ON THE CENTER OF A TRIANGULATED CATEGORY

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1 ON THE CENTER OF A TRIANGULATED CATEGORY HENNING KRAUSE AND YU YE Abtract. We dicu ome baic propertie of the graded center of a triangulated category and compute example ariing in repreentation theory of finite dimenional algebra. 1. Introduction The graded center of a triangulated category T with upenion functor Σ i a Z- graded ring. The degree n component conit of all natural tranformation from the identity functor Id to Σ n which commute modulo the ign ( 1) n with Σ. The graded center i the univeral graded commutative ring that act on T. For intance, the Hochchild cohomology HH (A) of an algebra A act on the derived category D(A) via a morphim HH (A) Z (D(A)) into the graded center. It eem that the firt ytematic ue of the graded center appear in work of Buchweitz and Flenner on the Hochchild cohomolgy of ingular pace [BF]. For related work of Lowen and van den Bergh in the etting of differential graded categorie we refer to [LV]. Block of finite group and their modular repreentation theory provide the context for recent work of Linckelmann on the graded center of table and derived categorie [Li]. Cloely related i the tudy of cohomological upport varietie which depend on the appropriate choice of a graded commutative ring acting on a triangulated category; ee [BIK]. In thi article, we prove ome tructural reult and provide complete decription of the graded center for ome mall example. The article i organized a follow. In 2, it i hown that for any abelian category A with enough projective object, there i an iomorphim of graded commutative ring Z (D b (A)) = Z (D b (Proj(A))). Here Proj(A) denote the full ubcategory of A coniting of all projective object and the iomorphim i given by retriction. In 3-4, we deal with derived categorie of hereditary categorie. Note that for a hereditary category, the derived category and the bounded derived category have the ame graded center. In 3, the category mod(r) of finitely generated module over a Dedekind domain R i conidered. We calculate Z (D(mod(R))) explicitly. A we how it relate cloely to the reidue field of all the maximal ideal of R. In 4, we conider the module category of a tame hereditary algebra and the category of coherent heave on a weighted projective line of non-negative Euler characteritic. We compute the graded center of their bounded derived categorie. Note that our method do not apply to wild cae. For a weighted projective line of wild type, we only get a ubalgebra of the graded center. The econd named author gratefully acknowledge upport by the Alexander von Humboldt Stiftung. He i upported in part by the National Natural Science Foundation of China (Grant No ). 1

2 2 HENNING KRAUSE AND YU YE In 5-6, we decribe the graded center of D b (mod(k[x]/x 2 )) and mod(k[x]/x n ) for n 2 repectively. 2. Morphim between graded center Definition 2.1. Let T be a triangulated category and Σ the upenion functor of T. We define a Z-graded abelian group Z (T ) = Z (T, Σ) a follow. For any n Z, let Z n (T ) denote the collection of all natural tranformation η : Id Σ n which atify ησ = ( 1) n Ση. The compoition of natural tranformation give Z (T ) the tructure of a graded commutative ring, and we call it the graded center of T. Graded commutative here mean that ηζ = ( 1) mn ζη for all η Z n (T ) and ζ Z m (T ). Remark 2.2. (1) The definition of the graded center Z (T ) make ene for any graded category, that i, an additive category equipped with an autoequivalence. In particular, the choice of the exact triangle of T i not relevant for Z (T ). (2) The degree zero part Z 0 (T ) i a ubring of the uual center Z(T ) of T, which by definition conit of all natural tranformation from the identity functor to itelf. Note that Z 0 (T ) = Z(T ) if Σ = Id. (3) The graded center Z (T ) need not be a et in general. However, it will be a et when the category T i mall. (4) For any object M in T we define the graded ring Ext T (M, M) by etting Ext n T (M, M) = Hom T (M, Σ n M) for any integer n. By definition there i a canonical graded ring morphim Z (T ) Ext T (M, M) mapping a natural tranformation η : Id Σ n to the morphim η M : M Σ n M. Following Rouquier [Ro] we et M 1 to be the full additive ubcategory of T which contain M and i cloed under finite direct um, ummand and the action of Σ, and for i 2 we define inductively M i a the full additive ubcategory of T coniting of all object iomorphic to direct ummand of object Z for which there exit an exact triangle X Y Z ΣX with X M 1 and Y M n 1. Now uppoe that M i an object of T with T = M d+1 for ome poitive integer d. We et N to be the kernel of the canonical morphim Z (T ) Ext T (M, M). It can be hown in thi cae that N i a nilpotent ideal atifying N 2d = 0; ee [Li] for a proof. In particular, Z (T ) i modulo nilpotent element a et. Let F : S T be an exact functor between triangulated categorie. An obviou quetion to ak i when the functor F induce morphim between Z (S) and Z (T ). Recently, Linckelmann gave an affirmative anwer for thi quetion in the cae that there exit a functor G: T S which i imultaneouly left and right adjoint to F and atifie ome further compatibility condition [Li]. The anwer for general F eem to be not known. The following propoition how that in ome very pecific ituation, for intance when F i fully faithful, we do obtain ome morphim between the graded center. Propoition 2.3. Let T be a triangulated category and S a full triangulated ubcategory. (1) The incluion functor i: S T induce a morphim of graded ring i : Z (T ) Z (S),

3 ON THE CENTER OF A TRIANGULATED CATEGORY 3 where i (η) X = η X for any η Z (T ) and X S. (2) The canonical functor π : T T /S induce a morphim of graded ring π : Z (T ) Z (T /S), where π (η) X = π(η X ) for any η Z (T ) and X T /S. Proof. The proof i routine. To check that π i well defined, one ue the fact that for any commutative diagram in T X α Z β f Y X Z f Y with Cone() and Cone( ) in S, we have γ (f/) = (f / ) α in the quotient category T /S, where we ue to denote the morphim whoe cone are in S. Up to now, little eem to be known about the propertie of the above morphim. For example, the quetion when i and π are urjective or injective i of pecial interet to u. Alo, one might tudy the induced morphim of graded ring (i, π ): Z (T ) Z (S) Z (T /S). Example 2.4. Let S T denote the direct product of two triangulated categorie S and T. We view S a a thick ubcategory of S T and the correponding quotient i equivalent to T. Then we have Z (T S) = Z (T ) Z (S) via the morphim (i, π ). For the ret of thi ection we focu on homotopy categorie and derived categorie. Firtly we introduce ome baic notation and convention. Let A be any additive category. We denote by C(A) the category of chain complexe in A. Recall that a chain complex in A i a equence of morphim in A X = ( X n d X n X n 1 ) with d X n d X n+1 = 0 for all n Z. A morphim of complexe i a chain map f : X Y coniting of a family of morphim f n : X n Y n in A with n Z uch that f n d X n+1 = d Y n+1 f n+1 for all n, that i, the diagram γ d X n+1 d X n+1 X n X n X n 1 f n+1 f n f n 1 Y n+1 d Y n+1 Y n d Y n Y n 1 commute. We denote by C + (A) the full ubcategory of C(A) which conit of all bounded below complexe, that i, the complexe X with X n = 0 for n 0. Similarly, we denote by C (A) and C b (A) the full ubcategory of bounded above complexe and complexe bounded in both direction, repectively. If moreover A i abelian, then for any integer n the n-th homology group H n (X) i by defintion Ker(d X n )/ Im(d X n+1 ), and any morphim f of complexe induce morphim of homology group H n (f): H n (X) H n (Y ) for all n Z.

