aa r X i v : . [ m a t h . AG ] J u l THICK SUBCATEGORIES ON CURVES
ALEXEY ELAGIN AND VALERY A. LUNTS
Abstract.
We classify thick subcategories
T ⊂ D b (coh C ) for smooth projectivecurves C over an algebraically closed field. Introduction
Let C be a smooth projective curve of genus g over an algebraically closed field k .We classify all finitely generated thick (triangulated) subcategories of D b (coh C ) .Namely we prove that all such subcategories T (if T 6 = 0 , D b (coh C ) ) are quiver-like , that is there is a finite quiver Q and an equivalence of categories D b ( Q ) ∼ → T where D b ( Q ) ⊂ D b ( Q ) is the full triangulated subcategory generated by the simplemodules corresponding to vertices (Theorem 4.4).We then classify the quivers Q which can be realized on curves in this way (The-orem 5.2). We also show that if Q and Q ′ are realizable quivers and there is anequivalence D b ( Q ) ≃ D b ( Q ′ )then Q ≃ Q ′ (Corollary 5.8).As a byproduct we obtain the following result (Propositions 4.8, 4.9): If g = 0 , , there are no infinite descending chains of finitely generated thick sub-categories of D b (coh C ) . On the other hand, if g ≥ , then there is an infinitedescending binary tree of such subcategories .This phenomenon should be compared with the case of 0 -dimensional schemes.Namely, let R be an artinian algebra. If R is a complete intersection, then there areno infinite descending chains of finitely generated thick subcategories of D b ( R − mod) ,see [CI]. On the other hand in [EL] simple examples of a non-complete intersection R are constructed, such that there exists a descending binary tree of such subcategories.The paper is organized as follows. Section 2 contains a brief reminder on triangu-lated categories and enhancements. In Section 3 we study quiver-like categories. In Section 4 we formulate our main observation: all thick subcategories on curves arequiver-like. In Section 5 we classify quivers which are realizable on curves (Definition4.5).2.
A reminder about generation and enhancements of triangulatedcategories
We fix a field k . All our categories are k -linear. References for triangulated anddg categories include [BK], [BN], [BLL], [Dr], [ELO], [Ke].If T is a triangulated category and X, Y are objects in T , we will freely use theequivalent notation for the corresponding space of morphismsHom( X, Y [ n ]) = Hom n ( X, Y ) = Ext n ( X, Y ) . If X = Y , then we also consider the graded algebraExt •T ( X, X ) = Ext • ( X, X ) = M n Ext n ( X, X ) . A triangulated category is Ext -finite if the space L n Hom(
X, Y [ n ]) is finite di-mensional for all objects X, Y .We say that a triangulated category T is non-split generated by a collection ofobjects X , . . . , X n if T is the smallest full triangulated subcategory of T whichcontains the objects X , . . . , X n . We denote this T = [ X , . . . , X n ] .We say that a triangulated category T is (split) generated by a collection of objects X , . . . , X n if T is the smallest full triangulated subcategory of T which containsthe objects X , . . . , X n and which is closed under direct summands in T . We denotethis T = h X , . . . , X n i .A thick subcategory of a triangulated category T is a full triangulated subcategorywhich is closed under direct summands in T .A category is Karoubian if it is idempotent complete, i.e. if every idempotentsplits. Note that a thick subcategory of a Karoubian triangulated category is alsoKaroubian. If a triangulated category T is closed under countable direct sums, thenit is Karoubian.For an abelian category A we denote by D ( A ) its (unbounded) derived categoryand by D b ( A ) its bounded derived category, these categories are triangulated. If A has countable direct sums then D ( A ) is Karoubian. Therefore all thick subcategoriesof D ( A ) are also Karoubian. HICK SUBCATEGORIES ON CURVES 3 If S is a dg category we denote by [ S ] its homotopy category. If the dg category S is pre-triangulated , then [ S ] is triangulated.An enhancement of a triangulated category T is a pre-triangulated dg category S together with an equivalence of triangulated categories [ S ] ∼ → T . This allows us toconsider an object X ∈ T as an object in the dg category S . Then we denote itsendomorphism dg algebra by R End( X ) := End S ( X ) . This dg algebra is well defined up to a quasi-equivalence and its cohomology algebrais H • ( R End( X )) = Ext •T ( X, X ) . For a dg algebra E we consider the triangulated category Perf( E ) . This is thethick subcategory in D ( E ) (= the derived category of right dg E -modules) whichis (split) generated by the dg E -module E . The triangulated category Perf( E ) isKaroubian and it has a natural enhancement. If dg algebras E and E ′ are quasi-isomorphic, then the triangulated categories Perf( E ) and Perf( E ′ ) are equivalent. Adg algebra is formal if it is quasi-isomorphic to its cohomology graded algebra.We will often use the following standard fact. Proposition 2.1.
