DG categories and exceptional collections

0.31 MB·English

by  Agnieszka Bodzenta

DG CATEGORIES AND EXCEPTIONAL COLLECTIONS
AGNIESZKABODZENTA
Abstract. In[BK]BondalandKapranovdescribehowtoassigntoafullexceptionalcollectiononavariety
X a DGcategory C such that the bounded derived category of coherent sheaves onX isequivalent to the
bounded derived category ofC. Inthispaper weshow that thecategory C has finitedimensionalspaces of
morphisms. We describe how it behaves under mutations and present an algorithm allowing to calculate
it for full exceptional collections with vanishing Extk groups for k > 1. Finally, we use it to describe an
exampleofanon-commutative deformationofcertain rationalsurfaces.
Introduction
Derived categories of coherent sheaves have been an object of vivid interest since their discovery in
3
1960’s. Full strong exceptional collections are one of the most important tools used to study them. As
1
0 proved in [B], such a collection on a smooth projective variety X provides an elegant description of the
2 category Db(Coh(X)) – it is equivalent to a derived category of modules over a quiver with relations. If an
exceptional collection σ is full but not strong the description becomes more complicated. In [BK] Bondal
n
a and Kapranovprovethatin this case the categoryDb(Coh(X)) is equivalentto a bounded derivedcategory
J of modules over some DG category C . It follows from the construction that C has finitely many objects.
σ σ
1 The DG category C is a full subcategory of an enhancement of Db(Coh(X)) – a DG category C such
σ
2
that H0(C) is equivalent to Db(Coh(X)). Lunts and Orlov in [LO] show that this enhancement is strongly
e

] unique. CanonicallytheDGcategoryC isasubcategoryofcomplexesofinjectivesheaves. Suchadescription
e
G
suggests that the category C has infinitely dimensional spaces of morphisms between objects and therefore
σ e
A it is inconvenient to work with.
h. In this paper, using the theory of A∞ categories, we show that Cσ can be in fact chosen to be finite and
oriented. It means that one can partially order objects of C in such a way that there are no nontrivial
t σ
a morphisms from C to C if C (cid:22) C . Furthermore, all spaces of morphisms in C are finite dimensional.
1 2 2 1 σ
m
Thus, C can be graphically presented as a DG quiver – a quiver with grading and differential on arrows.
σ
[ The set of full exceptional collections on X admits an action of the braid group. It is givenby mutations

defined in [B]. Using twisted complexes we lift this action to associated DG categories. It allows us to
2
v describe the category Cσ for any collection σ which can be mutated to a strong one.
8 Families of full exceptional collections on rational surfaces have been given in [HP]. Hille and Perling
4 also describe how to obtaina tilting object froma full exceptionalcollectionwith Extk groupsvanishing for
1
k >1. Using their idea of universalextensions we can calculate the DG categoryC for any full exceptional
6 σ
collection σ = hE ,...,E i with Extk(E ,E ) = 0 for k > 1. In particular, this condition is satisfied by full
. 1 n i j
5
exceptional collections of line bundles on smooth rational surfaces.
0
We calculate the category C for an exceptional collection on P2 blown up along a subscheme of degree
2 σ
1 twosupportedatonepoint. ItturnsoutthatfindingatripleMasseyproductofmorphismsinthiscollection
: is the most important part of the whole calculation.
v
Fullstrongexceptionalcollectionshavealsobeenusedin[AKO],[HI]and[P]todescribenon-commutative
i
X deformations of varieties. Knowledge of the DG categories of not necessarily strong exceptional collections
r canlead to a generalisationofthis idea. We presentan example ofa non-commutativedeformationfromP2
a blown up in two different points to P2 blown up along a subscheme of degree 2 supported at one point.
1. Enhanced triangulated categories
Letusrecalldefinitionsandfactsaboutenhancedtriangulatedcategoriesandexceptionalcollectionsafter
[B], [BK], [BLL] and [K2].
Exceptional collections and mutations. Let D be a C-linear triangulated category. An object E ∈ D
is calledexceptional if D(E,E)=C and D(E,E[i])=0 for i6=0. An orderedcollectionσ =hE ,...,E i of
1 n
exceptional objects is called an exceptional collection if D(E ,E [k])=0 for i>j and all k. An exceptional
i j
collection σ is strong if we also have D(E ,E [k])=0 for k 6=0 and all i, j. It is full if the smallest strictly
i j
full subcategory of D containing E ,...,E is equal to D.
1 n
Let hE,Fi be an exceptionalpair in D. Then hL F,Ei andhF,R Ei arealso exceptionalpairs for L F
E F E
and R E defined by means of distinguished triangles in D.
F
Theauthorissupported byaPolishMNiSWgrant(contract numberNN201420639).
Keywords and phrases. Exceptional collections,DGcategory, non-commutativedeformation.
1
2 A.BODZENTA
L F →Hom (E,F)⊗E →F →L F[1],
E D E
E →Hom (E,F)∗⊗F →R E →E[1].
D F
Here, Hom (E,F) denotes a complex of C-vector spaces with trivial differential; Hom (E,F) =
D D
D(E,F[k]). For an element E ∈ D and a complex V• the tensor product is defined by
EL⊗V• = dimVkE[−k].
k∈Z i=0
ForaneLxceptioLnalcollectionσ =hE1,...,Enithe i-thleftmutationLiσ andthei-thrightmutation Riσ
are defined by
L σ =hE ,...,E ,L E ,E ,E ,...,E i,
i 1 i−1 Ei i+1 i i+2 n
R σ =hE ,...,E ,E ,R E ,E ,...,E i.
i 1 i−1 i+1 Ei+1 i i+2 n
Mutations define an action of the braid group on the set of full exceptional collections.
