Derived categories and syzygies

Deriv ed categories and syzygies Jiaqun WEI ∗ Abstract W e in tro duce syzygie s for deriv ed categories and s tudy their prop erties. Using these, w e pro v e the deriv ed inv ariance of the follo wing classes of artin algebras: (1) syzygy-finite algebras, (2) Igusa-T o dorov alge bras, (3) AC alg ebras, (4) algebras satisfying the finitistic Ausland er conjecture, and (5) algebras s atisfying th e gen- eralized Auslander-Reiten conjecture. In particular, Go renstein CM-finite algebras are deriv ed in v arian ts. MSC2000: Primary 18E30 16E05 S econdary 18G35 16G10 16E65 Keywor ds : syzygy; deriv ed categ ory; derive d equ iv alence; Igus a-T od oro v algebra; CM-finite alge bra Auslander condition 1 In t ro duc tion Syzygies w ere in tro duced b y Hilb ert [8] in 1890. Now aday s, syzy gies are of general imp ortance in algebraic geometry , homology algebra a nd represen tation theory of groups and algebras etc.. The adv an tage o f syzygies is that they contain imp orta n t information of modules and they can test some homological dimens ions. F or instance, Zimmermann- Huisgen and her coauthors sho w ed that the structure of syzygies is imp ortant and p o w erful in understanding and compute v arious finitistic dimension of some alg ebras suc h as string algebras, monomial algebras and serial algebras [4, 20, 21, 22]. Monomial algebras and serial a lgebras are all syzygy-finite. By definition, syzygy-finite algebras means that there is an in teger s suc h that the class of a ll n -th syzygies , where n > s , is represen tation finite, or equiv alen tly , the n um b er of non- isomorphic indecomp os- able mo dules in the class is finite. Syzygy-finite alg ebras hav e nice homological prop erties. They satisfy a gro up of homological conj ectures related to the finitistic dimension con- jecture, for instance, Auslander conjecture (c.f. [5]) and Auslander-Reiten conj ecture [1] etc.. Recen tly , alg ebras of finite Cohen-Macaulay ty p e, or CM-finite algebras, a re more attractiv e, see for instance [11] and references therein. Note that syzygy-finite alg ebras ∗ Suppo rted b y the National Science F ounda tion of China (No.109 71099 ) 1 are clearly CM-finite. W e don’t know if CM-finite alg ebras are also sy zygy-finite, but for an imp orta nt class of algebras, that is, Gorenstein a lg ebras, they are the same. In t his pap er, w e extend the no t ion of syzygy to deriv ed categories and study their prop erties. As is w ell known, deriv ed categories are v ery imp orta n t in mo dern study of algebraic geometry and represe n tation theory of groups and algebras, and deriv ed equiv a- lences play an imp ortant role in the study . In particular, there are remark able conjectures concerning deriv ed equiv alences. F or instance, one f a mous conjecture in a lgebraic geom- etry , first made b y Bo ndal a nd Orlov , as serts that if X 1 and X 2 are biratio na l smo oth pro jectiv e Calabi-Y au v arieties o f dimension n , then there is an equiv alence b etw een their deriv ed catego r ies [2]. While in repres en tation theory of groups, the famous Ab elian D e- fect Group Conjecture of Brou ´ e claims that a blo c k algebra A of a finite group a lgebra and its Brauer corresp onden t B is derive d equiv alent provided that their common defect group is ab elian, see for instance [14]. Syzygies in deriv ed category is pow erful concerning deriv ed equiv alences, as w e show in this pap er. F or ex ample, using syzygies in derive d category , we prov e that deriv ed equiv alences preserv e the syzygy-finiteness of a lg ebras. This pro vides a n imp ortan t wa y to obtain syzygy-finite algebras, in particular, CM-finite algebras. The reader is r eferred to [19] for o ther w a ys to obtain syzygy-finite algebras. Moreo ve r, us ing syzygies in de- riv ed category , we also prov e that deriv ed equiv alences preserv e the following interesting classes of a lgebras: Igusa-T o do r ov a lgebras, A C-a lgebras, algebras satisfying the finitistic Auslander conj ecture and alg ebras satisfying the generalized Auslander-Reiten conjecture. Igusa-T o dorov algebras w ere intro duced in [18] in connection with the study of the finitistic dimension conjecture using Igusa-T o doro v functor. Suc h algebras hav e finite finitistic dimension. The class of Igusa-T o doro v algebras is large, including syzygy-finite algebras, alg ebras with represen tatio n dimension at most t hree, algebras with radical cub e zero and mo st algebras whic h w ere recen tly pro ved to ha v e finite finitistic dimension, see [18] for details. W e refer to [18, 19] fo r other metho ds to judge when an algebra is Igusa-T o dorov. A C-algebras ar e algebras satisfying Auslander’s condition. They are studied in detail in [3 ] recen tly . Auslander’s condition ( AC) for an alg ebra R asserts that f or every finitely generated left R -mo dule M there is an integer n = n M , called Auslander b ound o f M , suc h that Ext i R ( M , N ) = 0 for all i > n , whenev er N is a finitely generated left R -mo dule satisfying Ext i R ( M , N ) = 0 f or all but finitely many i . Auslander conjectured all finite dimensional algebras satisfy Auslander’s condition (c.f.