorphism. We introduce two groupoids Aand D
(that is, categories in which all morphisms are invertible). In both, the objects are
Abelian varieties. The morphisms in Aare isomorphisms between Abelian varieties
regarded as algebraic groups. The morphisms in Dare exact equivalences between
the derived categories of coherent sheaves on Abelian varieties; that is,
MorA(A, B):=Iso(A, B)and Mor
D(A, B):=Eq(A, B).
Theorem 5.1.7 provides a map from the set Eq(A, B) to the set Iso(A×
A, B ×
B),
taking an equivalence ΦEto the isomorphism fE. We consider the map Ffrom Dto
Athat assigns to an Abelian variety Athe variety A×
Aand acts on the morphisms
as described above.
Proposition 5.1.12. The map F:D−→ Ais a functor.
Proof. To prove the assertion, we need only show that Frespects composition of
morphisms. Consider three Abelian varieties A,B,andC.LetEand Fbe objects
of Db(A×B)andDb(B×C) respectively, such that the functors
ΦE:Db(A)−→ Db(B)andΦ
F:Db(B)−→ Db(C)
are equivalences. We denote by Gthe ob ject of Db(A×C) that represents the
composite of these functors.
The relation (10) gives an isomorphism Ad G∼
=Ad F◦Ad E. Hence, we see that
ΦJ(F)◦ΦJ(E)∼
=(Φ−1
SA◦Ad F◦ΦSA)◦(Φ−1
SA◦Ad E◦ΦSA)∼
=Φ−1
SA◦Ad G◦ΦSA∼
=ΦJ(G).
By Theorem 5.1.7, all the objects J(E), J(F), and J(G) are line bundles concentrated
on the graphs of the isomorphisms fE,fF,andfG, respectively. Thus, we obtain
the relation fG=fF·fE.
Corollary 5.1.13. Let Abe an Abelian variety and ΦEan auto-equi val ence of
the der ived ca tegory Db(A). Then the correspondence ΦE→ fEdefines a group
homomorphism
γA:AuteqDb(A)−→ Aut(A×
A).
Thus, there is a functor F:D−→ A. Our next ob jective is to describe this
functor. For this, we must determine which elements of Iso(A×
A, B ×
B)canbe
realized as fEfor some E, and also answer the question of when fE1=fE2holds
for two equivalences E1and E2.
Consider an arbitrary morphism f:A×
A−→ B×
B. It is convenient to represent
it as a matrix αβ
γδ
,
where αis a morphism from Ato B,βfrom
Ato B,γfrom Ato
B,andδfrom
Ato
B. Each morphism fdefines two other morphisms
fand
ffrom B×
Bto
A×
Ahaving the following matrix forms:
f=
δ
β
γαand
f=
δ−
β
−γα.
580 D. O. Orlov
We define the set U(A×
A, B ×
B) to be the subset of Iso(A×
A, B ×
B) consisting
of fsuch that
fcoincides with the inverse of f,thatis,
U(A×
A, B ×
B):=f∈Iso(A×
A, B ×
B)
f=f−1.
If B=A, then we denote this set by U(A×
A). We note that U(A×
A)isa
subgroup of Aut(A×
A).
Definition 5.1.14. We say that an isomorphism f:A×
A∼
−→ B×
Bis isometric
if it belongs to U(A×
A, B ×
B).
Proposition 5.1.15. For any equivalence ΦE:Db(A)∼
−→ Db(B)the isomorphism
fEis isometric.
Proof. Passing to the algebraic closure if necessary, we can assume that kis alge-
braically closed. To verify the equality
fE=f−1
E, it is enough to establish that
these morphisms coincide at closed points. Suppose that fEtakes (a, α)∈A×
A
to (b, β)∈B×
B. We must show that
fE(b, β)=(a, α), or, equivalently, that
fE(−b, β)=(−a, α).
The isomorphism fEis given by the Abelian subvariety X→A×
A×B×
B.
Hence, we must show that P(0,0,β,−b)⊗OX∼
=P(α,−a,0,0) ⊗OX, or, equivalently,
that
J:= P(−α,a,β,−b)⊗J(E)
is isomorphic to the sheaf J(E).
By Proposition 5.1.11, the functor given by Jis the composite of the functors
represented by the objects J(S(−a,−α)), J(E), and J(S(b,β)). Thus, Jcoincides with
J(E), where Eis the object of Db(A×B) representing the functor
Φ(b,β)◦ΦE◦Φ(−a,−α).
