Derived equivalences and stable equivalences of Morita type, I

Deriv ed equiv alences and stable equiv alences of Morita type, I. W ei Hu and Changchang Xi ∗ School of Mathematica l Sciences , Laborato ry of Mathematics and Complex Syste ms, B eijing Normal Uni ver sity , 10087 5 Beijing, People’ s R epubli c of China E-mail: hwxbest@1 63.com xicc@bn u.edu.cn Dedicate d to Pr ofes sor Shaoxue Liu on the occasion of his 80th Birthday Abstract For self-injective algebras, Rickard pr oved that each derived equi valence induces a stable equi valence of Morita type. For gene ral algebra s, it is unknown when a deriv ed equ iv alence implies a stable e quiv alence of Morita type. In this paper , we ﬁrst sho w that each derived equivalence F between the deriv ed cate gories of Artin algebras A and B arises n aturally a functo r ¯ F between their stable module categories, which can be used to comp are certain homo logical dimensions of A with that of B ; and then we giv e a sufﬁcient condition for the functor ¯ F to be an e quiv alence. Moreover , if we work with ﬁnite-dimensional algebras over a ﬁeld, then the sufﬁcient con dition guaran tees the existence of a stable equivalence of Morita type. In this way , we extend the classic r esult of Ricka rd. Furthermo re, we provide se veral inductiv e methods for constru cting those d erived equi valences that indu ce stable equiv alences of Morita type. It turns out that we may produce a lot of (usually not self- injective) ﬁnite-dimensional alg ebras which are both deri ved-equivalent and stably equiv alent of Morita type, thus they share many common in variants. 1 Introd uction As is well -kno wn, deri ved equi valen ce and s table equ iv alence of Morita type are two fund amental types of equi v- alence s in alg ebras and ca tego ries, and pl ay an imp ortant role in the mod ern repre sentati on theory of groups and algebr as, th ey transfer informatio n fr om one alge bra to anothe r , and pro vide a con venie nt b ridge between tw o dif feren t (deri ved or stable) categor ies. In particular , deri v ed equiv alences pre serv e many signiﬁcant in v arian ts; for e xample, the center of an algeb ra, the number of non-i somorphi c simple modules, the Hochsc hild coho- mology gro ups and Cartan determinants, whi le stab le equi v alenc es of Morita type, introd uced in around 1990 (see [3], for e xample) and app earing frequ ently in the block theory of ﬁnite grou ps, preserv e also many n ice in vari ants; for instance, the global, ﬁnitist ic, and represent ation dimensio ns [18] as wel l as th e representat ion types [7]. For self-injecti ve alge bras, the two notions are closely r elated to each other , this was reve aled by a well-kno w n result of R ickard [15], whi ch states that a deriv ed equi v alenc e between self-in jecti v e algebras in- duces always a stable equi v alence of Morita type. Moreov er , the remarkabl e A belian Defect Group Conjecture of Brou ´ e, which states that the module catego ries of a block algebra A of a ﬁnite group alg ebra and its Brauer corres ponden t B sh ould ha ve equ iv alent deriv ed cate gories if the ir common defect group is abelian (see [12]), makes the two con cepts more attract iv e and intimate. Ho we ver , for general ﬁnite-dimensio nal alg ebras, deri ved equi v alen ce and stable equ iv alence of Morita t ype se em to be comple tely dif ferent from each oth er; for example , a represe ntation -ﬁnite algebra may be deriv ed-eq ui v alent to a r epresen tation- inﬁnite algebra via a tilting modu le, and conseq uently they neith er are stably equi v alent of Morita type nor hav e the same represen tation dimension. Thus a natural question aris es: What kind of r elation ship between a deriv ed e qui v alenc e and a stable eq ui v alence ∗ Corresponding author . Email: xicc@b nu.edu.cn; Fax: 0086 10 58802136 ; T el.: 0086 10 58808 877. 2000 Mathematics Subject Classiﬁcation: 18E30,16G10;1 8G20,16D90. Ke ywo rds: derive d equiv alence, ﬁnitistic dimension, stable equi v alence, stable equiv alence of Morita type, tilting complex . 1 of Morit a type for general ﬁnite-dimensi onal alge bras could exist? In other wo rds, we consider the follo wing questi on: Question. When does a deriv ed equi v alence between two ﬁnite-dimensio nal (no t necess arily self-injec ti ve) algebr as A and B induce s a stable equi v alen ce of Morita type between them ? Thus, a positi v e answer to the abo v e quest ions would let us know more in var iants between alg ebras A and B . H o wev er , in the past time, litt le is known about this question. One ev en do es not known when a deri ved equi v alen ce induces a stable equi v alen ce for general ﬁnite-dimension al algebras. In the present paper , we shall pr ovid e s ome answers to this question. T o state our mai n result, l et us introduc e the notion of an almost ν -stable fun ctor . Sup pose F is a deri v ed equi v alence between two Artin algebra A and B , with the quasi -in verse functor G . Further , suppos e T • : · · · − → 0 − → T − n − → · · · − → T − 1 − → T 0 − → 0 − → · · · is a radic al tilting complex ove r A associ ated to F , and sup pose ¯ T • : · · · − → 0 − → ¯ T 0 − → ¯ T 1 − → · · · − → ¯ T n − → 0 − → · · · is a radical tilting comple x ov er B associated to G . The functor F is called almost ν -stable if add ( L − n i = − 1 T i ) = add ( L − n i = − 1 ν A T i ) , and add ( L n i = 1 ¯ T i ) = add ( L n i = 1 ν B ¯ T i ) , where ν A is the N akayama functor of A . Note that the summation s e xclu de o nly the term in d egre e 0 . If A an d B are self-injecti ve, ev ery deri ved equiv alence between A and B is almost ν -stable (by Proposition 3.8 below). S urpris ingly , e v en bey ond the class of s elf-inj ecti ve algebras there are plent y of almost ν -stable deri ved equi v alenc es, for exa mple, th e deri ve d equiv alences co nstruct ed in [6, Coroll ary 3.8] and in Propositio n 3.11 belo w . In f act, we shall gi ve a general machinery belo w to produce such deri v ed equi v alences . W ith this not ion in mind, our main result can be stated as follo ws. Theor em 1.1. Let A and B be Artin alge bra s, and let F be a d erived equivalen ce between A and B. Then: ( 1 ) F induces a functo r ¯ F f r om the stable module cate gory over A to that ov er B. ( 2 ) If F i s almost ν -s table , then the fu nctor ¯ F deﬁne d in ( 1 ) is an equivale nce. Furthermor e, if F is an almost ν -stabl e de rived e quival ence betwee n ﬁnite-dimensi onal algebr as A a nd B over a ﬁeld k , then ther e is a stable equivalenc e Φ of Morita type betwee n A and B such that Φ ( X ) ≃ ¯ F ( X ) for all objec ts X in the stable module cate gory over A. As a conseque nce