[5]). Ho w ev er, the conjecture fails b y coun terexamples firstly giv en in [10]. A revisited v ersion of Auslander conjecture, named the finitistic Auslander conjecture, asserts that the finitistic Auslander b ound of eve ry a lg ebra is finite [17]. Note that the finitistic Auslander conjecture implies the finitistic dimens ion conjecture. In [17], the generalized Auslander-Reiten conjecture is also form ulated, whic h asserts that for an algebra R , if M is a finitely generated left R -mo dule suc h that Ext i R ( M , M ⊕ R ) = 0 for all i ≥ n , then pro jectiv e dimension of M is at most n . In the sp ecial case n = 1 , it is just Auslander-Reiten conjecture [1]. Though w e w ork on artin algebras and finitely generated left mo dules throughout 2 this pap er, syzygies (resp ., cosyzygies ) defined here for deriv ed categories mak e sense for the deriv ed categor y of an y ab elian categories with enough pro jectiv e ( r esp., inje ctiv e) ob jects. It is expected that these notions can ha v e nice applications in more areas. The pap er is organized as follows . In Section 2 w e in tro duce notations used in the pap er and recall some basic facts on deriv ed categories. W e in tro duce syzygies for deriv ed categories and study their basic prop erties in Section 3. In Sections 4 a nd 5, w e prov e that deriv ed e quiv alences preserv e syzygy-finite algebras and Igusa-T o do r o v algebras resp ec- tiv ely . In Section 6, w e prov e that A C-algebras, algebras satisfying the finitistic dimension Auslander conjecture and algebras satisfying the generalized Auslander-Reiten conjecture also ha v e deriv ed in v ariance. 2 Preliminaries In this section, w e recall some basic definitions and facts whic h are nece ssary f or our pro ofs. Let R be an artin algebra, whic h means R is a finitely generated A -mo dule ov er a comm utativ e artin ring A . W e denote b y mo d R the catego ry of finitely generated left R - mo dules. The subcategory of mo d R consisting of pro jectiv e (resp., injectiv e) mo dules is denoted b y P R (resp., I R ). W e denote f g the comp osition of homomorphisms f : L → M and g : M → N . W e strengthen tha t we w ork on c hain complexe s. Let C be a class of R -mo dules. A (c hain) complex X ov er C is a set { X i ∈ C , i ∈ Z } equipp ed with a set of homomorphisms { d i X : X i → X i − 1 , i ∈ Z | d i +1 X d i X = 0 } . W e usually write X = { X i , d i X } . A chain ma p f b et w een complexes, say fro m X = { X i , d i X } to Y = { Y i , d i Y } , is a set of maps f = { f i : X i → Y i } such tha t f i d i Y = d i X f i − 1 . A complex X = { X i , d i X } is right (resp., left) b ounded if X i = 0 for all but finitely man y negativ e (resp., p ositiv e) in tegers i . A complex X is b ounded if it is b oth left and right b ounded, equiv alently , X i = 0 f or all but finitely many i . A complex X is homologically b ounded if all but finitely many ho mologies of X a re zero. Let I ⊆ Z b e an interv a l. W e sa y that a complex X = { X i , d i X } has a ho mological (resp., represen tation) inte rv al I , if the homologies H i ( X ) = 0 (resp., terms X i = 0) for all i 6∈ I . W e iden tify an R - mo dule with a complex concen tred on the 0-th term, i.e., a complex with represen tation in terv a l [0 , 0]. The category of all complexes ov er C with c hain maps is denoted b y C ( C ). The homotopical category of complexes o ve r C is denoted b y K ( C ). When C is an ab elian category , then the derived category of complexes ov er C is w ell defined and is denoted b y D ( C ). Th e subcategories of K ( C ) and D ( C ) consisting of bo unded (resp., righ t b ounded, left bounded) complexes are denoted b y K b ( C ) (resp., K − ,b ( C ), K + ,b ( C )) and D b ( C ) resp ectiv ely . Let I b e an interv al. W e denote by K r I ( C ) (res p., K h I ( C )) and D I ( C ) the sub category of K ( C ) and D ( C ) consisting of complexes with represen tatio n (resp ., homological) inte rv al I , resp ectiv ely . Similarly , w e denote by D I ( C ) the sub category o f D ( C ) consisting of complexes with h omological inte rv al I . F or instance, K r[0 , 0] ( P R ) is 3 just the sub category P R and D [0 , 0] ( R ) is just t he same as mo d R . W e simply write D ( R ) and D b ( R ) fo r D (mo d R ) a nd D b (mo d R ) resp ectiv ely . It is w ell kno wn that D b ( R ) ⊆ D ( R ), K b ( P R ) ⊆ K − ,b ( P R ), K b ( I R ) ⊆ K + ,b ( I R ) are all triangulated categories. F or basic results in tr ia ngulated categories and derive d categories, w e r efer to [7] and [16]. Moreo v er, D b ( R ) is equiv alen t to b oth K − ,b ( P R ) and K + ,b ( I R ) as tr iangulated catego ries. W e denote by [ − ] the shift f unctor on complexes. In f act, for a complex X , the complex X [1] is obtained f r om X b y shifting X to the left one degree. The notation add X denotes the class of all direct summands of finite sums of a complex X . Tw o algebras R a nd S are said to b e deriv ed-equiv alen t if D b ( R ) and D b ( S ) are equiv alen t as triangulated categories. Ric k ard [13 ] pro v ed that t wo algebras R and S are deriv ed-equiv alent if and only if there is a complex T ∈ K b ( P R ) with S ≃ Hom D ( R ) ( T , T ) suc h that Hom D ( R ) ( T , T [ i ]) = 0 for all i 6 = 0 and add T g enerates K b ( P R ) as a triangle category . Suc h complex is called a tilting complex. In fact, if F : D b ( R ) ⇄ D b ( S ) : G define an equiv a lence, then T := G ( S ) is just a tilting complex. In particular, if T is a tilting R - mo dule, then T induces a deriv ed equiv alence b et w een D b ( R ) and D b (End R T ) [7]. W e refer to [9] for recen t dev elopmen t on constructing deriv ed equiv a lences. F or a n in terv al I ⊆ Z , w e denote b y σ I ( X ) the brutal truncated complex whic h is obtained from the complex X b y replacing eac h X i , wh ere i 6∈ I , with 0. F or instance, if M is a complex in K r[ n, ∞ ) ( P R ), then σ [ n, ∞ ) ( M ) = M . Throughout the pap er, R stands for an artin algebra. Homomorphisms and isomor- phisms b et w een complexes alw a ys means in D ( R ). In particular, restricting to mo dule category , they are just the usual homomorphisms and isomorphisms in module category . 