By Corollary 5.1.10, this composite is isomorphic to the functor ΦE. This means
that the object Eis isomorphic to E, and hence J=J(E)∼
=J(E).
As a corollary of Theorem 5.1.7 and Proposition 5.1.15 we get the following
result.
Theorem 5.1.16. Let Aand Bbe two Abelian varieties over a field k.If the
derived categories of coherent sheaves Db(A)and Db(B)are equivalent as triangu-
lated categories,then there is an isometric isomorphism between A×
Aand B×
B.
The converse holds for Abelian varieties over an algebraically closed field of
characteristic 0, as proved in [38]. We give another proof of this fact in §5.3.
Corollary 5.1.17. For any Abelian variety Athere are only finitely many non-
isomorphic Abelian varieties whose derived categories of coherent sheaves are equiv-
alent to Db(A)(as triangulated categories).
Proof. It was proved in [26] that, for any Abelian variety Z, there are only finitely
many Abelian varieties up to isomorphism admitting an embedding in Zas Abelian
subvarieties. Applying this assertion to Z=A×
Aand using Theorem 5.1.16, we
obtain the desired result.
Derived categories of coherent sheaves 581
5.2. Objects representing equivalences, and groups of auto-equivalences.
It follows from Propositions 5.1.12 and 5.1.15 that there exists a homomorphism
from the group Auteq Db(A) of exact auto-equivalences to the group U(A×
A)
of isometric automorphisms. In this section we describe the kernel of this homo-
morphism. As we know from Proposition 5.1.11, all the equivalences Φ(a,α)[n]
belong to the kernel. We show that the kernel consists exactly of these. To prove
this, we need an assertion which is of independent interest: we prove that for an
Abelian variety, if a functor of the form ΦEis an equivalence, then the object Eon
the product is actually a sheaf, up to a shift in the derived category. We note that
this is false, for example, for K3 surfaces.
Lemma 5.2.1. Let Ebe an object on A×Bdefining an equivalence ΦE:Db(A)−→
Db(B). Consider the projection q:(A×
A)×(B×
B)−→ A×Band write Kfor
the direct image Rq∗J(E), where J(E)is the object defined in 5.1.6. Then Kis
isomorphic to the object E⊗(E∨|(0,0)), where E∨|(0,0) stands for the complex of
vector spaces which is the inverse image of the object R·Hom (E,OA×B)under the
embedding of the point (0,0) into the Abelian variety A×B.
Proof. Consider the Abelian variety
Z=(A×
A)×(A×A)×(B×B)×(B×
B)
and the object
H=p∗
1234SA⊗p∗
35E∨[n]⊗p∗
46E⊗p∗
5678S∨
B[2n].
It follows from Proposition 2.1.2 on composition of functors and from diagram (51)
that J(E)∼
=p1278∗H, and hence the object Kequals p17∗H. To evaluate the latter
object, we first consider the projection of Zto
V=A×(A×A)×(B×B)×B,
and denote it by v.Now,toevaluatev∗H, we recall that the functor ΦSAis the
composite of ΦPAand RµA∗,where
PA=p∗
14O∆⊗p∗
23P∈Db((A×
A)×(A×A)).
One sees readily that p134∗PA∼
=OTA[−n], where T⊂A×A×Ais the subvariety
isomorphic to Aand consisting of the points (a, 0,a). Next,takingintoaccountthe
equality µA(a1,a
2)=(a1,m(a1,a
2)), we can see that p134∗SAis also isomorphic to
OTA[−n]. We verify the equality p134∗S∨
B[2n]=OTBinthesameway.
Thus, we have
v∗H∼
=p∗
123OTA⊗p∗
24E∨⊗p∗
35E⊗p∗
456OTB
on V. Consider the embedding
j:A×A×B×B−→ Vgiven by (a1,a
2,b
1,b
2)→ (a1,0,a
2,0,b
1,b
2).
The object v∗His isomorphic to j∗M,where
M=(E∨|(0,0))⊗p∗
12O∆A⊗p∗
23E⊗p∗
34O∆B.
Finally, we see that K∼
=p14∗M∼
=(E∨|(0,0))⊗E.
582 D. O. Orlov
Proposition 5.2.2. Let Aand Bbe Abelian varieties and Ean object of Db(A×B)
such that the functor ΦE:Db(A)∼
−→ Db(B)is an exact equivalence.Then Ehas
only one non-trivial cohomology sheaf,that is,it is isomorphic to an object F[n],
where Fis a sheaf on A×B.