Enhanced triangulated categories.
Definition 1.1. A DG category is a preadditive category C in which abelian groups C(A,B) are endowed
with a Z-grading and a differential ∂ of degree one. The composition of morphisms
C(A,B)⊗C(B,C)→C(A,C)
is a morphism of complexes and for any object C ∈ C the identity morphism id is a closed morphism of
C
degree zero.
We assume that all DG categories are C - linear.
For an element x in a graded vector space we will denote by |x| the grading of x. By Ci(A,B) we will
denote morphisms of degree i.
A DG category C is ordered if there exists a partial order (cid:22) on the set of objects such that C(A,B)= 0
for B (cid:22)A. It is finite if the set of objects of C is finite and for any C ,C ∈C the vector space C(C ,C ) is
1 2 1 2
finite dimensional.
ToaDGcategoryCweassociatethreecategories. ThegradedcategoryCgrisobtainedfromCbyforgetting
the differentials onmorphisms while the homotopy category H(C) has the same objects as C and morphisms
given by the cohomology groups of morphisms in C. A further restriction to the zeroth cohomology gives a
preadditive category H0(C).
A morphism s : C →C′ in C is a homotopy equivalence if H(s) is an isomorphism. Then we say that C
and C′ are homotopy equivalent.
A DG functor between two DG categoriesC and C′ is an additive functor F :C →C′ which preservesthe
grading and differential on morphisms. DG functors between DG categories form a DG category.
Let F,G:C →C be DG functors. To construct a DG category DG-Fun(C ,C ) we put
1 2 1 2
DG-Fun(C ,C )k(F,G) to be the set of natural transformations t:Fgr →Ggr[k] (i.e. for C ∈ C we
1 2 1
have t ∈ Ck(F(C),G(C))). The differential ∂ is defined pointwise; for t : F(C) → G[k](C) we have
C 2 C
(∂(t)) =∂(t ):F(C)→G[k−1](C).
C C
The category of contravariant DG functors is denoted by DG-Funo(C,C′). A functor F : C → C′ is a
quasi-isomorphism if H(F):H(C)→H(C′) is an isomorphism.
ThecategoryDGVectC ofcomplexesofvectorspaceswithhomogenousmorphismsf :V• →W•,f(Vi)⊂
Wi+k and a differential
∂(f)i =∂ fi+1−(−1)|f|fi+1∂
W V
is a DG category.
A right DG module M over a DG category C is an element of DG-Funo(C,DGVectC). After [K2] we
define a derived category D(C) as a localization of H0(DG-Funo(C,DGVectC)) with respect to the class of
homotopy equivalences. By Db(C) we will denote the subcategory of D(C) formed by compact objects. The
Yoneda embedding givesafunctor h:C →Db(C)whichassignsto everyC ∈C afree module h =C(−,C).
C
For a DG category C we define the category C of formal shifts. The objects of C are of the form C[n]
whereC ∈C andn∈N. ForelementsC1[k]andCb2[n]ofC weputHomlCb(C1[k],C2[n])b=HomlC+n−k(C1,C2).
For appropriate sign convention see [BLL].
b
LetB,C be objects ofa DGcategoryC andlet f ∈C(B,C) aclosedmorphism. Assume that B[1] is also
an object of C; i.e. there exists an object B′ and closed morphisms t : B → B′, t′ : B′ → B of degree 1
and -1 respectively such that t′t = idB and tt′ = idB′. An object D ∈ C is called a cone of f if there exist
morphisms
B′ i // D p // B′, C j // D s // C,
such that
pi=1, sj =1, si=0, pj =0, ip+js=1.
DG CATEGORIES AND EXCEPTIONAL COLLECTIONS 3
It is proved in [BLL] that the cone of closed degree zero morphism is uniquely defined up to a DG
isomorphism.
One can formally add cones of closed morphisms to a DG category C by considering the category Cpre-tr
of one-sided twisted complexes over C.
Definition1.2. Aone-sidedtwistedcomplexoveraDGcategoryC isanexpression( n C [r ],q )where
i=1 i i i,j
Ci’s are objects of C, ri ∈Z, n≥0 and qi,j ∈C1(Ci[ri],Cj[rj]) such that qi,j =0 forLi≥j and ∂q+q2 =0.
One-sided twisted complexes over C form a DG category Cpre-tr with the morphism space between
C =( C [r ],q) and C′ = ( C′[r′],q′) given by the set of matrices f = (f ), f : C [r ] → C′[r′].
i i j j i,j i,j i i j j
With La differential defined as L
∂(f)=(∂f )+q′f −(−1)degffq
i,j
the category Cpre-tr is a DG category. We will denote its zeroth homotopy category H0(Cpre-tr) by Ctr.