3 Syzygies in deriv ed categor i es Recall that D b ( R ) is equiv alen t to K − ,b ( P R ) as triangulated categories. F or a complex M ∈ D b ( R ), a pro jective resolution of M is a complex P ∈ K − ,b ( P R ) suc h that P ≃ M in D ( R ) . In case M is an R - mo dule, P is just a usual pro jectiv e resolution of M . Definition 3.1 L e t M ∈ D b ( R ) and n ∈ Z . L et P b e a pr o je ctive r esolution of M . We say that a c omplex in D b ( R ) is an n -th syzygy of M , i f it is isomorphic to ( σ [ n, ∞ ) ( P ))[ − n ] in D ( R ) . In the c a se, the n -th syzygy of M is denote d b y Ω n D ( M P ) , or simply by Ω n D ( M ) if ther e is no d anger of c onfusion. Th us, the n -th syzygy of a complex M ∈ D b ( R ) dep ends on the choice of pro j ectiv e resolution of M . In this case that M is an R -mo dule and n > 0, the brutal truncated complex ( σ [ n, ∞ ) ( P ))[ − n ] is just the pro jectiv e resolution of the n -th syzygy of M defined b y P . Hence, syzygies o f M define d here coincide with the usual syzygies in mo dule category . W e leav e to the reader the en tire pro of of the following lemma. 4 Lemma 3.2 L e t M , N ∈ D b ( R ) and n, m, a, k , i b e inte gers. L et P b e a pr oje ctive r eso- lutions of M . (1) Ω n D ( M ) ∈ D [0 , ∞ ) ( R ) and Hom D ( R ) ( Q, Ω n D ( M )[ i ]) = 0 for any pr oje ctive mo dule Q and any inte ger i > 0 . (2) If M ∈ D [ a,k ] ( R ) for some a ≤ k , then Ω n D ( M ) is c ontaine d in D [0 , 0] ( R ) , D [0 ,k − n ] ( R ) , D [ a − n,k − n ] ( R ) for c ases n ≥ k , a ≤ n ≤ k , n ≤ a , r esp e ctively. In p articular, Ω k D ( M ) is isomorphic to an R -mo dule and M ≃ Ω a D ( M )[ a ] . (3) Ω n + m D ( M [ m ]) ≃ Ω n D ( M ) . (4) Ω n + m D ( M ) ≃ Ω m D (Ω n D ( M )) , fo r m ≥ 0 . (5) Ω n D ( M ) ⊕ Q is also an n -th syzygy of M , wher e Q is a pr oje ctive mo dule. (6) M ∈ K b ( P R ) if and only if any/some syzygy of M is a l s o in K b ( P R ) . (7) Ω n D ( M ⊕ N ) ≃ Ω n D ( M ) ⊕ Ω n D ( N )) . Let f b e a c hain map b et w een complexes, sa y from X = { X i , d i X } to Y = { Y i , d i Y } . Recall that the cone of f , denoted b y cone( f ), is a complex suc h that, for eac h i , (cone( f )) i = Y i ⊕ X i − 1 , d i cone( f ) = ( d i Y f i − 1 0 d i − 1 X ). It is w ell know n t hat there is a canoni- cal triangle X → f Y → cone( f ) → in D ( R ). Using the construction of cones, one obtains the fo llo wing canonical triangles in D ( R ), f o r any complex X and any n ≥ m : ( σ 1) ( σ [ n +1 , ∞ ) ( X ))[ − 1] → σ [ m,n ] ( X ) → σ [ m, ∞ ) ( X ) → , ( σ 2) ( σ [ m,n ] ( X ))[ − 1] → σ [ −∞ ,m − 1] ( X ) → σ ( −∞ ,n ] ( X ) → , and ( σ 3) ( σ [ n, ∞ ) ( X ))[ − 1] → X ( −∞ ,n − 1] → X → . Lemma 3.3 L e t M ∈ D b ( R ) . Then ther e i s a triangle ( Ω n +1 D ( M ))[ n − m ] → B → Ω m D ( M ) → , wh e r e B ∈ K r[0 ,n − m ] ( P R ) and n ≥ m . In p articular, for any n , ther e is a triangle Ω n +1 D ( M ) → Q → Ω n D ( M ) → with Q pr oje ctive. Pro of . Let P b e a pro jective resolution of M . Then the first triangle is obtained from ( σ 1) b y shifting, letting B = ( σ [ m,n ] ( P ))[ − m ] ∈ K r[0 ,n − m ] ( P R )) in the case. T aking m = n , w e then obtain the second triangle. ✷ Recall that tw o mo dules N , N ′ are pro jective ly equiv alent if N ⊕ P ≃ N ′ ⊕ P ′ for some pro jectiv e mo dules P , P ′ . Let M b e a mo dule. It is we ll kno wn that an y tw o n -th syzygies of M are pro jectiv ely equiv alen t. It is also the case f or syzygies in deriv ed category . W e sa y that t w o complexes N , N ′ are pro j ective ly equiv alen t if N ⊕ P ≃ N ′ ⊕ P ′ for some pro jectiv e mo dules P , P ′ . Prop osition 3.4 L et M ∈ D b ( R ) and P , P ′ b e two pr oje ctive r es olutions of M . Then Ω n D ( M P ) and Ω n D ( M P ′ ) ar e pr oje ctively e quivalent. 5 Pro of . W e pro v e b y induction on n . Assume tha t M ∈ D [ a,k ] ( R ) for some in tegers a < k . Then Ω n D ( M P ) ≃ M [ − n ] ≃ Ω n D ( M P ′ ) in case n ≤ a , b y Lemma 3.2 (2). No w w e consider the case n + 1. By the induction a ssumption, there are pro jectiv e mo dules Q, Q ′ suc h that Ω n D ( M P ) ⊕ Q ≃ Ω n D ( M P ′ ) ⊕ Q ′ . Then w e ha v e t riangles Ω n +1 D ( M P ) → P n ⊕ Q → Ω n D ( M P ) ⊕ Q , and Ω n +1 D ( M P ′ ) → P ′ n ⊕ Q ′ → Ω n D ( M P ′ ) ⊕ Q ′ , where P n and P ′ n are pro jectiv e, by Lemma 3.3. Since w e hav e a homomorphism b et w een the la st terms of t hese t w o tria ng les and Hom D ( R ) ( P n ⊕ Q, Ω n +1 D ( M P ′ )[1]) = 0 b y Lemma 3.2 (1) , there is the follo wing commu tativ e diagram: Ω n +1 D ( M P ′ ) ✲ P ′ n ⊕ Q ′ ✲ Ω n D ( M P ′ ) ⊕ Q ′ ✲ ❄ ❄ ❄ Ω n +1 D ( M P ) ✲ P n ⊕ Q ✲ Ω n D ( M P ) ⊕ Q ✲ Since the ho momorphism in the rig h t column is an isomorphis m, w e ha ve a canonical triangle Ω n +1 D ( M P ) → P n ⊕ Q ⊕ Ω n +1 D ( M P ′ ) → P ′ n ⊕ Q ′ → . Note a gain that P ′ n ⊕ Q ′ are pro jectiv e and that Ho m D ( R ) ( P ′ n ⊕ Q ′ , Ω n +1 D ( M P )[1]) = 0 b y Lemma 3.2 (1), so the ab ov e triangle splits. Hence w e obtain that Ω n +1 D ( M P ) ⊕ ( P ′ n ⊕ Q ′ ) ≃ ( P n ⊕ Q ) ⊕ Ω n +1 D ( M P ′ ). It f ollo ws that Ω n +1 D ( M P ) and Ω n +1 D ( M P ′ ) are pro jectiv ely equiv alen t. ✷ By the ab ov e result, w e see that the n - t h syzygy of M is unique up to pro jectiv ely equiv alences. By abuse o f language, w e sp eak of the n -sy zygy of a complex in D b ( R ). W e a lso ha v e t he following easy result. The pro of is left to the reader. Prop osition 3.5 (Dimension shifting) L et M ∈ D b ( R ) . Assume that N ∈ D [ c,d ] ( R ) for some inte gers c ≤ d . The n ther e is an isomorphism Hom D ( R ) (Ω n + m D ( M ) , N [ j ]) ≃ Hom D ( R ) (Ω n D ( M ) , N [ j + m ]) for any inte gers n, m, j such that m ≥ 1 and j > − c . The following result provide s a w ay to compare syzy gies for complexes in some trian- gles. Lemma 3.6 L e t M → B → N → b e a triangle in D b ( R ) . I f B ∈ K r( −∞ ,k ] ( P R ) for some inte ger k , then Ω k D ( M ) ≃ Ω k +1 D ( N ) . Henc e , Ω n D ( M ) ≃ Ω n +1 D ( N ) for al l n ≥ k . Pro of . Let i : M → B b e the homomorphism in the triangle. Assume tha t P is a pro- jectiv e resolution of M , then we ha ve a homomorphism f : P → B in K − ,b ( P R ) follo w ed from the equiv alence b etw een D b ( R ) and K − ,b ( P R ). Note that cone( f ) ∈ K − ,b ( P R ) and N ≃ cone( f ), i.e., cone( f ) is a pro jectiv e resolution of N . Since B ∈ K r( −∞ ,k ] ( P R ), it 6 is easy to see