bondal - representation of associative algebras and coherent sheaves

H3B. Axaa. HayK CCCP Math. USSR IzvestiyaCep. MateM. TOM 53(1989), JVfe 1 Vol. 34(1990), No. 1

REPRESENTATION OF ASSOCIATIVE ALGEBRASAND COHERENT SHEAVES

UDC512

A. I. BONDAL

ABSTRACT. It is proved that a triangulated category generated by a strong exceptionalcollection is equivalent to the derived category of modules over an algebra of homo-morphisms of this collection. For the category of coherent sheaves on a Fano variety,the functor of tightening to a canonical class is described by means of mutations ofan exceptional collection generating the category. The connection between mutabilityof strong exceptional collections and the Koszul property is studied. It is proved thatin the geometric situation mutations of exceptional sheaves consist of present sheavesand the corresponding algebra of homomorphisms is Koszul and selfconsistent.

Bibliography: 18 titles.

1. Introduction

The goal of this paper is to study interconnections of various categories with thecategories of representations of finite-dimensional associative algebras. The maintool is the notion of exceptional collections or, in a more general situation, that of asemi-orthogonal collection of admissible subcategories.

Let Ε be an exceptional object of an abelian category sf. This means thatExt' (E,E) — 0 for i > 0. Then, using E, one can construct a functor FE from thecategory sf into the derived category Db(mod-A) of representations of the algebraA = Hom(E,E):

FE(M) = RHom(E,M),

FE{M) is a complex of right Λ-modules. The functor F is extendable to the derivedfunctor DbF, from Dbsf to Db(mod-A). If Ε has sufficiently many direct summands,then DbFE is an equivalence of triangulated categories (Theorem 6.2).

As an example of J / , we may consider the category Sh(P") of coherent sheaves onthe projective space P". Beilinson has shown in [1] that if we set EQ = φ " = 0 ^ ( / ) ,then DbFEo is an equivalence of categories. Later, Drezet and Le Potier [9] andGorodentsev and Rudakov [8] constructed the whole series of exceptional vectorbundles that are obtained by successive mutations (nepecrpoHica) of the bundle Eo.It is convenient to regard EQ as the whole exceptional collection of bundles (f(i),and to introduce a condition of simplicity into the notion of an exceptional object:Hom(E, E) = C. Then the mutations inside the collection are interpreted as the ac-tions of the Artin braid group. Kapranov [ 12] has constructed exceptional collectionson quadrics, Grassmannians, and flag manifolds.

1980 Mathematics Subject Classification (1985 Revision). Primary 16A46, 16A64, 18F20.

© 1990 American Mathematical Society0025-5726/90 $1.00 + $.25 per page

23

24 Α. Ι. BONDAL

Projective modules over finite-dimensional associative algebras are another exam-ple of an exceptional collection. Mutations of such a collection generalize reflectionfunctors [4] and tilting modules [5], which are used in the representation theory ofquivers.

From the point of view of the theory of quivers, the investigation of exceptionalobjects can be motivated in the following way: This theory is engaged in a classi-fication of finite-dimensional associative algebras with tame representation theory.However, tame algebras are only a small isle in the ocean of wild algebras. Whatto do with the wild algebras? First of all one tries to describe all the simple non-variable representations. Among them, the exceptional objects occupy in importantplace from a general functorial point of view. Second, one tries to partition the set ofalgebras (or, more generally, differentially-graded algebras) into classes, depending onthe properties of mutability of exceptional collections, and, third, to define a notionof stability of algebra representations and classify the stable modules. The analogywith the theory of sheaves on P" indicates the complexity of the latter task.

The exceptional collection property as defined in [8] is not, generally speaking,preserved under mutations. Thus we must weaken it. In this form, it is successfullyused in every triangulated category.

If i»: 38 —> sf is an embedding of the subcategory generated by the elements ofan exceptional collection into the base category, then it will be shown in Theorem3.2 that the category 3§ is admissible, i.e. there exist right and left adjoint functors i[

and /*. These functors generalize the Beilinson resolvent [14] and represent a variantof the bar construction [12].

It was noted in [8] that the mutations of an exceptional collection generate ahelix. Theorem 4.1 shows that this is connected with the fullness of an exceptionalcollection. Note that by identifying the derived categories of coherent sheaves on amanifold with modules over an algebra, the tightening (nozycpyTKa) functor on thecanonical class transforms into the derived Nakayama functor or, as it is called inquiver theory, into the Coxeter functor.

We demonstrate further, through some examples, the way geometry and algebrarelate. Thus, for instance, representations of quivers which consist of two verticesand two arrows from one vertex into another correspond to the sheaves on P'. Aswe know, this is a time quiver, and its representations were described long ago byKronecker. The unique parameter determining indecomposable representations isalso the parameter on the projective line.

In §7 we define Koszul algebras with ordered projective objects and prove that theKoszul condition is equivalent to strong exceptionality of the dual collection, whichis constructed via mutations of projective modules and consists of irreducible objectstranslated along the derived category.

In §8 we investigate, from a purely algebraic point of view, the questions of preser-vation of the property of strong exceptionality under mutations. It turns out to benecessary to impose some homological conditions on the algebra of homomorphismsbetween the elements of the exceptional collection. We call the algebras satisfyingthese conditions self-consistent. Their independent study is obviously important onits own.

Finally, we prove in §9 that geometry furnishes examples of self-consistent alge-bras.

To conclude this Introduction, we point out an interesting connection of the afore-mentioned with the theory of perversion of sheaves. If a stratification of a manifold is

REPRESENTATIONS OF ASSOCIATIVE ALGEBRAS 25

such that all strata are contractible [13], then the triangulated category of complexeswith homology that is locally constant on stratas has a full exceptional collection.Using a known correspondence of sheaves, subject to stratification by Schubert cellsof the flag manifolds with modules over a semisimple Lie algebra, we obtain anexceptional collection in the category & [2], consisting of Verma modules.

This paper is dedicated to A. Grothendieck, for his sixtieth birthday.

2. Exceptional collections and mutations

Let stf be a triangulated category [6] and A and Β objects in sf; Hom(A,B) isa vector space over a field K. We introduce the following notation for the gradedcomplex of A"-vector spaces with trivial differential:

Horn \A, B) = ®Homk^(A,B)[-k];kez

here Hom^(/1, Β) = Ηοην(Λ, TkB), where Γ is a translation in the triangulated cat-egory J / , and the number in square brackets denotes that the space Ηοπν(Λ, TkB)has grading equal to k.

In the case when si is the derived category of an abelian category, Horn (A, B) isquasi-isomorphic to RHom(^,5).

DEFINITION. An object Ε is called an exceptional object if it satisfies the followingconditions:

Horn'\E, E) = 0 for i φ 0, Hom(£\ E) = K.

DEFINITION. An ordered collection of exceptional objects (E0,...,En) is called anexceptional collection in s/ if it satisfies the condition

Horn (Ej,Ek) = 0 for j > k.

We will call an exceptional collection of two objects an exceptional pair.The mutations of exceptional collections of sheaves on the projective space P"

have been determined in [8]. The following definition is a natural generalization tothe case of arbitrary triangulated categories:

DEFINITION. Let (E, F) be an exceptional pair. We define objects LEF and RFEwith the aid of distinguished triangles in the category s/:

LEF ^Hom{E,F)®E->F,

Ε -+Hom(£,.F)* ®F -* RFE; { '

here V[k] ® E, where V is a vector space, denotes an object equal to the direct sumof dim V copies of the object TkE. Under conjugation of vector spaces, gradingchanges the sign. A left (right) mutation of an exceptional pair τ = {E,F) is thepair LET - {LF,E) (respectively, REx = (F,RE)). Lower indices will be omittedwhenever this does not cause confusion.