Let C = ( C [r ],q ) be an object of Cpre-tr. The shift of C is defined as C[1] = ( C [r +1],−q ).
i i i,j i i i,j
The categoryLCpre-tr is closed under formal shifts. L
Let f : C → D be a closed morphism of degree 0 in Cpre-tr, where C =( n C [r ],q ), D =
i=1 i i i,j
( m D [s ],p ). The cone of f is a twisted complex C(f)=( n+mE [t ],u ), wLhere
i=1 i i i,j i=1 i i i,j
L L
C for i≤n,
E = i
i (cid:26) Di−n for i>n,
r +1 for i≤n,
t = i
i (cid:26) si for i>n,
q for i,j ≤n,
i,j
u = f for i≤n<j,
i,j i,j−n
 p for i,j >n.
i−n,j−n
The convolution functor Tot : (Cpre-tr)pre-tr → Cpre-tr establishes a quasi-isomorphism between
(Cpre-tr)pre-tr andCpre-tr. LetC =( ni Di[ri],qi )beobjectsofCpre-tr andletC =( n C [r ],q )bea
i j=1 j j jk i=1 i i ij
twistedcomplexin(Cpre-tr)pre-tr. ThLentheconvolutionTot(C)isequalto( n nij DLi[ri +r ],qi +q ).
i=1 j=1 j j i jk ij
The DG category C is pretriangulated if the embedding H0(C) → CtrLis anLequivalence. The category
H0(C) for a pretriangulated category is triangulated.
AtriangulatedcategoryD is enhanced ifit has anenhancement – a pretriangulatedcategoryC suchthat
D is equivalent to H0(C).
Recall, that an additive categoryis Karoubianif every projector splits. The categoryCtr needs not to be
Karoubian. As showed in [BLL] the category Db(C) is the Karoubization of Ctr.
A standard example of an enhanced triangulated category is the derived category D(A) of an abelian
category A with enough injectives I. Its enhancement is the category of complexes of injective sheaves
Kom(I).
With the above definitions we are ready to state the following theorem.
Theorem 1.3. [Theorem 1 of [BK]] Let C be a pretriangulated category, E ,...,E objects of C and C ⊂C
1 n
the full DG subcategory on the objects E . Then the smallest triangulated subcategory of H0(C) containing
ie e e
E ,...,E is equivalent to Ctr as a triangulated category.
1 n e
It follows that a full exceptional collection σ = hE ,...,E i in an enhanced triangulated category D
1 n
leads to an equivalence of D and Ctr for some DG category C . The category C is a subcategory of the
σ σ σ
enhancement of D and thus Hi(C (E,F))=D(E,F[i]) for any elements E, F of σ.
σ
Remark 1.4. If an enhanced triangulated category D is Karoubian,a full exceptionalcollection σ leads to
an equivalence of D and Db(C ).
σ
2. Properties of DG categories associated to full exceptional collections
FromnowonletX beasmoothprojectivevariety. LetDb(X)=Db(Coh(X))denotetheboundedderived
category of the category of coherent sheaves on X. It is an enhanced Karoubiantriangulated category – its
enhancement is givenby Komb(I) – the DG categoryofcomplexes of injective sheavesbounded frombelow
and with finitely many nonzero coherent cohomology groups.
Letσ =hE ,...,E ibeafullexceptionalcollectionofcoherentsheavesonX. ByTheorem1.3thereexists
1 n
a DG category C with objects E ,...,E such that Db(X) is equivalent to Ctr. As Db(X) is Karoubianthe
σ 1 n σ
same is true about Ctr and hence Ctr = Db(C ). As σ is an exceptional collection the category H(C ) is
σ σ σ σ
ordered and finite.
As two quasi-isomorphicDG categorieshave equivalent derivedcategories,the categoryC is determined
σ
up to a quasi-isomorphism.
4 A.BODZENTA
Remark 2.1. IfaDGcategoryC hasonlyzerothcohomologythenitisquasi-isomorphictoH(C)=H0(C).
Indeed, for C ,C ∈C let C(C ,C )= Cn(C ,C ) with differential
1 2 1 2 n∈Z 1 2
∂nL :Cn(C ,C )→Cn+1(C ,C )
C1,C2 1 2 1 2
and let C be a DG category such that ob C =ob C and
I I
C (C ,C )= Cn(C ,C )⊕ker∂0 .
I 1 2 1 2 C1,C2
Mn<0
Then the natural inclusion functor C →C is a quasi-isomorphism. Let us also set
I
J =Im(∂−1 )⊕( Cn(C ,C ))
C1,C2 C1,C2 1 2
Mn<0
for any C and C in C and consider the category C with ob C =ob C and
1 2 I/J I/J
C (C ,C )=C (C ,C )/J .
I/J 1 2 I 1 2 C1,C2
Then C is isomorphic to H(C) and the natural functor C →C is a quasi-isomorphism.
I/J I I/J
Itshowsthatifσ isastrongexceptionalcollectiontheDGcategoryC isquasi-isomorphictoanordinary
σ
category.
Finiteness. As the enhancement of D is Komb(I), calculating the category C requires taking injective
σ
resoultions. This suggest that the category C can have infinitely dimensional morphisms spaces. However,
σ
it is A -quasi-isomorphic to an ordered DG category with finite-dimensional morphisms between objects.
∞
A categories. We recall definitions after [K1] and [L-H].
∞
Definition 2.2. An A category A over C consists of
∞
• objects ob(A),
• for any two A,B ∈ob(A) a Z - graded C-vector space A(A,B),
• for any n≥1 and a sequence A ,A ,...,A ∈ob(A) a graded map:
0 1 n
m :A(A ,A )⊗...⊗A(A ,A )→A(A ,A )
n n−1 n 0 1 0 n
of degree 2−n such that for any n
(−1)r+stm (id⊗r⊗m ⊗id⊗t)=0.
r+1+t s
r+Xs+t=n
Notethatwhenthese formulaeareappliedto elementsadditionalsignsappearbecauseoftheKoszulsign
rule:
(f ⊗g)(x⊗y)=(−1)|x||g|f(x)⊗g(y).