4 4 HENNING KRAUSE AND YU YE The homotopy category K(A) ha the ame object a C(A). The morphim are the equivalence clae of the morphim in C(A) modulo the null-homotopic morphim, that i, thoe with component of the form d Y n+1 h n + h n 1 d X n for ome family of morphim h n : X n Y n+1 in A, n Z. The upenion functor (or hift functor) Σ of C(A) i defined by (ΣX) n = X n 1, d ΣX n = d X n 1 on the object and by (Σf) n = f n 1 on any morphim f. Clearly Σ i not only an autoequivalence but alo an automorphim of C(A). Moreover, Σ alo induce an automorphim of K(A) and K(A) admit a triangulated tructure with upenion functor Σ. Let D(A) denote the derived category of A, i.e., the localization of K(A) with repect to the quai-iomorphim. Note that D(A) i again a triangulated category with upenion functor Σ. One define K (A) and D (A) with {+, b, } in a imilar way. Now let A be an abelian category with enough projective object, and P be the full ubcategory coniting of all projective object. We denote by K +,b (P) the thick ubcategory of K + (P) which conit of bounded below complexe X with H n (X) = 0 for almot all n. Clearly, we have K b (P) K +,b (P). In ome cae, object in K b (P) are alo called perfect complexe. It i known that the following compoition of functor K +,b (P) = D b (A) K + (P) = K + (A) D + (A) induce equivalence K + (P) D + (A) and K +,b (P) D b (A) of triangulated categorie. The quotient category D g (A) = D b (A)/K b (P) i called the triangulated category of ingularitie of A, becaue it i an invariant of the ingularitie provided that A i the category of heave on ome variety. We know that D g (A) = 0 if and only if all object of A have finite homological dimenion. When A = Mod(A) for ome elf-injective ring A, then D g (A) i equavalent to the table module category Mod(A) of A. We are now in a poition to tate our main reult of thi ection. Theorem 2.5. Let A be an abelian category with enough projective object and P the full ubcategory coniting of all projective object. Then the embedding K b (P) K +,b (P) induce an iomorphim Z (K b (P)) Z (D b (A)) of graded commutative ring. To prove the theorem, we need ome preparation. For each n Z the n-th truncation functor ι n : C(A) C(A) i defined for a complex X by (ι n X) i = X i for i n and 0 for i > n, and d ιn X i = d X i for i n and 0 for i > n. Clearly, ι n end C(A) to C (A) and C + (A) to C b (A). Note that we have a natural morphim i n : ι n X X, and ometime we ue i n X to emphaize X. We have (in ) = id for n and 0 for > n. The following lemma i crucial in the proof of the main theorem.

5 ON THE CENTER OF A TRIANGULATED CATEGORY 5 Lemma 2.6. Let X C(P), Y C(A) and f : X Y be a chain map with H n (Y ) = 0 for n > 0. Then f i null-homotopic if and only if the compoition f i n : ι n X X Y i null-homotopic for ome n 0. Proof. One direction i clear ince the null-homotopic morphim form an ideal. Converely, uppoe that f i n i null-homotopic for ome n 0. To how that f i alo null-homotopic, it uffice to find a family {h n : X n Y n+1 n Z}, uch that f n = d Y n+1 h n + h n 1 d X n hold for all n. By applying the hift functor, one can aume without lo of generality that f i 0 i null-homotopic. Thu there exit a family of morphim in A, ay {h n : X n Y n+1, n 0}, uch that f n = d Y n+1 h n + h n 1 d X n for all n 0. Since f i a chain map, we have d Y 1 f 1 = f 0 d X 1 = d Y 1 h 0d X 1 + h 1d X 0 dx 1, and hence d Y 1 (f 1 h 0 d X 1 ) = 0, which implie that Im(f 1 h 0 d X 1 ) Ker(dY 1 ) = Im(dY 2 ), the lat equality hold becaue H 1 (Y ) = 0. Now X 1 i projective implie that f 1 h 0 d X 1 factor through d Y 2, i.e., there exit h 1 : X 1 Y 2 uch that f 1 = d Y 2 h 1 + h 0 d X 1, thu we get the required h 1. Now repeat the argument and the lemma follow. Propoition 2.7. Let t Z and η : Id Σ t be a natural tranformation for the category K b (P). Then η extend uniquely to a natural tranformation η : Id Σ t for the category K +,b (P). Proof. Firt we will contruct a morphim η X : X Σ t X for any X K +,b (P). The idea i to ue certain approximation. Since η i a natural tranformation for K b (P), we have for each n a morphim ζ n = η ι n X : ι n X Σ t ι n X. Now we fix a chain map ζ 0 : ι 0 X Σ t ι 0 X which i a repreentative of ζ 0. We can contruct inductively the repreentative ζ n of ζ n for all n 0, uch that ζ n+1 i = ζ n i for all n 0 and i n. In fact, uppoe that ζ n ha been contructed, and let ξ be any repreentative of ζ n+1 X. Conider the morphim j : ι n X ι n+1 X which i given by j m = id Xm for all m n and 0 otherwie. Since η i a natural tranformation, the diagram ι n X ζ n j ι n+1 X ξ Σ t ι n X Σt j Σ t ι n+1 X commute in the category K b (P), i.e., δ := ξ j Σ t j ζ n i null-homotopic. Explicitly, δ i = ξ i η i for i n and δ i = 0 for i n + 1. Now there exit a family of morphim {h i : (ι n X) i (Σ t ι n+1 X) i+1 i Z} with h i = 0 for i > n, uch that δ i = d Σt ι n+1 X i+1 h i + h i 1 d ιn X i. The family {h i } can be viewed a a family of morphim {h i : (ι n+1 X) i (Σ t ι n+1 X) i+1 i Z}, thu it give a null-homotopic morphim δ : ι n+1 X Σ t ι n+1 X, which atifie δ i = δ i for all i n. We are done by etting ζ n+1 = ξ δ. Now we define η X by ( η X ) n = ζn 0 for n 0 and ( η X ) n = ζn n for n > 0. We claim that η i a natural tranformation from Id to Σ t for the category K +,b (P). Note that by contruction, η X atifie the following condition: for any n 0, there exit a repreentative ζx n for η ι n X, which i given by (ζx n ) i = ( η X ) i for i n, and (ζx n ) i = 0 for i n + 1. In other word, η X i n X = Σt i n X η ι n X a chain map for all n 0, where i n X denote the natural morphim from ιn X to X a before.