A mutation of an exceptional collection σ = (Eo, ...,En) is defined as a mutationofa pair of adjacent objects in this collection:

R,a = (Eo,..., Ej-\, Ei+1,REj+lEi,Ei+2,... ,En),

L,a = {Eo,... ,Ej_uLEiEi+i,Ei,El+2,- ..,En).

It is convenient to view the object RE+{Ei as translation of Ej to the right in thecollection σ = (Eo,...,En). We can do mutations again in the mutated collectionΛ,+ισ-in particular, to translate Rj+\Ej further right. The result of multiple transla-tion of the object Ej in the collection σ will be denoted by RkE,, and the resultingcollection by Rka. Analogously for left mutations.

26 Α. Ι. BONDAL

ASSERTION 2.1. A mutation of an exceptional collection is an exceptional collection.

The proof is analogous to the one given in [7], with RHom replaced by Horn .Let (XQ, ..., Xn) be a collection of objects in si. Denote by (XQ, ...,Xn) the mini-

mal full triangulated subcategory containing the objects Xj. We will say that a collec-tion of objects (XQ,...,Xn) generates the category si if {XQ,...,Xn) coincides withs/.

LEMMA 2.2. If an exceptional collection (EQ, ...,En) generates a category sf, thenthe mutated collection also generates si.

PROOF. Let si = (E0,...,En) and 38 = (E0,...,Ei+l,REj,...,En). ThenRHom(£j,£j+ i) ' <g> Ei+l belongs to si, since si is closed under direct sums andshifts. Hence, by (1), REj e si, and so 38 c si. We can analogously establish thatsi <z38, s i n c e t h e c o l l e c t i o n ( E o , . . . , E n ) is o b t a i n e d f r o m (EQ, ..., E i + l , R E j , . . . , E n )by left mutation of the pair (Ei+\,REi).

Let us regard i?, and L, (i = 0,..., η - 1) as operations on the set of exceptionalcollections.

ASSERTION 2.3. a) Rj and L, are mutually inverse. RjLj = 1.b) Ri (respectiveloy £,,·) induce actions of the braided group of η strings:

RiRi+\Ri = Ri+lRiRi+\, LJLJ+\LJ = Lj+\LjLj+\.

PROOF. In essence, for a) we need to prove that if (E, F) is an exceptional pair,then RELEF = F, and in b) that if (E, F, (?) is an exceptional triple, then the resultof right translation of Ε consecutively through F and G is equivalent to translationthrough G and RQF. These claims have been proved in this form in [7]. Below wegive another proof of this statement based on mutations of categories.

3. Mutations of categories

This section is devoted to an exposition of a more general approach to the notionof mutations, not connected explicitly with exceptional collections. On the otherhand, functoriality of the construction will be used below for exceptional collections.

For the sake of a transparent presentation let us define mutations in the followingsituation. Let us assume that V is a vector space over Κ with a nonsingular (non-symmetric) bilinear form / . Consider the following grading on V: V = 0 V, (i -0,...,«), with the condition χ( Vj, Vt) = 0 for j > i and the restriction of / to Vj non-degenerate. Mutation means replacing the grading Vi by a new grading V/, where onlytwo components in the grading are changed: V[ = Vi+X and V/+l =±Vi+ir\Vi®Vi+u

where x ^ + i is the left orthogonal to Vi+l i.e. x € χ ^ + ι ο x(x,Vi+l) = 0. Theconditions on the grading are preserved under this operation. The homomorphismπ, which is the composition

n=poh:Vi®Vi+l~ V'M Θ Vi+l Λ V'M,

when restricted to Vt gives the isomorphism of the space with the bilinear form

Vj -̂+ V'+\- The image of a vector from Vj under this isomorphism is called itsmutation.

Now we replace the space V by a triangulated category si, Vj by a subcategory s/j,and the form / by Horn (?,?).

DEFINITION. TWO triangulated subcategories 38 and Ψ of a triangulated categorysi are called orthogonal if for all objects X e 38 and Υ eW we have Honv(X, Y)= 0.


DEFINITION. Let 38 be a triangulated subcategory of sf. The full subcategorygenerated by the objects 7 e i such that for every X € 38 we have Hom^ (X, Y) = 0is called the right orthogonal to 38 and is denoted by 38^. The left orthogonal isdefined analogously.

LEMMA 3.1. Let Ψ = ^ ± . The following statements are equivalent:a) 38 and<£ generates/.b) For every X ess? there exists a distinguished triangle Β —> X —• C, where B e J

and C e Ψ.c) For the inclusion functor z*: 38 —> J / fAere w α n'gA? adjoint functor Ϊ: srf ->> 38

i.e., for all Aes/ and fieJ

d) For the inclusion functor j*:W—>sf there exists a left adjoint functor j * : s/, i.e., for every A e Λ/ a«c? C e ?

PROOF. The equivalence of c) and d) has been proved in [6]. Let us assume thatc) holds. Let a: iX —• X be the image of the identity morphism id,!^ under theisomorphism

Hom&(i-X, i-X) =i Hom^{ιΧ,Χ),

occurring in the definition of an adjoint functor (for simplicity of notation, we iden-tify the objects fleJ1 and UB e si). Let us embed a in the distinguished triangle

i-X ^X^C.

The object iX belongs to 3§. Consequently, in order to show that c) implies b), it isnecessary to establish that C e ? . Let Β e 38 and apply the functor Hom(fi,?), tothe triangle just obtained. We have the following long exact sequence:

-• Hom(5, iX) ^ Ylom{B,X) -• Ηοΐη(β, C) -• Horn1 (B, iX) H ' 1 .

From adjointness of the functors f and /* the following isomorphisms follow:Hom{B,iX) -^ Hom(B,X), V5 e 38, VX e sf. Applying the functorial propertyof these isomorphisms in the second argument to the morphism a, we see that theseisomorphisms coincide with α»; α,[1] is an isomorphism of the same type as a,, onlyfor the object X[l]. Thus, a* and a«[l] are isomorphisms. Then the aforementionedexact sequence gives Hom(fi, C) = 0, and therefore C e W

We show now that b) implies c). Let X e sf; then we have the triangle Β -*X —* C. Let i X = B. Let us show that such a triangle is unique up to a uniqueisomorphism and the correspondence extends to a functor. Let B' —» X' —* C beanother such triangle and let / : X' —» X be a morphism. We show that there is aunique ψ: Β' —> Β making the following diagram commutative:

Β -• X • C

B' • X< • C "

Applying the functor Hom(Z?',?) to the first triangle, and taking into account thatHorn'{B1, C) = 0, we see that Yiom(B',B) = Uora{B',X). In that case ψ is thepreimage of / ο g under this isomorphism. This proves that the correspondence

28 Α. Ι. BONDAL

extends to a functor uniquely. If we set X' = X in the diagram and if / is theidentity morphism, then we obtain a unique isomorphism as ψ. This ensures thatthe functor is well-defined on the objects. The existence of the unique morphismC —> C is proved analogously.