An A algebra is an A category with one object.
∞ ∞
An A category A is ordered if there exists a partial order (cid:22) on the set ob(A) such that A(A,A′) = 0
∞
for A′ (cid:22)A. It is finite if ob(A) is a finite set and A(A,A′) is finite-dimensional for any A and A′.
The operation m gives for any A,B ∈ ob(A) a structure of a complex on A(A,B). The homotopy
1
category with respect to m is denoted by H(A).
1
The A category is minimal if the operation m is trivial.
∞ 1
Any DG category can be regarded as an A category with m given by the differential, m given by
∞ 1 2
composition and trivial m ’s for i>2.
i
For any set S there exists an A category C{S}. The objects of C{S} are elements of S and
∞
C if s =s ,
C{S}(s ,s )= 1 2
1 2 (cid:26) 0 otherwise.
All operations m in C{S} are trivial.
n
Definition 2.3. A functor of A categories F :A→B is a map F :ob(A)→ob(B)and a family of graded
∞ 0
maps
F :A(A ,A )⊗...⊗A(A ,A )→B(F (A ),F (A ))
n n−1 n 0 1 0 0 0 n
of degree 1−n such that
(−1)r+stF (id⊗r⊗m ⊗id⊗t)= (−1)pm (F ⊗...⊗F ),
r+1+t s r i1 ir
r+Xs+t=n i1+.X..+ir=n
where p=(r−1)(i −1)+(r−2)(i −1)+...+2(i −1)+(i −1).
1 2 r−2 r−1
Composition of functors is given by
(F ◦G) = F ◦(G ⊗...⊗G ).
n s i1 is
i1+.X..+is=n
DG CATEGORIES AND EXCEPTIONAL COLLECTIONS 5
A functor F :A→B is called an A -quasi-isomorphism if F is a quasi-isomorphism.
∞ 1
An A category A is strictly unital if for any object A∈ ob(A) there exists a morphisms 1 ∈A(A,A)
∞ A
of degree 0 such that for any object A′ in A and any morphisms φ ∈ A(A,A′), ψ ∈ A(A′,A) we have
m (φ,1 )=φ and m (1 ,ψ)=ψ. Moreover,for n6=2 the operation m equals 0 if any of its argument is
2 A 2 A n
equal to 1 .
A
In particular, for any set S the category C{S} is strictly unital.
The category A is homologically unital if there exist units for the homotopy category H(A).
Lefévre-Hasegawain [L-H] gives a connection between strictly and homologicaly unital categories.
Proposition 2.4. Minimal homolgically unital A category is A -quasi-isomorphic to a minimal strictly
∞ ∞
unital A category.
∞
Remark 2.5. AminimalA categoryisequaltoitshomotopycategory. Hence,anA -quasi-isomorphism
∞ ∞
F given by the above proposition satisfies F =id.
1
An A category A is augmented if there exists a strict unit preserving functor ǫ:C{ob(A)}→A. Then
∞
A decomposes as A=C{ob(A)}⊕A¯.
Minimal model. Any A category A is A -quasi-isomorphicto its homotopy category H(A).
∞ ∞
Theorem 2.6. [Kadeishvili[Ka]] If A is an A category, then H(A) admits an A category structuresuch
∞ ∞
that
(1) m =0 and m is induced from mA and
1 2 2
(2) there is an A -quasi-isomorphism A→H(A) inducing the identity on cohomology.
∞
Moreover, this structure is unique up to a non unique A -isomorphism.
∞
The A category H(A) is called the minimal model of A.
∞
Remark 2.7. LetC be a DG category. Its minimal modelH(C) is A -quasi-isomorphicto a strictly unital
∞
A category. Indeed,thecategoryC isstrictlyunitalandhencehomologicallyunital. Itshomotopycategory
∞
isalsohomologicallyunitalandProposition2.4guaranteesthatthereexistsastriclyunitalminimalcategory
A -quasi-isomorphic to H(C).
∞
The universal DG category. For any A category A there exists a DG category U(A) and an A -quasi-
∞ ∞
isomorphism A→U(A). To define the category U(A) we need the following definitions (see further [L-H]).
Definition 2.8. A DG cocategory B consists of
• the set of objects B ∈ob(B),
i
• for any pair of objects B ,B ∈ob(B) a complex of C-vector spaces B(B ,B ) with a differential dij
i j i j
of degree one and
• a coassociative cocomposition – a family of linear maps
∆:B(B ,B )→ B(B ,B )⊗B(B ,B ).
i j k j i j
Bk∈Xob(B)
These data have to satisfy the condition
∆◦d=(d⊗id+id⊗d)◦∆.
For any set S the A category C{S} is also a DG cocategory.
∞
Afunctor Φ betweenDGcocategoriesB andB′ preservesthe gradinganddifferentialsonmorphismsand
satisfies the condition
∆◦Φ=(Φ⊗Φ)◦∆.
A DG cocategory B is counital if it admits a counit – a functor η :B →C{ob(B)}. The category B
is coaugmented if it is counital and admits a coaugmentation functor ε : C{ob(B)} → B such that the
composition ηε is the identity on C{ob(B)}.