Let T be a Karoubian triangulated category which has an enhance-ment. Assume that T is generated by an object X , i.e. T = h X i . Consider the dgalgebra E = R End( X ) . Then there exists a natural equivalence of categories T ≃
Perf( E ) . Finally, for a ring R we denote by Mod − R (resp. mod − R ) the category of right R -modules (resp. finitely generated right R -modules).3. Quiver-like triangulated categories A quiver means a finite quiver, i.e. the set of vertices and arrows is finite.Let Q be a quiver with n vertices v , . . . , v n . Let k Q be the corresponding(hereditary) path algebra. Denote by D b ( Q ) = D b (Mod − k Q ) the bounded derivedcategory of right k Q -modules. Since the algebra k Q is hereditary, every object in D b ( Q ) is isomorphic to the direct sum of its cohomology. ALEXEY ELAGIN AND VALERY A. LUNTS
Let
R ⊂ k Q denote the radical of k Q , i.e. R is the 2 -sided ideal generated by allarrows. Denote by (mod − k Q ) R the abelian category of finitely generated (=finite-dimensional) R -torsion k Q -modules. Define D b ( Q ) ⊂ D b ( Q ) as the subcategory ofall finite complexes with cohomology in (mod − k Q ) R . The triangulated categories D b ( Q ) and D b ( Q ) are Karoubian.Let s , . . . , s n be the simple k Q -modules, corresponding to the vertices. For futurereference we record the following easy fact. Lemma 3.1.
In the above notation the following holds:(1) Every object in (mod − k Q ) R has a finite filtration with s i ’s as subquotients.The corresponding associated graded module is independent of the filtration and (3.1) K ((mod − k Q ) R ) ≃ M i Z [ s i ] . (2) Every object in D b ( Q ) is isomorphic to the direct sum of its cohomology, and K ( D b ( Q )) ≃ M i Z [ s i ] . (3) D b ( Q ) = h s , . . . , s n i = [ s , . . . , s n ] , i.e. D b ( Q ) is non-split generated by the s i ’s.Proof. (1) Let M ∈ (mod − k Q ) R , then the quotients of the (finite) filtration M ⊃ M · R ⊃ M · R ⊃ . . . are directs sums of s i ’s. The isomorphism (3.1) is given by thedimension vector of k Q -module. (2) follows from gldim( k Q ) = 1 . In (3), inclusions D b ( Q ) ⊃ h s , . . . , s n i ⊃ [ s , . . . , s n ] are obvious, while D b ( Q ) ⊂ [ s , . . . , s n ] followsfrom (1) and (2). (cid:3) We will not need the following lemma but include it here for the interested reader.
Lemma 3.2.
The natural functor
Ψ : D b ((mod − k Q ) R ) → D b ( Q ) is an equivalence.Proof. Note that both categories are Karoubian and generated by the object ⊕ s i .Moreover, Ψ( s i ) ≃ s i for any i and thus Ψ is essentially surjective. To check that Ψis fully faithful it suffices (using the standard devissage technique) to check that Ψinduces isomorphismsExt m (mod − k Q ) R ( s i , s j ) = Hom mD b ((mod − k Q ) R ) ( s i , s j ) → Hom mD b ( Q ) ( s i , s j ) = Ext m Mod − k Q ( s i , s j ) HICK SUBCATEGORIES ON CURVES 5 for any i, j and m ∈ Z . For m < m = 0 this holds since(mod − k Q ) R ⊂ Mod − k Q is a full subcategory. For m = 1 this holds by Yoneda’s de-scription of Ext groups since the subcategory (mod − k Q ) R ⊂ Mod − k Q is extension-closed. For m > m Mod − k Q ( s i , s j ) = 0 since k Q is hereditary, let us checkthat Hom mD b ((mod − k Q ) R ) ( s i , s j ) = 0 for m > f : s i → s j [ m ] in D b ((mod − k Q ) R ) has the form s i q ←− C • p −→ s j [ m ] , where C • is a bounded complex over (mod − k Q ) R , p, q are homomorphisms ofcomplexes and q is a quasi-isomorphism. We claim that there exists a complex P • = [ P − → P ] over (mod − k Q ) R and a quasi-isomorphism s : P • → C • . Then f = pq − = pss − q − = 0since ps = 0 (recall that m > P • = [ ¯ P − d −→ ¯ P ] be a resolution of s i by projective finitelygenerated k Q -modules. There exists a quasi-isomorphism ¯ s : ¯ P • → C • . Since C k are R -torsion modules, one can take N such that C k · R N − = 0 for k = 0 , − P := ¯ P / ( ¯ P · R N ) . By assumptions, ¯ s : ¯ P → C factors via P .Recall that H • ( ¯ P ) ≃ s i , hence d is injective (we will treat ¯ P − as a submodule in¯ P ) and ¯ P · R N ⊂ ¯ P · R ⊂ ¯ P − . Let P − := ¯ P − / ( ¯ P · R N ) . Then P • is quasi-isomorphic to ¯ P • (and to s i ). Clearly P • is a complex over (mod − k Q ) R . Also¯ s − ( ¯ P · R N ) ⊂ ¯ s − ( ¯ P − · R N − ) ⊂ ¯ s − ( ¯ P − ) · R N − ⊂ C − · R N − = 0 , hence ¯ s − : ¯ P − → C − factors via P − . Therefore, ¯ s factors via a quasi-isomorphism s : P • → C • . This concludes the proof of the claim and the lemma. (cid:3) Definition 3.3.
A triangulated category T is called quiver-like if there exists a finitequiver Q and an equivalence of triangulated categories D b ( Q ) ≃ T . We obtain the immediate consequence of Definition 3.3 and Lemma 3.1.