It is obvious that b) implies a). In order to prove the converse it is necessary tocheck that the full subcategory generated by the objects X which can be includedin the distinguished triangle Β —> X -+ C is closed with respect to shifts and theoperation of taking cones of morphisms. For the shifted object X[i] we have thetriangle B[i] -> X[i] -> C[i], where B[i] e 38 and C[i] e W, since 38 and Ψ aretriangulated categories. Let / : X —• X' be a morphism of objects of the desiredform. Then, as we established above, we have the following commutative diagram,which is unique up to a unique isomorphism:

C — f — + C • C,

1 1

t Iwhere Ch denotes the cone of a morphism h; C,C eW, and B,B' <E&. This diagramcan be closed using a distinguished triangle in the last column, by the generalizedoctahedron axiom ([3], p. 24). This triangle is the desired triangle. Reasoning ofthis type has been used in [3] in working with i-structures.

DEFINITION. 38 is an admissible subcategory of si if there are right and left adjointfunctors for the inclusion functor /»: 38 —> si. These adjoint functors will be denotedby z! and /*.

Let Ψ - &x and R&W = ±&. Then, according to Lemma 3.1, the inclusionfunctors j , : W —> si and j\ = r,: R&ff —> sf have respectively left and right adjointfunctors. Let us denote them by 7*: si —> 38 and r !: si -> R^'. Restricting j *to RggSi, we get the equivalence of the categories R^gW and W, since the inversefunctor will be rl restricted to Ψ.

DEFINITION. The category R&W, together with the functor R^ = r ! | ^ , R@: Ψ —>R,@ff, which realizes the transfer from the right orthogonal of an admissible category38 to the left one, is called the right mutation of W through £$.

Analogously the functor l^ = j*\R^W realizing the transfer from the left orthog-onal to the right is called the left mutation of R^W through &.

Let X be a topological space with a sheaf of rings @, let U an open subset of X, andF the closed complement of U. Let / and j be the embedding morphisms i: F —> Xand j : U -> X. Moreover, let us denote by J / = D+(X,<?),& = D+{F,<?) andΨ = D+(U,(?) the derived categories of the sheaves of if-modules over X, F, and Urespectively. Then 38 furnishes an example of an admissible subcategory of si, andW is the orthogonal complement. The corresponding functors have been described in[3], §1.4; the requirements for admissibility of the inclusion 38 —• si are equivalentto the gluing conditions ([3], 1.4.3.1-1.4.3.5), and the notation is compatible.

THEOREM 3.2. a) Let 38 = {E0,...,En) be a subcategory of si generated by anexceptional collection. Then £% is an admissible subcategory ofsi.


b) Set Ψ = SB1-·, then j*X = Ln+lX[n + 1], where Ln+lX is defined by induction:L°X = X, and

Lk+lX -» Hom(En_k,LkX) <g> En_k -> LkX A L*+1AT[1]. (3)

The notation is compatible with that adopted for mutations.PROOF. It follows easily from (3) by induction that L"+lX c W. By Lemma 3.1

it is necessary to show that every object X €sf embeds in the following triangle:

Β -> X -> Ln+]X[n + 1], where Be3S. (4)

We prove it by induction on the length of the collection. For η = -1 the claimis obvious. Assume that the claim has already been proved for the collection(Ει,..., En). For every l e j / we have

L"X[n-l]->B0-+X-^L"X[n], (5)

where Bo e (E\,. ,.,En). Let γ = a[n] ο β, where α is a morphism from the triangle(3) and β is from (5). Then we have the following commutative diagram ([3], p. 24):

φ —

1Β —

ϊ

— > 0 -

1— • χ -

•J

• φ [ ΐ ]

ί—γ—•* Ln+xX[n

UΒο > Χ —?-* LnX[n]

Since Bo and Φ belong to 38, the middle row of the diagram is a triangle of theform (4).

Let sfk, where k = 0,...,«, be a collection of admissible subcategories of srf withthe property that s/ι ns/j - 0 for / Φ j , Hom(j^,sfk) - 0 for / > k, and the sfk in totogenerate the category sf. Mutations of such a collection are determined in full anal-ogy with the case of a space with a nonsymmetric bilinear form. An example of sucha collection of categories may be obtained from exceptional collections, according toTheorem 3.2. To that end, it is necessary to break up the collection (Eo, ...,En) intosuccessive segments and consider the categories generated by exceptional objects inany one segment. If 38 = {E,,..., Ε j) (where j - i = k) is the category generatedby the elements of a segment and Ε is an exceptional object from the collection,with Ε e ±^ (i.e. Ε lies farther to the right than Ej in the collection), then definingLkE by (3) is compatible with the definition in §2, i.e. multiple left translation of anexceptional object through the exceptional collection (Ej,...,Ej) is a mutation L&Eof that object through the subcategory generated by the objects (Ej,...,Ej). Thismutation does not depend on the choice of the generating collection of the category38, which proves Assertion 2.3.

The mutation functor L·^ is a category equivalence; thus if the initial categoryis generated by an exceptional collection, then so will be the mutated category, andconsequently it will be admissible. This enables iterated mutations of such categories.

30 Α. Ι. BONDAL

It is possible to extend the class of admissible categories which preserve the prop-erty under mutations, using the formalism of Serre duality, but we will not need itin the sequel.

4. Helixes

It is interesting to consider the derived category Db(Sh(X)) of the category ofcoherent sheaves on a manifold X as an example of a triangulated category s/. Anexample of an exceptional collection on a projective space P" is a collection of sheaves&{i), i = 0,1,...,«. Mutations of this collection have been studied in [8].

Let (EQ,...,£•„) be an exceptional collection. We extend it to an infinite sequence(in both directions) of objects of s/ (£,·, i = -oo, . . . , +oo), defining by induction:

En+i = R"Ei_i and £_,· = L"En_i+l for i > 0. (6)

It has been shown in [8] that the exceptional collections of foliations on P m areconstructed through mutations of a collection {<f(i)} satisfying the condition E, =Ei+m+i(K), where the £,·, / = -oo, . . . , +oo, are taken in the above sense.

Such an infinite sequence was called there a helix. Let us extend its definition toan arbitrary manifold.

DEFINITION. A sequence Et (infinite in both directions) of objects of the derivedcategory Db(Sh(X)) of coherent sheaves on a manifold X of dimension m is called ahelix of period η if E, = Ei+n ® K[m - η + 1]. Here Κ is the canonical class and thenumber in square brackets denotes the multiplicity of the shift (per [8]) of an objectto the left viewed as a graded complex in Db(Sh(X)).

In the case of projective space P m the period η is equal to m + 1 and a shift in thederived category does not occur. For the quadric Qm, depending on the parity of thedimension, there is either a shift by one to the right (if m is even) or no shift (if mis odd).

Beilinson has shown with the aid of the diagonal resolvent that the collection <f(i)generates the category D*(Sh(P")), and Drezet, La Potier, Gorodentsev, and Rudakovhave shown that this is valid for mutated collections as well.

DEFINITION. An exceptional collection is called a thread of a helix if the sequenceconstructed by (6) is a helix of period η + 1.

Swan [17] has constructed exceptional collections on quadrics, and Kapranov hasdone it on Grassmannians, quadrics (independently), and flag manifolds [11], [12].We will show here that these collections are also threads of a helix.

THEOREM 4.1. Let (E0,...,En) be an exceptional collection of a foliation on amanifold X of dimension m with an abundant anticanonical class. Then the followingconditions are equivalent:

1) The collection E, generates the derived category Db{Sh(X)).2) The collection Et is a thread of a helix.

First we prove the following fact.