of the proof of Theorem 1.1, we hav e the follo wing facts on the homological dimension s of algeb ras. Cor ollary 1.2. Let A and B be Artin alg ebr as, and let F be a d erived equival ence b etween A and B. If add ( A Q ) = add ( ν A Q ) , then ( 1 ) gl . di m ( A ) 6 gl . dim ( B ) , ( 2 ) ﬁn . dim ( A ) 6 ﬁn . dim ( B ) , ( 3 ) dom . di m ( A ) > dom . dim ( B ) . wher e gl . d im ( A ) , ﬁn . dim ( A ) and dom . dim ( A ) stan d for the global, ﬁnitisti c and dominant dimensi ons of A, r espe ctively . Note th at if A and B are ﬁnite-dime nsional self-injec ti ve, we re-o btain the well-k no wn result [1 5] of Ric kard from Theor em 1.1: Deriv ed-e qui v alent self-injecti ve algeb ras are stably equi v alent of Morita type. M oreo ver , Theorem 1.1 allo ws us to obtain a lot of (usually not self-injecti ve) algebras which are both der iv ed- equi v alen t and stably equiv alent of Morita type. By the followin g corollary , we can eve n repeatedly constru ct deri ve d equi v alen ces satisfy ing the almost ν -stable conditio n. 2 Cor ollary 1.3. L et k be a ﬁeld , and let F be an a lmost ν -stable deriv ed equivalence between two ﬁnite- dimensio nal k-algebr as A and B. Then: ( 1 ) F or any ﬁ nite-di mensiona l self-injectiv e k-algebr a C, ther e is an almost ν -s table derive d equival ence between the two tenso r algebr as A ⊗ k C and B ⊗ k C . ( 2 ) Let ¯ F be the stable equival ence induced by F in Theor em 1.1. Then, for each A-module X , ther e is an almost ν -stab le derived equiva lence between the endomorp hism algeb ras End A ( A ⊕ X ) and E nd B ( B ⊕ ¯ F ( X )) . ( 3 ) If X is an A-module suc h tha t F ( X ) is isomorphic to a B-module Y , then th er e is an almos t ν -stable derive d equiva lence between the one-po int e xtensio ns A [ X ] and B [ Y ] . This paper is organ ized as follo ws. In Section 2, we shall recall some basic deﬁnitions and facts requir ed in proofs . In S ection 3, we ﬁrst sho w that ev ery deriv ed equi v alence F between two Artin algebra s A and B giv es rise to a functor ¯ F between their stab le module categ ories, a nd then gi v e a suf ﬁcient condition for the functor ¯ F to b e an equiv alence between stable module c ateg ories o v er Artin algebras. In Section 4, we deduce some proper ties of th e func tor ¯ F and then compar e homolo gical dimensio ns of A w ith that of B . In pa rticular , we get Corolla ry 1 .2. As a by-pro duct, we re-obta in the result th at a deri ved equi v alen ce pres erve s th e ﬁniteness of ﬁnitistic dimension. In Section 5, w e show that the condition giv en in S ection 3 is suf ﬁ cient for F to ind uce a stable equi v alenc e o f Morita type when we work with ﬁ nite-dime nsional algebras o ver a ﬁeld. In S ection 6 , we gi ve se vera l m ethods to constru ct indu cti vel y deri ve d equiv alences satisfy ing the almost ν -stable condition . Finally , in Section 7, we ex hibit a couple of examples to ex plain our points about the main result. 2 Pr eliminaries In this secti on, w e shall recall basic deﬁniti ons and fact s require d in our proofs. Let C be an additi ve categ ory . For two morp hisms f : X → Y and g : Y → Z in C , the composition of f with g is written as f g , which is a morph ism fro m X to Z . But for two functors F : C → D and G : D → E of cate gories, their compo sition is den oted by GF . For an o bject X in C , we denote by ad d ( X ) the full subcate gory of C consistin g of all direct summands of ﬁnite direct sums of copies of X . Through out this pape r , unles s speciﬁed othe rwise, all alg ebras will be Artin algeb ras ov er a ﬁxed commu- tati v e Artin ring R . All m odules will be ﬁnitely generated unita ry left modules. If A is a n Arti n algebra, the cate gory of all modules ov er A is denoted by A -mod; the full sub cate gory of A -mod con sisting of proj ecti ve (respe cti vel y , injecti ve) modules i s denoted b y A -proj (respecti ve ly , A -inj). W e denote by D the usual duality on A -mod. The duality Hom A ( − , A ) from A -proj to A op -proj is denot ed by ∗ , that is, for each projecti ve A - module P , the projecti ve A op -module H om A ( P , A ) is denoted by P ∗ . W e den ote by ν A the Naka yama fun ctor D Hom A ( − , A ) : A -pro j − → A -inj. The stable module categ ory A -mod of an algebra A is, by deﬁnition, an R -category in which objects are the same as the objects of A -mod and, for two objects X , Y in A -mod , their morphism set, denoted by Hom A ( X , Y ) , is the quoti ent of H om A ( X , Y ) modulo the homomorphisms that factorize through projecti ve m odules . T wo algebr as are said to be stably equiva lent if their stable module cate gories are equi v alent as R -categori es. For ﬁ nite-dime nsional algebras o ver a ﬁeld k , there is a special class of stable equi valen ces, n amely stable equi v alen ces of Morita type. R ecall that two ﬁnite-dimens ional algebra s A an d B ov er a ﬁeld k are call ed stably equiva lent of Morita type if there are two bimodu les A M B and B N A satisfy ing the follo wing properti es: ( 1 ) all of the one-s ided modules A M , M B , B N and N A are project iv e, and ( 2 ) there is an A - A -bimodule isomorphism A M ⊗ B N A ≃ A ⊕ U for a projec ti ve A - A -bimodu le U ; and there is a B - B -bimodul e isomorph ism B N ⊗ A M B ≃ B ⊕ V for a proj ecti ve B - B -bimodule V . If two ﬁnite -dimensio nal alg ebras A and B over a ﬁeld are stabl y equi v alen t of Morita type, then the functo r T N : A -mod − → B -mod deﬁned by B N ⊗ A − is an equi v alenc e. It is also called a stable equivale nce of Morita type . (Note that if we exten d the deﬁnition of a stable equi v alence of Morita type from ﬁnite-dimens ional 3 algebr as to R -project iv e Arti n R -alg ebras, then there i s an op en problem of whether T N could induce a stab le equi v alen ce on stable module categ ories, namely whether A U ⊗ A X is pro jecti v e for ev ery A -module X .) No w , we recal l some deﬁnitions rele v ant to deri v ed equi v alenc es. Let C be an add iti ve cate gory . A co mplex X • ov er C is a se quence of morphisms d i X between o bjects X i in C : · · · → X i − 1 d i − 1 X − → X i d i X − → X i + 1 d i + 1 X − → X i + 2 → · · · such that d i X d i + 1 X = 0 for all i ∈ Z . W e write X • = ( X i , d i X ) . For each complex X • , the brutal truncation σ < i X • is a subc omplex of X • such that ( σ < i X • ) k is X k for all k < i and zero otherwise . Similarly , we deﬁne σ ≥ i X • . The c ateg ory of comple xes o v er C is den oted by C ( C ) . The homotop y cate gory of comple x es ov er C is d enoted by K ( C ) . When C is an abelian categ ory , the deri ved cate gory of complex es ov er C is denoted by D ( C ) . The full subcate gor y