6 6 HENNING KRAUSE AND YU YE Now let X, Y K +,b (P) and f : X Y be any chain map. Aume that η X : X Σ t X and η Y : Y Σ t Y are arbitrary chain map with the property η X i n X = Σt i n X η ι n X and η Y i n Y = Σt i n Y η ι n Y for n 0. We will how that η Y f = Σ t f η X in K +,b (P). Note that in the cube below, the other five face are commutative by the contruction of η X, η Y, ι n X and ιn Y. X i n X ι n X i n Y ι n Y ηι n X Y f η ι n Y η X Σ t X Σ t ι n X η Y Σ t Y Σ t ι n Y By Lemma 2.6 it uffice to how that η Y f i n X = Σt f η X i n X for ome ufficiently large n. Thi i equivalent to howing that η Y i n Y ιn f = Σ t f Σ t i n X η ι n X. The left hand ide i Σ t i n Y η ι n Y ι n f, and ince η ι n Y ι n f = Σ t ι n f η ι n X and Σ t f Σ t i n X = Σt i n Y Σt ι n f, the equality hold. Thu by fixing uch η X for each X, we can extend η to the category K +,b (P). For the uniquene, we need only to take f = id X in the above argument. Thi complete the proof. Corollary 2.8. Let A and P be a before. Then Z t (K b (P)) = 0 for all t < 0, and therefore Z (K b (P)) and Z (D b (A)) are poitively graded. Proof. Suppoe that η i a natural tranformation from Id K b (P) to Σ t for ome t < 0. K b (P) We prove that η X = 0 by uing induction on the length of the upport of X, where the upport of X mean the interval [i, j], uch that i and j are repectively the minimal and maximal integer m with X m 0. Without lo of generality, we may aume that i = 0 and we ue induction on j. In the cae j = 0, clearly Hom K b (P)(X, Σ t X) = 0 for t < 0. Suppoe η X = 0 for all j m and uppoe X = ( 0 X m+1 X 0 0 ). By the ame argument a in the proof of Propoition 2.7, there i a repreentative ζ of η X, uch that ζ i = 0 for all i m, and now the aumption t < 0 implie that (Σ t X) m+1 = 0, which force that ζ m+1 = 0, thu ζ = 0 and hence η X = 0. With the above preparation, we can now prove the main theorem. Proof of Theorem 2.5. Fix η Z t (K b (P)). By Propoition 2.7, η extend uniquely to a natural tranformation η : Id Σ t for the category K +,b (P), and clearly i ( η) = η, where i i induced by the embedding i: K b (P) K +,b (P). By the ame argument a in the lat part in the proof of Propoition 2.7, one can how that ησ = ( 1) n Σ η, which implie that η Z n (K +,b (P)). Thi prove the urjectivity of i. The injectivity of i follow from the uniquene of the extenion. Remark 2.9. Suppoe there are enough injective object in A and denote by I the full ubcategory of injective. Then we have D b (A) = K,b (I) and the dual verion of the theorem ay that there i an iomorphim of graded center Z (K b (I)) = Z (D b (A)). Σ t f

7 ON THE CENTER OF A TRIANGULATED CATEGORY 7 3. Finitely generated module over Dedekind domain The following two ection are devoted to tudying the graded center of the derived category of ome hereditary categorie. We look at ome baic example and ue explicit calculation. Firt we dicu the derived category of the category of finitely generated module mod(r) for any Dedekind domain R. We tart with ome preparation. Let R be an arbitrary unitary ring and denote by Z(R) the center of R. Let z Z(R) and M mod(r). Then we have a morphim l z Hom R (M, M), which i given by l z (m) = z m. Thi i indeed a morphim of module ince z i in the center of R. Moreover, l z induce a natural tranformation from the identity functor to itelf for mod(r) a well a for D b (mod(r)). Now let H be a hereditary abelian category, that i, Ext i H (M, N) = 0 for any M, N H and i 2. Conider the derived category of H and oberve that any object X D(H) i iomorphic to i Z Σi (H i (X)). Here, Σ i the hift functor and H i (X) i viewed a a talk complex concentrated in degree zero. For a imple proof of thi, ee [Kr, 1]. We have the following eay lemma. Lemma 3.1. Left multiplication induce an injective ring homomorphim Z(R) Z 0 (D b (mod(r))). Moreover, if R i hereditary, then thi i an iomorphim. Proof. For a proof, we jut ue the fact that left multiplication give an iomorphim from Z(R) to the uual center of mod(r), i.e., the ring of natural tranformation from the identity functor to itelf, and that mod(r) i a full ubcategory of D b (mod(r)). Moreover, if R i hereditary, then all object of D b (mod(r)) are of the form i Z Σi M i with M i mod(r) viewed a a talk complex concentrated in degree zero. Now the lemma follow eaily. Remark 3.2. Note that the morphim in the lemma need not be an iomorphim; ee [Kü] or 5 below. Lemma 3.3. Let H be an arbitrary hereditary category. Then Z (D b (H)) i concentrated in degree 0 and 1. Moreover, the incluion D b (H) D (H) D(H) induce iomorphim of graded center Z (D b (H)) = Z (D (H)) = Z (D(H)). Proof. We have Hom D(H) (M, Σ m M) = Ext m H (M, M) = 0 for all M H and m 2, ince H i hereditary. Thu there i no nontrivial natural tranformation from Id to Σ m for the category D b (H) for m 2, and the firt part of the lemma follow. The lat aertion follow from the fact that any element in the graded center Z (D(H)) i uniquely determined by the retriction to the talk complexe. The minor difference between both derived categorie i that any object in D(H) i an infinite direct um of talk complexe while object in D b (H) can alway be written a finite direct um. Similarly, we have Z (D + (H)) = Z (D(H)). Due to the lemma, to tudy the graded center of the derived categorie of hereditary abelian categorie, we need only to conider the bounded one. Now uppoe that H = H 1 H 2, where H 1 and H 2 are full additive ubcategory of H, and we ue to indicate that any object of H i a direct um of an object of H 1 and an object of H 2, and Hom H (M 2, M 1 ) = Ext 1 H (M 1, M 2 ) = 0 for all M 1 H 1 and M 2 H 2. We et Σ H 1 to be the minimal additive ubcategory of D b (H) which contain H 1 and i cloed under Σ, in other word the ubcategory coniting of all