Corollary 3.4.
Let T be a quiver-like triangulated category with an equivalence Φ : D b ( Q ) → T . Put t i = Φ( s i ) . Then we have the following. ALEXEY ELAGIN AND VALERY A. LUNTS (1) For any indecomposable object B ∈ T there exists a sequence of objects B , . . . , B m such that B m = B [ d ] for some d ∈ Z , B = 0 , and for each i = 1 , . . . , m the object B i fits into an exact triangle B i − → B i → t j i → B i − [1] for some j i .(2) K ( T ) ≃ L j Z [ t j ] .(3) T = [ t , . . . , t n ] , i.e. T is non-split generated by the t j ’s. Lemma 3.5.
Let A be an abelian category in which for every object A there existsan injective resolution → A → I → I → . Let A , . . . , A n be objects in A such that Hom( A i , A j ) = δ ij · k and Ext s ( A i , A j ) = 0 for all i, j and s = 0 , . Then the dg algebra R End( ⊕ ni =1 A i ) is formal.The same holds for projective resolutions.Proof. Choose injective resolutions A j → I • j of length 1 . Then the dg algebra R End( ⊕ j A j ) is quasi-isomorphic to E := End( ⊕ j I • j ) = E − ⊕ E ⊕ E . Let e j ∈ E be the idempotent of the summand I • j . Then E = ⊕ i,j e i E e j , E = ⊕ i,j e i E e j and d sends e i E e j to e i E e j . Choose a subspace V = V ⊕ V of E ,where V = ⊕ j k e j ⊂ E , V i,j ⊂ e i E e j is any complement of d ( e i E e j ) ⊂ e i E e j ,and V = ⊕ i,j V i,j ⊂ E . Then V is a dg subalgebra of E and the inclusion V ⊂ E is a quasi-isomorphism.In case of projective resolutions the proof is similar. (cid:3) Corollary 3.6.
Let Q be a quiver and consider the category D b ( Q ) with the standardenhancement coming from the embedding D b ( Q ) ⊂ D b ( Q ) . Then the dg algebra R End( ⊕ s i ) is formal. Hence there is an equivalence of triangulated categories D b ( Q ) ≃ Perf(Ext • ( ⊕ s i , ⊕ s i )) where the graded algebra Ext • ( ⊕ s i , ⊕ s i ) is considered as a dg algebra with zero dif-ferential. HICK SUBCATEGORIES ON CURVES 7
Proof.
The formality of the dg algebra R End( ⊕ s i ) follows from Lemma 3.5 appliedto the abelian category A of all right k Q -modules and taking A i = s i .Because the category D b ( Q ) is Karoubian by Proposition 2.1 we get the equivalence D b ( Q ) ≃ Perf( R End( ⊕ s i )) . The last assertion then follows from the fact that the dg algebras Ext • ( ⊕ s i , ⊕ s i ) and R End( ⊕ s i ) are quasi-isomorphic. (cid:3) Definition 3.7.
Let T be a triangulated category with an enhancement. A collectionof objects { t , . . . , t n } is T is called vertex-like if the endomorphism dg algebra R End( ⊕ t i ) is formal, and in addition Hom( t i , t j ) = δ ij · k , and Hom p ( t i , t j ) = 0 for all i, j and p = 0 , . Remark 3.8.
It follows from Corollary 3.6 that the collection of objects { s , . . . , s n } in the category D b ( Q ) is vertex-like. Note that the dimension of the space Ext ( s i , s j ) is equal to the number of arrows from v j to v i . The next proposition gives a necessary and sufficient condition for a category to bequiver-like.
Proposition 3.9.
Let T be an Ext -finite triangulated category. The following con-ditions are equivalent.(1) T is quiver-like.(2) T ≃
Perf( E ) where E = E ⊕ E is a dg algebra with zero differential andsuch that E = k × . . . × k .(3) T is Karoubian, it has an enhancement and it is generated by a collection ofobjects { t , . . . , t n } that is vertex-like.Moreover, if T satisfies (3), then there exists a quiver Q and an equivalence Φ : D b ( Q ) → T such that Φ( s i ) = t i .Proof. (1) ⇒ (2) is contained in Corollary 3.6.(2) ⇒ (3). Since T ≃
Perf( E ) , it is Karoubian and has an enhancement. Let e , . . . , e n ∈ E be the idempotents corresponding to the factors in E = k × . . . × k .Then the right dg E -modules e i E are h-projective, they generate Perf( E ) , and the ALEXEY ELAGIN AND VALERY A. LUNTS dg algebra R End( M i e i E ) = R End( E ) = E is formal. In addition Hom( e i E, e j E ) = δ ij · k , Hom p ( e i E, e j E ) = 0 for all i, j and p = 0 , t i = e i E .(3) ⇒ (1). Consider the graded algebra E = Ext • ( ⊕ t i , ⊕ t i ) as a dg algebra withzero differential. Our assumptions imply that the category T is equivalent to thecategory Perf( E ) .Now define the quiver Q with vertices v , . . . , v n and the number of arrows from v i to v j equal to dim Hom( t j , t i [1]) . Let s , . . . , s n ∈ D b ( Q ) be the correspondingsimple modules. By construction, we have Ext • ( t i , t j ) ≃ Ext • ( s i , s j ) for all i, j . ByCorollary 3.6 we get D b ( Q ) ≃ Perf(Ext • ( ⊕ s i , ⊕ s i )) ≃ Perf( E ) ≃ T , i.e. T is quiver-like. This proves the implication (3) ⇒ (1) and also the last assertionof the proposition. (cid:3) Proposition 3.10.