ASSERTION 4.2. The functor Hom{En,l)* is representable in the subcategory(Eo, ...,£•„) generated by the objects Eit and the representing object is LnEn[n], i.e.

Hom{En,X)* ^ Hom(X,L"En[n]), (7)

where X e (Eo, . . . , E n ) .

PROOF. Let us consider the subcategory 38 = (E0,...,En-i). By Theorem 3.2,there is a triangle Β -> En -> LnEn[n], where fieJ and L"En[n] e 38*-. From the


triangle we see that

Horn (En,L"En) = Horn (En,EH) = K.

This means that Hom(En, L"En[k]) = 0 for k φ 0, and Hom(£n, L"En) = K. Let usconstruct a correspondence between Hom(En,X) and Hom(X,L"En[n]), associatingwith any two morphisms their composition:

Hom{En,X) ®Hom(X,LnEn[n]) -^Hom{En,LnEn[n]). (8)

Let us show that this correspondence is nonsingular for every X. By Theorem 3.2for X we have the triangle Γ - » Ι - > L"X[n], where K e J 1 . Using that triangle, wecan rewrite (8) in the following form:

Hom(En,L"X[n])®Hom(LnX[n],LnEn[n])-+Hom(En,LnEn[n]);

L"X[n] belongs to &-1. If we consider the exceptional collection (LnEn,Eo,...,£n_i),we can see that 38 ̂ = {LnEn) (see Lemma 6.1 for more details). The category(L"En) consists of direct sums and shifts of the object LnEn, since it is exceptional.Therefore we need only check the nonsingularity of (8) for X = L"En, and in thiscase it is obvious.

PROOF OF THEOREM 4.1. 1)=>2). Suppose that {E0,...,En) is a collection fromthe hypothesis of the theorem. By Proposition 4.2, L"En[n] is the representing objectof the functor Hom(En,X)* in the category Db{Sh(X)). But Serre duality gives

Hom{En,X)* ^ Hom{X,En ®K[m]).

It follows from the uniqueness of the representing object that L"En[n] = En{K)[m\;hence £L| = LnEn = En(K)[m - n]. This is a helicity condition for / = - 1 . Sinceevery successive subcollection of length η + 1 in the sequence (6) is exceptional, thehelicity follows for every /.

2)=>1). Assume that the collection (E0,...,En) satisfies condition 2) of the the-orem. By Lemma 2.2, mutations of objects £, belong to the subcategory j / ={EQ,...,En). Consequently, all elements of the helix belong to sf. Taking intoaccount the invariance of sf under shifts, we have Eo{pK) € s/ for all ρ e Z.

It follows from the existence of an adjoint functor (see §3) that every object Xof the initial category embeds into a distinguished triangle ΪΧ —> X —> Υ, whereίΧ € sf and RHom(y4, Y) = 0, for all A e si. Thus, it is enough to show that allsuch Υ are equal to zero; Υ is represented by a finite complex C of objects of theabelian category of coherent sheaves on X. Let us suppose that the anticanonical classis very ample, and let us embed X (with the aid of K) into projective space φ: X —• Ρ'.Then q>*(Ci) will be coherent sheaves on P', and the operator of tensoring by Κ goesinto the operator of tensoring by &{\). We have

= Η*

= Η* (ρ*,φ. (εζ ® C

For p > 0, all higher Η' (i > 0) will be equal to zero. That means thatRHom '{En(pK), C) can be calculated with the aid of the complex K{p):

ί ® C

32 Α. Ι. BONDAL

with the natural differential. But RHom'(E0(pK),C ) = 0, and hence the complexΚ is acyclic.

A theorem of Serre [16] states that the abelian category of sheaves Sh(P') onP' = P(F / + 1 ) is isomorphic to the quotient category of finitely generated gradedmodules over S'(V*) modulo the full subcategory of finite-dimensional modules.Moreover, the isomorphism assigns the graded module 0 p Ηϋ{&~ ® <f{p)) to thesheaf &', where we may assume that ρ » 0. Isomorphism of categories induces anisomorphism of the derived categories. Since the complex φρ^,0Κ(ρ) is acyclic, thecorresponding object of the derived category Db(S') is equal to zero; hence, by Serre'stheorem, φ*{Ε$ ® C ) = 0. Consequently C = 0. Let us point out, moreover, thatin the case when Κ is not very ample it must be replaced by a very ample multiplenK; then the arguments are analogous for the rest. This proves the theorem.

5. Quivers

A quiver Δ is a set consisting of vertices and arrows between them. We will beinterested infinite quivers, i.e. quivers for which the number of vertices and arrowsis finite. A path is a sequence of arrows in which the beginning of each followingarrow coincides with the ending of the previous one. The path length is the numberof arrows in the path. Composition of paths is defined to be the concatenated path(if it is defined). Formal linear combinations with coefficients in a field Κ form analgebra of paths Κ A with respect to the operation of composition of paths. Here thecomposition β ο a of paths a and β is considered to be zero if the beginning of βdoes not coincide with the end of a. Vertices correspond to singular paths of length0. They turn out to be projections in the algebra of paths.

If S c Κ A is a subset, then the quotient algebra of the algebra Κ A of paths modulothe ideal generated by S is called a quiver with relations. The generators of the idealmay be chosen in the form of linear combinations of paths with the same beginningand same end. We will denote by KAk the ideal in KA generated by the paths oflength greater than or equal to k.

Quivers with relations represent a large class of algebras.Let A be a finite-dimensional associative algebra over a field K. The algebra A is

called a basis algebra if ^4/rad^ is the direct sum of several copies of Κ (radA isthe radical of the algebra A). Every algebra A' is Morita-equivalent to a basis algebraA. This means that their categories of representations are equivalent. A theorem ofGabriel says that every basis finite-dimensional ΑΓ-algebra is a quiver Δ with relationsS. If the assumption S c Κ A2 is made, then the quiver is uniquely determined. Inwhat follows we will assume this condition satisfied.

EXAMPLE 5.1. The quiver Pn contains two vertices and η arrows from the firstinto the second. For example, P2: • =t ·.

EXAMPLE 5.2. The quiver An contains η vertices X\,...,Xn and η - 1 arrowsφ ι : Xj —• X i + \ , i = 1 , . . . , « - 1 . F o r i n s t a n c e , A 3 : • — • · — » · .

EXAMPLE 5.3. The quiver Sn contains η vertices X\,...,Xn and (n - \)n arrowsφ\: Xi -> XM (i = Ι , . . . ,/ i - \;j = 1,...,«). The relations are φ\+ιφΊ = φ%χψ\.For instance S3:

relations: ψιψ] = y/j<Pi, i,j = 1 , 2 , 3 .

We point out that A2 coincides with P{ and S2 coincides with P2.

REPRESENTATIONS O F ASSOCIATIVE ALGEBRAS 33

EXAMPLE 5.4.

relations: /Τ,α, = <5,77·, i,j G {1,2}.

Let A be the algebra of paths of a quiver Δ with relations S: A — KA/(S). Thenthe image Κ A1 under the canonical epimorphisms Κ A —> A is the radical of A. Thecomplement of that radical contains the paths of length 0. They are indexed by thevertices of the quiver Δ and will be denoted by pa e A, where α is a vertex of Δ. Theelements pa are orthogonal projections: ραρβ = pppa = 0 for α Φ β, and p\ = pa.