of K ( C ) and D ( C ) consi sting of bou nded comple xes o ver C is denoted by K b ( C ) and D b ( C ) , r especti vely . Moreo ve r , we denote by C − ( C ) the categ ory of complex es bounded abov e, and by K − ( C ) the homotopy cate gory of C − ( C ) . D ually , w e hav e the category C + ( C ) of comple x es bounde d below and the homotop y cate gor y K + ( C ) o f C + ( C ) . As usual, for a giv en Artin algebra A , we simply write C ( A ) for C ( A -mod ) , K ( A ) for K ( A -mod ) and K b ( A ) for K b ( A -mod ) . Similarly , we write D ( A ) and D b ( A ) for D ( A -mod ) a nd D b ( A -mod ) , respecti vely . It is well-kno w n that, for an Artin algebra A , K ( A ) and D ( A ) are triangu lated cate gori es. For basic results on triangu lated categori es, we refer to Happel’ s book [5]. Throu ghout this paper , we deno te by X [ n ] rather than T n X the ob ject obta ined from X by shif ting n ti mes. In partic ular , for a complex X • in K ( A ) or D ( A ) , the comple x X • [ 1 ] is obtaine d from X • by shift ing X • to the left by one deg ree. Let A be an A rtin algebra. A homomorphism f : X − → Y of A -modules is cal led a radic al map if, for any module Z and homomorphisms h : Z − → X and g : Y − → Z , th e co mpositio n h f g is not a n i somorphis m. A comple x over A -mod is called a radical complex if all of its dif feren tial maps are radical maps. Every comple x ov er A -mod is isomorphic in the h omotop y c ateg ory K ( A ) t o a radical c omplex . It is easy t o see t hat if two radica l comple x X • and Y • are isomorph ic in K ( A ) , then X • and Y • are isomorph ic in C ( A ) . T wo algebras A and B are said to be derived-e quivale nt if their deri ved catego ry D b ( A ) and D b ( B ) are equi v alen t as t riangul ated categ ories. In [1 3], Rickard pro v ed that two algebras are deri ved -equi v alen t if and only if there is a comple x T • in K b ( A -proj ) satisfyin g (1) Hom D b ( A ) ( T • , T • [ n ]) = 0 for all n 6 = 0, an d (2) add ( T • ) generat es K b ( A -proj ) as a triangu lated categ ory such that B ≃ End D b ( A ) ( T • ) . A comple x in K b ( A -proj ) sat isfying the abov e two condit ions is called a tilt ing comple x o ver A . By the condition (2), each indecompo sable projecti ve A -module is a direct summand of T i for some inte ger i . It is kno wn that, gi ven a deri v ed equiv alence F between A and B , there is a unique (up to iso morphism) tilting comple x T • ov er A suc h that F ( T • ) ≃ B . This comple x T • is call ed a tilting comple x associ ated to F . Let F be a deri ved equi v alen ce b etween two Artin algeb ras A and B , and let Q • be a tilting comple x assoc iated to F . W itho ut loss of generality , we may assume tha t Q • is radical and that Q i = 0 for i < − n and i > 0. T hen we ha v e the follo wing fact, for con venienc e of the reader , w e includ e here a proof. Lemma 2.1. Let A and B be two Artin alg ebr as, a nd let F and Q • be as above . If G : D b ( B ) − → D b ( A ) is a quasi- in verse of F , then ther e is a radical tilting comple x ¯ Q • associ ated to G of the following form: 0 / / ¯ Q 0 / / ¯ Q 1 / / · · · / / ¯ Q n / / 0 . Pr oof. Note that the tilting comple x Q • associ ated to F is radical and of the form: 0 / / Q − n / / · · · / / Q − 1 / / Q 0 / / 0 . Let ¯ Q • be a radical comp lex in K b ( B -proj ) such that ¯ Q • isomorph ic to F ( A ) in D b ( B ) . Then G ( ¯ Q • ) ≃ GF ( A ) ≃ A in D b ( A ) , which means that ¯ Q • is a tiltin g complex associated to G . Moreov er , on the one hand, we hav e Hom D b ( B ) ( ¯ Q • , B [ m ]) ≃ Hom D b ( A ) ( A , Q • [ m ]) = 0 4 for all m > 0, and cons equentl y ¯ Q • has zero terms in all neg ati ve de grees. On the other hand, we ha ve Hom D b ( B ) ( B , ¯ Q • [ m ]) ≃ Hom D b ( A ) ( Q • , A [ m ]) = 0 for all m > n and Hom D b ( B ) ( B , ¯ Q • [ n ]) 6 = 0. Thus th e comple x ¯ Q • has zero terms i n all de grees lar ger than n , and its n -th term is non- zero. The follo wing lemma will be used freq uently in our proofs. Again, we includ e here a proof for con ven ience of the reade r . Lemma 2.2. Let A be an arbitrar y ring, and let A -Mod be the cate gory of all left (not necess arily ﬁnitely gen erat ed) A-modul es. Suppose X • is a comple x over A -Mod bound ed abo ve and Y • is a comple x over A -Mod bound ed below . Let m be an inte ger . If one of the followin g two condit ions holds: ( 1 ) X i is pr ojective for all i > m and Y j = 0 for all j < m; ( 2 ) Y j is injec tive for all j < m and X i = 0 for all i > m, then θ X • , Y • : Hom K ( A -Mod ) ( X • , Y • ) → Hom D ( A -Mod ) ( X • , Y • ) induced by the localiza tion f unctor θ : K ( A -Mod ) → D ( A -Mod ) is an isomorp hism. Pr oof. For simplic ity , we write K = K ( A -Mod ) and D = D ( A -Mod ) . The cate gory of all left (no t nec- essaril y ﬁnitely generated) projecti ve A -modules is deno ted by A -P roj. By applying the shift functor , we may assume m = 0. Suppo se (1) is satisﬁed. First, we consid er the case X i = 0 for all i < 0. Let · · · / / P − 1 / / P 0 π / / X 0 / / 0 be a proj ecti ve resolutio n of X 0 with all P i being proj ecti ve A -modules. Then the follo wing complex , · · · / / P − 1 / / P 0 π d 0 X / / X 1 / / X 2 / / · · · , denote d by P • X , is in K ( A -Proj ) and bounded above since X i is projecti v e for all i > 0 by our assumption, and there is a qua si-isomor phism π • : P • X → X • : · · · / / P − 1   / / P 0 π   π d 0 X / / X 1 / / · · · 0 / / X 0 d 0 X / / X 1 / / · · · . W e claim that Hom K ( π • , Y • ) : Hom K ( X • , Y • ) − → Hom K ( P • X , Y • ) i nduced by π • is an isomorph ism. A ctually , if f • ∈ H om K ( P • X , Y • ) , then f 0 fact orizes through the m ap π : P 0 → X 0 . Suppose f 0 = π g 0 for some g 0 : X 0 → Y 0 . Let g i : = f i for all i > 0. T hen g • = ( g i ) is a chain map from X • to Y • and f • = π • g • . Consequentl y , the map Hom K ( π • , Y • ) is surject iv e. F urther , w e sho w that the map Hom K ( π • , Y • ) is injecti ve. In fact, if π • α • = 0 for a mo rphism α • in Hom K ( X • , Y • ) , then the re are m aps h i : X i → Y i − 1 for all integ er i ≥ 1 such that πα 0 = π d 0 X h 1 and α i = d i X h i + 1 + h i d i − 1 Y . Thus α 0 = d 0 X h 1 since π is an epimorphis m. Hence α • = 0, which implies that Hom K ( π • , Y • ) is injecti v e. It follo ws that Hom K ( π • , Y • ) is an isomorph ism. Note that π • induce s a commutati ve diagra m Hom K ( X • , Y • ) θ X • , Y • − − − − → Hom D ( X • , Y • ) ( π • , − )   y ( π • , − )   y Hom K ( P • X , Y • ) θ P • X , Y • − − − − → H om D ( P • X , Y • ) . 