8 8 HENNING KRAUSE AND YU YE complexe with homologie contained in H 1. Note that Σ H 1 i not a triangulated ubcategory in general. Thi will happen if H 1 i a thick ubcategory of H, i.e., H 1 i cloed under extenion, kernel and cokernel, and in thi cae, H 1 i alo a hereditary abelian category and Σ H 1 = D b (H 1 ). For a proof of thi, one ue again the fact that any object in D b (H) i a direct um of talk complexe. Since Σ i an autoequivalence of Σ H 1, we can alo define the graded center of Σ H 1 with repect to Σ, and denote it by Z (Σ H 1 ). Propoition 3.4. Let H = H 1 H 2 be a hereditary abelian category. Then the retriction map induce an iomorphim of abelian group Z 1 (D b (H)) = Z 1 (Σ H 1 ) Z 1 (Σ H 2 ). Proof. We produce an invere map. Firt oberve that any object in D b (H) can be written uniqueley a X 1 X 2 with X 1 Σ H 1 and X 2 Σ H 2. Let η 1 : Id Σ H 1 Σ Σ H 1 and η 2 : Id Σ H 2 Σ Σ H 2 be natural tranformation. Then we define η : Id D b (H) Σ D b (H) by etting η X1 X 2 to be the map (η 1 ) X1 (η 2 ) X2. We will how that η i indeed a natural tranformation. To thi end we need to check that for any morphim f : X Y in D b (H), we have Σf η X = η Y f. Since any object of D b (H) can be uniquely written a i Z Σi (M1 i M 2 i) with M 1 i H 1 and M2 i H 2, we need only to check the above compatibility for the morphim of the form f : Σ i M 1 Σ j M 2 and g : Σ i M 2 Σ j M 1 with M 1 H 1 and M 2 H 2. We claim that Σf η M1 = η M2 f = 0 and Σg η M2 = η M1 g = 0. In fact, ince H i hereditary, both ide will vanih unle j = i + 1 or j = i. If j = i + 1, the equalitie hold ince Ext 2 H (M, N) = 0 for all M, N H. Otherwie, if j = i, we have g = 0 and Ext 1 H (M 1, M 2 ) = 0. Now the aertion follow eaily, and thi complete the proof. Now we can begin the tudy of Z (D b (mod(r))) for a Dedekind domain R. A Dedekind domain i an integral domain uch that each ideal can be written a a finite product of prime ideal, or equivalently, a noetherian integrally cloed domain with Krull dimenion at mot one. Thi name wa given to uch ring in honor of R. Dedekind, who wa one of the firt to tudy uch ring in the The ring of algebraic integer of number field provide an important cla of Dedekind domain, which play a crucial role in algebraic number theory. The aumption on the Krull dimenion implie that each nonzero prime ideal of R i maximal, and that the category H = mod(r) i hereditary and any object M of mod(r) i a direct um of a torion-free module and a torion module. Any finitely generated torion-free module i projective, and any finitely generated torion module i a finite direct um of cyclic module. Moreover, by the Chinee Remainder Theorem, each cyclic module i a finite direct um of module of the form R/p l with p a maximal ideal of R and l 1. We have a decompoition of categorie H = H + H 0, where H + denote the full ubcategory of H coniting of all projective module and H 0 conit of all torion module. Note that H 0 i a thick ubcategory and hence hereditary. Moreover, there exit an Aulander-Reiten tranlation τ in H 0, which i by definition an autoequivalence of H 0 uch that there exit a natural iomorphim D Ext 1 H (X, Y ) = Hom H (Y, τx). Thi identity i uually called Serre duality and implie the exitence of Aulander-Reiten equence in H 0.

9 ON THE CENTER OF A TRIANGULATED CATEGORY 9 Let max(r) denote the et of all maximal ideal of R. Then all the indecompoable in H 0 are given by {R/p l p max(r), l 1}. Denote by H p the ubcategory of H 0 coniting of all p-torion module, i.e., the additive category generated by {R/p l l 1} which i alo abelian, hereditary and a thick ubcategory of H 0. Note that H p i τ- invariant and H 0 = p max(r) H p. One can how that H p i equivalent to the ubcategory of mod(r p ) which conit of torion R p -module, where R p i the localization of R at p; and H p i obviouly a full ubcategory of mod( ˆR p ), where ˆR p i the completion of R with repect to p, i.e., the invere limit lim R/p l. Note that we have an iomorphim R/p = R p /pr p of reidue field, and we denote it by k p. Fix an element x p \ p 2. Since R p i a dicrete valuation ring, the multiplication with x give u an iomorphim R/p l = p/p l+1 of R-module for any l 1. Thu we have in H p Aulander-Reiten equence and σ 1 p : 0 R/p R/p 2 R/p 0 σ l p : 0 R/p l R/p l 1 R/p l+1 R/p l 0 for all l 2, where the morphim from R/p l+1 to R/p l i the natural quotient map, and we ue the iomorphim R/p l = p/p l+1 induced by the multiplication with x. Thi ay that the Aulander-Reiten quiver of H p i a tube of τ-period 1. For the Aulander- Reiten equence for Dedekind domain, ee alo [AL], Example3.1. Note that we have a natural equivalence τ = Id Hp. It i eay to how that Ω(R/p ) = R p for any 1, where Ω: mod(r p ) mod(r p ) i the yzygy functor. We have a preentation of R/p 0 R p l x R p R/p 0, where l x denote the multiplication by x. Note that we have an iomorphim R/p = R p /p R p. Now the above exact equence induce an epimorphim Hom R (R p, R/p i ) Ext 1 R(R/p, R/p i ), and when i thi induce an iomorphim. Now it i eay to how that the Aulander- Reiten equence σp correpond to the compoition R p R/p = p 1 /p R/p. Moreover, all the Aulander-Reiten equence with tarting term R/p are given by the morphim of the form R p Soc(R/p ) R/p, which equal λσp for ome 0 λ k p. On the other hand, we identify Hom D b (H)(M, ΣN) = Ext 1 H (M, N) for any abelian category H and any object M, N H. Now we can write down the graded center of D b (H p ) explicitly by uing the notion of a trivial extenion. Let A be an arbitrary ring and M an A-A-bimodule. The trivial extenion ring of A by the bimodule M, denoted by T (A, M), i defined to be the ring whoe additive ring i A M with multiplication given by (a, m) (a, m ) = (aa, am + ma ) for all a, a A and m, m M. Note that T (A, M) can be identified with the ubring of the upper triangular ring ( ) A M 0 A

10 10 HENNING KRAUSE AND YU YE which conit of all the matrice with equal diagonal entrie. The trivial extenion ring i a poitively graded ring which i concentrated in degree 0 and 1 with T (A, M) 0 = A and T (A, M) 1 = M. Converely, let A = A 0 A 1 be an arbitrary poitively graded ring which i concentrated in degree 0 and 1. Then A = T (A 0, A 1 ) a graded ring, where the A 0 -bimodule tructure on A 1 i induced by the multiplication of A. If A i commutative and M an A-module, one can alo define the trivial extenion T (A, M), where M i viewed a an A-A-bimodule. Note that in thi cae T (A, M) i alway graded commutative. Propoition 3.5. Let R be a Dedekind domain, p a maximal ideal of R and k p the reidue field. Then a a graded ring, Z (D b (H p )) = T ( ˆR p, k p ), where k p i viewed a a imple ˆR-module. l Z,l 1 Proof. Note that element in ˆR p are by definition equence q = (q i ) i Z,i 0 with q i R/p i and atifying π i,j (q i ) = q j for all i > j, where π i,j : R/p i R/p j i the natural quotient map. Now the collection of morphim {l qi : R/p i R/p i, i N} determine uniquely an element in Z 0 (D b (H p )), where l qi i given by multiplication with q i, and it i eay to how thi correpondence give a bijection between ˆR p and Z 0 (D b (H p )), which mean that Z 0 (D b (H p )) = ˆR p. Now we conider the degree 1 component of the graded center. For any l Z, l 1, we define ηp l Z 1 (D b (H p )) by etting (ηp) l R/p = 0 for all l, (ηp) l R/p l = σp l and (ηp) l Σ i (R/p ) = ( 1) i Σ i (ηp) l R/p for all i,. To how ηp l Z 1 (D b (H p )), it uffice to how that ηp l i a natural tranformation. For thi, one need to check that for all i, j, m, n and f : Σ i R/p m Σ j R/p n, the equality Σf (ηp) l Σ i R/p m = (ηl p) Σ j R/pn f hold. Thi i clear ince both ide of the above equality vanih, unle f i an iomorphim, where we ue the fact that any σp l i given by an almot plit equence. The argument above how that if for each l, we fix an Aulander-Reiten equence, ay λ l ηp l for ome λ l k p, with tarting term R/p l, then we obtain an element Σ l λ l ηp l in Z 1 (D b (H p )). The infinite product make ene ince when applied to any object in D b (H p ) it become a finite um. Converely, let η Z 1 (D b (H p )). We claim that η R/p l correpond to either an Aulander-Reiten equence or i zero for all l. Thi can be done by induction on l. Clearly it hold for l = 1. Aume that it hold for all l n 1, we will how that it i alo true for l = n. From the exact equence 0 p n 1 /p n R/p n π R/p n 1 0 we obtain an exact equence Ext 1 R(R/p n, R/p) Ext 1 R(R/p n, R/p n ) f Ext 1 R(R/p n, R/p n 1 ) 0. It i eay to how that Ker(f) i one dimenional and panned by the Aulander- Reiten equence with tarting term R/p n. Since η i a natural tranformation, the map π : R/p n R/p n 1 yield Σπ η R/p n = η R/p n 1 π = 0, where the lat equality hold ince η R/p n 1 i given by ome Aulander-Reiten equence or zero. Thu η R/p n correpond to an Aulander-Reiten equence by the above argument. The propoition now follow.

