DERIVED CATEGORIES AND LIE ALGEBRAS

DERIVED CATEGORIES AND LIE ALGEBRAS JIE XIAO, FAN XU AND GUANGLIAN ZHANG Dedicated to Professor George Lusztig on the occasion of his 60th birthday Abstract. We associate topological spaces to the bounded derived category of the module category of a finite dimensional algebra with finite global dimension. It has a local structure of affine variety under the action of the algebraic group. We consider the constructible functions which reflect the stratifications of the orbit spaces. By using the convolution rule, we obtain the geometric realization of the Lie algebras arising from the 2-period version of the derived category. 1. Introduction 1.1. In the last thirty years of the twentieth century, there were two parallel fields in mathematics got extensively developed. One is the infinite dimensional Lie theory, in particular, the Kac-Moody Lie algebras. One is the representation theory of finite dimensional algebras, in particular, the representations of quivers. The close relation between the two subjects was discovered in a very early stage. Gabriel in [G] found that the quivers of finite representation type were given by the Dynkin graphs in Lie theory, and the dimension vectors provide the bijective correspondence between the isomorphism classes of indecomposable representations of the quiver and the positive root system of the semisimple Lie algebra. After the Gabriel theorem, a lot of progress on the connection between the representations of quivers or hereditary algebras and Lie algebras had been made, for example, by Bernstein- Gelfand-Ponomarev [BGP] and Dlab-Ringel [DR]. The final and most general result is the Kac theorem [K1] which extends to consider the quiver and the symmetric Kac-Moody algebra of arbitrary type. It states that the dimension vectors of indecomposable representations are exactly the positive roots, a unique indecomposable corresponds to each real root and infinitely many to each imaginary root; the multiplicity of the imaginary root, which is conjectured by Kac in [K2], is given by a geometric parameter in terms of representations of the quiver. A new progress on the Kac conjecture is by Crawley-Boevey and Van den Bergh in [CBV]. 1.2. Ringel in [R2] discovered his Hall algebra structure by giving an answer to the following fundamental question: how to recover the underlying Lie algebra structure directly from the category of representations of the quiver. The research was supported in part by NSF of China and by the 973 Project of the Ministry of Science and Technology of China. 2000 Mathematics Subject Classification. Primary 18E30, 17B37, 16G10; Secondary 16G20, 14L30, 17B67. Key words and phrases. Derived category, orbit space, constructible function, Lie algebra, Kac- Moody. 1

2 JIE XIAO, FAN XU AND GUANGLIAN ZHANG Let Q be a quiver, A = F q Q the path algebra of Q over F q : the finite field with q elements. Set P = {isoclasses of representations of Q}. For any α P choose V α to be a representative in the class α. Given three classes λ, α, β P, let g λ αβ be the order of the finite set {W V λ W = V β, V λ /W = V α }. By taking v = q and the integral domain Q(v), the (twisted) Ringel-Hall algebra H (A) can be defined to be a free Q(v)-module with basis {u λ λ P} and multiplication is given by u α u β = v α,β λ P g λ αβu λ for all α, β P. One may consider the subalgebra of H (A) generated by u i = u αi, for i I(= Q 0 ) where α i P is the isoclass of simple A-module at vertex i. The subalgebra is called the composition algebra and it is denoted by C (A). On the other hand, the index set I of simple A-modules together with the symmetric Euler form (, ) of A is a Cartan datum in the sense of Lusztig [L4]. For a Cartan datum (I, (, )), the quantized enveloping algebra U q defined by Drinfeld [Dr] and Jimbo [J] is associated with it. The positive part U q + is generated by E i, i I with subject to the quantum Serre relations. There is a usual way to define the generic form C (Q) of the composition algebra C (A) by considering the representations of Q over infinitely many finite fields. Then C (Q) is a Q(v)-algebra where v becomes a transcendental element over Q. Put u ( n) i C (Q), which is generated by u ( n) = u n i [n] i! for i I and n N and let C (Q) Z be the integral form of i, i I, n N over the integral domain Z = Z[v, v 1 ]. Also the quantum group U q + has the integral form U + Z, which is generated by E (n) i, i I, n N over Z. Then by Ringel [R1] and Green [Gr], the canonical map C (Q) Z U + Z by sending u( n) i to E (n) i for i I and n N leads to a Z-algebra isomorphism, if the two algebras share a common Cartan datum. Let ind P = {isoclasses of indecomposable representations of Q}. Then gαβ λ can be regarded as a function on q for α, β, λ ind P. In fact, Ringel in [R2] proved that gαβ λ is an integral polynomial on q when Q is of finite type. One can take the integral value gαβ λ (1) by letting that q tends to 1. Ringel [R2], for Q of finite type, prove that the u α, α ind P, spanned a Lie subalgebra of C (Q) q=1 with Lie bracket [u α, u β ] = (g λ αβ(1) g λ βα(1))u λ λ ind P for α, β ind P. This realized the positive part n + of the semisimple Lie algebra g. Of course, Ringel s approach also works for Q of arbitrary type. In general, there exists the generic composition Lie subalgebra L of C (Q) q=1 generated by u i, i I and ind P is no longer to index a basis of L. Now L is canonically isomorphic to the positive part n + of the symmetric Kac-Moody Lie algebra g. For a realization of the whole g, not just its positive part, Peng and Xiao in [PX3] have constructed a Lie algebra from a triangulated category with the 2-periodic shift functor T, i.e, T 2 = 1. If, specially, consider the 2-periodic orbit category of the derived category of a finite dimensional hereditary algebra, the Lie algebra obtained in [PX3] gives rise to the global realization of symmetrizable Kac-Moody algebra of arbitrary type. In [PX3] they consider the triangulated categories over finite fields. Replacing counting the order of the filtration set, they calculate the order of the orbit space of a triangle.