Let B be a coaugmented DG cocategory.Denote by B¯ a cocategory with the same objects as B and
morphisms B¯(B ,B )=ker ε.
i j
For an augemnted A category A one can define its bar DG cocategory B (A). Recall that as an
∞ ∞
augmented category A can be written as A¯⊕C{ob(A)}. Then B (A) = Tc(SA¯) is a tensor cocategory of
∞
the suspension of A¯. Here SA¯ denotes the category A¯ with a shift in a morphisms spaces (SA¯)n(A,A′) =
A¯n+1(A,A′). SA¯is not anA category,howeverthe operationsm inA¯define gradedmaps ofdegree1 in
∞ n
SA¯.
b :SA¯(A ,A )⊗...⊗SA¯(A ,A )→SA¯(A ,A ),
n n−1 n 0 1 0 n
b =−s◦m ◦ω⊗n,
n n
where s:V →SV is a suspension of a graded vector space V and ω =s−1.
6 A.BODZENTA
The cocategory B (A) has the same objects as A and
∞
B (A)(A,A′)=SA¯(A,A′)⊕ SA¯(A ,A′)⊗SA¯(A,A )
∞ 1 1
A1∈Mob(A)
⊕ SA¯(A ,A′)⊗SA¯(A ,A )⊗SA¯(A,A )⊕...
2 1 2 1
A1,AM2∈ob(A)
for A6=A′. In the case A=A′ we have
B (A)(A,A)=1 ⊕SA¯(A,A)⊕ SA¯(A ,A)⊗SA¯(A,A )
∞ A 1 1
A1∈Mob(A)
⊕ SA¯(A ,A)⊗SA¯(A ,A )⊗SA¯(A,A )⊕...
2 1 2 1
A1,AM2∈ob(A)
where the degree of 1 is zero. To simplify the notation we shall write (α ,...,α ) for α ⊗...⊗α . The
A n 1 n 1
differential in B (A) is given by
∞
d(α ,...,α )=
n 1
n n−k+1
(−1)|αl−1|+...+|α1|(α ,...α ,b (α ,...,α ),α ,...α ),
n l+k k l+k−1 l l−1 1
kX=1 Xl=1
for (α ,...,α )∈B (A)(A,A′). The cocomposition is given by
n 1 ∞
∆(α ,...,α )=
n 1
n−1
1A′ ⊗(αn,...,α1)+(αn,...,α1)⊗1A+ (αn,...,αl+1)⊗(αl,...,α1).
Xl=1
With these definitions B (A) is an augmented DG cocategory.
∞
Remark 2.9. Let A be an ordered and finite A category such that A¯(A,A)=0 for any object A∈obA.
∞
Then the DG cocategory B (A) also satisfies these conditions; i.e. is ordered, finite and B (A)(A,A)=0
∞ ∞
for any object A.
Analogously, to an augmented DG cocategory B via a cobar construction one can assign a DG category
Ω(B). Let B be a DG cocategorywith a differential d and cocomposition ∆. Its cobarDG categoryis equal
to T(S−1B¯). Here S−1B¯denotes the shift ofthe cocategoryB andT(S−1B¯) is the tensor DG categoryofit.
As before the morphisms spaces in T(S−1B¯) are given by
Ω(B)(B,B′)=S−1B¯(B,B′)⊕ S−1B¯(B ,B′)⊗S−1B¯(B,B )
1 1
B1∈Mob(B)
⊕ S−1B¯(B ,B′)⊗S−1B¯(B ,B )⊗S−1B¯(B,B )⊕...
2 1 2 1
B1,BM2∈ob(B)
for B 6=B′ and by
Ω(B)(B,B)=1 ⊕S−1B¯(B,B)⊕ S−1B¯(B ,B)⊗S−1B¯(B,B )
B 1 1
B1∈Mob(B)
⊕ S−1B¯(B ,B)⊗S−1B¯(B ,B )⊗S−1B¯(B,B )⊕...
2 1 2 1
B1,BM2∈ob(B)
for1 –amorphismindegreezero. ThecompositioninΩ(B)isdefinedbyconcatenationandthedifferential
B
∂ on the morphisms spaces is
∂ = 1⊗...⊗1⊗(d+∆)⊗1⊗...⊗1.
X
Remark 2.10. IfanorderedandfiniteDGcocategoryB satisfiesthe conditionB¯(B,B)=0,thenthesame
is true about Ω(B).
For an augmented A category A its universal DG category U(A) is defined as Ω(B (A)). There is a
∞ ∞
natural map A → U(A). Lefévre-Hasegawa proves in [L-H] that this map extends to a functor and is an
A -quasi-isomorphism. Moreover,foranA -quasi-isomorphismsφthefunctorU(φ)isaquasi-isomorphism
∞ ∞
of DG categories.
DG CATEGORIES AND EXCEPTIONAL COLLECTIONS 7
A modules.
∞
Definition2.11. AnA∞ moduleoveranA∞ categoryAisanA∞ functorM :A→DGVectC. Amorphism
of modules G:M →N is given by a family {GA : M0(A)→N0(A)}A∈obA of morphisms in DGVectC such
that the diagrams
M (A )Mn(αn−1⊗...⊗α0//M) (A )
0 0 0 n
GA0 GAn
N ((cid:15)(cid:15)A )Nn(αn−1⊗...⊗α0//)N (A(cid:15)(cid:15) )
0 0 0 n
commute for any n∈N, A ∈obA and α ∈A(A ,A ).
i i i i+1
The categoryofA modules overanA categoryAwillbe denotedasModstrictA. This notationagrees
∞ ∞ ∞
with the notation in [L-H].