11 ON THE CENTER OF A TRIANGULATED CATEGORY 11 Next we combine Lemma 3.1, Propoition 3.4 and that mod(r) = H + H 0 with H 0 = p max(r) H p. Thi give the following propoition. Note that H + conit of free module and hence Z 1 (Σ H + ) = 0. Propoition 3.6. Let R be a Dedekind domain and max(r) the et of all maximal ideal. Then a a graded ring Z (D b (mod(r))) = T (R, k p ), where each k p = R/p i viewed a a imple R-module. p max(r) l Z,l 1 4. Tame hereditary algebra and weighted projective line Thi ection deal with the derived category for ome further clae of hereditary categorie. We conider either the category of module mod(a) of a tame hereditary algebra A or the category Coh(X) for a weighted projective line X of non-negative Euler characteritic. Unfortunately, our method do not work for the wild cae. What we want to emphaize i that tube are of pecial importance in our calculation. Throughout thi ection, k denote an algebraically cloed field and all categorie conidered are aumed to be k-linear; therefore the graded center are k-algebra. Note that mot reult hold for an arbitrary bae field k; however the proof would require modification. We begin by tudying tube. The tube occurring in thi ection are different from the one for Dedekind domain and we will ue a different method to deal with them. Note that one can ue completed path algebra to unify the proof. Let C be a unierial hom-finite hereditary length k-category. Recall that a length category i an abelian category uch that any object ha a compoition erie of finite length. Note that a length category i alway a Krull-Remak-Schmidt category, i.e., any object can be written a a finite direct um of indecompoable and the endomorphim ring of any indecompoable object i local. Following [AR], a locally finite abelian category i called unierial if any indecompoable object of finite length ha a unique compoition erie. It follow from Theorem 2.13 in [Si], that any hom-finite length category i equivalent to the category of finite length comodule of ome baic coalgebra. And ince k i aumed to be algebraic cloed, any baic coalgebra i pointed, that i, it can be realized a a ubcoalgebra of certain path coalgebra of ome quiver. A quiver Q = (Q 0, Q 1,, t) i by definition an oriented graph, where Q 0 i the et of vertice, Q 1 the et of edge which are uually called arrow, and t are two map from Q 1 to Q 0 uch that for each arrow α, (α) and t(α) denote repectively the tarting vertex and the terminating vertex of α. A path in Q i a equence of arrow α 1 α 2 α n with t(α i ) = (α i+1 ) for 1 i n 1, (α 1 ) and t(α n ) are called the tarting vertex and terminating vertex repectively and n i the length. Each vertex v can be viewed a a path of length 0 which tart and terminate at v. It i well known that there i a path algebra and a path coalgebra tructure on the vector pace kq with bai coniting of all path in Q and the multiplication and comultiplication are given by compoing and plitting the path. Denote by kq a and (kq c,, ɛ) the path algebra and path coalgebra of Q repectively.

12 12 HENNING KRAUSE AND YU YE We denote the category of k-repreentation by Rep(Q) and the ubcategory of locally nilpotent repreentation by NRep(Q). A uual we denote the ubcategorie coniting of finite length object by rep(q) and nrep(q). It i well known that Rep(Q) i equivalent to the module category of the path algebra kq a and NRep(Q) i equivalent to the comodule category of the path coalgebra kq c. Let n, m Z {, + } with n m. We ue A [n,m] to denote the following quiver. The vertice are indexed by {i Z n i m} and for each n i m 1 there i exactly one arrow which tart at the vertex i and terminate at i + 1. Now denote the quiver A [,0], A [,+ ], A [0,+ ] and A [0,n] for any n 1 by A, A, A and A n repectively. Alo we denote by Z n the baic cycle of length n for any n 1, i.e., the quiver obtained from A n by gluing the vertice 0 and n. The following claification i a pecial cae of Theorem 2.10(i) in [CG]. Lemma 4.1. Let C be a unierial hereditary length k-category. Then C i equivalent to nrep(q) for ome quiver Q, where Q i a dijoint union of quiver of type A, A, A, A n or Z n. The idea of the proof i eay. A hown in [Si], C i equivalent to the category of finite length comodule of ome pointed coalgebra C, and that C i hereditary mean that C mut be a path coalgebra and hence C i given by ome nrep(q). The fact that C i unierial implie that for each vertex v Q 0, there i at mot one arrow tarting at v and at mot one arrow terminating at v, and hence the lemma follow. Now uppoe that Q i one of A, A, A and A n. Then the category nrep(q) i directed. More explicitly, any indecompoable object M nrep(q) i a tone, i.e., End C (M) = k and Ext 1 C (M, M) = 0. Thi ha the following conequence. Propoition 4.2. Z (D b (nrep(q))) = k for Q = A, A, A and A n. The only cae left i Q = Z n, where n 1 i a poitive integer. It i well known that nrep(z n ) i a tube of τ-period n. We need to fix ome notation. Denote by S i the imple repreentation with repect to the vertex i, and M [l] the indecompoable repreentation with ocle S i and of length l, for any i and l 1. In the category nrep(q), there i neither a nonzero projective nor a nonzero injective object, while in the category NRep(Q) there are enough injective object, and we denote by M [ ] i injective module with ocle S i. Note that {M [l] i or imply M i the indecompoable i 0 i n 1, 1 l } give a complete et of ioclae of indecompoable in NRep(Q). There i a monomorphim i l : M [l] M [l+1] and an epimorphim π [l] : M [l] M [l 1] 1 for any and l, and any morphim between the indecompoable i a linear combination of compoition of uch morphim. For convenience, we write again π a π, and we et M [l] = 0 and i l = π l = 0 for l 0. More generally, we can define monomorphim i l,t = i l+t 1 and epimorphim π l,t i l : M [l] = π t+1 l t+1 πl : M [l] We alo et i l, have the equality of morphim M [l+t], 0 n 1, l 1, t 1, M [l t] t, 0 n 1, l 1, 1 t l 1. to be the incluion M [l] M and i l,0 = π l,0 π l+1 i l = i l 1 1 πl : M [l] M [l] 1 = id [l] M. Note that we