that σ [ k +1 , ∞ ) (cone( f )) ≃ σ [ k +1 , ∞ ) ( P [1]), b y the construction of cone( f ). It follo ws that Ω k +1 D ( N ) ≃ ( σ [ k +1 , ∞ ) (cone( f )))[ − ( k + 1)] ≃ ( σ [ k +1 , ∞ ) ( P [1]))[ − ( k + 1 )] ≃ Ω k +1 D ( M [1]) ≃ Ω k D ( M ). b y Lemma 3.2 (3 ). Hence, the conclusion follo ws. ✷ F or general triangles, w e ha v e the follo wing res ult. Prop osition 3.7 L et L → M → N → b e a triangle in D b ( R ) . Then, for any inte ge r n , ther e exists a triangle Ω n D ( L ) → Ω n D ( M ) → Ω n D ( N ) → . Pro of . The pro of is giv en by induction on n . W e may assume that L, M , N ∈ D [ a,k ] ( R ) for some in tegers a ≤ k . In case n ≤ a , w e o bt a in that Ω n D ( L ) ≃ L [ − n ], Ω n D ( M ) ≃ M [ − n ] and Ω n D ( N ) ≃ N [ − n ], b y Lemma 3.2 (2). Hence w e hav e a triangle Ω n D ( L ) → Ω n D ( M ) → Ω n D ( N ) → by assumption. No w consider the case n + 1. By Lemma 3.3, w e hav e triangles Ω n +1 D ( L ) → B → Ω n D ( L ) → a nd Ω n +1 D ( N ) → C → Ω n D ( N ) → , where B , C are pro jectiv e. By t he in- duction assumption, there is a triang le Ω n D ( L ) → Ω n D ( M ) → Ω n D ( N ) → . Note that Hom D ( R ) ( C , (Ω n D ( L ))[1]) = 0 b y Lemma 3.2 (1), so we can obta in the fo llo wing triangle comm utativ e dia g ram for some M ′ ∈ D b ( R ). ❄ ❄ ❄ Ω n D ( L ) ✲ Ω n D ( M ) ✲ Ω n D ( N ) ✲ ❄ ❄ ❄ B ✲ B ⊕ C ✲ C ✲ ❄ ❄ ❄ Ω n +1 D ( L ) ✲ M ′ ✲ Ω n +1 D ( N ) ✲ Consider the triangle M ′ → B ⊕ C → Ω n D ( M ) → from the middle column in the diagram. Since B ⊕ C is a pro jectiv e R -mo dule, B ⊕ C ∈ K r[0 , 0] ( P R ). By Lemmas 3.6 and 3.2, w e obtain that Ω 0 D ( M ′ ) ≃ Ω 1 D (Ω n D ( M )) ≃ Ω n +1 D ( M ). Not e that M ′ ∈ D [0 , ∞ ) ( R ) follo w ed from the t r ia ngle Ω n +1 D ( L ) → M ′ → Ω n +1 D ( N ) → from the up ro w in the diagram, so Ω 0 D ( M ′ ) ≃ M ′ b y Lemma 3.2 (2). It f ollo ws that M ′ ≃ Ω n +1 D ( M ) and hence w e ha ve a triangle Ω n +1 D ( L ) → Ω n +1 D ( M ) → Ω n +1 D ( N ) → . ✷ There is also the notion dua l to syzygies. Recall tha t D b ( R ) is equiv a len t to K + ,b ( I R ) as triangula t ed cat ego ries. F or a complex M ∈ D b ( R ), an injectiv e resolution of M is a complex I ∈ K + ,b ( I R ) suc h that I ≃ M . In case M is an R -mo dule, I is just a usual injectiv e resolution of M . Using t he injectiv e resolution of a complex in D b ( R ), w e can define the notion o f cosyzy gies. Definition 3.1’ L et M ∈ D b ( R ) an d n ∈ Z . L et I b e an inje ctive r esolution of M . We say that a c omplex in D b ( R ) is an n -th c os yzygy of M , if it is is o m orphic to 7 ( σ ( −∞ ,n ] ( I ))[ − n ] in D ( R ) . In the c ase, the n -th c osyzygy of M is denote d by Ω D n ( M I ) , or simply by Ω D n ( M ) if ther e is no danger of c onfusion. W e state the dual results without pro ofs. Lemma 3.2’ Let M , N ∈ D b ( R ) and n, m, a, b, i b e in tegers. (1) Ω D n ( M ) ∈ D ( −∞ , 0] ( R ) and Hom D ( R ) (Ω D n ( M ) , Q [ i ]) = 0 for an y injectiv e mo dule Q and an y in teger i > 0. (2) If M ∈ D [ a,k ] ( R ), f o r some a ≤ k , then Ω D n ( M ) is con tained in D [0 , 0] ( R ), D [ a − n, 0] ( R ), D [ a − n,k − n ] ( R ) for corresp onding cases n ≤ a , a ≤ n ≤ k , n ≥ b resp ective ly . In part icular, Ω D a ( M ) is isomorphic to an R -mo dule and M ≃ Ω D k ( M )[ k ]. (3) Ω D n + m ( M [ m ]) ≃ Ω D n ( M ). (4) Ω D n + m ( M ) ≃ Ω D m (Ω D n ( M )) for m ≤ 0. (5) Ω D n ( M ) ⊕ Q is also an n -th cosyzygy of M , where Q is any injectiv e mo dule. (6) M ∈ K b ( I R ) if and o nly if an y/some cosyzygy of M is also in K b ( I R ). (7) Ω D n ( M ⊕ N ) ≃ Ω D n ( M ) ⊕ Ω D n ( N )). Lemma 3.3’ L et M ∈ D b ( R ) . T h en ther e is a triangle Ω D n ( M ) → B → (Ω D m − 1 ( M ))[ m − n ] → for som e B ∈ K r[ m − n, 0] ( I R ) , wher e n ≥ m . I n p articular, for a ny n , ther e is a triangle Ω D n ( M ) → I → Ω D n − 1 ( M ) → with I inje ctive. Prop osition 3.4’ L et M ∈ D b ( R ) an d I , I ′ b e two inje ctive r esolutions of M . Then Ω D n ( M I ) and Ω D n ( M I ′ ) ar e in j e ctively e quivalent. Prop osition 3.5’ (Dimension shifting) Assume that N ∈ D [ c,d ] ( R ) , wher e c ≤ d . Then ther e is an isomorphism Hom D ( R ) ( N , (Ω D n ( M ))[ j ]) ≃ Hom D ( R ) ( N , (Ω D n + m ( M ))[ j + m ]) for any inte gers n, m, j such that m ≥ 1 and j > − d . Lemma 3.6’ L et M → B → N → b e a triangle in D b ( R ) . If B ∈ K r[ a, ∞ ) ( I R ) for some inte ger a , then Ω D a ( M ) ≃ Ω D a +1 ( N ) . Henc e , Ω D n ( M ) ≃ Ω D n +1 ( N ) for al l n ≤ a . Prop osition 3.7’ Let L → M → N → b e a tr iangle in D b ( R ). Then there is a tria ng le Ω D n ( L ) → Ω D n ( M ) → Ω D n ( N ) → , f or any in teger n . Let us remark that one can define sy zygies ( r esp., cosyzygies) in the deriv ed category of any ab elian category with enough pro jectiv e (resp., injective ) ob jects. 4 Syzygy-finit e alge b ras Let C ⊆ D b ( R ). F or an in teger n , we denote b y Ω n D ( C ) the class of all n - t h syzygies of complexes in C . The class C is r epresen tation-finite pro vided that C ⊆ add M for some M ∈ D b ( R ), or equiv alen t ly , the n um b er of non- isomorphic indecomp osable direct summands of ob jects in C is finite. It is easy to see that if E ⊆ C a nd C is represen tation- 8 finite then E is a lso represe n tation-finite. W e say that C is n - syzygy-finite provided that S i ≥ n Ω i D ( C ) is represen tation-finite. By t hat C is syzygy-finite, w e mean that C is n - syzygy-finite for some n . It follo ws from Lemma 3.2 (3) that C is syzygy-finite if and only if C [ m ] is sy zygy-finite for a ny/some inte ger m . An artin algebra R is called syzygy-finite if the class mo d R is syzygy-finite. The follo wing algebras are known to be syzygy-finite. • Algebras of finite represen tation t yp e • Algebras of finite global dimens ion. • Monomial algebras [2 0]. • Left serial algebras [21]. • T orsionles s-finite algebras, c.f. [15], including: ◦ Algebras R with rad n R = 0 and A/ rad n − 1 A represen tation-finite. ◦ Algebras R with rad 2 R = 0. ◦ Minim al represe n tation-infinite algebras. ◦ Algebras stably equiv alent to hereditary algebras. ◦ Righ t glued algebras and left glued a lg ebras. ◦ Algebras of the form R / so c R with R a lo cal algebra of quaternion t yp e. ◦ Special biserial a lgebras. • Algebras p ossessing a left idealized extension whic h is torsionless-finite ( indeed 2- syzygy-finite) [18]. • Algebras p ossessing an ideal I of finite pro jectiv e dimension suc h that I ra d R = 0 and R/I is syzygy-finite [19]. The following result giv es a c haracterization of syzygy-finite alg ebras in term of derive d category . Theorem 4.1 An algebr a R is syzygy-fini te if and only if D [ a,k ] ( R ) is syzygy-finite for any/some inte gers a ≤ k . Pro of . Assume that D [ a,k ] ( R ) is syzygy-finite fo r some in tegers a ≤ k . Since (mo d R )[ a ] ⊆ D [ a,k ] ( R ), (mo d R )[ a ] is syzygy-finite and hence mo d R is syzygy-finite, i.e., R is a syzy gy- finite alg ebra. On the other hand, assume that R is syzygy-finite. F or an y in tegers a ≤ k , w e hav e Ω b D ( D [ a,k ] ( R )) ⊆ mo d R b y Lemma 3.2 (2). Since mo d R is syzygy-finite, w e see that Ω k D ( D [ a,k ] ( R )) is syzygy-finite. It follo ws t ha t D [ a,k ] ( R ) is syzygy-finite f rom Lemma 3.2 (4). ✷ Prop osition 4.2 L et C , E ⊆ D b ( R ) and B ⊆ K r( −∞ ,k ] ( P R ) for some b . Assume that, for any E ∈ E , ther e is a triangle C E → B E → E → in D b ( R ) with C E ∈ C and B E ∈ B . If C is syzygy-finite, then E is also syzygy-finite. Pro of . Assume that C is m -syzygy-finite, for some m . F or an y E ∈ E , consider the triangle C E → B E → E → in D b ( R ) with C E ∈ C and B E ∈ B . Since B E ∈ 9 K r( −∞ ,k ] ( P R ) b y assumptions, we hav e that Ω i D ( C E ) ≃ Ω i +1 D ( E ) for all i ≥ k , b y Lemma 3.6. Hence, S i ≥ n +1 Ω i D ( E ) is represen tation-finite, for some n ≥ k , if and o nly if the class { Ω i D ( C E ) | E ∈ E , i ≥ n } is represen t a tion-finite. Since { Ω i D ( C E ) | E ∈ E , i ≥ n } ⊆ S i ≥ n Ω i D ( C ) and the latter is represen tation- finite whenev er n ≥ m , w e obtain that S i ≥ n +1 Ω i D ( E ) is represen tation-finite f o r n = max { m, k } . It follows that E is syzygy- finite. ✷ W e need the follo wing result on basic prop erties of tilting complexes. Lemma 4.3 Assume that ther e is a n e quivalenc e F : D b ( R ) ⇄ D b ( S ) : G . L et T := G ( S ) . Assume that T ∈ K [ a,k ] ( P R ) for some inte g ers a ≤ k . Then (1) F ( D [ c,d ] ( R )) ⊆ D [ a − d,k − c ] ( S ) , for any c ≤ d . (2) G ( K r[ c,d ] ( P S )) ⊆ K r[ a + c,k + d ] ( P R ) , for any c ≤ d . Pro of . (1) F or any M ∈ D b ( R ) and an y i , w e ha v e isomorphisms: H i ( F ( M )) ≃ Hom D ( S ) ( S, F ( M )[ i ]) ≃ Hom D ( R ) ( G ( S ) , G ( F ( M ))[ i ]) ≃ Hom D ( R ) ( T , M [ i ]) . Since T ∈ K [ a,k ] ( P R ) and M ∈ D [ c,d ] ( R ), w e o btain that Hom D ( R ) ( T , M [ i ]) = 0 for i 6∈ [ a − d, k − c ]. The conclusion then follo ws. (2) T ak e an y B ∈ K r[ c,d ] ( P S ). Note that there are triangles Ω i +1 D ( B ) → P i → Ω i D ( B ) → fo r all intege rs i , where eac h P i is a pr o jectiv e S -mo dule, b y Lemma 3.3. Since B ∈ K r[ c,d ] ( P S ), we see that Ω d D ( B ) is (isomorphic to) a pro jectiv e S - mo dule and that B ≃ Ω c D ( B )[ c ] by Lemma 3.2 (2 ). Applying the functor G to these triangles, w e obtain triangles G (Ω i +1 D ( B )) → G ( P i ) → G (Ω i D ( B )) → . Note tha t G (Ω d D ( B )) , G ( P i ) ∈ add( G ( S )) = add T ⊆ K r[ a,k ] ( P R ), so w e obtain that G (Ω c D ( B )) ∈ K r[ a,k +( d − c )] ( P R ) from the ab o v e triangles, b y using the construction of cones. Consequen tly , we see tha t G ( B ) ≃ G ((Ω c D ( B ))[ c ]) ≃ ( G (Ω c D ( B )))[ c ] ∈ K r[ a + c,k + d ] ( P R ). ✷ The follo wing result show s that derive d equiv a lences preserv e syzy gy-finite classes. Prop osition 4.4 Assume that ther e is an e quivalenc e F : D b ( R ) ⇄ D b ( S ) : G . L et T := G ( S ) ∈ K r[ a,k ] ( P R ) and C ⊆ D b ( S ) . If C is syzygy-finite, then G ( C ) is als o syzygy- finite. Pro of . Since S i ≥ n Ω i D ( C ) is represen tation finite for some n , w e hav e some M ∈ D b ( S ) suc h that S i ≥ n Ω i D ( C ) ⊆ add M . C laim : G ( S i ≥ n Ω i D ( C )) is k -syzygy-finite. Pr o of . T ak e any C ∈ C . Note that , for an y i , there is a triangle Ω i +1 D ( C ) → C i → Ω i D ( C ) → , where C i is pro jectiv e, by Lemma 3.3. So, b y applying the functor G , w e obtain a tria ng le G (Ω i +1 D ( C )) → G ( C i ) → G (Ω i D ( C )) → . 10 Note that G ( C i ) ∈ add T ⊆ K r[ a,k ] ( P R ), so w e hav e that Ω j D ( G (Ω i +1 D ( C )) ≃ Ω j +1 D ( G (Ω i D ( C )) for all j ≥ k , by Lemma 3.6. It f o llo ws that Ω j D ( G (Ω i D ( C ))) ≃ Ω j − 1 D ( G (Ω i +1 D ( C ))) ≃ · · · ≃ Ω k D ( G (Ω j + i − k D ( C ))) for all j ≥ k . Hence, w e obtain that the class S j ≥ k Ω j D ( G ( S i ≥ n Ω i D ( C ))) = S j ≥ k S i ≥ n Ω j D ( G (Ω i D ( C ))) is represen tation finite if and only if the class S j ≥ k S i ≥ n Ω k D ( G (Ω j + i − k D ( C ))) = S j ≥ b Ω k D ( G ( S i ≥ n Ω j + i − k D ( C ))) is represen- tation finite. Since S i ≥ n Ω i D ( C ) ⊆ add M , w e see that the last class is con tained in the class S j ≥ b Ω k D ( G (add M )) = Ω k D ( G (add M )) and hence is represen tatio n finite. The claim then follo ws. No w ta ke an y C ∈ C . Note that there is some m C suc h tha t C [ − m C ] ≃ Ω m C D ( C ) by Lemma 3.2 (2), so w e hav e a tria ngle (Ω n D ( C ))[ n − 1] → B → C → by L emma 3.3 , where B ∈ K [ m C ,n − 1] ( P S ) and n ≥ m C . Then w e obtain a tria ngle G ((Ω n D ( C ))[ n − 1]) → G ( B ) → G ( C ) → , by applying the functor G . By Lemma 4.3 (2), G ( B ) ∈ K [ a + m C ,k + n − 1] ( P R ) ⊆ K ( −∞ ,k + n − 1] ( P R ). Note that G (Ω n D ( C )) is k -syzygy-finite follow ed from the claim, so G ((Ω n D ( C ))[ n − 1]) = G (Ω n D ( C ))[ n − 1] is ( k + n − 1 )-syzygy-finite, by L emma 