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H3B. Axaa. HayK CCCP Cep. MateM. TOM 53(1989), J f 1 Ve

Math. USSR Izvestiya Vol. 34(1990), No. 1

REPRESENTATION OF ASSOCIATIVE ALGEBRAS AND COHERENT SHEAVESUDC512 A. I. BONDAL ABSTRACT. It is proved that a triangulated category generated by a strong exceptional collection is equivalent to the derived category of modules over an algebra of homomorphisms 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 of an exceptional collection generating the category. The connection between mutability of strong exceptional collections and the Koszul property is studied. It is proved that in the geometric situation mutations of exceptional sheaves consist of present sheaves and 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 the categories of representations of finite-dimensional associative algebras. The main tool is the notion of exceptional collections or, in a more general situation, that of a semi-orthogonal collection of admissible subcategories. Let be an exceptional object of an abelian category sf. This means that Ext' (E,E) 0 for i > 0. Then, using E, one can construct a functor FE from the category sf into the derived category Db(mod-A) of representations of the algebra

A = Hom(E,E): FE(M) = RHom(E,M),FE{M) is a complex of right -modules. The functor F is extendable to the derived functor DbF, from Dbsf to Db(mod-A). If 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 on the projective space P". Beilinson has shown in [1] that if we set EQ = " = 0 ^ ( / ) , then DbFEo is an equivalence of categories. Later, Drezet and Le Potier [9] and Gorodentsev and Rudakov [8] constructed the whole series of exceptional vector bundles 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 actions of the Artin braid group. Kapranov [ 12] has constructed exceptional collections on quadrics, Grassmannians, and flag manifolds.1980 Mathematics Subject Classification (1985 Revision). Primary 16A46, 16A64, 18F20. 1990 American Mathematical Society 0025-5726/90 $1.00 + $.25 per page23

24

. . BONDAL

Projective modules over finite-dimensional associative algebras are another example of an exceptional collection. Mutations of such a collection generalize reflection functors [4] and tilting modules [5], which are used in the representation theory of quivers. From the point of view of the theory of quivers, the investigation of exceptional objects can be motivated in the following way: This theory is engaged in a classification of finite-dimensional associative algebras with tame representation theory. However, tame algebras are only a small isle in the ocean of wild algebras. What to do with the wild algebras? First of all one tries to describe all the simple nonvariable representations. Among them, the exceptional objects occupy in important place from a general functorial point of view. Second, one tries to partition the set of algebras (or, more generally, differentially-graded algebras) into classes, depending on the properties of mutability of exceptional collections, and, third, to define a notion of stability of algebra representations and classify the stable modules. The analogy with 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 successfully used in every triangulated category. If i: 38 > sf is an embedding of the subcategory generated by the elements of an exceptional collection into the base category, then it will be shown in Theorem 3.2 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 variant of the bar construction [12]. It was noted in [8] that the mutations of an exceptional collection generate a helix. Theorem 4.1 shows that this is connected with the fullness of an exceptional collection. Note that by identifying the derived categories of coherent sheaves on a manifold with modules over an algebra, the tightening (nozycpyTKa) functor on the canonical class transforms into the derived Nakayama functor or, as it is called in quiver theory, into the Coxeter functor. We demonstrate further, through some examples, the way geometry and algebra relate. Thus, for instance, representations of quivers which consist of two vertices and two arrows from one vertex into another correspond to the sheaves on P'. As we know, this is a time quiver, and its representations were described long ago by Kronecker. The unique parameter determining indecomposable representations is also the parameter on the projective line. In 7 we define Koszul algebras with ordered projective objects and prove that the Koszul condition is equivalent to strong exceptionality of the dual collection, which is constructed via mutations of projective modules and consists of irreducible objects translated along the derived category. In 8 we investigate, from a purely algebraic point of view, the questions of preservation of the property of strong exceptionality under mutations. It turns out to be necessary to impose some homological conditions on the algebra of homomorphisms between the elements of the exceptional collection. We call the algebras satisfying these conditions self-consistent. Their independent study is obviously important on its own. Finally, we prove in 9 that geometry furnishes examples of self-consistent algebras. To conclude this Introduction, we point out an interesting connection of the aforementioned 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 complexes with homology that is locally constant on stratas has a full exceptional collection. Using a known correspondence of sheaves, subject to stratification by Schubert cells of the flag manifolds with modules over a semisimple Lie algebra, we obtain an exceptional 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 objects in sf; Hom(A,B) is a vector space over a field K. We introduce the following notation for the graded complex of A"-vector spaces with trivial differential:

Horn \A, B) = Homk^(A,B)[-k]; kezhere Hom^(/1, ) = (, TkB), where is a translation in the triangulated category J / , and the number in square brackets denotes that the space (, TkB) has grading equal to k. In the case when si is the derived category of an abelian category, Horn (A, B) is quasi-isomorphic to RHom(^,5). DEFINITION. An object is called an exceptional object if it satisfies the following conditions: Horn'\E, E) = 0 for i 0, Hom(\ E) = K.DEFINITION. An ordered collection of exceptional objects (E0,...,En) exceptional collection in s/ if it satisfies the condition

is called an

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 to the case of arbitrary triangulated categories: DEFINITION. Let (E, F) be an exceptional pair. We define objects LEF and RFE with the aid of distinguished triangles in the category s/:

LEF ^Hom{E,F)E->F, -+Hom(,.F)* F -* RFE;

{

'

here V[k] E, where V is a vector space, denotes an object equal to the direct sum of dim V copies of the object TkE. Under conjugation of vector spaces, grading changes the sign. A left (right) mutation of an exceptional pair = {E,F) is the pair LET - {LF,E) (respectively, REx = (F,RE)). Lower indices will be omitted whenever this does not cause confusion. A mutation of an exceptional collection = (Eo, ...,En) is defined as a mutation ofa 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 the collection = (Eo,...,En). We can do mutations again in the mutated collection ,+-in particular, to translate Rj+\Ej further right. The result of multiple translation of the object Ej in the collection will be denoted by RkE,, and the resulting collection by Rka. Analogously for left mutations.

26

. . 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 minimal full triangulated subcategory containing the objects Xj. We will say that a collection of objects (XQ,...,Xn) generates the category si if {XQ,...,Xn) coincides with s/. LEMMA 2.2. If an exceptional collection (EQ, ...,En) the mutated collection also generates si. generates a category sf, then

PROOF. Let si = (E0,...,En) and 38 = (E0,...,Ei+l,REj,...,En). Then RHom