Proof. Consider the projection
q:(A×
A)×(B×
B)−→ A×B
and write qfor its restriction to the Abelian subvariety Xwhich is the support of
the sheaf J(E) and the graph of the isomorphism fE. By Theorem 5.1.7, J(E)equals
i∗(L), where Lis a line bundle on X.WewriteKfor the object R·q∗J(E)=R·q
∗L.
The morphism qis a homomorphism of Abelian varieties; set d=dimKer(q).
Then dim Im(q)=2n−d, so that the cohomology sheaves Hj(K) are trivial for
j/∈[0,d].
On the other hand, by Lemma 5.2.1, Kis isomorphic to E⊗(E∨|(0,0)).
After shifting Ein the derived category if necessary, we can assume that the
rightmost non-zero cohomology sheaf of Eis H0(E). Let H−i(E)fori0be
the leftmost non-zero cohomology sheaf of E,andHk(E∨) the highest non-zero
cohomology sheaf of E∨. Replacing Eby T∗
(a,b)Eif necessary, we can assume that
the point (0,0) belongs to the support of Hk(E∨). Since the support of Ecoincides
with the support of K, it follows that the supports of all cohomology sheaves E
belong to Im(q). In particular, we have the inequality codim Supp H−i(E)d.
Hence, the cohomology sheaf of the object (H−i(E))∨[−i] of degree less than i+d
is trivial.
The canonical morphism H−i(E)[i]−→ Einduces a non-trivial morphism
E∨−→ (H−i(E))∨[−i].
Since the indices of the non-trivial cohomology sheaves of the second object belong
to the ray [i+d, ∞), we see that ki+d,where,asabove,Hk(E∨) is the highest
non-zero cohomology sheaf of E∨. Thus, the object
K=E∨|(0,0) ⊗E(52)
has non-trivial cohomology sheaf with the same index ki+d. On the other hand,
we already know that all the cohomology sheaves Hj(K) are trivial for j/∈[0,d].
This is only possible if i= 0. Thus, the ob ject Ehas only one non-trivial cohomology
sheaf, with index 0, and hence it is isomorphic to a sheaf.
We now consider the case B∼
=A.LetEbe a sheaf on A×Asuch that ΦEis an
auto-equivalence. We want to describe all the sheaves Efor which fEis the identity
map, that is, its graph Xis the diagonal in (A×
A)×(A×
A). Thus, the object
K=E∨|(0,0) ⊗E=R·q∗J(E)
is of the form ∆∗(M), where Mis an object on Aand ∆: A−→ A×Ais the
diagonal embedding.
Derived categories of coherent sheaves 583
We assume first that (0,0) belongs to the support of E. Hence, E∨|(0,0) is a
non-trivial complex of vector spaces. Then the condition K=∆
∗(M) implies the
existence of a sheaf Eon Asuch that E∼
=∆∗(E). Hence, ΦE(·)∼
=E⊗(·). Since
ΦEis an auto-equivalence, Eis a line bundle. One sees readily that the condition
fE= id can only hold if E∈Pic0(A).
If (0,0) does not belong to Supp E, we replace Eby the sheaf E:= T(a1,a2)∗E
in such a way that its support contains (0,0). It follows from Proposition 5.1.11
that fE=fE. As shown above, there is an isomorphism E∼
=∆∗(E), where
E∈Pic0(A). Hence, E∼
=T(a1−a2,0)∗∆∗(E). We thus obtain the corollary.
Proposition 5.2.4. Let Abe an Abelian variety.The kernel of the homomorphism
γA:AuteqDb(A)−→ U(A×
A)
consists of the auto-equivalences of the form Φ(a,α)[i]=Ta∗(·)⊗Pα[i], and hence
is isomorphic to the group Z⊕(A×
A)k,where (A×
A)kis the group of k-poi nts
of the Abelian variety A×
A.
Corollary 5.2.4. Let Aand Bbe t wo Abe lian vari eti es and E1and E2objects
on the product A×Bthat define equivalences between their derived categories of
coherent sheaves.In this case if fE1=fE2,then
E2∼
=Ta∗E1⊗Pα[i]
for some k-point (a, α)∈A×
A.