Assume that the field k is algebraically closed. Let A be anabelian ( k -linear) category such that any object A ∈ A has an injective resolution oflength . Assume that the category D b ( A ) is Karoubian. Let T ⊂ D b ( A ) be afinitely generated Ext -finite thick subcategory. Assume there exists a linear function r : K ( T ) → Z , such that for any nonzero F ∈ T ∩ A one has r ([ F ]) > . Then the category T satisfies the condition (3) in Proposition 3.9, and hence it is quiver-like.The same holds for projective resolutions.Proof. Note that A is hereditary and thus any object in T is a direct sum of itscohomology. It follows that one can choose a finite set of generators in T belongingto A ⊂ D b ( A ) . Take any family A , . . . , A n ∈ A of nonzero objects generating T such that(1) P i r ([ A i ]) is the minimal possible;(2) the number n is the maximal possible among all families with the fixed P i r ([ A i ]) .(such a family exists because r ([ A ]) > A ∈ T ∩ A ). Note that A i ≇ A j for i = j . We claim that the family { A , . . . , A n } is vertex-like. HICK SUBCATEGORIES ON CURVES 9
First we check that there are no morphisms between A i ’s except for scalar multipli-cation. Let f : A i → A j be a morphism. Denote K := ker f, I := im f, C := coker f .Since A is hereditary, the complex Cone ( f ) in T is quasi-isomorphic to K [1] ⊕ C .Since T is thick we get that K, C ∈ T . Also we get I ∈ T and h A i , A j i = h K, I, C i . If i = j we have r ([ A i ]) + r ([ A j ]) = r ([ K ]) + r ([ I ]) + r ([ I ]) + r ([ C ]) = ( r ([ K ]) + r ([ I ]) + r ([ C ])) + r ([ I ]) . Replacing A i , A j with K, I, C we get a generating family with the smaller P i r ([ A i ])unless I = 0 , it contradicts to condition (1). Hence f = 0 . If i = j we get r ([ A i ]) = r ([ K ]) + r ([ I ]) = r ([ I ]) + r ([ C ]) . Replacing A i with K, I we get a generating family with the same P i r ([ A i ]) andwith the bigger number of objects unless I = 0 or K = 0 , it contradicts to condition(2). Hence f = 0 or K = 0 . Similarly, replacing A i with I, C we see that I = 0or C = 0 . Thus, if f = 0 then K = C = 0 and f is an isomorphism. We provedthat for each i the endomorphism algebra End A i is a finite-dimensional division k -algebra. Since k is algebraically closed, End A i = k .Clearly, Hom s ( A i , A j ) = 0 for s = 0 , R End( ⊕ i A i ) isformal by Lemma 3.5. Therefore { A , . . . , A n } is a vertex-like collection. Also T isKaroubian, hence it satisfies the condition (3) of Proposition 3.9. (cid:3) Corollary 3.11.
Let A be a hereditary k -algebra over an algebraically closed field.Let T = h M , . . . , M n i ⊂ D b (Mod − A ) be any thick subcategory generated by finite-dimensional (over k ) modules. Then T is quiver-like.Proof. We use Proposition 3.10. Namely, we take A = Mod − A and r to be thefunction induced by the dimension of a module over k . (cid:3) Corollary 3.12.
Let T be a quiver-like triangulated category. Then any finitelygenerated thick subcategory T ′ ⊂ T is also quiver-like.Proof. We may assume that T = D b ( Q ) for a quiver Q and use Corollary 3.11 with A = k Q . (cid:3) Proposition 3.13.