DERIVED CATEGORIES AND LIE ALGEBRAS 3 By a hard work, they obtained a Lie ring g (q 1) over Z/(q 1) for the prime powers q = F q. Then they performed their work over finite field extensions of arbitrarily large order and construct a generic Lie algebra which is similar to the generic composition Lie subalgebra done by Ringel in [R3]. A transcendental Lie algebra was finally obtained. 1.3. Quickly after the work of Ringel [R1], people realized that a geometric setting of Ringel-Hall algebra is possible by using the convolution multiplication (see [Sch] and [L1]). Let Q be a quiver and α = i I a ii N[I] a dimension vector. We fix a I-graded space C α = (C ai ) i I. Then E α = Hom C (C a s(h), C a t(h) ) is an affine space. Set h:s(h) t(h) G α = Π i I GL(a i, C). For any (x h ) E α and g = (g i ) G α, we define the action g (x h ) = (g t(h) x h g 1 s(h) ). For any Q-representation M with dimm = α, let O M E α be the G α -orbit of M. For an algebraic variety X over C, a subset A of X is said to be constructible if it is a finite union of locally closed subsets. A function f : X C is constructible if it is a finite C-linear combination of characteristic functions 1 O for constructible subsets O. We define M Gα (Q) to be the space of constructible G α -invariant functions E α C, and let M G (Q) = α NI M G α (Q). Let inde α (Q) to be the constructible subset of E α consisting of all points x which correspond to indecomposable Q-representations, and let indm Gα (Q) to be the space of constructible G α -invariant functions over inde α. We may regard as indm Gα (Q) = {f M Gα (Q) suppf inde α }, and indm G (Q) = α R +indm Gα (Q), where, by Kac theorem, R + is the positive root system of the Kac-Moody Lie algebra corresponding to Q. The space M G (Q) = α NI M G α (Q) cab be endowed with the associative algebra structure by the convolution multiplication: 1 O1 1 O2 (y) = χ(f y O 1O 2 ) for any G α -invariant constructible set O 1 and G β -invariant constructible set O 2 with α, β NI, where F y O 1O 2 = {x O 2 M(x) M(y) and M(y)/M(x) O 1 } and χ(x) denotes the Euler characteristic of the topological space X. As in [Rie] and [DXX], it can be proved that the space indm G (Q) has a Lie algebra structure under the usual Lie bracket [1 O1, 1 O2 ] = 1 O1 1 O2 1 O2 1 O1. Applying this setup to the case Q being a tame quiver, Frenkel-Malkin-Vybornov [FMV] gave an explicit realization of the positive parts of affine Lie algebras. 1.4. The great progress is made by Lusztig, who apply the Hall algebras in a geometric setting to study the quantum groups (see [L1] and [L2]) and the enveloping algebras (see [L5]). The canonical bases of the quantum groups and the semicanonical bases of the enveloping algebras were originally constructed in terms of representations of quivers. However Lusztig [L2] has pointed out that a more suitable choice is the preprojective algebra, which is given by the double quiver of Q with the Gelfand-Ponomarev relations. Further progress in this direction is the

4 JIE XIAO, FAN XU AND GUANGLIAN ZHANG study of Nakajima [N] on his quiver varieties, which leads to a geometric realization of the representation theory of Kac-Moody algebras. Inspired by Ringel s work on Hall algebras and Lusztig s geometric approach to quantum groups, the aim of this paper is to give a global and geometric realization of the Lie algebras arising from the derived categories, which is a generalization of our earlier work [PX3]. 1.5. If we consider the module category of A = CQ/J, we have the algebraic variety E d (Q, R) for A-modules with a fixed dimension vector d and it is a G-variety where G = G d (Q) is a reductive group. According to the work of C.de Concini and E.Strickland in [CS] and M.Saorin and B.Huisgen-Zimmermann in [SHZ], this geometry can be generalized to over the chain complexes of A-modules. Section 2 is devoted to do this. The set C b (A, d) of all complexes of A modules with the dimension vector sequences d and the set P b (A, e) of all projective complexes with the projective dimension vector sequence e can be endowed with the affine variety structures. It is natural to consider the quotient spaces Q b (A, d) and QP b (A, d) for d K 0 (D b (A)) where the equivalent relations to define the quotient are induced by the quasi-isomorphisms in the homotopy category. Our main results in Section 2: (1) the topological spaces of Q b (A, d) and QP b (A, d) are homeomorphic canonically; (2) the topological structure of Q b (A, d) and QP b (A, d) is invariant under the derived equivalent; (3) the topological space QP b (A, d) is obtained from the affine variety by the action of algebraic groups and a family of transition operators t α, where all t α are induced by the contractible complexes and the property of t α is easily controlled. Therefore the main point in Section 2 is that, we can regard the G d, T -invariant geometry in P b (A, d) as the moduli space in which the orbits index the isomorphism classes of objects in the derived category. In Section 3, we consider the set of the equivalence classes of G e -invariant constructible functions over P b (A, e) for any e dim 1 (d). An equivalence class can be viewed as a G d, T -invariant constructible function over P b (A, d) and the convolution multiplication between the equivalence classes can be defined. The concept of support-bounded is important for us. We prove that the derived equivalent functors send the G d, T -invariant support-bounded constructible functions to G d, T -invariant support-bounded constructible functions again. Our main theorem in Section 3 is that our convolution rule is well-defined for the multiplication of G d, T -invariant support-bounded constructible functions. For this aim we need to define the quasi Euler characteristic of the orbit spaces induced by triangles in the triangulated category. The theorem of Rosenlicht [Ro] for the algebraic group action on varieties is crucial for us. It is interesting to compare the quasi Euler characteristic with the naïve Euler characteristic in [Jo1, Section 4.3]. Section 4 is just to transfer the results in Section 3 to the 2-periodic orbit categories of the derived categories. Section 5 is used to verify the Jacobi identity. In [PX3], by counting the Hall numbers FXY L for the triangles of the form X L Y X[1], it has been proved that the Jacobi identity can be deduced from the octahedral axiom of the triangulated categories. However we need to prove that the correspondences among the various orbit spaces in the derived categories are actually given by the algebraic morphisms of algebraic varieties. We think this geometric method is more transparent to reflect the hidden symmetry in the derived category. Additionally, we get the two properties which is unknown in [PX3]. Firstly the proper assumption