5 Since π • is a quasi-isomor phism, the righ t vert ical map is an isomorphis m. W e ha ve sho wn that the left vert ical map is an isomorphism. Moreov er , since P • X is a comple x in K ( A -P roj ) and bounded abov e, the map θ P • X , Y • is an isomor phism. I t follo ws that θ X • , Y • : Ho m K ( X • , Y • ) − → Hom D ( X • , Y • ) is an isomorph ism. No w , let X • be an arbitra ry complex satisfyin g the condition (1). Then there is a distingu ished triangle σ < 0 X • [ − 1 ] − → σ ≥ 0 X • p − → X • − → σ < 0 X • in K . This triangle can be vie wed as a distingu ished triangle in D . Applying the fun ctors Hom K ( − , Y • ) and Hom D ( − , Y • ) to the triangle, we ha ve an ex act commutati v e diagram Hom K ( σ < 0 X • , Y • ) − − − − → Hom K ( X • , Y • ) K ( p , − ) − − − − → Hom K ( σ ≥ 0 X • , Y • ) − − − − → Hom K ( σ < 0 X • [ − 1 ] , Y • )   y θ X • , Y •   y θ σ ≥ 0 X • , Y •   y θ σ < 0 X • [ − 1 ] , Y •   y Hom D ( σ < 0 X • , Y • ) − − − − → Hom D ( X • , Y • ) D ( p , − ) − − − − → Hom D ( σ ≥ 0 X • , Y • ) − − − − → Hom D ( σ < 0 X • [ − 1 ] , Y • ) . By our assu mption, we hav e Y i = 0 for all i < 0, and th erefore Hom K ( σ < 0 X • , Y • ) = 0. Note that σ < 0 X • is isomorph ic in D to a comple x P • 1 in K ( A -Proj ) such that P i 1 = 0 for all i ≥ 0. It follo ws that H om D ( σ < 0 X • , Y • ) ≃ Hom D ( P • 1 , Y • ) ≃ Hom K ( P • 1 , Y • ) = 0. Thus both m aps K ( p , − ) and D ( p , − ) are injec ti ve. Note tha t σ ≥ 0 X • is a comple x satisfyin g the condit ion (1) and the terms of σ ≥ 0 X • in all neg ati ve d egre es are zero. By the ﬁrst part of the proof, we see that the map θ σ ≥ 0 X • , Y • is an isomorphis m. It follo ws that θ X • , Y • is injecti ve. In particular , the map θ σ < 0 X • [ − 1 ] , Y • is inje cti ve. B y the Fiv e Lemma (see [17, Exercis e 1.3.3, p.13], the map θ X • , Y • is surjecti v e. Thus θ X • , Y • is an isomorph ism. The proof for the situat ion (2) proceed s dually . In the followin g, we point out a relationsh ip betwee n the N akayama functor and a deri ved equi v alence . Let F : D b ( A ) − → D b ( B ) be a der iv ed equiv alence between Artin algebras A and B ov er a commutati v e A rtin ring R . By [13, T heorem 6.4], F induc es an equi v alence from K b ( A -proj ) to K b ( B -proj ) . Not e that the Nakayama functo r ν A : A -p roj − → A -inj induce s a functor fro m K b ( A -proj ) to K b ( A -inj ) , which is again denoted by ν A . When A and B are ﬁnite-dimens ional alge bras ove r a ﬁeld k , it is kno wn that F ( ν A P • ) ≃ ν B F ( P • ) in D b ( B ) for each P • ∈ K b ( A -proj ) [15]. W e shall e xtend this to Artin alg ebras by usi ng the no tation of Ausla nder -Reiten triang le. Recall that a distin guishe d trian gle X f − → M g − → Y w − → X [ 1 ] in D b ( A ) is called an Auslan der -Reiten triang le if (AR1) X and Y are indecompo sable, (AR2) w 6 = 0, and (AR3) if t : U − → Y is not a split epimorph ism, then t w = 0. For a giv en object Y in D b ( A ) , if there is an Auslander -R eiten triang le X f − → M g − → Y w − → X [ 1 ] , then it is uniqu e up to isomorph ism [5 , Propositio n 4.3, p.33]. The ﬁrst statement of the follo wing lemma is essent ially due to Happel (see [5, Theorem 4.6, p.37]), here we shal l modify the origin al proof to deal with Artin algebras . Lemma 2 .3. L et A be a n Artin alg ebr a over a co mmutative Artin ring R. Then, for each i ndecompo sable object P • in K b ( A -proj ) , ther e is an Au slande r- Reiten triangle ( ν A P • )[ − 1 ] − → L • − → P • − → ν A P • in D b ( A ) . Furthermor e, w e have F ( ν A P • )) ≃ ν B F ( P • ) in D b ( B ) . Pr oof. Note that there is an in vertible natu ral transf ormation α P : D Hom A ( P , − ) − → Hom A ( − , ν A P ) for each projec ti ve A -module P . This induce s an in vertible natural tran sformatio n of two functors from D b ( A ) to R -mod. α P • : D Hom D b ( A ) ( P • , − ) − → H om D b ( A ) ( − , ν A P • ) . 6 Recall that D = Hom R ( − , J ) with J an inject iv e en velope of the R -module R / rad ( R ) . Let ψ be a non-zero R - module homomorphism from End D b ( A ) ( P • ) / rad ( End D b ( A ) ( P • )) to J . W e deﬁne φ : End D b ( A ) ( P • ) − → J to be the composi tion of the canon ical surject iv e homomorphism End D b ( A ) ( P • ) − → End D b ( A ) ( P • ) / rad ( End D b ( A ) ( P • )) with ψ . Then φ is non- zero and v anishe s on rad ( End D b ( A ) ( P • )) . Thus we ha ve a dist inguish ed tria ngle ( ν A P • )[ − 1 ] − − − − → L • − − − − → P • α P • ( φ ) − − − − → ν A P • , where L • [ 1 ] is the mapping con e of the map α P • ( φ ) . Clearly , this tria ngle sati sﬁes the conditio ns (AR1) an d (AR2). Let f : X • − → P • be a m orphis m which is not a split epimorp hism. Then we ha ve a commutati v e diagra m D Hom D b ( A ) ( P • , P • ) α P • − − − − → Hom D b ( A ) ( P • , ν A P • ) ( f ∗ , − )   y   y ( f , − ) D Hom D b ( A ) ( P • , X • ) α P • − − − − → H om D b ( A ) ( X • , ν A P • ) , where f ∗ : Hom D b ( A ) ( P • , P • ) − → Hom D b ( A ) ( P • , X • ) is induced by f . S ince f is not a split epimorph ism and P • is indec omposabl e, we ﬁnd that, for each morphism g : P • − → X • , the compositio n g f is in rad ( End D b ( A ) ( P • )) . By the deﬁnition of φ , we ha ve ( f ∗ φ )( g ) = φ ( g f ) = 0 for all g ∈ H om D b ( A ) ( P • , X • ) , that is, f ∗ φ = 0. It follo ws that f α P • ( φ ) = α P • ( f ∗ φ ) = 0. This pro v es (AR 3). No w , for the indec omposabl e objec t P • in K b ( A -proj ) , there is an Auslan der -Reiten triangl e ( ∗ ) ( ν A P • )[ − 1 ] − → L • − → P • − → ν A P • in D b ( A ) . Since F ( P • ) is in K b ( B -proj ) and indecompo sable, there is also an Auslande r- Reiten triangle ( ν B F ( P • ))[ − 1 ] − → L ′• − → F ( P • ) − → ν B F ( P • ) in D b ( B ) . Further , if we apply the functo r F t o ( ∗ ), we get another Auslander -Reit en triangle F ( ν A P • )[ − 1 ] − → F ( L • ) − → F ( P • ) − → F ( ν A P • ) in D b ( B ) . By the uniqueness of the Auslan der -Reiten triangle associat ed t o the giv en comple x F ( P • ) , we see that ν B F ( P • ) is isomorp hic to F ( ν A P • ) in D b ( B ) . Finally , let us re mark that, gi ven a functor F : C → D , if we ﬁx a n object F X in D for each o bject X in C such that F X ≃ F ( X ) , th en there i s a f unctor F ′ : C → D s uch that F ′ ≃ F and F ′ ( X ) = F X for e very X in C . Actually , let s X denote the isomorp hism from F X to F ( X ) , and we deﬁne F ′ ( f ) : = s X F ( f ) s − 1 Y for each f : X → Y . Then this F ′ is a desir ed functor . 