13 ON THE CENTER OF A TRIANGULATED CATEGORY 13 for all and l. The yzygy functor Ω 1 i given by Ω 1 (M [l] ) = M l, Ω 1 (i l ) = π l and Ω 1 (π l ) = id M l. In the cae n = 1, the ubcript i omitted for implicity. Lemma 4.3. Let Q = Z n. Then Z 0 (D b (nrep(q))) = k[[ξ]], where ξ i the natural tranformation from the identity functor to itelf, which i given by ξ M [l] = i l n,n n πl,n : M [l] M [l]. It i eay to check that ξ i a natural tranformation. Alo the infinite um m 0 λ mξ m give a natural tranformation, where λ m k for all m. Oberve that thi doe make ene, becaue the um i indeed a finite um when applied to any object in nrep(q). To how that thi give all the natural tranformation, one jut ue the fact that {ξ m M [l], m 0} pan End C (M [l] ) for any M [l] nrep(q). Clearly, we have an exact equence 0 M [l]. Thi induce an epimorphim Hom C (M r [m] M [l] i l, π M,l M l 0 for any, M l ) Ext 1 [m] C (M r, M [l] ), which i In particular, we can identify Hom C (M [l], M l ) with an iomorphim when m l. Ext 1 [l] C (M, M [l] ). The following lemma i needed to decribe the degree 1 component of the graded center of D b (nrep(q)). Lemma 4.4. Let Q = Z n and n 1. If n 2, then Z 1 (D b (nrep(q))) = 0; if n = 1, then a a k-vector pace, Z 1 (D b (nrep(q))) = k η l, l Z,l 1 where η l i given by (η l ) M [l] = i 1, π l,l 1 and (η l ) M [a] = 0 for a l. Proof. Firt we conider the cae n 2. Fix η Z 1 (D b (nrep(q))). We how that η [l] M = 0 for all and l by uing induction on l. Clearly η [1] M = 0 for all, ince there i no elf extenion for the imple object if n 2. Now aume that the aertion hold for l 1. Applying the naturality of η to the injection i l 1 : M [l 1] M [l], we get the equality η [l] M i l 1 = Σi l 1 η [l 1] M = 0. We claim that thi equality hold only if η [l] M = 0. Otherwie, if η [l] M : M [l] M l i nonzero, then the dimenion of Im(η [l] M ) i at leat n ince M [n] l i the minimal ubmodule of M l with the ame top a M [l], thu the dimenion of Im(η [l] M i l 1 ) i at leat n 1 and hence nonzero (here we ee the difference between n = 1 cae and n 2 cae), and now we ue the iomorphim Hom C (M [l 1], M l ) = Ext 1 [l 1] C (M, M [l] ) to get that the left hand ide of the above equality i nonzero, thi introduce a contradiction. Now we aume that n = 1. Note that any Aulander-Reiten equence with tarting term M [l] i given by a nonzero multiple of η l. Now we can ue the ame argument M [l] a in the proof of Propoition 3.5. Combining Lemma 4.3 and 4.4, we get the following.

14 14 HENNING KRAUSE AND YU YE Propoition 4.5. Let Q = Z n and n 1. If n 2, then Z (D b (nrep(q))) = k[[ξ]] i a graded k-algebra concentrated in degree 0; if n = 1, we have an iomorphim Z (D b (nrep(q))) = T (k[[ξ]], k) l Z,l 1 of graded algebra, where k i viewed a the unique imple k[[ξ]]-module on which ξ act trivially. Moreover, we have an iomorphim of graded algebra T (k[[ξ]], k) = k[[ξ]][η]/(η 2 ), l Z,l 1 where ξ i of degree 0 and η i of degree 1. Remark 4.6. In cae that n = 1, we know that nrep(z 1 ) i equivalent to the category of finite dimenional nilpotent k[x]-module, which i jut the category of finitely generated (x)-torion module over the Dedekind domain k[x]. Thu Lemma 3.5 applie and we get the ame reult. More generally, one can conider the completed path algebra kẑn of the quiver Z n. Then nrep(z n ) i equivalent to the category of finite dimenional nilpotent module over kẑn, and the center of kẑn i iomorphic to k[[x]]. Remark 4.7. Combining Lemma 4.1 and Propoition 4.2 and 4.5, we have now a decription of Z (D b (C)) for any unierial hereditary length k-categroy C. Next we conider the category of finite dimenional module over finite dimenional hereditary k-algebra. Since k i aumed to be algebraically cloed, we need only conider the path algebra. Now let Q be a finite, connected quiver without oriented cycle and A = kq the path algebra. Note that in thi cae, the center of the algebra i the field k. Firt we conider the finite type cae. Propoition 4.8. Let Q be a quiver uch that the path algebra kq i of finite repreentation type. Then Z (D b mod(kq)) = k. Proof. The proof i almot the ame a the one of Propoition 4.2. If A i of finite repreentation type, then any indecompoable A-module M i a tone. In particular, Hom D b (A)(M, Σ 1 M) = Ext 1 A (M, M) = 0. Next we conider the tame cae. Let τ be the Aulander-Reiten tranlation in mod(a). The Aulander-Reiten quiver of mod(a) conit of the preprojective part, the preinjective part and the regular part. Recall that a A-module M i preprojective if and only if τ n M = 0 for ufficiently large n; and M i preinjecitve if and only if τ n M = 0 for ufficiently large n. Module without preprojective and preinjective ummand are called regular module. Denote by P, R and I the full ubcategory of preprojective module, regular module and preinjective module repectively. We have the decompoition mod(a) = P R I. Let η Z 1 (D b (mod(a))), ince preprojective and preinjective module have no elf-extenion, we get the following eay lemma by applying Propoition 3.4. Lemma 4.9. Let Q be a quiver uch that the path algebra kq i of tame repreentation type, and let R denote the full ubcategory of mod(kq) coniting of regular module. Then Z 1 D b (mod(kq)) = Z 1 (D b (R)).