5.3. Semi-homogeneous vector bundles. In the previous sections we showed
that an equivalence ΦEfrom Db(A)toDb(B) induces an isometric isomorphism of
varieti es A×
Aand B×
B. In this section we assume that the field kis algebraically
closed and char(k) = 0. Under these assumptions, using the technique of [30] and
the results of [7], we will show that every isometric isomorphism f:A×
A−→ B×
B
can be realized in this way. The fact that the existence of an isometric isomorphism
between the varieties A×
Aand B×
Bimplies the equivalence of the derived
categories Db(A)andDb(B) was proved in [38]. Thus, we will give another proof
of this fact.
We first recall that every line bundle Lon an Abelian variety Dgives a map
ϕLfrom Dto
Dthat sends a point dto the point corresponding to the bundle
T∗
dL⊗L−1∈Pic0(D). This correspondence defines an embedding of NS(D)into
Hom(D,
D). Moreover, it is known that the map ϕ:D−→
Dbelongs to the image
of NS(D) if and only if ϕ=ϕ.
Semi-homogeneous bundles on an Abelian variety allow us to generalize the
above phenomenon as follows. To every element of NS(D)⊗Qone assigns a cor-
respondence on D×
D, and every such correspondence is obtained from a semi-
homogeneous bundle (see Proposition 5.3.6 and Lemma 5.1.10 below).
We first recall the definitions of homogeneous and semi-homogeneous bundles on
Abelian varieties and some facts concerning them.
Definition 5.3.1. A vector bundle Eon an Abelian variety Dis homogeneous if
T∗
d(E)∼
=Efor every point d∈D.
584 D. O. Orlov
Definition 5.3.2. We say that a vector bundle Fon an Abelian variety Dis
unipotent if there is a filtration
0=F0⊂F1⊂···⊂Fn=F
such that Fi/Fi−1∼
=ODfor all i=1,...,n.
The following proposition characterizes homogeneous vector bundles.
Proposition 5.3.3 ([28], [30]). Let Ebe a vector bundle on an Abe lian variety D.
Then the following conditions are equivalent :
(i) Eis homogeneous,
(ii) there exist line bundles Pi∈Pic0(D)and unipotent bundles Fisuch that
E∼
=i(Fi⊗Pi).
Definition 5.3.4. A vector bundle Eon an Abelian variety Dis said to be semi-
homogeneous if for every point d∈Dthere exists a line bundle Lon Dsuch that
T∗
d(E)∼
=E⊗L. (We note that, in this case, the bundle Lbelongs to Pic0(D).)
We recall that a vector bundle on a variety is simple if its endomorphism algebra
coincides with the field k.
The following assertion was proved in [30].
Proposition 5.3.5 ([30], Theorem 5.8). Let Ebe a sim ple vect or bund le on an
Abelian variety D.Then the fol lowing conditions are equivalent :
(1) dim Hj(D, End (E)) = n
jfor any j,j=0,...,n,
(2) Eis a semi-homogeneous bundle,
(3) End (E)is a homogeneous bundle,
(4) there exists an isogeny π:Y−→ DandalinebundleLon Ysuch that
E∼
=π∗(L).
Let Ebe a vector bundle on an Abelian variety D.Wewriteµ(E)forthe
equivalence class det(E)
r(E)in NS(D)⊗ZQ. To every element µ=[L]
l∈NS(D)⊗ZQ,
and hence to each bundle E, we can assign a correspondence Φµ⊂D×
Dgiven by
Φµ=Im
"D(l,ϕL)
−−−−→D×
D#,whereϕLis the well-known map from Dto
Dthat
takes dto the point corresponding to the bundle T∗
dL⊗L−1∈Pic0(D). If the
bundle is a line bundle L, we obtain the graph of the map ϕL:D−→
D.Wewrite
q1and q2for the projections of Φµto Dand
Drespectively.
The paper [30] contains a complete description of all simple semi-homogeneous
bundles.
Proposition 5.3.6 ([30], 7.10). Let µ=[L]
l,where [L]is the equivalence class of
abundleLin NS(D)and la positive integer.Then
(1) there is a simple semi-homogeneous vector bundle Ewith slope µ(E)=µ;
(2) every simple semi-homogeneous vector bundle Ewith slope µ(E)=µis of
the form E⊗Mfor some line bundle M∈Pic0(D);
(3) we have the equalities r(E)2=deg(q1)and χ(E)2=deg(q2).
The following assertion enables one to characterize all semi-homogeneous vector
bundles in terms of simple bundles.