A left ^-module, i.e. a vector space V over the field Κ with the left action of thealgebra A is called a representation of a quiver. The action of orthogonal projectionsdecomposes V into a direct sum: V = ®apaV. Let us write Va = paV. Then anarrow from the vertex α into the vertex β determines a linear operator Va -> Vp. Itbecomes clear from this that our definition of a representation of a quiver coincideswith the traditional one, where to every vertex a there corresponds a vector spaceVa, and to every arrow from a to β a morphism Va —> Υβ so that the relations S aresatisfied.

Representations of the quiver Δο ρ ρ obtained from Δ by inverting the arrows corre-spond to left modules over A, while the relations are written in the reverse order.

An ordered quiver with relations is a quiver in which the vertices are ordered andthe beginning of every arrow has an index smaller than the end (with the exception ofthe arrows of order 0). The quivers in Examples 5.1-5.4 are ordered. In Example 5.4this is achieved by placing the top and bottom vertices in an arbitrary order betweenleft and right vertices.

Let Δ be an ordered quiver with vertices X$,...,Xn, and let /?, be the projectioncorresponding to Xt in the algebra A of paths of Δ. In what follows, we will findit convenient to consider right modules over A. Let us denote the category theyform by Λ-mod. Every representation V of A has a decomposition V = φ ρ GjV,where G(V = Vpt. Let us denote by St the representation for which GjV = 0 forj φ i and (7, V = K, and all the arrows are represented by the zero morphisms. Themodules 5, (/ = 0,1,...,«) give a description of all the irreducible representationsof A. Indeed, every module V has a filtration by modules FkV = @^G,V. Thequotient FkV\Fk~x V is a direct sum of copies of S^. Projective modules of A aresubmodules of A as right module over itself, and are of the form Pk = pkA. Thedecomposition A = φ £ />, holds. Moreover,

A = ΗΟΠΙΛ(Λ,Λ) = ΗΟΠΙΛ 0 Ρ , , 0 ? , = 0(Hom(/>/,P,·))-ij

This equality allows for an interpretation of the arrows of a quiver as morphismsbetween projective modules. In particular, Hom(P(-,P,·) = 0 for / > j .

It is easy to check that GiPk — pkApi = 0 for I > k, and GkPk is a one-dimensionalspace. This enables one to construct the following exact sequence:

0^Fk-ip ρ S o. (9)

34 Α. Ι. BONDAL

LEMMA 5.5. Let V be a right Α-module, and assume that GiV — Ofor i > k. ThenVe(P0,...,Pk).

We recall that (Po> · · ·»Pk) is the triangulated subcategory generated by the subob-jects Po,...,Pk in ^'(mod-^).

PROOF (by induction). For k =: 0, the statement is obvious. If the statement hasbeen proved for k = s-1, then, for every V, Fs~lV e <Po, · · ·, Ps-\)· For V satisfyingthe hypotheses of the lemma for k = s, we have the exact sequence

where GSV is viewed as a direct sum of copies of Ss. But it follows from (9) thatSk e(PQ,...,Ps). Thus Ve(P0,...,Ps).

LEMMA 5.6. Let 38 = (Po,...,Pk-\) be a subcategory of Db(mod-A), and leti: 38 L - » J / be the inclusion functor. Then

i*Pk = Sk=LkPk[k]. (10)

PROOF. Consider the exact sequence (9). In it Fk~lPk belongs to 38 by Lemma5.5. We may interpret that sequence as a triangle in Z)6(mod-^). By Lemma 3.1, it isenough to verify that Sk e 38 L. Since the P, are projective modules, Extj (Pj,Sk) = 0for j φ 0. It remains to show that ΗΟΓΠ(.Ρ,·,5*) = 0 for i < j .

Every such homomorphism gives a collection of mappings on graded components:GjPi —• GjSk- However, Sk has only the kth component, and GkPt = 0 for / < k.Thus, all the morphisms are equal to zero. By Theorem 3.2, we obtain (10).

6. Functors into the category Db(mod-A) connected with an exceptional collection

Again, let si be an arbitrary triangulated category.DEFINITION. . An exceptional collection (£Ό, ...,Εη) of objects of the category J /

satisfying the conditions Horn*(£,·,£)) = 0 for all / and j , with k Φ 0, is called astrong exceptional collection.

A n e x a m p l e o f a s t r o n g e x c e p t i o n a l c o l l e c t i o n is (<?,<?(I),.. .,(f(n)) o n P " .

L E M M A 6.1. Suppose that stf is generated by a (not necessarily strong) exceptionalcollection (Eo, ...,En). Set φ = (Eo, ...,Ek) and 38 = (Ek+l, ...,En). Then Ψ = 33^.

PROOF. Obviously Ψ is orthogonal to 38 on the right. Let X e 38 ̂ and leti-.Ψ —• J / be the inclusion functor. Then we have the triangle i'X —> X —> Z,Ζ e Ψ^. Also, Ζ G 38^-, since i'X and X belong to 38L. Consequently Ζ isorthogonal to all the generating objects £, and hence equal to 0.

Write Ε = 0 £ £, and A = Hom(£, E); A is the algebra of paths of a finite orderedquiver with relations. This quiver contains η + 1 vertices, and for the projectivemodules of this algebra the following isomorphisms hold:

Honw(Ei,Ej) ^ HomA(Pj,Pj).

THEOREM 6.2. Assume that the bounded derived category sf = Db(Sh(X)) of co-herent sheaves on a smooth manifold X is generated by a strong exceptional collection(E0,...,En). Then ssf is equivalent to the bounded derived category Db(mod-A) ofright finite-dimensional modules over the algebra A.

PROOF. For every object Υ e J / , fix its finite flat quasiprojective resolvent I(Y),i.e. a finite complex of the flat quasicoherent sheaves quasi-isomorphic to Y. Letus construct a functor Φ from A into D^(mod-A)—the bounded derived category


of complexes of infinite-dimensional right modules over A with finite-dimensionalhomology. Since the categories D^(mod-A) and Db(mod-A) are equivalent, Φ willbe exactly the functor we are interested in. Set Φ(Υ) = RHom(E,I(Y)) with thenatural right action A = Hom(£', E) on that complex.

We show that Φ is a category equivalence; Φ(£,) is a complex of Λ-modules withhomology Ηί(Φ(Εί)) = 0 for j φ 0, and Η°(Φ(Εί)) is isomorphic to Pt. It followsthat Φ{Εί) is quasi-isomorphic to P, in the category D$(mod-A).

In order to prove that Φ is strictly full functor, i.e. that it determines the isomor-phism

Ηοτη{Χ,Υ)^Ηοτη(Φ(Χ),Φ(Υ)) (11)

for all X and Υ in sf, we will use induction. First of all we point out that (11) holdswhen X and Υ are elements of the collection {is,·}, since is, goes to P,. Since {is,·} is astrong exceptional collection, (11) can be extended to the set of shifts of the is,·. Butthe category si is generated by this set, with one operation of the cone, because theshift X[i] of the object X embedded in the triangle A -+ X —• Β is itself embeddablein the triangle A[i] -» X[i] -> B[i].

Now, by induction, we may assume that the objects X and Υ are embeddable inthe triangles A ^> X —> Β and C —> Υ —> D, where (11) has been already establishedfor the pairs (A, C), (B, C), {A, D), and (B, D). For the homomorphisms, there is thefollowing commutative diagram with exact rows and columns:

Τ Τ ΐ• Hom(i4, C) -Η. Ηοΐη(Λ,Γ) -• Hom(^,£>) - f · · ·

ΐ Τ Τ• Hom(.Y,C) -+ Ηοτη(Χ,Υ) -• Hom{X,D) - f · · ·

τ τ τ• Hom(B, C) -f Hom(B, Y) -> Hom(B, D) -> • • •

ΐ Τ ΤThe same diagram can be constructed for the triangles Φ{Α) —• Φ(Χ) —» Φ(Β) and

Φ{0) -»Φ(7) —» Φ(Ο). The functor Φ gives a morphism of these diagrams, and itis an isomorphism for the underlying spaces. By the exactness of rows and columns,it follows that Φ gives an isomorphism for Hom(X, Y). This proves that the functorΦ is strictly full.