DERIVED CATEGORIES AND LIE ALGEBRAS 5 in [PX3] is not necessary, in fact, it is easy to give examples such that dimx = 0 for some nonzero indecomposable X in D b (A). Secondly we show that the Lie algebras arising from the 2-periodic orbit categories of the derived categories always possess the symmetric invariant form in the sense of Kac [K3], which is essentially non-degenerated. Section 6 is to apply the construction to the 2-periodic orbit categories of the derived categories of representations of quivers, particularly, tame quivers. This gives rise to a global realization of the symmetric (generalized) Kac- Moody algebras of arbitrary type. In particular, an explicit realization of the affine Lie algebras. In the last section, we indicate a method to replace the group actions by the actions of reductive groups. Our original algebraic groups G e may not be reductive. 1.6. Finally we should mention recent advances by Toën [T] and Joyce [Jo2]. Toën defined an associative algebras, called the derived Hall algebra, associated to a dg category over a finite field. A direct proof for Toën s theorem is given in [XX]. Joyce considered a new Ringel-Hall type algebra consisting of functions over stacks associated to abelian categories. Their results can be viewed as improvements of the Ringel-Hall type algebra with respect to categorification and geometrization. However, it is difficult to define an analog of the derived Hall algebra over the complex field (see [L6] [N]) or an analog for the 2-period version of a derived category (see [T] and [XX]). Hence, it is still an open question to define an associative multiplication which induces the Lie bracket in this paper and supplies the realization of the corresponding enveloping algebra. 2. Topological spaces associated to derived categories 2.1. Given an associative algebra A over the complex field C, in this paper, we always assume that A is both finite dimensional and finite global dimensional. By a result of P.Gabriel ([G]) the algebra A is given by a quiver Q with relations R (up to Morita equivalence). Let Q = (Q 0, Q 1, s, t) be a quiver, where Q 0 and Q 1 are the sets of vertices and arrows, respectively, and s, t : Q 1 Q 0 are maps such that any arrow α starts at s(α) and terminates at t(α). For any dimension vector d = (d i ) i Q0, we consider the affine space over C E d (Q) = α Q 1 Hom C (C ds(α), C dt(α) ) Any element x = (x α ) α Q1 in E d (Q) defines a representation (C d, x) where C d = i Q 0 C di. A relation in Q is a linear combination r i=1 λ ip i, where λ i C and p i are paths of length at least two with s(p i ) = s(p j ) and t(p i ) = t(p j ) for all 1 i, j r. For any x = (x α ) α Q1 E d and any path p = α 1 α 2 α m in Q we set x p = x α1 x α2 x αm. Then x satisfies a relation r i=1 λ ip i if r i=1 λ ix pi = 0. If R is a set of relations in Q, then let E d (Q, R) be the closed subvariety of E d (Q) which consists of all elements satisfying all relations in R. Any element x = (x α ) α Q1 in E d (Q, R) defines in a natural way a representation M(x) of A = CQ/J with dimm(x) = d, where J is the admissible ideal generated by R. We consider the algebraic group G d (Q) = GL(d i, C), ı Q 0

6 JIE XIAO, FAN XU AND GUANGLIAN ZHANG which acts on E d (Q) by (x α ) g = (g t(α) x α g 1 s(α) ) for g G d and (x α ) E d. It naturally induces the action of G d (Q) on E d (Q, R). The induced orbit space is denoted by E d (Q, R)/G d (Q). There is a natural bijection between the set M(A, d) of isomorphism classes of C-representations of A with dimension vector d and the set of orbits of G d (Q) in E d (Q, R). So we may identify M(A, d) with E d (Q, R)/G d (Q). The geometrization of A-modules with dimension vector d can be carried over, in the same spirit, to the complexes. First we consider the category of complexes C(A). Its objects are sequences M = (M n, n ) of finite dimensional modules and their homomorphisms (1)... n 1 Mn n n+1 n+2 Mn+1 Mn+2... such that n+1 n = 0 for all n. A morphism φ : M M between two complexes is a sequence of homomorphisms φ = (φ n : M n M n) n Z such that the following diagram is commutative. (2)... n 1 n n+1 n+2 Mn Mn+1 Mn+2... φ n φ n+1 φ n+2... n 1 M n n M n+1 n+1 M n+2 n+2... One says that such a morphism is homotopic to zero if there are homomorphisms σ n : M n M n 1 such that φ n = σ n+1 n + n 1σ n for all n Z. The factor category K(A) of C(A) modulo the ideal of morphisms homotopic to zero is called the homotopic category of A-modules. For each n the n-th homology of a complex is defined as H n (M ) = Ker n /Im n 1. Obviously, a morphism φ of complexes induces homomorphisms of homologies H n (φ ) : H n (M ) H n (M ) and if φ is homotopic to zero, it induces zero homomorphisms of homologies. One call a morphism φ in C(A) or in K(A) quasi-isomorphism if the induced morphisms H n (φ ) are isomorphisms for all n. Now the derived category D(A) is defined to be the category of fractions K(A)[N 1 ], where N is the set of all quasi-isomorphisms, which is obtained from K(A) by inversing all morphisms in N. One calls a complex right bounded (left bounded, bounded, respectively) if there is n 0 such that M n = 0 for n > n 0 ( there is n 1 such that M n = 0 for n < n 1, or there are both, respectively). The corresponding categories are denoted by C (A), K (A), D (A) ( by C + (A), K + (A), D + (A), or by C b (A), K b (A), D b (A), respectively). In this paper we mainly deal with the bounded situation. The category A-mod of finite dimensional A-modules can be naturally embedded into D(A) (even in D b (A)): a module M is identified with the complex M such that M 0 = M and M n = 0 for n 0. A complex P = (P n, n ) is called projective if all P n are projective A-modules. Since the category A-mod has enough projective objects, one can replace, when considering right bounded homotopic and derived category, arbitrary complexes by projective ones. We denote by P (A) and by P b (A) the full subcategories of C (A) and C b (A) which consist of right bounded and bounded projective complexes, respectively. Actually, we have D (A) K (P (A)) P (A)/I, where I is the ideal of morphisms homotopic to zero (see[gm]). Moreover, every finite dimensional A-module M has a projective cover, i.e., an epimorphism p M : P (M) M