3.2 (3) . Hence, by Prop osition 4.2, we ha v e t ha t G ( C ) is ( k + n )-syzygy-finite. ✷ No w we prov e that deriv ed equiv a lences preserv e syzy gy-finite algebras. Theorem 4.5 Assume that R, S ar e derive d e quivalent algebr as. If S is syzygy-finite, then R is also syzygy-finite. Pro of . By assumption, there is an equiv alence F : D b ( R ) ⇄ D b ( S ) : G . Assume that T := G ( S ) ∈ K r[ a,k ] ( P R ). Then F (mo d R ) = F ( D [0 , 0] ( R )) ⊆ D [ a,k ] ( S ), b y Lemma 4.3 (1). If S is syzygy-finite, then D [ a,k ] ( S ) is syzygy-finite b y Theorem 4.1. It fo llows that F (mo d R ) is syzygy-finite. Hence, w e o btain tha t mo d R = G ( F (mo d R )) is syzygy-finite, b y Prop osition 4.4. Th us, R is syzygy-finite. ✷ F or example, if R is deriv ed equiv alen t to a minimal represen ta t ion-infinite alg ebra or a monomial alg ebra, then R is syzygy-finite b y the a b o v e theorem. In particular, R is CM-finite in the case. Note tha t deriv ed equiv alences preserv e Gorenstein alg ebras and tha t syzygy-finiteness coincides with CM-finiteness for Gorenstein algebras, w e obtain the follo wing corollary . Corollary 4.6 Assume that R, S ar e derive d e quivalent al g ebr as. If S is Gor enstein CM- finite, then R is also Gor enstein CM-finite. 5 Igusa-T o do ro v algebras Recall that an art in algebra R is called n - Igusa-T o dor ov pro vided that there exists a fixed R -mo dule V a nd a nonnegativ e integer n suc h that, for any M ∈ mo d R , there is 11 an exact sequence 0 → V 1 → V 0 → Ω n D M → 0 with V 0 , V 1 ∈ add V [18]. A remark able prop ert y of Igusa-T o dorov algebras t ha t they satisfy the finitistic dimension conjecture. The follo wing algebras are Igusa-T o dorov . • Algebras with r a dical cub e zero. • Algebras with r epresen tation dimension at most three. • Syz ygy-finite algebras. • Algebras which are endomorphism algebras of mo dules o v er represen tation-finite algebras [18]. • Algebras with an ideal I of finite pro jectiv e dimensin suc h that I r a d 2 R = 0 (or I 2 rad R = 0) and R/I is syzygy-finite [19]. • Algebras p ossessing a left idealized extension whic h is 2 - syzygy-finite [18]. • Algebras whic h are endomorphism algebras of pro jectiv e mo dules o v er 2-Igusa- T o dorov algebras [1 8]. • Algebras with an ideal I of finite pro jectiv e dimensin suc h that I rad R = 0 and R/I is Igusa-T o doro v [1 9]. Let C b e a subclass of D b ( R ). W e say that C is relative hereditary provided that there is a complex V ∈ D b ( R ) suc h that, for any M ∈ C , there is a tr ia ngle V 1 → V 0 → M → with V 0 , V 1 ∈ add V . W e say that C is a n n -Igusa-T o dorov class, for some integer n , pro vided that Ω n D ( C ) is relativ e hereditary . It is easy to see that C is an Igusa-T o doro v (resp., r elativ e hereditary) class if and only if C [ n ] is a n Igusa-T o dorov (resp., relativ e hereditary) class for any /some n . It is also ob vious that if E ⊆ C and C is Igusa-T o dorov (resp., relativ e hereditary) then E is also Igusa-T o doro v (resp., relativ e hereditary). Lemma 5.1 L e t C ⊆ D b ( R ) . Then C is an Igusa-T o dor ov class if and only if Ω n D ( C ) i s an Igusa-T o dor ov class for any/some n . Pro of . W e first prov e that if Ω m D ( C ) is relativ e hereditary , t hen Ω n D ( C ) is also relative hereditary , for any n ≥ m . In fact, by definition there is a complex V ∈ D b ( R ) such that, fo r any M ∈ Ω m D ( C ), there is a triangle V 1 → V 0 → M → with V 0 , V 1 ∈ add V . By Prop osition 3.7, w e ha v e a tria ngle Ω n − m D ( V 1 ) → Ω n − m D ( V 0 ) → Ω n − m D ( M ) → . Note that Ω n − m D ( V 1 ) , Ω n − m D ( V 0 ) ∈ Ω n − m D (add V ) is indep enden t of M , and Ω n D ( C ) = Ω n − m D (Ω m D ( C )) f o r n ≥ m b y Lemma 3.2 (4), so w e obtain that Ω n D ( C ) is relativ e hereditary , for an y n ≥ m , b y definition. No w assume that C is an m -Igusa-T o dorov algebra. Then Ω m D ( C ) is relativ e heredi- tary . Let n b e an in teger, and take an inte ger t > max { 0 , m − n } , t hen w e ha v e that Ω t D (Ω n D ( C )) = Ω n + t D ( C ) is relativ e hereditary , by the ab o v e argumen t. It follows that Ω n D ( C ) is an Igusa-T o dorov algebra b y definition. Conv ersely , assume that Ω n D ( C ) is a t - Igusa- T o dorov algebra, for some in tegers n, t , then Ω n + t D ( C ) is relative hereditary . Hence C is an Igusa-T o dorov algebra by definition. ✷ The follo wing result giv es a c haracterization of Igusa-T o doro v algebras in term of deriv ed categories. 12 Theorem 5.2 An algebr a R i s Igusa-T o dor ov if and only if D [ a,k ] ( R ) is an Igusa-T o dor ov class for any/some a ≤ k . Pro of . If R is an Igusa-T o dorov algebra, then mo d R is a n Igusa-T o dorov class b y defi- nition. Not e that mo d R = Ω k D ( D [ a,k ] ( R )), so w e further hav e that D [ a,k ] ( R ) is an Igusa- T o dorov class, by Lemma 5.1. Con v ersely , assume that D [ a,k ] ( R ) is an Igusa-T o dorov class, then mo d R is also an Igusa-T o dorov class b y Lemma 5.1 again. Th us there is a fixed complex V ∈ D b ( R ) and an integer v such that, for an y M ∈ mo d R , there is a triangle V 1 → V 0 → Ω v D ( M ) → with V 1 , V 0 ∈ add V . Assume that V ∈ D [ c,d ] ( R ) for some c ≤ d , then Ω n D ( V ) ∈ mo d R for all n ≥ max { d, 0 } , b y Lemma 3.2 (2). By Prop osition 3.7 and Lemma 3.2 (4), there is a triangle Ω n D ( V 1 ) → Ω n D ( V 0 ) → Ω n D (Ω v D ( M ))( ≃ Ω n + v D ( M )) → . T ak e the in teger n such that n ≥ max { 0 , d, − v } , then all terms in the last tria ng le are R -mo dules. Hence, w e ha v e an exact sequence 0 → Ω n D ( V 1 ) → Ω n D ( V 0 ) → Ω n + v D ( M ) → 0. Note that Ω n D ( V 1 ) , Ω n D ( V 0 ) ∈ Ω n D (add V ) ⊆ mo d R and n + v are indep enden t of M , so we obtain that R is an Igusa- T o dorov algebra by definition. ✷ The follo wing result show s that derive d equiv a lences preserv e Igusa-T o doro v classes. Prop osition 5.3 Assume that