Db{mod-A) is equivalent to the homotopy category of projective modules whichare direct sums of P,. Therefore, it is generated by the modules Pt. But the P, belongto the image of Φ. Since D*(mod-^) and D^(mod-A) are equivalent, the surjectivityon objects follows.

REMARK. If sf = Db(W) is a bounded derived category of an abelian categoryΨ with sufficiently many injective objects, then the statement remains valid and isproved in the same way (without the use of the unnecessary equivalence betweenDb(mod-A) and Db(mod-A)).

It would be interesting to prove the theorem in the case when szf is an arbitrarytriangulated category, not furnished with the structure of a derived category. Diffi-culties here arise with the attempt to define the value of the functor on morphisms.In constructing the realization functor for /-structures in [3], analogous complica-tions are overcome by introducing additional external data such as a filtered derivedcategory.

It would be even more important to prove an analogue of Theorem 6.2 for arbitraryexceptional collections. Already here, it is not entirely clear how to recover the

36 Α. Ι. BONDAL

differential graded algebra of homomorphisms of the elements of the collection fromthe inner structure of triangulated category.

We show by examples of coherent sheaves on manifolds how to realize the equiv-alence of categories.

EXAMPLE 6.3 Let s# = D6(Sh(P')). Let (<f, @(1)) be a strong exceptional collectionand A = Hom(<^ Θ (f (l),<f θ <f (1)); A is the algebra of paths of the quiver P2 fromExample 5.1. This quiver consists of two vertices and two arrows from the firstvertex into the second. The identity endomorphisms in <? and <f(l) correspondto the projections p\ and p2. Indecomposable (right) modules of algebra A weredescribed by Kronecker and are well known. They correspond to roots in the latticeof weights {λχ,λϊ), where λ χ is the multiplicity of the irreducible representations 5Ίand λ2 is the multiplicity of the representation S2 in the Jordan-Holder compositionseries. Real roots are weights of the form (n,n + 1) and (n + \,n), where η > 0.To these correspond complexes of coherent sheaves on P 1, respectively <f(n) and&{-n) [I] (according to Theorem 6.2). Imaginary roots are weights (η,η),η > 0.To every such root there corresponds a one-dimensional family of indecomposablerepresentations Vn>x, where χ € Pl. To the module Vn<x corresponds the sheaf of jetsup to (n — l)st order, inclusive, at the point x.

We can show that, for every quiver without relations (and this corresponds tothe property that the homological dimension of A equals 1), all the indecomposableobjects of the category Db(mod-A) are equivalent, up to a shift, to pure modules (i.e.to the complexes of the form 0 —> Μ —> 0).

EXAMPLE 6.4. Let us consider an exceptional collection {&,...,(f(n)) inDb(Sh(Pn)) and its associated algebra A = QuHom(<?{i),&(j)), which is the al-gebra of paths of the quiver in Example 5.3.

EXAMPLE 6.5. Let Q be a nonsingular quadric in P 3 . It is known that Q isisomorphic to P 1 x P 1 . The collection (<f,(f(0, l),*f ( l ,0),^( l , 1)) maybe consideredas an exceptional collection. Here &(i,j) = @{ΐ) &(?(j). Mutations of this collectionhave been studied in [18]. The algebra of Example 5.4 corresponds to it. Exceptionalcollections on quadrics of arbitrary dimension have been found in [17] and [11].

7. Koszul algebras

Let (E0,...,En) be a strong exceptional collection. Consider the algebra A =Q)"j=0Hom(Ej,Ej). It is the algebra of paths of an ordered quiver with relationsand η + 1 vertices. Let us denote by Ak the subspace of A generated by the pathsof length k; then A becomes a graded algebra: A = AQ φ Α\ θ Αι θ · · ·; Ao is thesubalgebra that is a sum of one-dimensional subalgebras: Ao = φ ο &Pi where p,is the ith orthogonal projection; the Aj are equipped with the structure of an Ao-bimodule. Let us assume that the arrows of the quiver exist only between adjacentvertices. This is equivalent to the condition that A is generated by Ao and Αχ. Inother words, A^ = Αχ ®Α0 A\ ® · · • ® Αχ /Ik, where the /& are subspaces of relationswith the structure of an Λο-bimodule.

An algebra A of the form just described is called quadratic, if all the relations Ik aregenerated by I2 € Α χ ®Αο Α ι. The generating elements of I2 are linear combinations ofpaths of length 2 with the same beginning and end. Formally, the quadratic propertyis expressed in the form of the following inclusion:

lk <Z h ® A \ ® - - • ® A\-\ h Αχ ® · · · ® Αχ ® 7 2,Ao AQ

where every summand contains k - 1 factors and is considered to be a subspaceof̂ f*.


For a given algebra A, we can define a dual algebra Β; Β is the quadric algebrafor which Bo = A0,B\ — A*, J2 — 1^, where Ji c B\ ®AQ Bx = A\ ® A*, and theorthogonal space is considered with respect to the natural pairing.

The space Κ = B* ®Αο A is provided with the structure of a complex. For everyΛο-bimodule V, set V'> — PiVpj. It is clear that A\j Φ 0 only for / = j + 1, sincethe elements of Α ι are the arrows of the quiver. Let us consider arbitrary bases e\ inthe spaces A\+lj and the dual bases ξ-f in the spaces B{^+1. Let us also consider thefollowing operator d: Κ —> Κ:

where /(£/')* is the operator conjugate to the operator of left multiplication by ζ\ inΒ and l(ej) is the operator of left multiplication by ej in A. The operator d has theproperty d2 = 0 [15].

DEFINITION. . The complex Κ = Β* ® A equipped with the differential d is calleda Koszul complex.

The differential preserves the ̂ 0-bimodule structure of the complex K; thereforethe K''J are invariant with respect to d. The spaces K1' are one-dimensional for all/, and the differential d on them is equal to zero.

DEFINITION. An algebra A is called Koszul if the homology groups of the complexΚ are isomorphic to 0[j Kli.

The differential preserves the structure of a right Λ-module. Together with theaction of AQ on the left, this enables the decomposition of Κ into a sum of complexesof ^-modules: Κ = φ Κ', where

Κ' = φ Kl-i = 0 Bmi-k ® Ak'J = 0 B*'<k ® Pk;

here the i\ are protective ^-modules: P^ = PkA, and the K' are graded complexes ofA -modules, with grading index equal to k:

• B*u~2 ® P/_2 -» β* ' · ' " 1 ® /»/_! -^ P/ -f 0.

The Koszul property implies that the complexes K' are exact in all members, exceptPi, where the homology is one-dimensional and isomorphic to P\'. This means thatwe have the exact sequence

> B*u~2 ® Pj_2 -+ B*iJ~' ® />,·_ ι - P, - S,- - 0, (12)

where S, is an irreducible module, corresponding to the /th vertex.

ASSERTION 7.1. An algebra A is Koszul if and only if Extk (Sj,Sj) = 0 for k Φ i-j.