DERIVED CATEGORIES AND LIE ALGEBRAS 7 such that P (M) is projective and Ker p M rad P (M), the radical of P (M). Therefore, we can only consider minimal or radical projective complexes P = (P n, n ) with the property: P n is projective and Im n rad P n+1 for all n. Let rad P (A) be the full subcategory of P (A) which consist of minimal projective complexes. Since every projective complex in P (A) is quasi-isomorphic to a minimal projective complexes, we have D (A) rad P (A)/I, where I is the ideal of morphisms homotopic to zero. One immediately checks that a morphism φ between minimal projective complexes induces an isomorphism in D (A) if and only if φ itself is an isomorphism in rad P (A). If we further assume that the global dimension of A is finite, then we have D b (A) rad P b (A)/I, since any bounded complex has a bounded projective resolution. 2.2. Let the algebra A = CQ/J be of finite global dimension and the admissible ideal J is given by a set R of relations in Q. For a dimension vector d we understand as d : Q 0 N. We set the Q 0 -graded C-space C d = j Q 0 C d(j). For a sequence of dimension vectors d = (, d 1, d 0, d 1, ) with only finite many non-zero entries, we define C b (A, d) to be the subset of (see [SHZ]) Hom C (V d i, V d i+1) i Z E di (Q, R) i Z which consists of elements (x i, i ) i, where x i E di (Q, R) and M(x i ) = (C d i, xi ) is the corresponding A-module and i Hom C (C d i, C d i+1) is a A-module homomorphism from M(x i ) to M(x i+1 ) with the property i+1 i = 0. In fact, (M(x i ), i ) i, or simply denoted by (x i, i ) i, is a complex of A-modules and d is called its dimension vector sequence. The group i Z G d i (Q) acts on C b (A, d) via the conjugation action (g i ) i (x i, i ) i = ((x i ) gi, g i+1 i g 1 i ) i where the action (x i ) gi was defined as in Section 2.1. Therefore the orbits under the action correspond bijectively to the isomorphism classes of complexes of A-modules. We fix a set P 1, P 2,, P l to be a complete set of indecomposable projective A-modules (up to isomorphism). Let P b (A) be the full subcategory of C b (A) which consists of projective complexes P = (P i, i ) such that each P i has the decomposition l P i = e i jp j. j=1 We denote by e(p i ) the vector (e i 1, e i 2,, e i l ). The sequence, denoted by e(p ), (, e(p 1 ), e(p 0 ), e(p 1 ), ) is called the projective dimension sequence of P. As defined as in [JSZ], for a fixed projective dimension sequence e = (, e i, ), we define P b (A, e) to be the subset of i Z Hom A (P i, P i+1 ) = i Z Hom A ( l e i jp j, j=1 l j=1 e i+1 j P j ) which consists of elements ( i : P i P i+1 ) i Z such that i+1 i = 0 for all i Z, where e i = e(p i ) for i Z. Note that P b (A, e) is also an affine variety in a natural way. Since it is a closed subset of C b (A, d) for the dimension vector sequence d = (d i ), where d i = dimp i, we can regard P b (A, e) as a subvariety of C b (A, d).

8 JIE XIAO, FAN XU AND GUANGLIAN ZHANG The algebraic group G e (Q, R) = i Z Aut A (P i ) acts on P b (A, e) by conjugation (g i ) i ( i ) i = (g i+1 i g 1 i ) i. It is known from [JSZ] that two projective complexes in P b (A, e) are in the same orbit if and only if they are quasi-isomorphism (also see Lemma 2.3). Let rad P b (A) be the subcategory of P b (A) which consists of all bounded minimal projective complexes, i.e. such that Im i rad P i+1 for all i Z. In the way similar to P b (A), we can define rad P b (A, e) as the subset of P b (A, e) consisting of all minimal projective complex with fixed projective dimension sequence e. It is closed under the group action. 2.3. In this subsection we consider the topological structure which are endowed with C b (A), P b (A) and rad P b (A). Let K 0 (D b (A)), or simply by K 0, be the Grothendieck group of the derived category D b (A), and dim : D b (A) K 0 (D b (A)) the canonical surjection. It induces a canonical surjection from the abelian group of dimension vector sequences to K 0, we still denote it by dim. We have known that the set C b (A, d) of all complex of fixed dimension vector sequence d in C b (A) is an affine variety. Define C b (A, d) = C b (A, d) d dim 1 (d) for d K 0. In general, we have two common ways to define topology over C b (A, d) when we want that the topology over C b (A, d) is naturally induced by the topology over C b (A, d). One is weak topology, the other is strong topology. Here we prefer the latter, i.e. we choose the following set as topology base of closed subset: B = { d dim 1 d C d C d is the closed set of C b (A, d)} Moreover, we also define the quotient space Q b (A, d) = C b (A, d)/, where x y if and only if the corresponding complexes M(x) and M(y) are quasi-isomorphic to each other, i.e., they are isomorphic in D b (A). The topology of Q b (A, d) is quotient topology, i.e., let π : C b (A, d) Q b (A, d) be the canonical surjection, U is an open (closed) set of Q b (A, d) if and only if π 1 (U) is an open (closed) set of C b (A, d). We also define the orbit space of a set in C b (A, d). Let S be a subset of C b (A, d). Its orbit space is defined as O(S) = {y y x for some x S}. Then for any x C b (A, d) the closure of its orbit is O(x) = G d (y) d dim 1 (d) y C b (A,d), y x where G d (y) means the closure of G d (y) in C b (A, d). By an abuse of notation, we may define the global affine variety of C b (A) as C b (A) = C b (A, d) d K 0

DERIVED CATEGORIES AND LIE ALGEBRAS 9 and the quotient space Q b (A) = Q b (A, d). d K 0 One of the main theorem in this section is to prove that the topological spaces Q b (A, d) and Q b (A) are invariant under derived equivalence. Similarly, we can define the topology over P b (A). For any projective dimension sequence e, we know that P b (A, e) is a subvariety of C b (A, d) for some dimension vector sequence d. Also dim induces a surjection from the abelian group of projective dimension sequences to K 0. We may define P b (A, d) = P b (A, e) e dim 1 (d) for any d K 0. Then P b (A, d) has the natural topological structure, which closed subsets are those B = { B e B e is a closed set of P b (A, e)}. e dim 1 (d) We also have the quotient space QP b (A, d) = P b (A, d)/, where x y in P b (A, d) if and only if the corresponding projective complexes P (x) and P (y) are quasi-isomorphic, i.e., they are isomorphic in D b (A). The topology for QP b (A, d) is the quotient topology from P b (A, d). For any x P b (A, d), then the orbit is O(x) = {y P b (A, d) y x}. A complex X = (X i, i ) i is called contractible if the induced homological groups H i (X ) = ker i+1 /Im i = 0 for all i Z. It is easy to see that any contractible projective complex is isomorphic to a direct sum of shifted copies of complexes of the form... 0 0 P f P 0 0... with P is a projective A-module and f is an automorphism. We call an element x P b (A, e) contractible if the corresponding projective complex is contractible. Now we may define a family of operators on P b (A, e). For any x P b (A, e), we define t + x (y) = x y P b (A, e 1 + e 2 ) for any y P b (A, e 2 ) and { t x (y) = z, if y = x z, not defined, otherwise where x y is the element in P b (A, e 1 + e 2 ) such that P (x y) = P (x) P (y). Then t x t + x = Id and t + x t x = Id on the defined points. We set and T e = {t ± x x P b (A, e) is contractible} T = e dim 1 (0) T e