Let Q be a quiver with two vertices , such that for any i, j ∈{ , } there is at least one arrow from i to j . Then the category D b ( Q ) has aninfinite descending binary tree of thick subcategories. Moreover one can find such atree with the following additional property: if T and T are two elements of this treewhich are not located one above the other, then T ∩ T = 0 .Proof. Let a : 1 → , b : 2 → , c : 2 → k Q -module M (1) as follows: M (1)1 = k x, M (1)2 = k y with x · b = y and all other arrowsin Q acting by zero. Define another right k Q -module M (2) as M (2)1 = k α ⊕ k β,M (2)2 = k γ ⊕ k δ with the nontrivial action of the arrows given by α · a = β , β · b = γ , γ · c = δ . Then one checks thatHom( M ( i ) , M ( j ) ) = δ ij · k and Ext ( M ( i ) , M ( j ) ) = 0 for all i, j ∈ { , } . We conclude (using Lemma 3.5) that the modules M (1) , M (2) form a vertex-like set. Itfollows then from Proposition 3.9 and Corollary 3.4 that h M (1) , M (2) i = [ M (1) , M (2) ] .Consequently, the thick subcategory h M (1) , M (2) i is strictly smaller than D b ( Q ) (be-cause, for example, all objects in h M (1) , M (2) i have even-dimensional cohomology).Moreover, by Proposition 3.9 we get an equivalence h M (1) , M (2) i ≃ D b ( Q ′ ) wherethe quiver Q ′ also satisfies the assumptions of the present proposition. We can iter-ate the process, which then gives an infinite descending chain of thick subcategoriesof D b ( Q ) . To construct a required descending binary tree of subcategories we canproceed as follows.For convenience let us describe the modules M (1) , M (2) constructed above by thediagrams M (1) : • b → • M (2) : • a → • b → • c → • Let us similarly define the right A modules M (3) : • a → • a → • b → • c → • c → • M (4) : • a → • a → • a → • b → • c → • c → • c → • One checks that(3.2) Hom( M ( i ) , M ( j ) ) = δ ij · k and Ext ( M ( i ) , M ( j ) ) = 0 HICK SUBCATEGORIES ON CURVES 11 for all i, j ∈ { , , , } . Indeed, for Hom this can be done by hands, for Ext use χ ( M ( i ) , M ( j ) ) = i · j · χ ( S ⊕ S , S ⊕ S ) = ij (2 − |{ arrows in Q }| ) < . Hence the thick subcategory h M (3) , M (4) i ⊂ D b ( Q ) is also quiver-like.We claim that the categories h M (1) , M (2) i and h M (3) , M (4) i have zero intersec-tion. Assume the converse. Since any object in D b ( Q ) is a direct sum of its co-homology, it follows that there exists a nonzero indecomposable k Q -module L ∈h M (1) , M (2) i ∩ h M (3) , M (4) i . By Corollary 3.4, every indecomposable k Q -module in h M (1) , M (2) i (resp. in h M (3) , M (4) i ) has a filtration with subquotients M (1) , M (2) (resp. M (3) , M (4) ). Therefore, if M (resp. N ) is an indecomposable k Q -module in h M (1) , M (2) i (resp. in h M (3) , M (4) i ), then Hom( M, N ) = 0 by (3.2). In particular,Hom(
L, L ) = 0 and thus L = 0 , a contradiction.It is clear that we can now iterate the process to construct a descending binarytree of quiver-like categories with the required properties. (cid:3) Thick subcategories on curves
In this section we assume that C is a smooth projective connected curve over analgebraically closed field k . Our goal is to classify thick subcategories in D b (coh C ) . Lemma 4.1.
Let
T ⊂ D b (coh C ) be a thick subcategory which contains a nonzerovector bundle and a nonzero torsion sheaf. Then T = D b (coh C ) .Proof. Let
V, T ∈ T be a vector bundle and a torsion sheaf respectively. Let x ∈ Supp( T ) . It is easy to see that the skyscraper sheaf O x is in T .Choose a line bundle L and a surjection V → L . This gives a short exact sequenceof vector bundles 0 → E → V → L → . Choose a surjection L → O x and denote by V (1) ⊂ V the kernel of the composition V → L → O x . Thus V (1) ∈ T and we obtain a short exact sequence0 → E → V (1) → L ( − x ) → . Iterating this process we get for any n > → E → V ( n ) → L ( − nx ) → V ( n ) ∈ T . For n >> L ( nx ) ∈ T for some n . Itis then easy so see that L ( nx ) ∈ T for all n ∈ Z . Let F ∈ coh C . We can find an exact sequence of coherent sheaves0 → K → ⊕ L ( nx ) → ⊕ L ( mx ) → F → . The two middle terms and in T and the category coh C is hereditary. Hence F ∈ T as a direct summand of Cone ( ⊕ L ( nx ) → ⊕ L ( mx )) . Therefore coh C ⊂ T . Itfollows that T = D b (coh C ) . (cid:3) We obtain the immediate corollary.
Corollary 4.2. If T ⊂ D b (coh C ) is a thick subcategory, T 6 = 0 , D b (coh C ) , thenexactly one of the following holds:(1) Every object in T has torsion cohomology.(2) Every object in T has torsion free cohomology.Proof. Indeed, every object in D b (coh C ) is the direct sum of its cohomology. So itremains to apply Lemma 4.1. (cid:3) Definition 4.3.
We will say that a thick subcategory
T ⊂ D b (coh C ) is proper if T 6 = 0 , D b (coh C ) . We call T torsion (resp. torsion-free ) in case (1) (resp. (2)) inCorollary 4.2 holds. Now we can formulate our main observation.
Theorem 4.4.
Every finitely generated thick proper subcategory
T ⊂ D b (coh C ) isquiver-like.Proof. We consider the two cases of Definition 4.3.Case 1: T is torsion. In this case we may apply Proposition 3.10 with A = qcoh C and the function r : K ( T ) → Z induced by the dimension (over k ) of a torsion sheaf.Case 2: T is torsion-free. In this case we again apply Proposition 3.10 with A = qcoh C , but take the function r : K ( T ) → Z to be induced by the rank of avector bundle. (cid:3) Definition 4.5.
A quiver Q is called realizable if the category D b ( Q ) is equivalentto a thick finitely generated subcategory of D b (coh C ) for a smooth projective curveover an algebraically closed field k . In the next section we are going to classify the realizable quivers. For now let usgive some examples.
HICK SUBCATEGORIES ON CURVES 13
Definition 4.6.
We denote by Q m the quiver with one vertex and m loops. Lemma 4.7.