15 ON THE CENTER OF A TRIANGULATED CATEGORY 15 Recall that for a tame quiver, the regular part of the Aulander-Reiten quiver i a dijoint union of tube, and there are neither morphim nor extenion between different tube, i.e., R = t T R t, where T i an index et for all the tube, R t = nrep(zp(t) ) and p(t) denote the τ-period of R t. Each tube i an abelian ubcategory and we have D b (R) = t T Db (R t ). Applying Propoition 4.5 and 3.4, we get the following. Propoition Let Q be a tame quiver, and T 1 the index et of all homogeneou tube. Then Z (D b (mod(kq))) = T (k, k). t T 1 m 0 Next we conider the weighted projective line over the field k. Recall that a weighted projective line X i defined through the attached category Coh(X) of coherent heave, which i a mall k-category atifying certain axiom. Thi concept wa introduced by Geigle and Lenzing in [GL] to tudy the interaction between preprojective module and regular module for tame hereditary algebra. For a definition we refer the reader to [Le, 10], where one can alo find mot reference about thi ubject. Weighted projective line play an important role in the claification of hereditary categorie. By a theorem of Happel [Ha], any connected, Ext-finite, hereditary abelian k-category which ha a tilting complex i derived equivalent either to the category mod(a) for ome finite dimenional hereditary algebra A or the category Coh(X) for ome weighted projective line X. Firt we recall ome baic fact. Let X be a weighted projective line and H = Coh(X) the category of coherent heave. The category H ha Serre duality, i.e., there exit an equivalence τ : H H and a natural iomorphim D Ext 1 H (X, Y ) = Hom H (Y, τx). We denote by H 0 the full ubcategory coniting of all object of finite length. Then H 0 i a hereditary abelian ubcategory and H 0 = x C U x for ome index et C, where U x i a tube with finite τ-period p(x). Member in C are called the point of H. Note that there are only finitely many point with p(x) > 1. We denote by H + the ubcategory coniting of all object without a imple ubobject. Object of H + are called vector bundle. Any indecompoable object of H i either of finite length or a vector bundle. There i a linear form rk : K 0 (H) Z, called rank, which i τ-invariant, vanihe on object of H 0 and take poitive value on object of H +. Object of H + of rank one are called line bundle, and by definition H contain a line bundle. For any vector bundle E, we have a filtration E 0 E 1 E r = E with the line bundle factor E i /E i 1, where r = rk(e). For any line bundle L and any point x C, S U x dim k Hom H (L, S) = 1, where S run through all imple object in U x. Clearly we have H = H + H 0, and therefore any nonzero morphim between line bundle i a monomorphim. Now we conider the graded center of D b (H). Note that one can define the Euler characteritic χ H for H. If χ H > 0, then H i derived equivalent to the category mod(a) for ome finite dimenional tame hereditary algebra A, and in thi cae, the graded center ha been computed. Firtly we have the following eay lemma. Lemma Let X be a weighted projective line. Then Z 0 (D b (Coh(X))) = k. Proof. We denote Coh(X) by H a before. Since H contain a line bundle, we chooe one and denote it by L. Let η : Id H Id H be a natural tranformation. To prove the lemma, it uffice to how that if η L = 0, then η = 0.

16 16 HENNING KRAUSE AND YU YE Now aume that η L = 0. Let x C be an arbitrary point, and S U x the imple object with Hom H (L, S) 0. Note that uch S exit and i unique. By Propoition 4.3, η Ux = 0 if and only if η S [mr+1] = 0 for all m 0, where r = p(x) i the τ-period and S [mr+1] i the object in U x with ocle S and of length mr + 1. By uing induction we have dim k Hom H (L, S [mr+1] ) = m + 1 for any m 0. We claim that there exit an epimorphim from L to S [mr+1]. Otherwie, all morphim will factor through S [(m 1)r+1], and hence dim k Hom H (L, S [mr+1] ) = dim k Hom H (L, S [(m 1)r+1] ) = m, which give a contradiction. Let f : L S [mr+1] be an epimorphim. Since η i a natural tranformation, we have η S [mr+1] f = f η L = 0, and hence η S [mr+1] = 0. Now we have hown that η H0 = 0. Converely, uing a imilar argument, one can how that if η H0 = 0, then η N = 0 for any line bundle N. Since any vector bundle E ha a filtration with line bundle factor, we get, uing the five lemma, that η E = 0. Thi complete the proof. Combined with Propoition 3.4, Lemma 4.4 and Propoition 4.5, we obtain the following embedding of algebra. Lemma Let X be a weighted projective line, H = Coh(X) and C 1 the et of point of τ-period 1. Then the algebra Z = T (k, x C 1 m 0 k) i iomorphic to a ubalgebra of Z (D b (H)). In the tubular cae, i.e., χ H = 0, we have H = q Q { } H q, where for each q we have H q = H 0. In fact one can define the lope for object of H, and roughly peaking, for any q Q, H q i jut given by object of lope q, and H = H 0. With thi decompoition of categorie, we have the following propoition. Propoition Let X be a weighted projective line of Euler characteritic 0, H = Coh(X) and C 1 the et of point of τ-period 1. Then Z (D b (H)) = T (k, k). q Q { } x C 1 m 0 5. The graded center of D b (mod(k[x]/(x 2 ))) In thi ection, we will tudy the ring of dual number, which i by definition the k-algebra A = k[x]/(x 2 ), where k i an arbitrary bae field. Set C = mod(a) and P the full ubcategory of C coniting of projective module. One ha a complete decription of the indecompoable object of D b (C) = K +,b (P), and therefore one can write down the element in Z (D b (C)) explicitly. By Theorem 2.5, we need only to conider the category K b (P). The indecompoable object in K b (P) are well undertood, for example ee [Kü]. They are given by {A n m < m n < }, where A n m i the complex 0 }{{} A n x A x x A }{{} m 0, that i, (A n m) i = A for m i n and 0 otherwie, and d An m i = x for all m < i n 1, where we ue x to denote the multiplication map l x. If we allow n to take the value, then we get all indecompoable object in K +,b (P), in fact A m = Σ m S in the derived category, where S i the imple A-module, regarded a a talk complex

17 ON THE CENTER OF A TRIANGULATED CATEGORY 17 concentrated in degree zero. Note that ΣA n m = A n+1 m+1. The following lemma i baic for our computation. Lemma 5.1. Let < m n <, < m n <. If (m, n) (m, n ), then Hom K b (P)(A n m, A n m ) i at mot one dimenional. The morphim between indecompoable object in K b (P) are linear combination of compoition of the following four clae of morphim: (a) πm n,n : A n m A n m for m n n, (πm n,n ) m = x and (πm n,n ) i = 0 i m; (b) πm,m n : A n m A n m for m m n, (π n m,m ) i = 1 m i n; (c) i n,n m : A n m A n m for m n n, (i n,n m ) i = 1 m i n; (d) i n m,m : A n m A n m for m m n, (i n m,m ) m = x, and (i n m,m ) i = 0 i m. The morphim in the lemma look a follow. (a) 0 (b) 0 n n {}}{ A {}}{ A n {}}{ A 1 0 m {}}{ A 0 A A 0 m {}}{ A 0 A A 0 (c) 0 A 0 }{{} A }{{} A n n (d) 0 A 0 0 }{{} A n A 1 1 x }{{} A m x 0 m {}}{ A 0 A 1 }{{} A m A }{{} m 0 The proof i traightforward and left to the reader. For any m n <, the pace Hom K b (P)(A n m, A n m) i two dimenional, and we denote the morphim i n m,m = πm n,n by x n m. Now let η : Id K b (P) Id K b (P) be a natural tranformation. Clearly, η i uniquely given by ome datum {µ n m, λ n m k, < m n < } with η A n m = µ n m 1 + λ n mx n m. Propoition 5.2. Let η : Id K b (P) Id K b (P) be a natural tranformation and {µ n m, λ n m} the correponding datum. (1) We have µ n m = µ n m for any m, m, n and n. Converely, any datum of the form {µ, λ n m k, < m n < } arie a the datum of ome natural tranformation η : Id K b (P) Id K b (P) by etting η A n m = µ+λ n mx n m for any m and n. (2) If η Z 0 (K b (P)), then λ n m = λ n+r m+r for any m, n and r, and any element in Z 0 (K b (P)) i obtained in thi way.