3 Stable equiv alences induc ed by deriv ed equi valences In this section, we shall ﬁrst construc t a funct or ¯ F : A -mod − → B -mod bet ween the stable m odule categori es of two A rtin algebras A and B from a giv en deriv ed equi v alence F : D b ( A ) − → D b ( B ) , and then gi ve a suf ﬁcient condit ion to ensure that the functor ¯ F is an equi v alence . In Section 5, w e shall see a stronger conclusio n when we wor k with ﬁnite-dimensi onal algebras instead of general Artin algebras. Let us ﬁrst recall some notio ns and notatio ns. Let A be an Artin algeb ra. The homotopy cate gory K b ( A -proj ) can be consider ed as a triangula ted fu ll subcat egor y o f D b ( A ) . Let D b ( A ) / K b ( A -proj ) be the V erdie r quotien t o f D b ( A ) by the subcateg ory K b ( A -proj ) (for the deﬁnition, w e refer the reader to the excel lent book [11]). Then 7 there is a canonical functor Σ ′ : A -mod − → D b ( A ) / K b ( A -proj ) obtained by composing the natural embedding of A -mod into D b ( A ) with t he quotient func tor fro m D b ( A ) to D b ( A ) / K b ( A -proj ) . Clearly , Σ ′ ( P ) is i somorphi c to zero f or each p rojecti ve A -module P , s o Σ ′ fact orizes thr ough the natur al func tor A -mod − → A -mod. T his gi ves rise to an add iti ve functor Σ : A -mod − → D b ( A ) / K b ( A -proj ) . Rickard [14] pro ved that Σ is an equ iv alence provi ded that the algeb ra A is se lf-inje cti ve. But for an a rbitrary algebr a, this i s no longer true in genera l; for i nstanc e, if A is a non- semisimple Artin algebra of ﬁnite global dimensio n, the n the quotient catego ry D b ( A ) / K b ( A -proj ) is zero, and therefo re the functor Σ is a zero functor which can not be an equi v alen ce. Let A a nd B be two Artin algebras. Suppose F : D b ( A ) − → D b ( B ) is a deriv ed e qui v alenc e between A an d B . Then F induces an equiv alence b etween the quotien t categorie s D b ( A ) / K b ( A -proj ) and D b ( B ) / K b ( B -proj ) . For simplicity , we denote th is i nduced equi v alence also by F . Thus, if A and B ar e s elf-inje cti ve, then A -mod and B -mod are equi v alen t. H o wev er , this is not true in general for arbitrary ﬁnite-dimensio nal algebr as, namely we canno t ge t an equiv alence of stable module cate gories from a giv en deri v ed equiv alence in general. Ne ve rtheless , we may ask if there is any “good” funct or ¯ F : A -mod − → B -mod induce d by F , which could be a possib le candid ate for a stable equi valenc e under certa in addition al condition s, and woul d co ver the most interesting kno wn situa tions. In the follo w ing, we shall constru ct an additi ve fun ctor ¯ F : A -mod − → B -mod from F such that the diagram A -mod Σ − − − − → D b ( A ) / K b ( A -proj ) ¯ F   y   y F B -mod Σ − − − − → D b ( B ) / K b ( B -proj ) of additi ve funct ors is commutati ve up to natural isomorphisms. Furthermor e, we shall construct a possible candid ate for the in verse of ¯ F u nder an additio nal conditi on. From no w on, A and B a re Art in R -algebras, F is a d eri ve d equi valenc e betwee n A and B with t he quasi- in verse G . Let Q • be a tilting comple x ove r A associate d to F of the follo wing form: Q • : 0 / / Q − n / / · · · / / Q − 1 / / Q 0 / / 0 such that all diffe rentials are radic al maps. By Lemma 2.1, the re is a tilting complex ¯ Q • associ ated to G of the form ¯ Q • : 0 / / ¯ Q 0 / / ¯ Q 1 / / · · · / / ¯ Q n / / 0 with all dif feren tials being radical maps. W e deﬁne Q = L n i = 1 Q − i and ¯ Q = L n i = 1 ¯ Q i . Lemma 3.1. Let X be an A-module. T hen F ( X ) is isomorphic i n D b ( B ) to a radical comple x ¯ Q • X of the following form 0 / / ¯ Q 0 X / / ¯ Q 1 X / / · · · / / ¯ Q n X / / 0 , with ¯ Q i X ∈ add ( B ¯ Q ) for all i = 1 , 2 , · · · , n. Mor eove r , the complex ¯ Q • X of this form is unique up to isomorphis ms in C b ( B ) . In particu lar , if X is pr ojective , then ¯ Q • X is isomorp hic in C b ( B ) to a comple x in add ( ¯ Q • ) . Pr oof. Let H i be the i -th homology functor on complex es. F irst of all, we hav e H i ( F ( X )) ≃ H om D b ( B ) ( B , F ( X )[ i ]) ≃ Hom D b ( A ) ( Q • , X [ i ]) = 0 for all i > n and all i < 0, which m eans that F ( X ) has no homology in neg ati ve de grees and deg rees larg er than n . Clearly , we may assume that X is indecompo sable. If X is proj ecti ve , then X is is omorphic to a dir ect su mmand of A . Consequently , F ( X ) is isomorp hic in D b ( B ) to a direct summand L • of the complex ¯ Q • . Since all terms of ¯ Q • in positi ve degrees are in add ( B ¯ Q ) , all terms of L • in positi ve de grees are in add ( B ¯ Q ) . This shows that for ev ery projecti v e A -module P , the comple x F ( P ) is isomorph ic in D b ( B ) to a comple x with all of its terms of positi v e de grees in add ( B ¯ Q ) . Now we sho w 8 that i f P • is a complex in K b ( A -proj ) with P i = 0 for a ll i > 0, then F ( P • ) is isomorphic in D b ( B ) to a complex in which a ll o f i ts terms in positi ve degrees belong to add ( B ¯ Q ) . In fact, if P • has only o ne no n-zero t erm, th en w e may write P • = P [ t ] for a projecti ve A -module P and a non-ne gati v e integer t . In this case , F ( P • ) is isomorphic to a direc t summand of ¯ Q • [ t ] in which all terms in positi ve degr ees are in add ( B ¯ Q ) , as desired . No w , we assu me that P • has at least two non-zero terms. Then there is an in teg er s < 0 s uch that the brutal truncation s σ < s P • and σ ≥ s P • ha ve less non-zero terms t han P • does. By induction, the complex es F ( σ < s P • ) and F ( σ ≥ s P • ) are respec ti vely isomor phic to c omplex es Y • and Z • in K b ( A -proj ) , such t hat th eir terms in all p ositi v e degr ees are in add ( B ¯ Q ) . Since P • is t he mapp ing con e of t he ma p d s − 1 P from σ < s P • to σ ≥ s P • , the complex F ( P • ) is isomorph ic to the map ping con e of a chain map fr om Y • to Z • , and conseq uently all of i ts terms in positi v e deg rees lie in add ( B ¯ Q ) . No w , suppose that X is an ar bitrary indecompo sable A -module and P • = ( P i , d i ) is a mini mal projecti ve resolu tion of X . W e denote by Ω n ( X ) the n -th syzygy of X , and by P • 1 the comple x 0 / / P − n + 1 / / P − n + 2 / / · · · / / P 0 / / 0 . Then we ha ve a dis tinguis hed triangle in D b ( A ) Ω n ( X )[ n − 1 ] / / P • 1 / / X / / Ω n ( X )[ n ] . From this trian gle one gets the follo wing disting uished triangl e in D b ( B ) : F ( Ω n ( X ))[ n − 1 ] / / F ( P • 1 ) / / F ( X ) / / F ( Ω n ( X ))[ n ] . The complex P • 1 is in K b ( A -proj ) and all the terms of P • 1 in positi ve degre es are zero. Hence F ( P • 1 ) is isomorphic to a comple x Q • 1 in K b ( B -proj ) with Q i 1 in add ( B ¯ Q ) for all i > 0. S ince Ω n ( X ) is an A -module, the complex F ( Ω n ( X )) has no homology in all deg rees lar ger than n . Thus the comple x F ( Ω n ( X )) is isomorphic in D ( B ) to a c omplex P • 2 ∈ K − ( B -proj ) with zero