For any n , the quiver which is the disjoint union of n copies of thequiver Q is realizable on any curve. In fact any torsion category (Definition 4.3)supported at n distinct points is equivalent to D b ( Q ⊔ . . . ⊔ Q ) = D b ( Q ) ⊕ n . Proof.
Let p be a point on a smooth curve C . Then the sky-scraper sheaf O p ∈ D b (coh C ) is a vertex-like object withHom( O p , O p ) ≃ Ext ( O p , O p ) ≃ k . Hence by Proposition 3.9 the thick subcategory hO p i ⊂ D b (coh C ) is equivalent to D b ( Q ) . Now it is clear that the thick subcategory T = hO p , . . . , O p n i for n different points p , . . . , p n ∈ C is equivalent to D b ( Q ⊔ . . . ⊔ Q ) = D b ( Q ) ⊕ n . It remains to note that hO p i contains no proper thick subcategories and thus anythick finitely generated torsion subcategory in D b (coh C ) is of the form hO p , . . . , O p n i for some points p , . . . , p n . (cid:3) Proposition 4.8.
Let C be a curve of genus g and let T ⊂ D b (coh C ) be a thickfinitely generated proper subcategory.(1) If g = 0 , then T ≃ D b ( Q ) or T ≃ D b ( Q ⊔ . . . ⊔ Q ) . The first case occursif T is torsion-free and the second one occurs when T is torsion.(2) If g = 1 , then T ≃ D b ( Q ⊔ . . . ⊔ Q ) .(3) If g = 0 or g = 1 , the category T contains only finitely many distinct thicksubcategories.Proof. (1) Let g = 0 . Assume that T is torsion-free. Since every vector bundleon P is a direct sum of line bundles O ( n ) , it is easy to see that T = hO ( n ) i forsome n and hence T ≃ D b (mod − k ) ≃ D b ( Q ) . If on the other hand T is torsion, it follows from Lemma 4.7 that T ≃ D b ( Q ⊔ . . . ⊔ Q ) . (2) Let g = 1 and let F ∈ T be a nonzero indecomposable object. Then thereexists an autoequivalence Ψ of D b (coh C ) such that Ψ( F ) is a torsion sheaf.It follows that the category Ψ( T ) is torsion. Then again by Lemma 4.7 we findthat T ≃ Ψ( T ) ≃ D b ( Q ⊔ . . . ⊔ Q ) .(3) This follows from (1) and (2) and the fact that the categories D b ( Q ) and D b ( Q ) have no proper thick subcategories. (cid:3) It contrast to Proposition 4.8, thick subcategories of curves of genus g ≥ Proposition 4.9. (1) Let C be a curve of genus g ≥ . Then there exists aninfinite descending binary tree of thick subcategories in D b (coh C ) with the followingproperty: if elements T , T of this tree are not located one above the other, then T ∩ T = 0 . (2) For any n ≥ the quiver Q n is realizable on a curve of genus g = n .Proof. (1) Let L , L be distinct line bundles of degree 0 on C . Then for all i, j ∈ { , } we have Hom( L i , L j ) = δ ij · k and Ext s ( L i , L j ) = 0 for s = 0 , {L , L } is a vertex-likecollection, and the thick subcategory hL , L i ⊂ D b (coh C ) is equivalent to D b ( Q ) ,where Q is the quiver with vertices v , v and dim Ext ( L j , L i ) arrows from v i to v j for any i, j ∈ { , } . Note that for any i, j ∈ { , } by the Riemann-Roch formuladim Ext ( L i , L j ) = g ( C ) − δ ij > , hence by Proposition 3.13 the category D b ( Q ) has an infinite descending binary treeof thick finitely generated subcategories with the required property.(2) It suffices to take any line bundle L on C . The thick subcategory hLi ⊂ D b (coh C ) is equivalent to D b ( Q g ) where g is the genus of C . (cid:3) Realization of quivers on curves and uniqueness problems
In this section a curve means a smooth projective connected curve over an alge-braically closed field k . We complete the problem of classification of proper finitelygenerated thick subcategories of curves which we started in the previous section.In view of Theorem 4.4, Lemma 4.7, and Proposition 4.8 it remains to answer thefollowing questions: HICK SUBCATEGORIES ON CURVES 15 Q1 . Which quivers Q are realizable by torsion-free subcategories on curves of genus g ≥ Q2 . Suppose the quiver Q is realizable. Is then Q determined uniquely by thecategory D b ( Q ) ?We start with question Q1 .First let us summarize the relevant results from the previous sections. Proposition 5.1.