18 18 HENNING KRAUSE AND YU YE (3) A an algebra, Z 0 (K b (P)) = T (k, r 0 k), where T (k, r 0 k) i viewed a a graded algebra concentrated in degree 0. Proof. (1) Apply the naturality of η to get i n,n m ηm n = ηm n i n,n m and πm,m n η n m = ηm n πm,m n. From thi follow that µ n m = µ n m for any m, m, n and n. Converely, for any datum of the form {µ, λ n m k, < m n < }, we claim that the above contructed η i indeed a natural tranformation. In fact, one can eaily how that the equalitie f ηm n = ηm n f hold for thoe morphim f : A n m A n m lited in Lemma 5.1. Now uing the fact that K b (P) i a Krull-Remak-Schmidt category and any morphim i ome linear combination of compoition of morphim lited in Lemma 5.1, we get that η Y f = f η X hold for any morphim f : X Y in the category K b (P). Thu η i a natural tranformation. (2) Ue the fact that by definition η Z 0 (K b (P)) if and only if Ση = ησ, and thi i equivalent to the requirement that λ n m = λ n+1 m+1 for any m and n. (3) i an eay conequence of (2). In fact we can explicitly write down the element in Z 0 (K b (P)). For any r 0, let η r Z 0 (K b (P)) denote the natural tranformation obtained by etting (η r ) A n m = x n m for m n = r and 0 otherwie. Thu a vector pace Z 0 (K b (P)) = k 1 r 0 k η r. By direct computation, the multiplication atifie η r η r = 0 for any r and r and the iomorphim in (3) follow. Now we conider the natural tranformation from the identity functor to Σ t for any poitive integer t > 0. Note that Hom K b (P)(A n m, Σ t A n m) = 0 for any m, n with n < m+t, and in the cae n m + t, the morphim pace i one dimenional with bai element ft;m n = i n,n+t m+t πm,m+t. n Let ζ : Id D b (P) Σ t be a natural tranformation; it i uniquely determined by the datum {λ n t;m, n m + t}, where ζ A n m = λ n t;mft;m. n Applying the naturality of ζ to the morphim im n,n and πm,m n, one get f n t;m = ft;m n and n. Thu we get the following lemma. for any m, m, n Lemma 5.3. Let t > 0. All natural tranformation from Id K b (P) to Σ t form a one dimenional k-pace with a bai element ζ t, where ζ t i given by (η t ) A n m = ft;m n for all n m + t and 0 otherwie. Moreover, the multiplication atifie ζ t ζ t = ζ t+t for any t, t > 0 and ζ t η r = η r ζ t = 0 for any t > 0 and r 0, where the η r are given a in the proof of Propoition 5.2. Note that Σζ t = ( 1) t ζ t Σ if and only if either char(k) = 2 or char(k) 2 and t i even. Combined with the lat lemma and Propoition 5.2, we get the graded center of D b (mod(a)). Propoition 5.4. Let k be an arbitrary bae field. Then a a graded algebra, Z (D b (mod(k[x]/(x 2 )))) = T (k[ζ], r 0 k), where k i identified with k[ζ]/(ζ) a a k[ζ]-module, Z 0 (D b (mod(k[x]/(x 2 )))) = T (k, r 0 k),

19 ON THE CENTER OF A TRIANGULATED CATEGORY 19 and ζ i of degree 2 if char(k) 2, and of degree 1 if char(k) = The graded center of the table category mod(k[x]/(x n )) Another important cla of triangulated categorie are the table categorie of elfinjective algebra. We calculate the graded center in ome pecial cae, namely for the algebra of the form k[x]/(x n ) with n 2. Thee calculation are baed on the fact that the indecompoable object and their morphim are well undertood. Note that ome algebra of the form k[x]/(x n ) are Brauer tree algebra, and we refer to [KL] for the calculation of the graded center of their table module categorie. Let A = k[x]/(x n ) with k an arbitrary bae field. It i well known that A i unierial and that all the indecompoable object in mod(a) are of the form A l = A/x l A = x n l A with 1 l n. There are epimorphim πr l = l x l r : A l A r for l r and monomorphim i r l : A l A r for l r. For any l and r, Hom A (A l, A r ) ha a bai {f l,r = i r π l 1 min(l, r)}. Moreover, the yzygy functor Ω i given by Ω(A l ) = A n l and Ω(f l,r ) = πn r n in n l = f n l,n r n r l+ for all 1 l n 1, r, l. Now let C = mod(a) be the table category. One know that C i a triangulated category with upenion functor Σ = Ω 1 = Ω. In particular, we have Ω 2 = Id C in C. The indecompoable object in C are given by A l = A/x l A = x n l A with 1 l < n, and f l,r = 0 if and only if l + r n. Conequently, Hom C (A l, A n l ) = Hom A (A l, A n l ). For any elf-injective ring R, let Z(R) denote the graded center of R. There i a canonical morphim from Z(R) to Z 0 (mod(r)). A we will how below, thi map i not injective in general. The more intereting quetion i that whether it i urjective. In the cae A = k[x]/(x n ), the anwer i ye. In fact, for an arbitrary unierial elf-injective algebra, all natural tranformation from the identity functor to itelf for the table category come from the center of the algebra. Propoition 6.1. Let A = k[x]/(x n ) with n 2 and C = mod(a). Then Z 0 (C) = k[x]/(x [ n 2 ] ), where [ n 2 ] denote the maximal integer which i no larger than n 2. Proof. Note that A [ n 2 ] i of pecial importance ince End C (A [ n 2 ] ) i of maximal dimenion among the indecompoable object. Let η be a natural tranformation from Id C to Id C. We will how that η i uniquely determined by η A[ n 2 ]. We fix ome a A uch that η A[ n = l 2 ] a, where l a i given by the multiplication with a a before. Since η i a natural tranformation, we have π[ l n 2 ] η A l = l a π[ l n 2 ] for l > [ n 2 ] and n i[ 2 ] l η Al = l a i [ n 2 ] l for any l < [ n 2 ]. Now it i eay to how that η A l = l a for any l, ince the olution of the equation above are unique. Therefore we have an epimorphim from A to Z 0 (C), and eay computation how that l x [ n = 0 in C. 2 ] Next we will compute the natural tranformation from the identity functor to Ω. The following lemma i eay. Lemma 6.2. Let ζ : Id C Ω be a natural tranformation. Then for any 1 l < n, we l,n l have ζ Al = λ l f 1 for ome λ l k. And converely, any family {λ l, 1 l < n} induce l,n l a natural tranformation ζ by etting ζ Al = λ l f 1 for any l. Moreover, ζ Z 1 (C) if and only if λ l = λ n l for any l.