Derived categories of coherent sheaves 585
Proposition 5.3.7 ([30], 6.15, 6.16). Every semi-homogeneous vector bundle F
with slope µhas a filtration
0=F0⊂F1⊂···⊂Fn=F
such that Ei=Fi/Fi−1are simple semi-homogeneous vector bundles with the same
slope µ.Every simple semi-homogeneous bundle is stable.
The next two lemmas concerning semi-homogeneous bundles are direct corollaries
of the above assertions, and will be useful in what follows.
Lemma 5.3.8. Two simple semi-homogeneous bundles E1and E2with the same
slope µare either isomorphic or orthogonal to each other ;that is,either E1=E2
or
Exti(E1,E2)=0 and Exti(E2,E1)=0 for all i.
Proof. It follows from Proposition 5.3.6 that E2∼
=E1⊗M, and hence Hom (E1,E2)
is a homogeneous bundle. By Proposition 5.3.3, every homogeneous bundle can be
represented as the sum of unipotent bundles twisted by line bundles in Pic0(D).
Therefore, either all the cohomology Hom (E1,E2) vanishes, and hence the bundles
E1and E2are orthogonal, or Hom(E1,E2) admits a non-zero section. In the last
case we obtain a non-zero homomorphism from E1to E2. However, these two
bundles are stable and have the same slope. Thus, every non-zero homomorphism
is actually an isomorphism.
Lemma 5.3.9. Let Ebe a simple semi-homogeneous vector bundle on an Abelian
variety D.Then T∗
d(E)∼
=E⊗Pδif and only if (d, δ)∈Φµ.
Proof. We first show that for any point (d, δ)∈Φµthere is an isomorphism T∗
d(E)∼
=
E⊗Pδ. Indeed, set l=r(E)andL=det(E). By definition of Φµ, we know that we
can express (d, δ)=(lx, ϕL(x)) for some point x∈D.SinceEis semi-homogeneous,
there is a line bundle M∈Pic0(D) such that
T∗
x(E)∼
=E⊗M.(53)
Equating determinants, we obtain the equality T∗
x(L)∼
=L⊗M⊗l. By definition
of the map ϕL, this means that PϕL(x)=M⊗l. On the other hand, iterating the
equality (53) ltimes, we obtain
T∗
lx(E)∼
=E⊗M⊗l=E⊗PϕL(x).
Hence, T∗
d(E)∼
=E⊗Pδbecause (d, δ)=(lx, ϕL(x)).
Now for the converse. We introduce the subgroup Σ0(E)⊂
Dgiven by the
condition
Σ0(E):=δ∈
DE⊗Pδ∼
=E.(54)
Since Eis semi-homogeneous, End (E) is homogeneous by Proposition 5.3.5. Thus,
End (E) can be represented as a sum i(Fi⊗Pi), where all the Fiare unipotent.
Hence, H0(End (E)⊗P)=0foratmostr2line bundles P∈Pic0(D). That is,
the order of the group Σ0(E) does not exceed r2. On the other hand, it is known
586 D. O. Orlov
that q2(Ker(q1)) ⊂Σ0(E). Hence, we obtain the equalities ord Σ0(E)=r2and
q2(Ker(q1)) = Σ0(E).
We now assume that T∗
d(E)∼
=E⊗Pδfor some point (d, δ)∈D×
D.Consider
apointδ∈
Dsuch that (d, δ)∈Φµ. As already proved above, there is then an
isomorphism T∗
d(E)∼
=E⊗Pδ. Hence, E⊗P(δ−δ)∼
=E, and thus δ−δ∈Σ0(E).
However, since Σ0(E)=q2(Ker(q1)), it follows that the point (0,δ−δ) belongs to
Φµ. Thus, the point (d, δ) also belongs to Φµ.
Let fbe an isometric isomorphism. We now present a construction which shows
how to construct from fan object Eon the product such that Edefines an equiv-
alence of derived categories and for which fEcoincides with f.
Construction 5.3.10. We fix an isometric isomorphism f:A×
A−→ B×
Band
write Γ for its graph. As above, we represent the isomorphism fin the matrix form
f=xy
zw
.
Suppose that y:
A−→ Bis an isogeny. In this case one assigns to the map fan
element g∈Hom(A×B,
A×
B)⊗ZQof the form:
g=y−1x−y−1
−y−1wy−1.