Let C be a curve and let T ⊂ D b (coh C ) be a proper thick sub-category which is torsion-free (Definition 4.3). Then(1) T is quiver-like.(2) T = h E , . . . , E n i , where { E , . . . , E n } is a vertex-like collection of vectorbundles on C .(3) We have T = [ E , . . . , E n ] and K ( T ) = L i Z [ E i ] .(4) Every indecomposable object in T is of the form F [ m ] , where F is a vectorbundle that has a filtration with subquotients being E i ’s.Vice versa, a vertex-like collection of vector bundles { E , . . . , E n } generates atorsion-free proper thick subcategory of D b (coh C ) .Proof. (1) Follows from Theorem 4.4. Then (2),(3),(4) follow from Proposition 3.9and Corollary 3.4.For the last assertion: we know that h E , . . . , E n i ⊂ D b (coh C ) is a quiver-likesubcategory of D b (coh C ) . Now Corollary 3.4 implies that all indecomposable objectsof h E , . . . , E n i are (shifted) vector bundles on C . Hence it is a proper torsion-freethick subcategory of D b (coh C ) . (cid:3) Let C be a curve of genus g . For a vector bundle E on C let r ( E ) and d ( E )denote respectively its rank and degree. For vector bundles E, F put χ ( E, F ) = χ ( C, E, F ) = dim Hom(
E, F ) − dim Ext ( E, F ) . By a version of Riemann-Roch formula we have(5.1) χ ( E, F ) = r ( E ) r ( F )(1 − g ) + r ( E ) d ( F ) − r ( F ) d ( E ) . A finite quiver Q with vertices v , . . . , v n is determined by a square matrix A =( a ij ) ∈ M n × n ( Z ) with nonnegative entries a ij ≥ a ij is the number of arrows from v i to v j . Put Q = Q ( A ) . Recall (Proposition 3.9) that the quiver Q ( A ) is realized by a torsion-free category on a curve C if and only if there exists avertex-like collection of vector bundles { E , . . . , E n } on C such that(5.2) dim Ext ( E j , E i ) = a ij . By (5.1) the equation (5.2) is equivalent to the equation(5.3) a ij = r j r i ( g − − r j d i + r i d j + δ ij (= − χ ( E j , E i ) + δ ij )where r i = r ( E i ) and d i = d ( E i ) . This gives us a necessary condition for the quiver Q ( A ) to be realized on a curve of genus g . Actually this condition is also sufficient.The following theorem answers question Q1 above. Theorem 5.2.
Let g ≥ and let A = ( a ij ) ∈ M n × n ( Z ) be a matrix with nonnega-tive entries. Then the quiver Q ( A ) is realized by a torsion-free category on a givencurve C of genus g if and only if the following holds: there exists a collection ofintegers ( r , . . . , r n , d , . . . , d n ) ∈ Z n> × Z n , such that a ij = r j r i ( g − − r j d i + r i d j + δ ij for each pair ( i, j ) .Proof. We already explained the “only if” direction. For the “if” direction we willprove the following: given a set of integers ( r , . . . , r n , d , . . . , d n ) ∈ Z n> × Z n suchthat for each pair ( i, j )(5.4) r j r i ( g − − r j d i + r i d j + δ ij ≥ C of genus g there exists a vertex-like collection of vectorbundles { E , . . . , E n } with d ( E i ) = d i and r ( E i ) = r i . Choose vector bundles E i with d ( E i ) = d i and r ( E i ) = r i . For i = j the condition (5.4) means that χ ( E j , E i ) ≤ E, F aregeneric vector bundles of given rank and degree on a curve C of genus ≥ χ ( E, F ) ≤ E, F ) = 0 . Also a generic vector bundle E (of given rankand degree) on C is stable (see [NR, Prop. 2.6]) and thus End( E ) = k . It followsthat for a generic choice of vector bundles E i with degree d i and rank r i we havedim Hom( E j , E i ) = δ ij and the collection { E , . . . , E n } is vertex-like. (cid:3) Recall that the slope of a vector bundle E is d ( E ) /r ( E ) . HICK SUBCATEGORIES ON CURVES 17
Proposition 5.3.
Let C be a curve of genus g ≥ and let ( r , . . . , r n , d , . . . , d n ) ∈ Z n> × Z n . Then there exists a vertex-like collection of vector bundles { E , . . . , E n } on C with d ( E i ) = d i and r ( E i ) = r i if and only if for all ≤ i, j ≤ n we have (5.5) (cid:12)(cid:12)(cid:12)(cid:12) d i r i − d j r j (cid:12)(cid:12)(cid:12)(cid:12) g − . Proof.
Indeed the inequality (5.4) for pairs ( i, j ) and ( j, i ) with i = j is equivalentto the condition (5.5). Now the proposition follows by the same argument as in theproof of Theorem 5.2. (cid:3) Remark 5.4.
If a quiver Q ( A ) for a matrix A = ( a ij ) is realizable by a torsion-freecategory on a curve of genus g ≥ , then for all pairs of indices { i, j } at least one ofthe numbers a ij , a ji is positive. In particular, the quiver Q ( A ) is connected (comparewith Proposition 4.8 for g = 1 case). Indeed, this follows from Theorem 5.2. Some uniqueness and non-uniqueness results.Proposition 5.5.