te rms in all de grees larger t han n . It fol lo ws that P • 2 [ n − 1 ] has zer o terms in all degrees lar ger than 1. Hence F ( X ) is isomor phic to the mapping cone con ( µ ) of a map µ from P • 2 [ n − 1 ] to Q • 1 , and all the terms of con ( µ ) in positi v e de grees are in ad d ( B ¯ Q ) . N ote that F ( X ) has zero homology in all neg ati ve degrees and degrees large r than n . Thus con ( µ ) has the same property . Hence con ( µ ) is isomorph ic in D ( B ) to a radical complex 0 / / ¯ Q 0 X / / ¯ Q 1 X / / · · · / / ¯ Q n X / / 0 with ¯ Q i X ∈ add ( B ¯ Q ) for all i = 1 , 2 , · · · , n . Suppose U • and V • are two rad ical complex es of the form in Lemma 3.1 such that both U • and V • are isomorph ic to F ( X ) in D b ( B ) . Then U • and V • are isomorphic in K b ( B ) by Lemma 2.2. Since U • and V • are radica l comple xes, we kno w that U • and V • are isomor phic as comple xes. If X is projecti ve, then X ∈ add ( A ) and F ( X ) ∈ ad d ( F ( A )) . Since F ( A ) ≃ F G ( ¯ Q • ) ≃ ¯ Q • in D b ( B ) , we see that F ( X ) is i somorphi c in K b ( B -proj ) to a complex Y • ∈ add ( ¯ Q • ) . Thus ¯ Q • X is iso morphic in D b ( B ) to the Y • . Since Y • is a compl ex with the properties in Lemma 3.1, we hav e Y • ≃ ¯ Q • X as comple xes by the uniquenes s of ¯ Q • X . This sho ws that ¯ Q • X ∈ add ( ¯ Q • ) . Thus Lemma 3.1 is pro ved. Dually , we ha v e the follo wing lemma. Lemma 3.2. Let Y be a B-module. Then G ( Y ) is iso morphic in D b ( A ) to a rad ical comple x Q • Y of the form 0 / / Q − n Y / / · · · / / Q − 1 Y / / Q 0 Y / / 0 with Q − i Y ∈ add ( ν A Q ) fo r all i = 1 , 2 , · · · , n. Mor eov er , the compl e x Q • Y of th is for m is unique up to isomorphis ms in C b ( A ) . 9 Remark. One can eas ily see th at if X ≃ Y ⊕ Z in A -mod, then th e comple x ¯ Q • X deﬁned in Lemma 3.1 is isomorph ic in C b ( B ) to the direct sum of ¯ Q • Y and ¯ Q • Z . Similarly , if U ≃ V ⊕ W in B -mod, then the complex Q • U deﬁned in Lemma 3.2 is isomorph ic in C b ( A ) to the direct sum of Q • V and Q • W . The nex t lemma is useful in our proofs. Lemma 3 .3. L et A be an Artin alge bra , and let f : X − → Y be a ho momorphism between two A-mod ules X a nd Y . Suppo se P • is a comple x in K b ( A ) with P i pr ojective for all i ≥ 0 and injectiv e for all i < 0 . If f factorizes in D b ( A ) thr ough P • , then f factorizes thr ough a pr oje ctive A -module . Pr oof. There is a distin guishe d trian gle σ < 1 P • [ − 1 ] − → σ ≥ 1 P • b − → P • a − → σ < 1 P • in D b ( A ) . Note that Hom D b ( A ) ( σ ≥ 1 P • , Y ) ≃ Hom K b ( A ) ( σ ≥ 1 P • , Y ) = 0. T hus, if f = gh for a morphis m g : X − → P • and a morph ism h : P • − → Y , then bh = 0, and consequen tly h fact orizes thr ough a , say h = ah ′ for h ′ : σ < 1 P • − → Y . This means that f = gah ′ and factorizes through σ < 1 P • . Thus we may assume P i = 0 for all i > 0 and con sider the followin g distinguishe d triangle σ < 0 P • [ − 1 ] / / P 0 u / / P • v / / σ < 0 P • in D b ( A ) . Now , we suppo se f = gh for g : X − → P • and h : P • − → Y . Note that the co mplex σ < 0 P • is in K b ( A -inj ) by our assumption. Hence Hom D b ( A ) ( X , σ < 0 P • ) ≃ Hom K b ( A ) ( X , σ < 0 P • ) = 0, and conseque ntly gv = 0. T his implies that g fa ctorize s through P 0 , that is, g = g ′ u for a morp hism g ′ : X − → P 0 . S ince A -mod is fully embedded in D b ( A ) , the morph isms g ′ and uh are A -module homomorphisms, and therefore f = gh = g ′ ( uh ) , which fact orizes through the projec ti ve A -module P 0 . No w we deﬁne the fu nctor ¯ F . Pick an A -module X , by Lemma 3.1, we know that F ( X ) is isomorphic in D b ( B ) to a radica l comple x ¯ Q • X of the form 0 / / ¯ Q 0 X / / ¯ Q 1 X / / · · · / / ¯ Q n X / / 0 with ¯ Q i X ∈ add ( B ¯ Q ) for al l i = 1 , 2 , · · · , n . From n o w o n, for each A -mod ule X , we choose (on ce a nd f or a ll) such a complex ¯ Q • X . For each homomorphism f : X − → Y , we denote by f the image of f under the canonica l surjec ti ve map from Hom A ( X , Y ) to Hom A ( X , Y ) . Pro position 3.4. Let F : D b ( A ) − → D b ( B ) be a deriv ed e quivale nce between Artin algebr as A and B. Then ther e is an additive functo r ¯ F : A -mod − → B -mod sending X to ¯ Q 0 X suc h that the follo wing diagr am of the additi ve functors A -mod Σ − − − − → D b ( A ) / K b ( A -proj ) ¯ F   y   y F B -mod Σ − − − − → D b ( B ) / K b ( B -proj ) is commutati ve up to natur al isomorphis ms. Pr oof. B y the remark at the end of Section 2 and Lemma 3.1, w e may assume that F ( X ) is just ¯ Q • X for each A -module X , where ¯ Q • X is the complex that w e hav e ﬁxed abov e. Let ¯ Q + X denote the complex σ ≥ 1 ¯ Q • X . Then we ha ve a dist inguis hed triangle in D b ( B ) : ¯ Q + X i X / / F ( X ) π X / / ¯ Q 0 X α X / / ¯ Q + X [ 1 ] . 10 For e ach homomorph ism f : X − → Y of A -modules X and Y , there is a commutati ve diagram ¯ Q + X i X − − − − → F ( X ) π X − − − − → ¯ Q 0 X α X − − − − → ¯ Q + X [ 1 ]   y a f   y F ( f )   y b f   y a f [ 1 ] ¯ Q + Y i Y − − − − → F ( Y ) π Y − − − − → ¯ Q 0 Y α Y − − − − → ¯ Q + Y [ 1 ] . The map a f exi sts becaus e i X F ( f ) π Y belong s to Hom D b ( B ) ( ¯ Q + X , Q 0 Y ) ≃ Hom K b ( B ) ( ¯ Q + X , Q 0 Y ) = 0. Since B -mod is fully embed ded in D ( B ) , the morphism b f is a homomor phism of modules. If we ha ve another A -modul e homomorph ism b ′ f such that π X b ′ f = F ( f ) π Y , then π X ( b f − b ′ f ) = F ( f ) π Y − F ( f ) π Y = 0 and b f − b ′ f fact orizes throug h ¯ Q + X [ 1 ] . By Lemm a 3.3, the B -modu le ho momorphism b f − b ′ f fact orizes throu gh a p rojecti ve B -module. Thus, for each A -module homomorph ism f in Hom A ( X , Y ) , the m orphis m b f in Hom B ( ¯ Q 0 X , ¯ Q 0 Y ) is unique ly determin ed by f . Suppose f : X − → Y and g : Y − → Z are two homomorphi sms of A -modules, we can see tha t F ( f g ) π Z = π X b f g and F ( f g ) π Z = π X ( b f b g ) . By the uni queness of b f g , we hav e b f g = b f b g . Moreo ver , if X is a projec ti ve A -module, then F ( X ) ≃ ¯ Q • X and ¯ Q • X ∈ add ( ¯ Q • ) by th e proof of Lemma 3.1. In particu lar , ¯ Q 0 X is projec ti ve. Thus, if f fac torizes through a projecti v e modu le P , say f = gh w ith g ∈ Hom A ( X , P ) and h ∈ Hom A ( P , Y ) , t hen b f fact orizes through a projecti ve B -module s ince b f = b gh = ( b gh − b g b h ) + b g b h and since both b gh − b g b h and b g b h fact orize through projecti v e B -modules. For e ach A -module X , we deﬁne ¯ F ( X ) = ¯ Q 0 X . Note that ¯ Q 0 X is, up to isomorphis ms, uniquely determined by X (see Lemma 3.1). F or each homomorphi sm f in Hom A ( X , Y ) , we set ¯ F ( f ) = b f . T hen the