The morphism G : M → N of modules is an A -quasi-isomorphism if G is a quasi-isomorphism of
∞ A
complexes for any A∈obA.
The derived category D (A) of an A categoryA is defined as a localizationof the categoryModstrictA
∞ ∞ ∞
with respect to the class of A -quasi-isomorphisms. For A -quasi-isomorphic A categories the derived
∞ ∞ ∞
categories are equivalent.
Remark 2.12 (see [L-H]). For a DG category C the derived categories D(C) and D (C) are equivalent.
∞
It follows that for any augmented A category A the derived category D (A) is equivalent to the derived
∞ ∞
category of the universal algebra D(U(A)).
Theorem 2.13. Let C be a DG category with finitely many objects. Assume that H(C) is an ordered and
finite graded category such that H(C)(C,C)=C for any C ∈C. Then there exists an ordered and finite DG
category C such that D(C) is equivalent to D(C).
Proof. Theeorem2.6guaranteesexistenceofameinimalmodelofC –anA categoryH(C). Byremark2.7we
∞
can assume that H(C) is strictly unital. As H(C)(C,C)=C for any C the category H(C) is an augmented
oriented A category. We have
∞
D=D(C )=D (H(C )=D(U(H(C )).
σ ∞ σ σ
Remarks 2.9 and 2.10 show that the category U(H(C )) is ordered and finite. (cid:3)
σ
Corollary 2.14. LetX beasmoothprojective varietyandletσ =hE ,...,E ibeafullexceptional collection
1 n
on X. There exists an oriented, finite DG category C such that Db(X) is equivalent to Db(C ).
σ σ
Proof. By theorem 1.3 there exists a DG category Cfsuch that Db(X)=Db(C ). As sheavesfE ,...,E are
σ σ 1 n
exceptional and the category Coh(X) is Ext-finite, the category C satisfies conditions of theorem 2.13 and
σ
there exists a DG category C such that categories D(C ) and D(C ) are equivalent. As Db(C )⊂D(C ) is
σ σ σ σ σ
the subcategory of compact objects it follows that Db(C )=Db(C ). (cid:3)
f σ gσ
The category C is A -quasi-isomorphic to the category C angd hence Hk(C (E ,E ))=Extk (E ,E ).
σ ∞ σ σ i j X i j
TosimplifythenotationwewilldenotebyC anorderedfiniteDGcategoryassociatedtoafullexceptional
f σ f
collection σ.
Action of the braid group. We have seen that the braid group acts on the set of exceptional collection.
This action lifts to associated DG categories.
Twisted complexes provide a description of the categories C and C by means of C . To see it we
Liσ Riσ σ
need to define a tensor product of a twisted complex with a complex of vector spaces.
LetC beafiniteDGcategory,C ∈Cpre-tr beatwistedcomplexandletV• beafinitedimensionalcomplex
of vector spaces with the differential ∂i :Vi →Vi+1. C⊗V is defined as ( C[−i]⊕dimVi,q ) ∈
{i|Vi6=0} i,j
(Cpre-tr)pre-tr. The morphisms q are induced by the differential ∂i tensorLed with the identity on C and
i,i+1
q =0 for j 6=i+1.
i,j
Now, let C,D ∈Cpre-tr be twisted complexes. There exist closed morphisms of degree 0
φ:C⊗Hom (C,D)→D,
Cpre-tr
ψ :C →Hom (C,D)∗⊗D.
Cpre-tr
The morphism φ : C[−i]⊕dimHomi(C,D) → D is given by morphisms of degree i between C and D; ψ is
i,0
defined analogously. Now we define new twisted complexes over C
L D =Tot(C(φ)[−1]),
C
R C =Tot(C(ψ)),
D
where C(φ) denotes the cone of φ.
8 A.BODZENTA
Let σ = hE ,...,E i be a full exceptional collection on a smooth projective variety X and let
1 n
C be the category described in the theorem 2.13. Let denote by E ,...,E objects of C . We
σ 1 n σ
define two full subcategories of Cσpre-tr; CσLi with objects E1,...,Ei,LEiEi+1,Ei+2,...En and CσRi with
E ,...,E ,R E ,E ,...,E .
1 i−1 Ei+1 i i+1 n
Proposition 2.15. Let σ = hE ,...,E i be a full exceptional collection on X and let C be a finite DG
1 n σ
category with objects E ,...,E with HkC (E ,E ) = Extk (E ,E ) and such that Db(X) is equivalent to
1 n σ i j X i j
Db(Cσ). Then the categories CσLi and CσRi satisfy analogous conditions for the collections Liσ and Riσ
respectively.
Proof. As the category C is finite, mutations of twisted complexes over C are well defined. Furthermore,
σ σ
the categoryDb(X)isequivalentto Ctr. Under this equivalenceL E correspondstoL E . Hencethe
σ Ei i+1 Ei i+1
category CLi is the DG category described by theorem 1.3. By construction it is also finite. (cid:3)
σ
3. Calculating the category C
σ
First, let us define universal extensions after [HP].