ther e is an e quivalenc e F : D b ( R ) ⇄ D b ( S ) : G . L et T := G ( S ) ∈ K r[ a,k ] ( P R ) and C ⊆ D b ( S ) . If C is an Igusa-T o dor ov class, then G ( C ) is also an Igusa-T o dor ov class. Pro of . Assume t hat C is an n -Igusa-T o doro v class for some n , then there is a fixed complex V ∈ D b ( R ) suc h that, for an y C ∈ C , there is a t riangle V 1 → V 0 → Ω n D ( C ) → with V 1 , V 0 ∈ add V . Hence, we also ha ve a triangle G ( V 1 ) → G ( V 0 ) → G (Ω n D ( C )) → , b y applying the functor G . Similarly as in the pro o f of Prop osition 4.4, there is a triangle G ((Ω n D ( C ))[ n − 1]) → G ( B ) → G ( C ) → with G ( B ) ∈ K r( −∞ ,k + n − 1] ( P R ). So, w e ha v e tha t Ω k + n D ( G ( C )) ≃ Ω k + n − 1 D ( G ((Ω n D ( C ))[ n − 1])) ≃ Ω k + n − 1 D ( G ((Ω n D ( C )))[ n − 1]) ≃ Ω k D ( G ((Ω n D ( C ))), b y Prop osition 3.6 and Lemma 3.2 (3). Note that w e ha v e a triangle Ω k D ( G ( V 1 )) → Ω k D ( G ( V 0 )) → Ω k + n D ( G ( C )) → , by Prop osition 3.7 a nd the ab ov e ar gumen t. Since b + n and Ω k D ( G ( V 1 )) , Ω k D ( G ( V 0 )) ∈ Ω k D ( G (add V )) are indep enden t of C , so w e obtain that Ω k + n D ( G ( C )) is relativ e here ditary , i.e., G ( C ) is a ( k + n )-Igusa-T o dorov class. ✷ No w we obtain the following imp ortan t result on Igusa-T o doro v alg ebras. Theorem 5.4 Assume that R, S ar e de ri v e d e quivalent algebr as. If S is an Igusa-T o dor ov algebr a, then R is also an Igusa-T o dor ov algebr a. Pro of . The pro of is similar to that of Theorem 4.5. Namely , if the equiv alence is giv en b y F : D b ( R ) ⇄ D b ( S ) : G , then w e can obtain t ha t F (mo d R ) is an Igusa-T o dorov class prov ided that S is an Igusa- T o doro v algebra. Hence, w e obtain that mo d R = 13 G ( F (mo d R )) is an Igusa-T o dorov class, b y Prop osition 5.3. Th us, R is an Igusa- T o doro v algebra by Theorem 5.2. ✷ F or instance, if R is deriv ed equiv alen t to an algebra with r a dical cub e zero, or an algebra with represen tation dimension at most three, then R m ust b e a n Igusa-T o doro v algebra. 6 Auslander ’ s co ndition Let M ∈ D b ( R ) and C ⊆ D b ( R ). W e define the C -Auslander b ound of M t o b e the min- imal integer m , or ∞ if suc h minimal integer do esn’t exist, suc h that Hom D ( R ) ( M , N [ i ]) = 0 for all i > m , whenev er N ∈ C satisfies t ha t Hom D ( R ) ( M , N [ i ]) = 0 for a ll but finitely man y i . Let E ⊆ D b ( R ). The global C -Auslander b ound of E is the suprem um of all C - Auslander b ounds of ob jects in E . The finitistic C -Auslander b ound of E is the supremu m of all C -Auslander bounds of ob jects in E whose C -Auslander b ound is finite. In case that M is an R - mo dule and C = E = mo d R , t he notions g iven here coincide with the usual ones in mo dule categories [3, 17], i.e., Auslander b o und of M , globa l Auslander b ound o f the algebra R and the finitistic Auslander b ound of the alg ebra R , resp ectiv ely . W e hav e the follo wing easy o bserv a t io n. Lemma 6.1 L e t M ∈ D b ( R ) and C ⊆ D b ( R ) . Then M has finite C - Auslander b ound if and only if M [ m ] has finite ( C [ n ]) -Au slander b ound f o r any/some inte gers m, n . It is a lso easy to se e that if a complex M has finite C -Auslander b ound, then M also has finite E - Auslander b ound for any E ⊆ C . An algebra R is called a n A C-algebra provide d t hat ev ery R - mo dule has finite (mo d R )- Auslander b o und [3]. Auslander conjecture asserts that all algebras are AC-algebras. Ho w eve r, the conjecture f a ils in general [10,12]. W e refer to [3] f o r the list of A C-a lgebras. In [17], t he author suggests a revisited v ersion of Auslander conjecture, named the finitistic Auslander conjecture, whic h asserts that the finitistic Auslander b ound of ev ery alg ebra is finite. Note that the finitistic Auslander conjecture implies the finitistic dimension conjecture. W e hav e the follo wing c haracterization of A C-algebras in term of deriv ed cat ego ries. Theorem 6.2 (1) R is a n AC-algebr a if and only if every c omplex in D b ( R ) has fini te ( D [ c,d ] ( R )) -Auslander b ound for any/some inte gers c ≤ d . (2) The glob al Auslander b ound of R is finite if and only i f , for any/some inte gers a ≤ k and c ≤ d , the glob al ( D [ c,d ] ( R )) -Auslander b ound of D [ a,k ] ( R ) is fi nite. (3) T he finitistic Auslander b ound of R is finite if and only if, for any/some inte gers a ≤ k and c ≤ d , the finitistic ( D [ c,d ] ( R )) -Auslander b ound of D [ a,k ] ( R ) is finite. 14 Pro of . (1) The if- pa rt. Assume that ev ery complex in D b ( R ) has finite ( D [ c,d ] ( R ))- Auslander b ound for some integers c ≤ d . Then ev ery complex has finite ((mo d R )[ c ])- Auslander b ound since (mo d R )[ c ] ⊆ D [ c,d ] ( R ). It follo ws that ev ery complex, in particular ev ery R -mo dule, has finite (mo d R )-Auslander b ound b y Lemma 6.1. Hence, R is an A C- algebra. The only-if part. T ak e an y M ∈ D b ( R ) and any in tegers c ≤ d .. Assume that M ∈ D [ a,k ] ( R ) for some in tegers a ≤ k , then w e hav e that Ω k D ( M ) is an R - mo dule and M ≃ Ω a D ( M )[ a ], b y Lemma 3.2 (2). Now tak e a n y N ∈ D [ c,d ] ( R ), then w e o btain t ha t Hom D ( R ) ( M , N [ j ]) ≃ Hom D ( R ) (Ω a D ( M )[ a ] , N [ j ]) ≃ Hom D ( R ) (Ω a D ( M ) , N [ j − a ]) j − k > − c ≃ Hom D ( R ) (Ω k D ( M ) , N [ j − k ]) for all j > k − c , by dimension shifting in Prop osition 3.5. No w note that Ω k D ( M ) is an R -mo dule and N = Ω D d ( N )[ d ] b y Lemma 3.2’ (2), s o w e further obtain that Hom D ( R ) (Ω k D ( M ) , N [ j − k ]) ≃ Hom D ( R ) (Ω k D ( M ) , (Ω D d ( N ))[ j − k + d ]) j − k + c> 0 ≃ Hom D ( R ) (Ω k D ( M ) , (Ω D c ( N ))[ j − k + c ]), for all j > k − c , by dimension shifting in Prop osition 3.5’. It follo ws that Hom D ( R ) ( M , N [ j ]) ≃ Hom D ( R ) (Ω k D ( M ) , (Ω D c ( N ))[ j − k + c ]), for all j > k − c . Hence, if Hom D ( R ) ( M , N [ j ]) = 0 for all but finitely man y j , then w e hav e that Hom D ( R ) (Ω k D ( M ) , (Ω D c ( N ))[ i ]) = 0 f or all but finitely many i . Since R is an A C-alg ebra and Ω k D ( M ) , Ω D c ( N ) ⊆ mo d R , there is a n in teger l ≥ 0, indep enden t of Ω k D ( N ), suc h that Hom D ( R ) (Ω k D ( M ) , (Ω D c ( N ))[ i ]) = 0 for all i > l . It follow s from the ab ov e isomorphism that Hom D ( R ) ( M , N [ j ]) = 0 fo r a ll j suc h t ha t j − k + c > l and j > k − c , i.e., for all j > k − c + l . Not e that k − c + l is indep enden t of N , so w e obtain that M has finite ( D [ c,d ] ( R ))-Auslander b ound. (2) and (3 ). The pro ofs are similar to (1) , just note that Auslander b ounds are unique in b oth cases. ✷ Then, w e obtain an imp ortan t prop ert y of AC-algebras and algebras satisfying the finitistic Auslande r conjecture. Theorem 6.3 Assume that R , S ar e derive d e quivalent algebr as . (1) If S is an A C algebr a, then R is also an A C algebr a. (2) I f the glob al Au slander b ound of S is finite, then the glob al Auslander b ound of R is also finite. (3) I f the finitistic Auslander b ound of S is finite, then the fi n itistic A uslander b ound of R is also finite. Pro of . (1) T ak e any M , N ∈ mo d R . If Ho m D ( R ) ( M , N [ i ]) = 0 for all but finitely many i , then Hom D ( S ) ( F ( M ) , F ( N )[ i ]) = 0 for all but finitely man y i . Note tha t F ( N ) ∈ F (mo d S ) ⊆ D [ a,k ] ( S ) f o r some fixed integer a ≤ k , by Lemma 4.3. Since S is an AC 15 algebra, w e hav e that F ( M ) ∈ D b ( S ) has finite ( D [ a,k ] ( S ))-Auslander b ound b y Theorem 6.3. Then there is an in teger m , indep enden t of N , suc h that Hom D ( S ) ( F ( M ) , F ( N )[ i ]) = 0 for all i > m . It follo ws that Hom D ( R ) ( M , N [ i ]) = Ho m D ( R ) ( G F ( M ) , G F ( N )[ i ]) = 0 fo r all i > m , i.e., M has finite (mod R )-Auslander b o und. Hence, R is an AC -algebra. (2) and (3). The pro ofs are similar. ✷ No w w e turn to algebras satisfying the generalized Auslander-Reiten conjecture. W e note that an equiv alen t statemen t of the generalized Auslander-Reiten conjecture is that if M is an R -mo dule suc h that Hom D ( R ) ( M , ( M ⊕ R )[ i ]) = 0 for all but finitely man y i , then M is of finite pro jectiv e dimension, i.e., M is (isomorphic to) a complex in K b ( P R ). Lemma 6.4 The fol low ing c onditions ar e e quivale n t for a c omplex M ∈ D b ( R ) . (1) Hom D ( R ) ( M , ( M ⊕ R )[ i ]) = 0 for al l but finitely many i . (2) Hom D ( R ) (Ω n D ( M ) , (Ω n D ( M ) ⊕ R )[ i ]) = 0 for any/so me inte ge r n and al l but finitely many i . (3) Hom D ( R ) ( M , ( M ⊕ T )[ i ]) = 0 for any/so m e tilting c o mplex T and al l but finitely many i . Pro of . ( 1 ) ⇔ (2) Note that M = Ω m D ( M )[ m ] fo r some integer m , so w e obtain that Hom D ( R ) (Ω m D ( M ) , (Ω m D ( M ) ⊕ R )[ i ]) = 0 for all but finitely many i if and only if Hom D ( R ) ( M , ( M ⊕ R )[ i ]) = 0 for all but finitely man y i . By Prop osition 3.5, there is a triangle Ω j +1 D ( M ) → P j → Ω j D ( M ) → with P j pro jectiv e, for any j . So, one can easily ch ec k that Hom D ( R ) (Ω j D ( M ) , (Ω j D ( M ) ⊕ R )[ i ]) = 0 for all but finitely man y i if and only if Hom D ( R ) (Ω j +1 D ( M ) , (Ω j +1 D ( M ) ⊕ R )[ i ]) = 0 for a ll but finitely man y i . The conclusion then follo ws. (1) ⇔ (3) If T is a tilting complex, then R generates T and T generates R in K b ( P R ), b oth via finitely man y steps. Also note that Hom D ( R ) ( B , M [ i ]) = 0 for all but finitely man y i , whenev er B ∈ K b ( P R ). It follow s that Hom D ( R ) ( M , ( M ⊕ T )[ i ]) = 0 for all but finitely man y i if and only if Hom D ( R ) ( M , ( M ⊕ R )[ i ]) = 0 for all but finitely man y i . ✷ No w w e can pro vide a deriv ed v ersion of the generalized Auslander-Reiten conjecture. Theorem 6.5 An algebr a R satisfies the g ener alize d A uslander-R eiten c onje ctur e if and only if, for an y M ∈ D b ( R ) such that Hom D ( R ) ( M , ( M ⊕ T )[ i ]) = 0 for any/some tilting c omplex T a n d al l but finitely many i , it holds that M ∈ K b ( P R ) . Pro of . The sufficien t part follo ws from Lemma 6.4. The necessary part. No te that there is some n such tha t Ω n D ( M ) is an R -mo dule, by Lemma 3.2 (2). If M satisfies that Hom D ( R ) ( M , ( M ⊕ R )[ i ]) = 0 for some tilting complex and all but finitely many i , then w e hav e that Ho m D ( R ) (Ω n D ( M ) , (Ω n D ( M ) ⊕ R )[ i ]) = 0 for all but finitely man y i , by Lemma 6.4. Since R satisfies the generalized Auslander-Reiten conjecture, w e obta in the Ω n D ( M ) ∈ K b ( P R ). It follo ws that M ∈ K b ( P R ), b y Lemma 3.2 (6) . ✷ 16 Then, w e can show that deriv ed equiv alences preserv e generalized Auslander-Reiten conjecture. Theorem 6.6 Assume that R, S ar e derive d e quivalen t alg e br as. If S sa tisfi e s the gener- alize d Auslander-R eiten c onje ctur e , then R also satisfies the gener ali z e d Auslander-R eiten c onje ctur e. Pro of . T ak e an y M ∈ D b ( R ) suc h that Hom D ( R ) ( M , ( M ⊕ R )[ i ]) = 0 for a ll but finitely man y i . Then Hom D ( S ) ( F ( M ) , ( F ( M ) ⊕ F ( R ))[ i ]) = 0 for all but finitely man y i . Note that F ( R ) is a tilting complex in D b ( S ) and that S satisfies the generalized Auslander- Reiten conjecture, so w e obtain that F ( M ) ∈ K b ( P S ) b y Theorem 6.5. It follow s that M ≃ G F ( M ) ∈ K b ( P R ). Hence R satisfies the generalized Auslander-Reiten conjecture b y Theorem 6.5 again. ✷ References [1] Au slander M. and Reiten I., O n a generalized v ersion of the Nak ay ama Conjecture, Pro c. AMS. 52 (197 5), 69-74. [2] Bond al A. and O rlo v D., Semiorthogonal decomp osition for algebraic v arieties, arXiv: alg-ge om/95060 12 . 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Jiaqun Wei Scho ol of Mathematics Scie n c e, Nanjing Normal University, Nanjing 210046 , China Email: weijiaqun@njnu.e du.cn 18