abov e discussio ns sho w that ¯ F is well-deﬁned on Hom-sets and that ¯ F is a functor from A -mod to B -mod. N ote that ¯ F is additi ve since F is additi ve . T o ﬁnish the proof of the lemma, it remain s to sho w that π X : F ( X ) − → ¯ F ( X ) is a n atural isomorphism in the quotie nt categ ory D b ( B ) / K b ( B -proj ) . Tha t the morphism π X is an isomorph ism follo ws from the fact that ¯ Q + X is isomorphic to th e zero object in D b ( B ) / K b ( B -proj ) . Clearly , π X is natural in X since we ha ve a commutati ve diagra m F ( X ) π X − − − − → ¯ Q 0 X F ( f )   y   y b f F ( Y ) π Y − − − − → ¯ Q 0 Y in the quoti ent categ ory D b ( B ) / K b ( B -proj ) . It is appropriat e to introduce a name for the functor ¯ F . Giv en a deriv ed equi v alence F , the functor ¯ F constr ucted in Propositi on 3.4 is called a stable functo r of F th rougho ut this paper . Pro position 3.5. If add ( A Q ) = add ( ν A Q ) , then ther e is an additiv e functor ¯ G : B -mod − → A -mod sending U to Q 0 U suc h that the following diagr am of the additive functors B -mod Σ − − − − → D b ( B ) / K b ( B -proj ) ¯ G   y   y G A -mod Σ − − − − → D b ( A ) / K b ( A -proj ) is commutati ve up to natur al isomorphis ms. Pr oof. The idea of the proof of Propositio n 3.5 is similar to that of Proposition 3.4. W e just outline the ke y points of the const ruction of ¯ G . 11 For a B -module U , by Lemma 3.2, G ( U ) is isomorphic in D b ( A ) to a compl ex Q • U such that Q i U ∈ add ( ν A Q ) for all i < 0 and Q j U = 0 for all j > 0. By the remark at the end of Section 2, we can assume that G ( U ) is just Q • U . Let Q − U denote the comple x σ < 0 Q • U . W e ha ve a distingu ished triangle in D b ( A ) : Q − U [ − 1 ] β U / / Q 0 U λ U / / G ( U ) γ U / / Q − U . No w if g : U − → V is a homomorphis m of B -modules, then we hav e a commutati v e diagram Q − U [ − 1 ] β U − − − − → Q 0 U λ U − − − − → G ( U ) γ U − − − − → Q − U   y v g [ − 1 ]   y u g   y G ( g )   y v g Q − V [ − 1 ] β V − − − − → Q 0 V λ V − − − − → G ( V ) γ V − − − − → Q − V . The exis tence of u g follo ws from the fact that the morphism λ U G ( g ) γ V belong s to Hom D b ( A ) ( Q 0 U , Q − V ) ≃ Hom K b ( A ) ( Q 0 U , Q − V ) = 0. Since A -mod is fully embedded in D ( A ) , the map u g can be chose n to be an A - module homo morphism. Moreo ver , if u ′ g : Q 0 U − → Q 0 V is anothe r morphis m such that u ′ g λ V = λ U G ( g ) , then ( u g − u ′ g ) λ V = 0 and u g − u ′ g fact orizes through Q − V [ − 1 ] . S ince add ( A Q ) = add ( ν A Q ) , all the terms of the complex Q − V [ − 1 ] are proje cti ve- injecti ve. By Lemma 3.3, the morphism u g − u ′ g fact orizes thr ough a projecti v e module . Thus, for each B -module homomor phism g , the morphi sm u g in Hom A ( Q 0 U , Q 0 V ) is uniqu ely determine d by g . As in the proof o f Pro positio n 3.4, we can sho w tha t t he comp osition o f t wo morphisms is preser ved , namely u gh = u g u h for all g ∈ Hom B ( U , V ) and all h ∈ Hom B ( V , W ) . Moreo ver , if P is a projec ti ve B -module, then Q • P is isomorphic in D b ( A ) to a complex Q • 1 in add ( Q • ) . Since add ( A Q ) = add ( ν A Q ) , the co mplex Q • 1 is of the form in Lemma 3.2 . By the uniqueness of Q • P , we ha ve an isomorphis m Q • P ≃ Q • 1 in C b ( A ) . Hence Q 0 P ≃ Q 0 1 and Q 0 P is a projecti v e A -module. Thus, if g : U − → V fact orizes through a projecti ve B -module P , that is, g = st for s : U − → P and t : P − → V , then u g = u st = ( u st − u s u t ) + u s u t fact orizes through a projecti ve A -module. This sho ws that the map g 7→ u g is well-deﬁne d. No w , we d eﬁne ¯ G ( U ) : = Q 0 U for each B -module U and ¯ G ( g ) : = u g for each morphism g i n B -mod. Note that Q 0 U is, up to isomorphisms, uniquely determined by U (see Lemma 3.2). Thus we obtain an additi v e functo r ¯ G from B -mod to A -mod. Moreove r , the map λ U is a natural isomorp hism in the quotient category D b ( A ) / K b ( A -proj ) since Q − U is in K b ( A -proj ) . Pro position 3.6. Suppose a dd ( A Q ) = add ( ν A Q ) . Let ¯ F and ¯ G be th e f unctor s cons tructed i n Pr oposit ion 3.4 and Pr oposition 3.5, re spectiv ely . Then the compositio n ¯ G ¯ F is naturall y isomorphic to the identity functor 1 A -mod . In parti cular , ¯ G is dense , and the re strictio n of ¯ G to Im ( F ) is full. Pr oof. For each A -module X , we may a ssume that F ( X ) is t he compl ex ¯ Q • X deﬁned in L emma 3.1. For each B -module U , we assu me that G ( U ) is the complex Q • U deﬁned in Lemma 3.2. W e set ¯ Q + X = σ ≥ 1 ¯ Q • X and Q − U = σ < 0 Q • U . T hen all the terms of Q − U are proje cti ve- injecti ve since add ( A Q ) = add ( ν A Q ) . By deﬁnition, we hav e ¯ F ( X ) = ¯ Q 0 X for ea ch A -modul e X (see Propo sition 3.4), and ¯ G ( U ) = Q 0 U for ea ch B -modul e U (see Proposit ion 3.5). Thus, for each A -module X , there is a disting uished triangle ¯ Q + X i X / / F ( X ) π X / / ¯ F ( X ) α X / / ¯ Q + X [ 1 ] in D b ( B ) , and a disting uished triangle Q − ¯ F X [ − 1 ] β ¯ F X / / ¯ G ¯ F ( X ) λ ¯ F X / / G ¯ F ( X ) γ ¯ F X / / Q − ¯ F X 12 in D b ( A ) . Applyin g G to the ﬁrst triangl e, we obtain the follo wing commutati ve diagram in D b ( A ) G ¯ Q + X Gi X − − − − → GF ( X ) G π X − − − − → G ¯ F ( X ) G α X − − − − → G ¯ Q + X [ 1 ]   y q X [ − 1 ]   y η X      y q X Q − ¯ F X [ − 1 ] β ¯ F X − − − − → ¯ G ¯ F ( X ) λ ¯ F X − − − − → G ¯ F ( X ) γ ¯ F X − − − − → Q − ¯ F X . The existe nce of η X follo ws from the fact that G ( π X ) γ ¯ F X belong s to Hom D b ( A ) ( GF ( X ) , Q − ¯ F X ) ≃ Hom D b ( A ) ( X , Q − ¯ F X ) ≃ Hom K b ( A ) ( X , Q − ¯ F X ) = 0. Since GF is naturally isomorphic to the identity functor 1 D b ( A ) , there is a natu- ral morphism ε X : X − → GF ( X ) in D b ( A ) for each A -module X . Let θ X be the composi tion ε X η X . Then θ X : X − → ¯ G ¯ F ( X ) is an A -module homomorphism since A -mod is fully embedded in D b ( A ) . W e clai m that θ X is a natura l map in A -mod. Indeed, for any A -module homomorp hism f : X → Y , by the proof of Propos ition 3.4, we h a ve a homomorp hism b f : ¯ F ( X ) − → ¯ F ( Y ) of B -module s such that π X b f = F ( f ) π Y in D b ( B ) . By the p roof of Proposi tion 3.5, there is a homomo rphism u b f : ¯ G ( ¯ F ( X )) − → ¯ G ( ¯ F ( Y )) of A -modules such that u b f λ ¯ F Y = λ ¯ F X G ( b f ) in D b ( A ) . Thus, we ha ve in D b ( A ) : ( θ X u b f − f θ Y ) λ ¯ F Y = ( ε X η X u b f − f ε Y η Y ) λ ¯ F Y = ( ε X η X u b f − ε X GF ( f ) η Y ) λ ¯ F Y = ε X ( η X u b f λ ¯ F Y − GF ( f ) η Y λ ¯ F Y ) = ε X ( η X λ ¯ F X G ( b f ) − GF ( f ) η Y λ ¯ F Y ) = ε X ( G ( π X ) G ( b f ) − GF ( f ) G ( π