The element gdefines a certain correspondence on (A×B)×(
A×
B). One sees
readily that the condition that fis isometric implies the equality g=g.This
means that gin fact belongs to the image of NS(A×B)⊗ZQunder its canonical
embedding into Hom(A×B,
A×
B)⊗ZQ(see, for example, [32]). Hence, there exists
µ=[L]
l∈NS(A×B) such that Φµcoincides with the graph of the correspondence g.
Proposition 5.3.6 tells us that for any µone constructs a simple semi-homogeneous
bundle Eon A×Bwith slope µ(E)=µ.
We show presently that the functor ΦEfrom Db(A)toDb(B) is an equivalence
and fE=f. However, first let us compare the graphs Γ and Φµ.Ifapoint
(a, α, b, β) belongs to Γ, then
b=x(a)+y(α),
β=z(a)+w(α),and hence α=−y−1x(a)+y−1(b),
β=(z−wy−1x)(a)+wy−1(b).
Since fis isometric, we have the equality (z−wy−1x)=−y−1. Thus, a point
(a, α, b, β) belongs to the graph Γ if and only if (a, −α, b, β) belongs to Φµ.There-
fore,
Φµ=(1
A,−1
A,1B,1
B)Γ.
In particular, since fis an isomorphism, it follows that the projections of Φµto
A×
Aand B×
Bare isomorphisms.
Derived categories of coherent sheaves 587
Proposition 5.3.11. Le t Ebe the semi-homogeneous bundle on A×Bconstructed
from an isometric isomorphism fas just described.Then the functor ΦE:Db(A)→
Db(B)is an equivalence.
Proof.WewriteEafor the restriction of Eto the fibre {a}×B. By Theorem 2.1.5,
to prove that the functor ΦEis fully faithful, it is enough to show that all the
bundles Eaare simple and mutually orthogonal for distinct points.
First note that by Proposition 5.3.6, the rank of Eis equal to the square root of
the degree of the map Φµ−→ A×B,thatis,deg(β).
Since Eis semi-homogeneous, we see at once that all the bundles Eaare also
semi-homogeneous. Moreover, the slope µ(Ea) of the restriction is equal to δβ−1∈
NS(B)⊗Q⊂Hom(B,
B)⊗Q. For brevity we denote δβ−1by ν, considering it as
an element of NS(B)⊗Q. By Proposition 5.3.6, there is a simple semi-homogeneous
bundle Fon Bwith the given slope µ(F)=ν. Obviously, in this case the map Φν
is Im"
A(β,δ)
−−−→B×
B#.Sincefis an isomorphism, it follows that
A(β,δ)
−→ B×
Bis
an embedding. Hence, applying Proposition 5.3.6 again, we obtain the equality
r(F)=deg(β)=r(Ea). Thus, the two bundles Fand Eaare semi-homogeneous
and have the same slope and the same rank. Moreover, the bundle Fis simple. It
follows from Propositions 5.3.7 and 5.3.6, (2) that Eais also a simple bundle.
Next, it follows from Lemma 5.3.8 that for two points a1,a
2∈A, the bundles
Ea1and Ea2are either orthogonal or isomorphic. Suppose that they are isomorphic.
Since Eis semi-homogeneous, it follows that
T∗
(a2−a1,0)E∼
=E⊗P(α,β)(55)
for some point (α, β)∈
A×
B. In particular, we obtain
Ea2⊗Pβ∼
=Ea1∼
=Ea2.
Hence, Pβ∈Σ0(Ea) (see (54)).
By Lemma 5.3.9 and Proposition 5.3.6, the orders of the groups Σ0(E)and
Σ0(Ea)areequaltor2. We claim that the natural map σ:Σ
0(E)−→ Σ0(Ea)is
an isomorphism. Indeed, otherwise there would exist a point α∈
Asuch that
E⊗Pα∼
=E.Then(0,α
,0,0) ∈Φµby Lemma 5.3.9. This contradicts the fact
that the projection Φµ−→ B×
Bis an isomorphism.
Now if σis an isomorphism, there is a point α∈
Asuch that E⊗P(α,β)∼
=E.
It follows from (55) that
T∗
(a2−a1,0)E∼
=E⊗P(α−α,0).
By Lemma 5.3.9 this means that the point (a2−a1,α−α,0,0) belongs to Φµ.