PROOF, a) Let the algebra be Koszul. It has been shown in Lemma 5.6 thatHom(Pk,Sj) = 0 for k < j . By Lemma 5.5, Sk e(P0,..., Pk), and thus Hom(Pk,Sj) =0 for k > j . To compute the Ext's, we apply the functor Hom(?,Sy) to tne pro-jective resolvent of the module S1, obtained from the sequence (12). We get thatExt*(S/.S/) = 0 for k φ i - j , and Ext1"7'(S.-.Sy) = B'-J.

b) Suppose that, for instance, the /th complex of the form (12) is not exact. Sup-pose the expression B*''i®Pj stands at the first place from the right where the complexis not exact. Consider the corresponding segment of the complex (12):

• Bmi-J-2 ® Pj-i — B*iJ~' ®Pj-X^B*''J ® p,. -» . . . . (13)

38 Α. Ι. BONDAL

This complex decomposes into a sum of complexes K''s of graded componentswith respect to the right action of AQ:

• B * i J ~ l ® PJZ}'S - + B * u ® P f - > • • · .

For s = j,j — I this complex is exact at Pj. This follows from the quadric propertyof the algebra B. Let us choose the smallest 5 such that K''s is not exact at the placewe are interested in. Then s < j — 1. Let H''s be the homology space of the complexat the jth place. Then we can extend the complex (13) to a complex of projectivemodules, exact in all places, where the resolvent will have the following form at the7'th and (j - l)st places:

• Ρ Θ Β*'']~ι <g> Pj-χ φ / / j ' 1 ® Ps -» B*'-J ®Pj^ • S,• -• 0 .

Here Ρ is a sum of projective modules of the form Pt, where t <s < j — \ . Using thisresolvent to compute the E\tl~j+l (Si, Ss), we conclude that they are equal to (H'j's)*.Since s < j - 1, the claim has been proved.

It has been shown earlier (Lemma 5.6) that Sk = LkPk[k]. Thus,

Homk(V Pi, LJPj) = Homk+i~j (VPiVlUPjU]) = Extk+i~j (Si,Sj).

Using Theorem 6.2, we get

COROLLARY 7.3. The Koszul property of the algebra of homomorphisms of a strongexceptional collection {Ej} is equivalent to strong exceptionality of the collection{L'Ei}.

DEFINITION. The collection {L"En,Ln~lEn-\,...,E0} is called the left dual

collection of the collection {Eo, En}. Analogously, the right dual collection is{En,RE^u...,R"E0}.

8. Self-consistent algebras and mutations of strong exceptional collections

It would be interesting to determine under what conditions a collection preservesexceptionality under mutations. In the case of sheaves on projective spaces the mu-tability of a collection has been studied in [8].

Let us consider the infinite sequence (6), constructed on the strong exceptionalcollection σ = (EQ, .. .,En). We will call it a helix Sa. Let us call a mutation of acollection admissible if the collection obtained as a result of the mutation is a strongexceptional collection. A helix is called admissible if every thread of the helix is astrong exceptional collection. Following [8], mutations of a helix are denned to besimultaneous mutations of all objects with the mutual distance equal to the period.A helix mutation is admissible if the result of mutation is an admissible helix.

LEMMA 8.1. For an admissible helix S, the following conditions are equivalent:a) All mutations of S of the form Rk

E, where Ε is an element of the helix, areadmissible.

b) All mutations ofS of the form Lk

E are admissible.c) In every thread of the helix, mutations inside the thread of the form RE and LE,

where Ε is an element of the thread, are admissible mutations of the collection.

In helix mutations, it is sufficient to restrict ourselves to shifts by a distance smallerthan the period of the helix, because of periodicity.

PROOF. The equivalence of a) and b) follows from periodicity. Also, it is obviousthat a) and b) imply c).


Now let i ? | be a mutation of a helix S and let τ be a thread that contains RkE;ifΕ does not belong to τ, then R"E belongs to τ and the mutation RkE may be viewedas L"~kR"E, i.e. as a left mutation inside the thread. This proves that a) followsfrom c).

CONJECTURE. If one of the conditions of the lemma is fulfilled, then all mutationsof the helix are admissible.

It is clear that the conditions of Lemma 8.1 are necessary for this.We reformulate c) for an exceptional collection in terms of the homomorphism

algebra of that collection.Analogously to Koszul complex Κ = Β* <g> A of §7, we can define three more

complexes connected with the algebra A:

KX=A®B*, K2 = A®B, K3 = B®A.

Differentials in those complexes are given by the following formulas (in notation of§7):

di = Ε '(*/) ® '«/)·, d2 = Σ r ^ ) ® ι(Φ> ^ = Σ '(Φ ® zw')·The complex K\ may be interpreted as a Koszul complex for the algebra Aopp, and

K2 as the complex K3 for Aopp.Let us consider the complex K2 = A ® B. It is bigraded, since it is an Λο-bimodule:

K2 = φΚ^, K2

j = PjK2pj, and the K'2J are graded complexes with respect to the

differential d2:

0 — A i J -> A'-J-1 <8> BJ~lJ -Η. • A'-0 0 B°'J -f 0.

DEFINITION. An algebra A is called co-Koszul if the complexes K2

j for i Φ j areexact at all places except the last {A1-0 <g> B0·-*).

We specially point out that these requirements do not extend to the diagonal com-plexes κ γ.

DEFINITION. An algebra A is called self-consistent if A and Aopp are Koszul andco-Koszul.

This means that the complexes K'j and K'{j are exact for / φ j , and K2

j (re-

spectively K'jJ) may be nonexact only at the ends (respectively at the beginnings) for

i Φ j -

ASSERTION 8.2. The following statements are equivalent:a) The homomorphism algebra A of a strong exceptional collection is self-consistent.b) Mutations of the form Rk

E and LE inside σ are admissible.

PROOF. Following Theorem 6.2, we may identify £, with P,. Let us assume thatthe algebra is Koszul. Then, according to §7,

BjJ = Horn (L·Pi, UPj) = Horn (L'-jPi, Pj) = Hom*{Pi,R'-jPj)

= Hom(Ri-J-lPJ,Pi)*.

In particular, Hom'(R'~J~lPj,Pi) is concentrated as a complex in the zero-component. From this it follows easily that the complexes KJ

2 — (&jK2

J repre-sent the objects R"~'Pj, just as, in §7, K' represented L'Pi = Sj[-i]. Admissibilityof a mutation means the absence of Horn' for / Φ 0 between Pj and R"~'Ph where/' φ j . Computing them with the aid of the complexes K'2 and K', we immediately

40 Α. Ι. BONDAL

get the equivalence of the statements in the lemma. However we have assumedthe Koszul property. Consequently, it remains to prove that b) implies the Koszulproperty. To that end, according to Corollary 7.3 it is necessary to establish thatHomk(LiPhUPj) = 0 for fc ^ 0. We have

Horn {L·Pi, LjΡj) = Horn (Ll~JPt,Pj),

and this by condition b) implies the desired result.The quiver algebras in Examples 5.1 and 5.3 are self-consistent. The quiver An,

for η > 2, gives an example of a Koszul algebra that is not self-consistent.By all evidence, the complexes K!

2·' are an interesting characteristic of an algebra.For the collection {<f(i)} on P", they are exact everywhere except the end. In thegeneral case, the place where they are not exact (if there is only one) must have theform of a quantum super-grading.

We go back now to the geometric case.