10 JIE XIAO, FAN XU AND GUANGLIAN ZHANG whose action on P b (A, d) is partially defined. We have defined the algebraic group G e acts on P b (A, e). Let G d = e dim 1 (d) which action on P b (A, d) is partially defined too. For d K 0 we consider the set, denoted by G d, T, which consists of all operators generated by operators in G d and T under the ordinary composition. Obviously G d, T is a groupoid. The actions of G d, T induces an equivalence relation in P b (A, d). For x, y P b (A, d) we write x y to mean that they belong to the same G d, T -orbit. We have the following easy facts. Lemma 2.1. In P b (A), any projective complex can be uniquely decomposed (up to isomorphism) into the direct sum of a minimal projective complex and a contractible projective complex. Proof. See [BD] or [JSZ]. Lemma 2.2. Let f : P Q be a morphism between two minimal projective complex in P b (A). Then f is a quasi-isomorphism if and only if f is an isomorphism. Proof. See [JSZ]. Lemma 2.3. Let f : P Q be a morphism in P b (A) and e(p ) = e(q ). Then f is a quasi-isomorphism if and only if f is an isomorphism. Proof. See [JSZ]. By these facts, we know that in P b (A, d) the relation coincides with the relation. So we have G e QP b (A, d) = P b (A, d)/ = P b (A, d)/ G d, T. Let π : P b (A, d) QP b (A, d) be the canonical projection. The topology for QP b (A, d) is the quotient topology from P b (A, d). Now we consider radp b (A), first Proposition 2.4. The set of minimal projective complex with fixed projective dimensional sequence e forms a affine subvariety radp b (A, e) of P b (A, e) with the algebraic group action. Proof. For any complex P P b (A, e) with the form as follows: P i i P i+1 we note P radp b A (A, e) if and only if rada P i id i A rada P i+1 is zero map for any i Z. This shows radp b (A, e) is constructible subset of P b (A, e), hence a subvariety. Put for any d K 0. we remark: radp b (A, d) = e dim 1 (d) radp b (A, e)

DERIVED CATEGORIES AND LIE ALGEBRAS 11 Remark 2.5. Different from P b (A), there doesn t exist the operator T e acting on it, so any orbit of radp b (A, d) is the orbit in radp b (A, e) under the action of the group G e. 2.4. The aim of this subsection is to build the connection between the above first and second topology spaces. Lemma 2.6. Suppose the following diagram is a pullback of A-module, and g 1 is surjective, f 1 X Y f 2 g 1 Z g 2 W then we have the following properties hold: (1) Kerf 1 = Kerg2 ; Y (2) Imf 1 = W Img 2 ; (3) there exists the exact sequence: 0 X Y Z W 0. Proof. The first and third statement just follow the definition of pullback, refer to [ARS]. For the second statement, we use the first statement again to get an isomorphism:kerf 2 = Kerg1. By this, the conclusion follows. For any dimension vector sequence d = (d i ) i Z, we construct for any complex M in C b (A, d) a so-called free resolution, that is a complex F of finitely generated free A-modules, such that F is quasi-isomorphic to M. Assume dim C A = n. Let M be a complex with the following form: (3) r 1 0... M r 2 Mr 1 r Mr 0 which dim C M i = d i for any i Z. Here, if d i = (d j i ) j Q 0, then d i = j Q 0 d j i. Since M r has dimension d r, we have the surjective map: π r : A dr M r. Along the differential r and the above π r, we form the pullback X r 1 : (4) ˆ r... M r 2 X r 1 A d r 0 id ˆπ r π r... M r 2 M r r 1 Mr 0 Depending on Lemma 2.6, we have the following exact sequence: 0 X r 1 A dr M r 1 M r 0 The dimension of X r 1, denoted by l r 1, is d r 1 + d r (n 1). Similarly, we have the surjective map: π r : A lr 1 X r 1, and ˆπ r is also surjective by the lemma. So we can also form the pullback X r 2 as showed in the following diagram:

12 JIE XIAO, FAN XU AND GUANGLIAN ZHANG (5)... M r 3 X r 2 ˆ r 1 A lr 1 A lr 0 ˆπ r 1... M r 3 M r 2 X r 1 ˆ r π r A dr 0... M r 3 r 2 M r 2 r 1 M r 1 r M r 0 Inductively, we get a complex of free A-module F as follows: (6)... A li 1 ˆ i π i A l i... A lr 0 which l i = d i + (d i+1 + + d r )(n 1), in particular, l r = d r. This shows there exists a projective dimension sequence e determined by dimension vector sequence d such that F P b (A, e). Moreover, This complex is quasi-isomorphism to M. First, Alr H r (F ) = Im ˆ = r π r Im ˆ = r Im r This follows from that π r is surjective and the above lemma. In general, for i < r, H i (F ) = Ker ˆ i+1 π i+1 Im ˆ = Ker ˆ i+1 π i+1 i π i Im ˆ = (ˆπ i+1 π i+1 ) 1 (Ker i+1 ) i (ˆπ i+1 π i+1 ) 1 = H i (M ). (Im i ) In fact, we can construct the new complex from any place of a complex and two complexes are quasi-isomorphic to each other as the following diagram. Alr ˆπ r M r π r (7) X i 1 A di M i+1 M i 1 M i M i+1 where d i = dim C M i and X i 1 is the pullback. In this way, we obtain a map f : C b (A, d) P b (A, e) by mapping any complex to its free resolution. Proposition 2.7. The map f is a morphism of varieties from C b (A, d) to P b (A, e). In order to prove the proposition, we need to introduce the following lemmas as in ([CBS]) and ([Rich]). Let k be an algebraic closed field, d N. Then k d is the fixed d dimensional vector space. Let and Surj(k d, k d1 ) = {φ φ : k d k d1 surjective linear map } Inj(k d2, k d ) = {θ θ : k d2 k d injective linear map } V ses (d 1, d 2 ) = {(θ, φ) θ Inj(k d2, k d ) and φ Surj(k d, k d1 ) such that d = d 1 + d 2 and φθ = 0}.