Universal extensions. Let E, F be objects of a C-linear abelian category. Note that End(Ext1(E,F))=
Ext1(E,F ⊗ Ext1(E,F)∗). As id ∈ End(Ext1(E,F)) there exists a distinguished element id of
Ext1(E,F ⊗Ext1(E,F)∗).
e
E¯ – the universal extension of E by F is definedas the extensionof E by F ⊗Ext1(E,F)∗ corresponding
to id∈Ext1(E,F ⊗Ext1(E,F)∗). E¯ fits into the short exact sequence:
e
(1) 0 // F ⊗Ext1(E,F)∗ //E¯ // E //0.
This short exact sequence gives the long exact sequence
0 // Hom(E,F) //Hom(E¯,F) //Hom(F,F)⊗Ext1(E,F) // ....
It shows that Ext1(E¯,F) = 0 and the groups Extn(E¯,F) and Extn(E,F) are isomorphic for n > 1 if
Extn(F,F)=0 for n>0.
Moreover,if Extn(F,E)=0 for n≥0, Hom(F,F)=C and Extq(F,F)=0 for q >0 then the long exact
sequence
0 // Hom(F,F)⊗Ext1(E,F)∗ //Hom(F,E¯) // Hom(F,E) //...
shows that Hom(F,E¯)=Ext1(E,F)∗.
Thus, if F is exceptional and Extn(F,E) = 0 for all n, the objects E¯, F and morphisms between them
determine E – as a cone of the canonical morphism
F ⊗Hom(F,E¯) can //E¯ // E.
Assume that (E,F) is an exceptional pair in an enhanced triangulatedcategorywith the enhancement C
and denote by C the DG subcategory of C with objects E¯ and F. Then the cone of the canonicalmorphism
e
F ⊗Hom(F,E¯)→F¯ in Cpre-tr corresponds to E. Hence, as in the case of mutations, knowing the DG
e
subcategory of C with objects E¯ and F we can calculate the DG subcategory with objects E and F.
Theorem 3.1.eLet σ = hE ,...,E i be a full exceptional collection on a smooth projective variety X such
1 n
that Extk(E ,E ) = 0 for k > 1 and any i, j. The DG category C can be calculated by means of universal
i j σ
extensions.
Proof. Following [HP] we define E (1) = E (2) = ... = E (i) = E and for j > i we put E (j) to be the
i i i i i
universal extension of E (j−1) by E .
i j
0 //E ⊗Ext1(E (j−1),E )∗ // E (j) // E (j−1) //0.
j i j i i
Then F = n E (n) is a tilting object – it generates Db(X) and Exti(F,F) = 0 for i > 0. Moreover,
i=1 i
as the Ei’s areLexceptional and there are no morphisms from Ej to Ei(j −1), the objects E1(n),...,En(n)
determine E ’s. Thus, as described above, the endomorphisms algebra of n E (n) determines the DG
i i=1 i
structure of the collection (E1,...,En). L (cid:3)
4. Exceptional collections of line bundles on rational surfaces
After [HP] we describe full exceptional collections of line bundles on rational surfaces.
DG CATEGORIES AND EXCEPTIONAL COLLECTIONS 9
Constructionandmutations. LetX bearationalsurface. ThenX isablowupofP2oroftheHirzebruch
surface F . There is a sequence of blow-ups
a
X =X πt // X πt−1 //... π1 // X ,
t t−1 0
where π has an exceptional divisor E and X is either the projective plane P2 or the Hirzebruch surface
i i 0
F .
a
Let R denote preimages of the divisor E to X for k ≥ i. Then R ,...,R are mutually orthogonal of
i i k 1 t
self intersection -1.
FullexceptionalcollectionsoflinebundlesonX canbeobtainedbyanaugmentationfromanexceptional
collectiononX . Todescribe the augmentationwe identify a line bundle L onX with its pull backvia π ’s
0 i j
and denote them by the same letter.
Let σ = hL ,...,L i be an exceptional collection on X . The augmentation of σ is
1 s i
σ′ =hL (R )...,L (R ),L ,L (R ),L ,...L i – an exceptional collection on X .
1 i+1 k−1 i+1 k k i+1 k+1 s i+1
Hille and Perling in [HP] proved that all the above described collections are full and have no nontrivial
Ext2 groups between elements. Moreover, mutations allow to present every one of them in the following
form.
Proposition4.1. Anyexceptionalcollection oflinebundles onX obtainedviaaugmentationcanbemutated
to hO,O(R ),...,O(R ),L ,...,L i, where hO,L ,...,L i is an exceptional collection on X .
t 1 1 r 1 r 0
Proof. As O = (π ...π )∗(O ) and the line bundle O (R ) is a pullback of O (E ), the projection
X i t Xi X i Xi i
formula gives
C q =0,
Hq(X,O(R ))=Hq(X ,O(E ))=
i i i (cid:26) 0 q 6=0,
C q =0,1,
Hq(X,O(2R ))=Hq(X ,O(2E ))=
i i i (cid:26) 0 q 6=0,1,
Hq(X,O(−R ))=Hq(X ,O(−E ))=0.
i i i
Short exact sequences
0 // O(−R ) //O //O // 0,
i Ri
0 //O //O(R ) // O (R ) //0
i Ri i
show that
C q =0,
Extq(O,O )=
Ri (cid:26) 0 q 6=0,
C q =1,
Extq(O (R ),O)=
Ri i (cid:26) 0 q 6=1.