Let
C, C ′ be curves, g ( C ) , g ( C ′ ) ≥ . Let E , . . . , E n and E ′ , . . . , E ′ n ′ be two vertex-like families of vector bundles on C and C ′ respectively.Assume that Φ : h E , . . . , E n i → h E ′ , . . . , E ′ n ′ i is an equivalence between the corresponding quiver-like categories. Then n = n ′ andthere exist a permutation σ ∈ S n and m ∈ Z such that Φ( E i ) ≃ E ′ σ ( i ) [ m ] for all i = 1 , . . . , n .Proof. By Proposition 5.1, the Grothendieck group K ( h E , . . . , E n i ) is freely gener-ated by the classes [ E ] , . . . , [ E n ] and similarly for K ( h E ′ , . . . , E ′ n ′ i ) . The equiva-lence Φ induces an isomorphism K ( h E , . . . , E n i ) ≃ K ( h E ′ , . . . , E ′ n ′ i ) , which im-plies that n = n ′ .For any i the object Φ( E i ) is indecomposable and so by Proposition 5.1 we have(5.6) Φ( E i ) ≃ F i [ m i ]for some vector bundle F i on C ′ and m i ∈ Z .The equation (5.1) implies that r ( E i ) (1 − g ( C )) = χ ( C, E i , E i ) = χ ( C ′ , F i [ m i ] , F i [ m i ]) = χ ( C ′ , F i , F i ) = r ( F i ) (1 − g ( C ′ )) . It follows that the ratio r ( F i ) /r ( E i ) =: r/r ′ does not depend on i , where we denote r = p g ( C ) − , r ′ = p g ( C ′ ) − g ( C ) , g ( C ′ ) ≥ F i ] = X j b ij [ E ′ j ] ∈ K ( h E ′ , . . . , E ′ n ′ i )for some b ij ∈ Z > . Moreover, the matrix B = ( b ij ) is invertible over Z . It followsthat r ( F i ) = X j b ij r ( E ′ j ) , r · r ( E i ) = X j b ij r ′ · r ( E ′ j ) . Denote s i := r · r ( E i ) and s ′ i := r ′ · r ( E ′ i ) . We have now X i s i = X ij b ij s ′ j = X j ( s ′ j X i b ij ) > X j s ′ j , because P i b ij > B is non-degenerate and b ij ∈ Z > for all i, j ). Similarlywe have P j s ′ j > P i s i . It follows that P i b ij = 1 for any j and thus B is apermutation matrix.Hence [ F i ] = [ E ′ σ ( i ) ] in K ( h E ′ , . . . , E ′ n i ) for some σ ∈ S n and all i . It followsfrom Proposition 5.1 that Φ( E i ) ≃ E ′ σ ( i ) [ m i ] . Now Lemma 5.6 implies that all m i in(5.6) are equal. (cid:3) Lemma 5.6.
Let C be a curve of genus g ≥ . Let E , . . . , E n be vector bundleson C such that for some integers m i the objects { E [ m ] , . . . , E n [ m n ] } form a vertex-like collection. Then m i = m j for all i, j .Proof. It suffices to prove that m = m . Formula (5.1) implies that χ ( E , E ) + χ ( E , E ) = 2(1 − g ) r ( E ) r ( E ) < . It follows that at least one of χ ( E , E ) , χ ( E , E ) is negative. Assume χ ( E , E ) < ( E , E ) = 0 . By our assumption(5.8) Ext i ( E [ m ] , E [ m ]) = 0 implies that i = 1 . Equations (5.7) and (5.8) imply that m = m . (cid:3) HICK SUBCATEGORIES ON CURVES 19
Remark 5.7.
Lemma 5.6 also holds for g = 0 , but it fails for g = 1 : if L , L aredistinct line bundles of the same degree, then for any m the objects L and L [ m ] are orthogonal and hence the collection {L , L [ m ] } is vertex-like. The following is an answer to question Q2 above. Corollary 5.8.
Let
Q, Q ′ be quivers. Assume that Q is realizable and there is anequivalence D b ( Q ) ≃ D b ( Q ′ ) (hence Q ′ is also realizable). Then Q ≃ Q ′ .Proof. Assume that the quiver Q is realized on a curve of genus g . We considerseveral cases.Case 1: g = 0 and Q is realized by a torsion-free category. Then by Proposition4.8 we know that D b ( Q ) ≃ D b ( Q ) ≃ D b (mod − k ) . It follows that Q = Q = Q ′ .Case 2: g = 1 or Q is realized by a torsion category. Then by Proposition 4.8and Lemma 4.7 there exists an equivalence of categoriesΨ : D b ( Q ) ∼ → D b ( Q ⊔ . . . ⊔ Q ) . Comparing the K -groups of these categories we find that the two quivers have thesame number of vertices, say n . Let s , . . . , s n ∈ D b ( Q ) (resp. s ′ , . . . , s ′ n ∈ D b ( Q ⊔ . . . ⊔ Q ) ) be the collection of simple modules corresponding to vertices. Note thatthe objects s ′ i ∈ D b ( Q ⊔ . . . ⊔ Q ) are characterized (up to a shift) by the propertythat End( s ′ i ) = k . It follows that there exists a permutation σ ∈ S n such that foreach i Ψ( s i ) = s ′ σ ( i ) [ m i ] for some m i ∈ Z . Moreover, D b ( Q ⊔ . . . ⊔ Q ) is the orthogonal sum of its subcategories h s ′ i i . Itfollows that D b ( Q ) is also the orthogonal sum of its subcategories h s i i . Therefore Q ≃ Q ⊔ . . . ⊔ Q and similarly Q ′ ≃ Q ⊔ . . . ⊔ Q .Case 3: g > Q is realized by a torsion-free category. In this case anisomorphism Q ≃ Q ′ follows from Proposition 5.5. (cid:3) Remark 5.9.
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Institute for Information Transmission Problems (Kharkevich Institute), Moscow,RUSSIA; National Research University Higher School of Economics, Russian Feder-ation
E-mail address : [email protected] Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
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