DERIVED CATEGORIES AND LIE ALGEBRAS 13 The set mod k (A, d) means the A-module structure on k d, i.e, m mod k (A, d) means that m : A End k (k d ) is an algebra homomorphism. Let V ses A (d 1, d 2 ) = {(m, (θ, φ)) mod k (A, d) V ses (d 1, d 2 ) Kerφ is a submodule of (k d, m)}. Given a d 1 -ruple I = (i 1, i d1 ), let Surj(k d, k d1 ) I be the subset of Surj(k d, k d1 ) consisting of the matrices which doesn t belong to Surj(k d, k d1 ) I for any lexical order I < I and whose the I-th rows are linear independent. Then Surj(k d, k d1 ) = I Surj(k d, k d1 ) I. For any A Surj(k d, k d1 ) I, we can uniquely write down the d (d d 1 ) matrix B such that AB = 0. For example, if I = (1, 2,, d 1 ), then A = (A 1, A 2 ) such that A 1 is a invertible matrix. Hence, B = ( A 1 1 A 2 I ). It is clear that B Inj(k d2, k d ). In general, we have Lemma 2.8. Let d = d 1 + d 2. Then there exists a morphism of varieties λ : Surj(k d, k d1 ) Inj(k d2, k d ) such that (λ(φ), φ) V ses (d 1, d 2 ). Lemma 2.9. Let µ : VA ses mod k (A, d 1 ) mod k (A, d 2 ) be the map sending (m, (θ, φ)) to (m 1, m 2 ) where m 1 = φmφ 1 : A End k (k d1 ) and m 2 = θ 1 mθ : A End k (k d2 ) are the module structures on k d1 and k d2 respectively. Then the map µ is a morphism of varieties. Now we come to prove our proposition. Proof. Without loss of generality, we assume d = (d 1,, d r ) for some r N. We set d(1) := d. By Lemma 2.8 and 2.9, the above construction of free resolution induces a morphism of varieties: f 1 : C b (A, d(1)) C b (A, d(2)) where d(2) = (d 2 1,, d2 r ) satisfies d2 i = d i for i < r 1 and d 2 r = dimadr and d 2 r 1 = dimadr +d r 1 d r. Inductively, we obtain a chain of morphisms of varieties C b (A, d(1)) f 1 C b (A, d(2)) f 2 C b (A, d(r)) By the construction, d(r) is determined by d and dima. Let f = f r 1 f 1. Then we deduce a morphism from C b (A, d(1)) to C b (A, d(r)) satisfying the image of any complex under f is a projective complex. We complete the proof of Proposition 2.7. 2.5. We have the following results. Theorem A As the topological spaces we defined, the morphism f induces a homeomorphism of quotient spaces from Q b (A) to QP b (A).

14 JIE XIAO, FAN XU AND GUANGLIAN ZHANG Proof. We have the following canonical commutative diagram: (8) P b (A) π P QP b (A) i C b (A) π C i Q b (A) where π P and π C are canonical projections. By the topology as defined above, i is continuous if and only if π C i is continuous. By Proposition 2.7, we also have the following commutative diagram: (9) C b (A) π C Q b (A) f P b (A) π P f QP b (A) Similarly, f is continuous if and only if π P f is continuous. So, by Proposition 2.7, both of i and f are continuous maps. Moreover, we have proved that f i = 1 QP b (A) and i f = 1 Q b (A). The theorem is proved. Let us recall Rickard s results ([Rick1] and [Rick2]) on Morita theory of derived categories. First, a tilting complex T over A is an object in K b (P ) which satisfies the following conditions: 1. For any i 0, Hom Db (A)(T, T [i]) = 0; 2. The category add(t ) generates K b (P ) as a triangulated category. For convenience, we shall consider the right A-module temporarily and A-B-bimodule means module for A op C B in the following part of this section. Rickard proved the following result. Theorem 2.10. For any two finite dimensional basic C-algebras A and B, they are derived equivalent if and only if there is a tilting complex T over A such that the endomorphism ring End Db (A)(T ) op is isomorphic to B. Moreover, the method of Rickard implies the further results. If A and B are derived equivalent by a functor F, then the functor F induces a derived equivalence between D b (A op C A) and D b (B op C A). The image of A as A op C A module under this functor is a complex, say, in D b (B op C A). The functor F induces also a derived equivalence between D b (B op C B) and D b (A op C B). The image of B as B op C B module under this functor is a complex, say Θ, in D b (A op C B). Then L B : D b (B) D b (A) is an equivalence of triangulated categories. It has two quasi-inverses and RHom D b (A)(X, ) : D b (A) D b (B) L B Θ : D b (A) D b (B) Theorem B Let F : D b (A) D b (B) be a functor of derived equivalence, then F induces a homeomorphism f: Q b (A) Q b (B).

DERIVED CATEGORIES AND LIE ALGEBRAS 15 Proof. According to Theorem A, it is sufficient to prove that F induces a homeomorphism between QP b (A) and QP b (B). We consider as a B op C A-projective complex: (10) j... j j+1... where j is B op C A-projective module. For any x P(B, e 1), P (x) is the corresponding B-projective complex with the following form: (11)... P i P i P i+1... where P i is B-projective module. Define P (f(x)) is the following complex(see [Rick2]): (12)... i+j=n P i B j d n i+j=n+1 P i B j... where d n = i P id+( 1) i id i and i+j=n P i B j is A-projective module. We set e 2 to be the A-projective dimension sequence of P (f(x)). Then we have the map f induced by F. f : P(B, e 1 ) P b (A, e 2 ) By the above construction, it is clearly a morphism of varieties. In the same way, we can give the morphism g : P b (A, e 3 ) P b (A, e 4 ) where P (g(y)) = P (y) A Θ, which is induced by G = A Θ. Because F and G are quasi-inverse to each other, P (fg(y)) = P (y) and P (gf(y)) = P (x), where = means quasi-isomorphism. We have the following commutative diagram: (13) Q b (A) P b (A) π A = QP b (A) g P(B) π B ḡ QP b (B) = Q b (B) Here ḡ, induced by g, is continuous just because π B g is continuous and π A is the quotient map. Similarly, We have the following commutative diagram: (14) P b (A) π A f P(B) π B Q b (A) = QP b (A) f QP b (B) = Q b (B) Here f, induced by f, is continuous just because π A f is continuous and π B is quotient map. The equations fḡ = id Qb P (A) and ḡ f = id Qb P (B) show both are homeomorphisms.