ThecollectiononX obtainedviaaugmentationisoftheformhL (R ),...,L (R ),L ,L (R ),L ,...L i,
1 t i−1 t i i t i+1 s
where L ’s are pull backs of line bundles on X .
j t−1
Equality
Hom(L ,L (R ))=Hom(O,O(R ))=C
i i t t
and the short exact sequence
0→L →L (R )→O (R )→0
i i t Rt t
show that this collection can be mutated to hL (R ),...,L (R ),O (R ),L , L ,...L i.
1 t i−1 t Rt t i i+1 s
Then
Hom(L (R ),O (R ))=Hom(O(R ),O (R ))=Hom(O,O )=C
k t Rt t t Rt t Rt
and the exact sequences
0→L →L (R )→O (R )→0
k k t Rt t
provide further mutations to hO (R ),L ,...,L i.
Rt t 1 s
The collection hL ,...,L i is a pull back of a collection on X and it again has the form
1 s t−1
hL′(R ),...,L′ (R ),L′,L′(R ),L′ ,...,L′ i for some k. As before, it can be mutated to
1 t−1 k−1 t−1 k k t−1 k+1 s−1
hL′(R ),...,L′ (R ), O (R ),L′,...,L′ i and then to hO (R ),L′,...,L′ i.
1 t−1 k−1 t−1 Rt−1 t−1 k s−1 Rt−1 t−1 1 s−1
Continuing, we can mutate the collection on X to hO (R ),...,O (R ), O,L ,...,L i. And this
Rt t R1 1 1 r
collection can be mutated to hO,O(R ),...,O(R ), L ,...,L i. (cid:3)
t 1 1 r
10 A.BODZENTA
Example. Let X be the blow up of P2 along the degree 2 subscheme supported at a point x . The Picard
0
group of X is generated by divisors E , E and H with intersection form given by
1 2
E2 =−2, E2 =−1, H2 =1,
1 2
E E =1, E H =0.
1 2 i
In the notation of the previous section we have
R =E +E , R =E .
1 1 2 2 2
The collection hO ,O (E ),O (E +E ),O (H),O (2H)i is full and exceptional with a quiver
X X 2 X 1 2 X X
β¯
)) δ1 //
O α // O (E ) β // O (E +E ) γ1 //O (H) δ2 //O (2H)
X X 2 X 1 2 γ2 55// X δ3 // X
η
with β¯ in degree 1 and relations
δ γ =δ γ , δ η =δ γ βα,
1 2 2 1 1 3 1
δ η =δ γ βα, β¯α=0,
2 3 2
γ β¯=0, γ β¯=0.
1 2
Let D denote a strict transform of a line on P2 such that D ∩E 6= ∅. The morphism γ in the above
2 2 2 2
quiveriszeroalongD +E . αhaszerosalongE andβ –alongE . Thesethreemorphismsaredetermined
2 2 2 1
uniquely up to a constant. On the other hand, γ is zero along D – a strict transform of a line on P2 such
1 1
that D ∩E 6=∅. The divisor D is not determined uniquely and one can change γ in the above quiver by
1 1 1 1
adding some multiplicity of γ .
2
Note that morphisms form OX to OX(H) are pull backs of sections of OP2(H). As the pull back of x0 in
X is E ∪E , a sectionofO (H)onX is either zero onbothE andE (possibly with some multiplicities)
1 2 X 1 2
oritis nonzerooneverypointofE andE . Sections havingzerosalongE andE arelinearcombinations
1 2 1 2
of γ βα and γ βα.
1 2
The morphisms η in the quiver is any section of O (H) which is nonzero on E and E . Hence, it can
X 1 2
be changed by adding any linear combination of γ βα and γ βα.
1 2
To obtain the relations given, we note that Hom(O ,O (H)) = Hom(O (H),O (2H)). Then we put
X X X X
δ ↔γ βα, δ ↔γ βα and δ ↔η.
1 1 2 2 3
To sum up, α, β and γ are determined uniquely. Other morphisms can change by
2
γ γ +aγ
1 1 2
η η+bγ βα+cγ βα
1 2
for a,b,c∈C. These morphism determine δ ’s in such a way that the above relations are satisfied.
i
InordertocalculatetheDGquiverofthiscollection,wehavetocalculatetheendomorphismsalgebraofa
collectionhO ,V,O (E +E ),O (H),O (2H)i,where V isdefined bya non-trivialextensionofO (E )
X X 1 2 X X X 2
by O (E +E ).
X 1 2
0 // O (E +E ) φ1 //V φ2 // O (E ) // 0.
X 1 2 X 2
The collection of sheaves hO ,V,O (E +E ),O (H),O (2H)i is no longer exceptional. However, we
X X 1 2 X X
can still draw the quiver of morphisms between objects. Then we can present O (E ) as a complex
X 2
{O (E +E )⊗Hom(O (E +E ),V)→V}
X 1 2 X 1 2
and calculate the DG quiver of the collection hO ,O (E ),O (E +E ), O (H),O (2H)i
X X 2 X 1 2 X X
The quiver of (hO ,V,O (E +E ),O (H),O (2H)i is
X X 1 2 X X
η
(( δ1 //
O ζ // V βφ2// O (E +E ) O (H) δ2 // O (2H)
X oo X 1 2 88 X δ3 // X
φ1 ι1 88
ι2
Here ζ : O →V is such a morphism that φ ζ = α and ι : V →O (H) are such that ι φ = γ . Then
X 2 i X i 1 i
ι φ βφ ζ =γ βα and ι φ βφ ζ =γ βα.
1 1 2 1 2 1 2 2