9. Strong exceptionality in the geometric case

Let us denote by D-° the full subcategory in Db(Sh(X)) generated by the complexeswith homology concentrated in the positive dimensions. D-° is denned analogously.

Let us consider an exceptional pair (E,F) of objects of Db(Sh(X)).

LEMMA 9.1. a) Let Ε be a sheaf [ο Ε e D^° Π D^°) and F e D^°. Then LEF e

b) Let F € D^° Π D^° and Ε e D^°. Then RFE e D^°.

The proof follows immediately from considering the long exact sequence of thehomology functor applied to the triangles determining mutations (1).

Let J / = Db(Sh{X)), where I is a manifold of dimension n, and let σ =(£Ό,...,£„) be the exceptional collection consisting of sheaves and generating s/.We point out that the length of the collection is one greater then the dimension ofthe manifold.

ASSERTION 9.2. Mutations of the collection σ also consist of pure sheaves (i.e. com-plexes concentrated in the zero-component of the grading).

PROOF. Since the collection σ generates a category, by Theorem 4.1, S = Sa isa helix in the sense of §4, i.e. a shift by the period of the helix tightens the sheafinto the canonical class (it is not possible to shift in the derived category, since theperiod is one greater than the dimension of the manifold). Thus, all the elementsof the helix are pure sheaves. Using Lemma 9.1, we easily get that the iterated leftmutations Lk

sEt of the objects of the helix belong to D-°, and that the repeated rightmutations Rk

sEi belong to D-°. But it follows from the definition of a helix thatR^Ej - Ln

s~kEi+n+\. This means that RkEi e D-° Π D^°, i.e. that it is a pure sheaf.

THEOREM 9.3. The collection σ is a strong exceptional collection.

PROOF. It is possible to attain, through mutations, that every two elements of thecollection appear in succession without changing those elements. By Assertion 9.2 thenew collection also consists of sheaves; therefore it is sufficient to verify the absenceof higher Horn's between adjacent elements of the exceptional collection.

Let (E,F) be an exceptional collection. Then Horn'(E,F) = Horn (F,RFE)*.Hence, if Homk{E,F) φ 0 for k > 0, then Hom~k(F,RFE) φ 0. Again, accordingto Proposition 9.2, F and RFE are pure sheaves and have no negative Horn's—acontradiction.


From 9.2 and 9.3, we derive the following:

COROLLARY 9.4. The mutations σ are strong exceptional collections.

According to §8, the algebra of homomorphisms of the elements of the collectionis self-consistent and, in particular, Koszul.

Besides the collection {&{i)} on P", Theorem 9.3 is also applicable to exceptionalcollections on odd-dimensional quadrics [17], [11]. It is necessary to modify theformulation of the theorem in order to be able to use it on even-dimensional quadricsand Grassmannians, where exceptional collections also exist [10].

Let s/i, i = 0,...,«, be a collection of subcategories generating Db(Sh(X)), whereX is a manifold of dimension n. Assume that H o m ( ^ , j ^ ) = 0 for / > j and thatevery category is generated by a collection of mutually orthogonal exceptional sheaves(E[,... ,-Ej): Horn (E'p,E^) = 0 for all ρ and q. These data may be represented asan exceptional collection, completely decomposed into segments by the orthogonalsheaves occurring in one segment.

The categories sft are admissible by Theorem 3.2, and therefore we can mutatethem.

THEOREM 9.5. Under the formulated hypotheses, the collection {Ey. V/,j} is astrong exceptional collection. Its strong exceptionality is preserved under mutationsthrough the categories s/j.

The proof is analogous to that of Theorem 9.3.REMARKS a) The number of categories is one greater than the dimension of the

manifold.b) Under mutations through the subcategory s#i the exceptional objects occurring

in the collection will be the pure sheaves.c) The claim of the theorem is no longer true if we mutate through some of the

exceptional objects occurring in one category £/,.The collections for quadrics and Grassmannians, as has been shown in [12], satisfy

the conditions of the theorem, from which it immediately follows that the correspond-ing algebras are Koszul and self-consistent.

In conclusion I would like to thank A. L. Gorodentsev, Μ. Μ. Kapranov, and I.A. Panin for numerous useful discussions, and also A. N. Rudakov and A. N. Tyurinfor their attention and encouragement.

Received 29/MAR/88

BIBLIOGRAPHY

1. A. A. Beilinson, Coherent sheaves on P" and problems in linear algebra, Funktsional. Anal, i Prilo-zhen. 12 (1978), no. 3, 68-69; English transl. in Functional Anal. Appl. 12 (1978).

2. Alexandre [Α. Α.] Beilinson and Joseph Bernstein, Localisation de g-modules, C. R. Acad. Sci. ParisSer. I Math. 292 (1981), 15-18.

3. A. A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analyse et Topologie sur les EspacesSingulier. I (Proc. Colloq., Luminy, 1981), Asterisque, vol. 100, Soc. Math. France, Paris, 1982.

4. I. N. Bernshtein [Joseph Bernstein], I. M. Gel'fand, and V. A. Ponomarev, Coxeter functors andGabriel quivers, Uspekhi Mat. Nauk 28 (1973), no. 2 (170), 19-33; English transl. in Russian Math.Surveys 28 (1973).

5. Sheila Brenner and M. C. R. Butler, Generalizations of the Bernstein-Gelfand-Ponomarev reflectionfunctors, Representation Theory. II (Proc. Second Internat. Conf., Ottawa, 1979), Lecture Notes in Math.,vol. 832, Springer-Verlag, 1980, pp. 103-169.

6. J. L. Verdier, Categories derivees Quelques resultats (etat 0), Seminaire de Geometrie Algebrique duBois-Marie (SGA 4±), Lecture Notes in Math., vol. 569, Springer-Verlag, 1977, pp. 262-311.

42 Α. Ι. BONDAL

7. A. L. Gorodentsev, Mutations of exceptional foliations on P", Izv. Akad. Nauk SSSR Ser. Mat. 52(1988), 3-15; English transl. in Math. USSR Izv. 32 (1989).

8. A. L. Gorodentsev and A. N. Rudakov, Exceptional vector bundles on projective spaces, Duke Math.J. 54(1987), 115-130.

9. J.-M. Drezet and J. Le Potier, Fibres stables et fibres exceptionnels sur P 2 , Ann. Sci. Ecole Norm.Sup. (4) 18(1985), 193-243.

10. Μ. Μ. Kapranov, On the derived category of coherent sheaves on Grassmann manifolds, Izv. Akad.Nauk SSSR Ser. Mat. 48 (1984), 192-202; English transl. in Math. USSR Izv. 24 (1985).

11. , Derived category of coherent bundles on a quadric, Funktsional. Anal, i Prilozhen. 20 (1986),no. 2, 67; English transl. in Functional Anal. Appl. 20 (1986).

12. , On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92(1988), 479-508.

13. Robert MacPherson and Kari Vilonen, Elementary construction of perverse sheaves, Invent. Math.84 (1986), 403-435.

14. Christian Okanek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projectivespaces, Birkhauser, 1980.

15. Stewart B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39-60.16. Jean-Pierre Serre, Faisceaux algebriques coherents, Ann. of Math. (2) 61 (1955), 197-278.17. Richard G. Swan, K-theory of quadric hyperswfaces, Ann. of Math. (2) 122 (1985), 113-153.18. A. N. Rudakov, Exceptional vector bundles on a quadric, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988),

788-812; English transl. in Math. USSR Izv. 33 (1989).

Translated by R. M. DIMITRIC