16 JIE XIAO, FAN XU AND GUANGLIAN ZHANG 3. Constructible functions over derived categories 3.1. In the last section, we have defined the topological spaces for the derived categories. This kind of topology supplies many properties similar to that of affine variety of modules for a fixed dimension vector. At the same time, there exist some essential differences. We may consider the degeneration as in [JSZ] for our situation. For any X and Y in K b (P b (A)) we denote by X Y if there is a distinguished triangle Y X Z Z Y [1] for some Z K b (P(A)). On the topological side, we denote by X top Y if Y G d, T (X) = O(X) in P b (A, d) where O(X) is the orbit of X under the action of G d, T. In order to avoid confusion, we use the following different notation: X top Y if and only if Y G e X = O e (X) in P b (A, e) for a fixed projective dimension sequence e where O e (X) = O(X) P b (A, e). We can rewrite Theorem 1 and Theorem 2 in [JSZ] as follows: Theorem 3.1. For any X and Y in K b (P(A)), then X Y if and only if X top Y. Indeed, under the topology defined in Section 2.3, we have O e (X) = O(X). e Proposition 3.2. The orbit of any complex is locally closed in P b (A, d). Proof. For a fixed projective dimension sequence e, let x P b (A, e). Then the orbit O e (y x ) is locally-closed subset in P b (A, e ) where y x is obtained by applying a operator in T d to x and the projective dimension sequence of y x is e. That is, O e (y x ) = C e (y x ) D e (y x ) where C e (y x ) and D e (y x ) are open and closed subset in P b (A, e ) respectively. So we have, for the orbit of x in P b (A, d), O(x) = e O e (y x ) = e C e (y x ) D e (y x ) = ( e C e (y x )) ( e D e (y x )) According to our topology, it is a locally-closed subset in P b (A, d). We recall the following property which is proved in [JSZ]: A complex P (x) corresponding to x P b (A, e) is partial tilting complex if Hom K b (A)(P (x), P (x)[1]) = 0. Then O e (x) = {y P b (A, e) y x} is open in P b (A, e). Furthermore, the orbit O(x) in P b (A, d) is also open if the corresponding complex P (x) is a partial tilting complex. Proposition 3.3. Any point in QP b (A, d) is not closed, that is, any orbit is not closed in P b (A, d). Proof. Take any projective A-module P and t C, we define the complex C(t) = (P i, i ) i Z by P 0 = P 1 = P and other P i = 0 for i 0, 1; 0 = t and other i = 0 for i 0. If t 0, then C(t) is contractible but C(0) is not quasi-isomorphic to zero. Moreover, C(1) top C(0). Similarly, For any x P b (A, d), P (x) is the corresponding complex, P (x) C(1) top P (x) C(0), i.e.the point corresponding to

DERIVED CATEGORIES AND LIE ALGEBRAS 17 P (x) C(0) is in the closure of the point corresponding to P (x) C(1)(it is quasi isomorphic to P (x)), but not in its orbit. This shows the orbit of x is not closed. By this proposition, we can construct an infinite sequence of non-trivial degenerations: P (x) top P (x) C(0) top P (x) C(0) C(0) 3.2. We recall the definition of constructible functions and results on the Euler characteristic of complex algebraic variety. Then we define the quasi Euler characteristic for the quotient space of a complex algebraic variety. Definition 3.4. A function f on a complex algebraic variety X is called constructible if f is the sum of finite terms with the form as m i 1 Oi, where m i C and 1 Oi takes values 1 on each point of the constructible subset O i and 0 otherwise. We denote by M(X) the set of all constructible functions on algebraic variety X with values in C. The set M(X) is naturally a C-linear space. Let G be an algebraic group acting on X. Then we denote by M G (X) the subspace of M(X) consisting of all G-invariant functions. Let χ denote Euler characteristic in compactly-supported cohomology. Let X be an algebraic variety and O a constructible subset as the disjoint union of finitely many locally closed subsets X i for i = 1,, m. Define χ(o) = m i=1 χ(x i). We note that it is well-defined. We will use the following properties: Proposition 3.5 ([Rie] and [Jo1]). Let X, Y be algebraic varieties over C. Then (1) If the algebraic variety X is the disjoint union of finitely many constructible sets X 1,, X r, then r χ(x) = χ(x i ). i=1 (2) If ϕ : X Y is a morphism with the property that all fibers have the same Euler characteristic χ, then χ(x) = χ χ(y ). In particular, if ϕ is a locally trivial fibration in the analytic topology with fibre F, then χ(x) = χ(f ) χ(y ). (3) χ(c n ) = 1 and χ(p n ) = n + 1 for all n 0. We recall the definition of pushforward functor from the category of algebraic varieties over C to the category of Q-vector spaces (see [Mac] and [Jo1]). Let φ : X Y be a morphism of varieties. For f M(X) and y Y, define φ (f)(y) = c Q cχ(f 1 (c) φ 1 (y)) Theorem 3.6 ([Di],[Jo1]). Let X, Y and Z be algebraic varieties over C, φ : X Y and ψ : Y Z be morphisms of varieties, and f M(X). Then φ (f) is constructible, φ : M(X) M(Y ) is a Q-linear map and (ψ φ) = (ψ) (φ) as Q-linear maps from M(X) to M(Z). In order to deal with orbit spaces, we need to consider geometric quotients. Definition 3.7. Let G be an algebraic group acting on a variety X and φ : X Y be a G-invariant morphism, i.e. a morphism constant on orbits. The pair (Y, φ) is called a geometric quotient if φ is open and for any open subset U of Y, the

18 JIE XIAO, FAN XU AND GUANGLIAN ZHANG associated comorphism identifies the ring O Y (U) of regular functions on U with the ring O X (φ 1 (U)) G of G-invariant regular functions on φ 1 (U). The following result due to Rosenlicht [Ro] is essential to us. Lemma 3.8. Let X be a G-variety, then there exists a open and dense G-stable subset which has a geometric G-quotient. By this Lemma, we can construct a finite stratification over X. Let U 1 be an open and dense G-stable subset of X as in Lemma 3.8. Then dim C (X U 1 ) < dim C X. We can use the above lemma again, there exists a dense open G-stable subset U 2 of X U 1 which has a geometric G-quotient. Inductively, we get a finite stratification X = l i=1 U i where U i is a G-invariant locally closed subset and has a geometric quotient, l dim C X. We denote by φ Ui the geometric quotient map on U i. Depending on this stratification, we define the quasi Euler characteristic of orbit space of X under the group action of G as follows: χ(x/g) = l χ(φ Ui (U i )) i=1 Here we use the same notation as Euler characteristic. Indeed, when G = id, then the quasi Euler characteristic is the Euler characteristic. So the overlapping notation should not cause any confusion. It is well-defined by the following observation. Lemma 3.9. Let X be a G-variety. If X = l i=1 U i and X = m j=1 V j are two finite stratifications satisfying that U i and V j are G-invariant locally closed subsets and have geometric G -quotients, then l m χ(φ Ui (U i )) = χ(φ Vj (V j )) i=1 Proof. First, the above Lemma tells us that the quotient topologies and the Zariski topologies as varieties coincidence for φ Ui (U i ) and φ Vj (V j ) respectively. This implies that φ Ui (U i V j ) = φ Vj (U i V j ). For any U i, φ Ui : U i φ Ui (U i ) is a morphism between complex algebraic varieties, hence image of constructible subset is constructible. This shows φ Ui (U i V j ) is constructible subset of φ Ui (U i ) since U i V j is constructible subset of U i. We get a partition φ Ui (U i ) = m j=1 φ U i (U i V j ). By Proposition 3.5, we have χ(φ Ui (U i )) = m j=1 χ(φ U i (U i V j )). Therefore, we have l χ(φ Ui (U i )) = i=1 l i=1 j=1 m χ(φ Ui (U i V j )) = j=1 l m m χ(φ Vj (U i V j )) = χ(φ Vj (V j )). i=1 j=1 In order to compute the Euler characteristic, we need the following two lemmas: Lemma 3.10. Let X be a G-variety. If X/G is the disjoint union of finitely many constructible subsets Z 1,, Z r, then χ(x/g) = r i=1 χ(z i). j=1