Since the projection Φµ−→ B×
Bis an isomorphism, we again obtain the equality
a2−a1= 0. Thus, for two distinct points a1and a2the corresponding bundles Ea1
and Ea2are orthogonal. Hence, the functor ΦE:Db(A)−→ Db(B) is fully faithful.
For the same reason, the adjoint functor ΨE∨is also fully faithful. Hence, ΦEis an
equivalence.
588 D. O. Orlov
Proposition 5.3.12. Le t Ebe the semi -homogeneous bundle constructed from an
isometric isomorphism f:A×
A−→ B×
Bas described above.Then fE=f.
Proof.WewriteXfor the graph of the morphism fE. It follows from Corol-
lary 5.1.10 that the point (a, α, b, β) belongs to Xif and only if
Tb∗E⊗Pβ∼
=T∗
aE⊗Pα,
which is equivalent to the equality
T∗
(a,b)E∼
=E⊗P(−α,β).
Hence, by Lemma 5.3.9 we see that X=(1
A,−1
A,1B,1
B)Φµ,whereµ=µ(E)is
the slope of E. On the other hand, by Construction 5.3.10, the graph Γ of fis also
equal to (1A,−1
A,1B,1
B)Φµ. Thus, the isomorphisms fEand fcoincide.
When constructing a bundle Efrom an isomorphism f, we assumed that the
map y:
A−→ Bis an isogeny. If this is not the case, then we represent fas
the composite of two maps f1∈U(A×
A, B ×
B)andf2∈U(A×
A)forwhichy1
and y2are isogenies. One sees readily that this is always possible. Now for any
map fiwe find the corresponding object Ei, consider the composite of the functors
ΦEi, and take the object representing it. The assertions proved in this section
and in the previous ones can be collected in the form of the following theorems.
Theorem 5.3.13. Let Aand Bbe two Abelian varieties over an algebraically closed
field of characteristic 0. Then the bounded derived categories of coherent sheaves
Db(A)and Db(B)are equivalent as triangulated categories if and only if there is
an isometric isomorphism f:A×
A∼
−→ B×
B.
Theorem 5.3.14. Let Abe an Abelian variety over an algebraically closed field of
characteristic 0. Then the group of exact auto-equivalences of the derived category
Auteq Db(A)fits in the following exact sequence of groups:
0−→ Z⊕(A×
A)k−→ Auteq Db(A)−→ U(A×
A)−→ 1.
Thus, the group Auteq Db(A) has a normal subgroup (A×
A)kwhich consists of
the functors of the form Ta∗(·)⊗Pα,where(a, α)∈A×
A. The quotient by this
subgroup is a central extension of U(A×
A)byZ.
This central extension is described by a 2-cocycle, a formula for which can be
found in [37].
Example 5.3.15. Consider an Abelian variety Afor which the endomorphism ring
End(A) is isomorphic to Z. Then the N´eron–Severi group NS(A) is isomorphic
to Z.WriteLand Mfor generators of NS(A)andNS(
A) respectively. The compos-
ite ϕM◦ϕLequals N·idAfor some N>0. In this case the group U(A×
A)coincides
with the congruence subgroup Γ0(N)⊂SL(2,Z). Next, let Bbe another Abelian
variety such that B×
Bis isomorphic to A×
A. One sees readily that every iso-
morphism of this kind is isometric. The Abelian variety Bcan be represented as the
image of some morphism A(k·id,mϕL)
−−−−−−−→A×
A. We can assume that gcd(k, m)=1.
Derived categories of coherent sheaves 589
We wr ite ψfor the morphism from Ato Bdefined in this way. The kernel of ψis
Ker(mϕL)∩Ak.Sincegcd(k, m)=1,wehaveinfactKer(ψ)=Ker(ϕL)∩Ak.On
the other hand, we have an inclusion Ker(ϕ)⊂AN. Thus, without loss of generality
we can assume that kis a divisor of N.Everydivisorkof Ninduces an Abelian
variety of the form B:= A/(Ker(ϕL)∩Ak). Obviously, two distinct divisors of N
give non-isomorphic Abelian varieties. Moreover, one sees readily that an embed-
ding of Bin A×
Asplits if and only if gcd(k, N/k) = 1. Hence, the number of
Abelian varieties Bsuch that Db(B)Db(A)equals2
s,wheresis the number
of prime divisors of N.
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Steklov Institute of Mathematics
Russian Academy of Sciences
E-mail address: orlov@mi.ras.ru
Received 05/FEB/03
Translated by IPS(DoM)
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