DERIVED CATEGORIES AND LIE ALGEBRAS 19 Proof. Let π : X X/G be the natural quotient map. It is clear that π 1 (Z i ) is constructible subset of X, so we can construct a stratification over π 1 (Z i ). Those stratifications for all i = 1,, r are disjoint with each other. They give rise to a G-invariant stratification over X. By Lemma 3.9 and Proposition 3.5, we deduce the identity. Lemma 3.11. Let X and Y be two algebraic varieties under the actions of the algebraic groups G and H, respectively. For the following commutativity diagram which is induced by a morphism of varieties ϕ : X Y, (15) X ϕ Y π X π Y X/G φ Y/H where π X and π Y are the canonical quotient maps. If φ is surjective and all fibres for the map φ have the same quasi Euler characteristic χ, then χ(x/g) = χ(y/h) χ Moreover, if there exists a finite stratification of Y/H = Y i such that all fibres for the map φ have the same quasi Euler characteristic χ i over any stratum Y i, then χ(x/g) = χ(y i ) χ i := χ(φ 1 (ȳ)). i ȳ Y/H Proof. Let X = m i=1 U i be a finite stratification satisfying that U i are G-invariant locally closed subsets and have geometric G-quotients. Then ϕ(u i ) is H-invariant constructible subset of Y. In a similar way as above, we can construct a finite stratification ϕ(u i ) = mi j=1 V ij over ϕ(u i ) such that V ij has geometric H-quotient. Moreover, π Vij (V ij ) is locally closed subset of π Y ϕ(u i ) by quotient topology. Hence U ij := (φπ X ) 1 (π Vij (V ij )) = (π Y ϕ) 1 (π Vij (V ij )) is locally closed subset of U i. The geometric quotient over U i induces the geometric quotient over U ij. Therefore we get the morphism between two geometric quotients φ ij : π Uij (U ij ) π Vij (V ij ) such that all fibres have the same Euler characteristic. Hence, using Proposition 3.5, we obtain χ(π Uij (U ij )) = χ(π Vij (V ij )) χ. Depending on this identity, we have χ(x/g) = i χ(π Ui (U i )) = ij χ(π Uij (U ij )) = ij χ(π Vij (V ij )) χ = χ(y/h) χ The rest part of the lemma follows the above conclusion. In particular, we frequently use the following corollary. Corollary 3.12. Let X, Y be algebraic varieties over C under the actions of an algebraic groups G. These actions naturally induce an action of G on X Y. Then χ(x G Y ) = χ(x/g y ) y Y/G where G y is the stabilizer in G of y Y and X G Y is the orbit space of X Y under the action of G.

20 JIE XIAO, FAN XU AND GUANGLIAN ZHANG 3.3. For any G d, T -invariant constructible subsets O 1 and O 2 of P b (A, d 1 ) and P b (A, d 2 ) respectively, we define the subset O 1 O 2 of P b (A, d 1 + d 2 ) to be the set of z P b (A, d 1 + d 2 ) such that there exists a triangle P (y) P (z) P (x) P (x)[1] in D b (A) where x O 1 and y O 2. By the octahedral axiom, we have (O 1 O 2 ) O 3 = O 1 (O 2 O 3 ). Inductively, we can define O 1 O 2 O s for all s > 1. They are still G d, T -invariant constructible subset. A G d, T -invariant constructible subset O of P b (A, d) is called support-bounded if there are finitely many G d, T orbits O X1, O X2,, O Xs such that O O X1 O X2 O Xs Proposition 3.13. A G d, T -invariant constructible subset O of P b (A, d) is supportbounded if and only if there exists a projective dimension sequence e such that O = G d, T (O P b (A, e)). In the following, we call this e a support projective dimension sequence of O. Proof. If O is support-bounded, then O O X1 O X2 O Xs. We may assume the projective dimension sequence of X i is e i. For any x O, by using the cone structure of distinguished triangle, we may assume that P (x) is quasi-isomorphic to a projective complex with projective dimension sequence e = e 1 + e 2 + + e s. Then x G d, T (O P b (A, e)). Conversely, suppose O = G d, T (O P b (A, e)) for some projective dimension sequence e = {(e i 1, e i 2,, e i l )}. Since O is G d, T -invariant, it is enough to prove that there exist X 1,, X n such that O P b (A, e) O X1 O Xn. Let P i = l j=1 ei j P j be the canonical decomposition of projective A-modules. Let x O P e and the corresponding projective complex P (x) be of the form: 0 P 1 d 1 P 2 d 2 P dm 1 d r P dm 0. We consider the complex M 1 to be given in the following form: 0 0 P 2 d 2 P m 1 d r P m 0. Then we have the triangle: P 1 M 1 [1] P (x) P 1 [1] where P 1 means the stalk complex given by the projective A-module P 1. Then x O M1[1] O P 1 [1]. Here M 1 [1] is a projective complex which has less nonzero terms than that of P (x). Repeating the above step on M 1 [1] and using induction, we finally get x O P m [m] O P 1 [1]. It verifies our claim. The following proposition shows the property of support-bounded is invariant under derived equivalence. Proposition 3.14. If two algebras A and B are derived equivalent, then this equivalent functor F induces the map f as in Theorem B sending constructible sets of support bounded in P b (A) to constructible sets of support bounded in P b (B).