Y )) = ε X ( G ( π X ) G ( b f ) − G ( F ( f ) π Y )) = ε X ( G ( π X ) G ( b f ) − G ( π X b f )) = 0 . This implies that the map θ X u b f − f θ Y fact orizes through Q − ¯ FY [ − 1 ] . It follo ws by Lemma 3.3 tha t θ X u b f − f θ Y fact orizes through a projecti ve module . Note that u b f = ¯ F ¯ G ( f ) . T hus θ X ¯ F ¯ G ( f ) − f θ Y = 0 in A -mod and θ X is natura l in X . T o ﬁnish the proof, we hav e to sho w that θ X is an isomorphism in A -mod for each A -module X . Clearly , we can a ssume that X is an in decompo sable non-project iv e A -module. Using th e method si milar to that in the proo f of Lemma 3.1, we can prov e that G ( ¯ Q + X ) is iso morphic in D b ( A ) to a radical comple x Q • 1 in K b ( A -proj ) with Q i 1 ∈ add ( A Q ) for all i ≤ 0. Since bo th X and G ¯ F ( X ) ha ve no homology in positi v e degr ees, the complex G ( ¯ Q + X ) has no homology in degrees greater than 1, and therefore Q i 1 = 0 fo r all i > 1. No w we may form the followin g commutati ve diagram in D b ( A ) : Q • 1 φ X − − − − → X λ − − − − → con ( φ X ) − − − − → Q • 1 [ 1 ]   y s   y ε X   y t   y s [ 1 ] G ( ¯ Q + X ) Gi X − − − − → G F ( X ) G π X − − − − → G ¯ F ( X ) − − − − → G ( ¯ Q + X )[ 1 ]   y η X    ¯ G ¯ F X λ ¯ F X − − − − → G ¯ F ( X ) , where s is an isomorph ism from G ( Q + X ) to Q • 1 , and where φ X = sG ( i X ) ε − 1 is a chain map, and where λ is induce d by the canonical m ap λ 0 from X to Q 1 1 ⊕ X deﬁned by the mapping cone. Since Q i 1 is in add ( A Q ) 13 for all i ≤ 0 and ze ro for all i > 1, the mapp ing cone con ( φ X ) h as terms in add ( A Q ) for all ne gati v e de grees and zero for all positi ve de grees. Note that a dd ( A Q ) = add ( ν A Q ) and t he A -mod ule Q is projec ti ve-i njecti ve. Consequ ently , the terms of con ( φ X ) in all neg ati ve de grees are projec ti ve- injecti ve. N ote that G ( ¯ F ( X )) ≃ Q • ¯ F ( X ) by our assumption. Thus, by de ﬁnition (see Lemm a 3.2), all terms of Q • ¯ F ( X ) in nega ti ve degrees are project iv e- injecti ve. Moreo ver , both con ( φ X ) and Q • ¯ F ( X ) ha ve zero terms in al l positi ve degree s. By Lemma 2.2 (2) , we ha ve Hom D b ( A ) ( con ( φ X ) , Q • ¯ F ( X ) ) = Hom K b ( A ) ( con ( φ X ) , Q • ¯ F ( X ) ) . Since the two morphisms ε X and s are both isomorph ism in D b ( A ) , the morphism t is also an isomorphi sm in D b ( A ) . Hence t is an iso morphism from con ( φ X ) to Q • ¯ F ( X ) in K b ( A ) . Moreov er , since X is ind ecomposa ble and non -projec ti ve, the complex con ( φ X ) is a radical comple x. Thus, t he chain map t is actually an i somorphi sm bet ween con ( φ X ) and Q • ¯ F X in C b ( A ) , and the mor phism t 0 : ( con ( φ X )) 0 = Q 1 1 ⊕ X − → Q 0 ¯ F X in de gree zero is an i somorphi sm of A -modules . From the abo ve commutati v e diagram, we hav e θ X λ ¯ F X − λ t = 0 in D b ( A ) . By Lemma 2.2 (2), we see that θ X λ ¯ F X − λ t is null-h omotopic in C b ( A ) . This means that θ X − λ 0 t 0 fact orizes through the projecti ve A -mod ule Q − 1 ¯ F X . Hence θ X = λ 0 t 0 is an isomorph ism in A -mod since λ 0 and t 0 both are isomorp hisms in A -mod. Remark. W ith out the condition add ( A Q ) = add ( ν A Q ) in Propositi on 3.5, we can similarly deﬁne a functo r ¯ G ′ : B -mod − → A -mod , as w as do ne in Propos ition 3.4. But t he disad va ntage of using ¯ G ′ is th at we do not kno w any b eha vior of the compositio n of ¯ F wit h ¯ G ′ . W e say that a deriv ed equi v alen ce F between Artin algebras A and B is almost ν -stabl e if add ( A Q ) = add ( ν A Q ) and add ( B ¯ Q ) = add ( ν B ¯ Q ) . The follo wing theorem sho ws that the almost ν -stab le conditio n is suf ﬁcient for ¯ F t o be an equi v alen ce. Theor em 3.7. Let A and B be two Artin R-algeb ras, and let F : D b ( A ) − → D b ( B ) be a derived equivalen ce. If F is almost ν -stable , then the stable functor ¯ F i s an equival ence . Pr oof. Since F is almost ν -stable, we ha ve add ( A Q ) = add ( ν A Q ) . By Proposition 3.6 , we ha v e ¯ G ¯ F ≃ 1 A -mod . Since F is almost ν -stable, we also hav e a dd ( ¯ Q ) = add ( ν ¯ Q ) . W ith a proof similar to that of Prop ositio n 3.6, we can show that ¯ F ¯ G is natural ly isomorphi c to the iden tity functor 1 B -mod . T hus ¯ F a nd ¯ G are equi v alence s of cate gories. Theorem 3.7 gi ves rise to a method of get ting stable equi v alence s from deri ved equi v alen ces. In Section 5, we shal l prov e that, for ﬁnite -dimensio nal algebr as, one e v en can get a stable equi v alence of Morita type, which has man y pleasant propert ies (see [3], [18], [19] and the reference s therein). In the foll o wing, we shall de v elop some pro perties of almost ν -st able f unctor s, which will be u sed in Section 5. Let A E be a direct sum of all those non-i somorphi c in decompos able p rojecti ve-inject iv e A -module s X that ha ve the proper ty: ν i A X is agai n projecti ve -injecti ve for ev ery i > 0. T he A -module A E is unique up to isomor- phism, and i t is called the m aximal ν -stable A -module. Similarly , we h a ve a maximal ν -stable B -module B ¯ E . The follo wing result sho w s that an almost ν -stable functor is closely related to the maximal ν -stable modules. Pro position 3.8. The followin g are equ ivalent : ( 1 ) F is almost ν -stab le, that is, add ( ν A Q ) = add ( A Q ) and add ( ν B ¯ Q ) = add ( B ¯ Q ) . ( 2 ) A Q ∈ add ( A E ) and B ¯ Q ∈ add ( B ¯ E ) . ( 3 ) A Q and ν B ¯ Q ar e pr ojective-in jective . Pr oof. Clearly , we hav e ( 1 ) ⇒ ( 2 ) ⇒ ( 3 ) . No w we sho w that ( 3 ) implies ( 1 ) . Assume that A Q is injecti ve . By Lemma 3.2, G ( B ) is isomorph ic in D b ( A ) to a radical complex Q • B = ( Q i B , d i ) w ith Q i B in add ( ν A Q ) for all i < 0. In part icular , Q i B is projecti v e-inje cti ve for all i < 0. Since G ( B ) ≃ Q • , the comple xes Q • and Q • B are isomorph ic in D b ( A ) . Since A Q is injecti ve by assumptio n, all the terms of Q • in nega ti ve degre es are injecti ve. 14 By L emma 2 .2, the co mplex es Q • and Q • B is i somorphi c in K b ( A ) . Since bo th Q • and Q • B are radical, the y are also isomorphic in C b ( A ) . In particula r , we hav e Q i ≃ Q i B for all i < 0, an d therefore A Q : = L − n i = − 1 Q i ≃ L − n i = − 1 Q i B ∈ add ( ν A Q ) . Since A Q a nd ν A Q h as the same numbe r of is omorphis m classes of i ndecompo sable direct summands, we hav e add ( A Q ) = add ( ν A Q ) . Similarly , we hav e add ( B ¯ Q ) = add ( ν B ¯ Q ) . This prove s ( 3 ) ⇒ ( 1 ) . Remark: From Lemma 3.8, we can see that ev ery de ri ved eq ui v alence between two s elf-inje cti ve Artin algebr as is almost ν -stabl e. T hus, we can re-obt ain the result [14, Corollary 2.2] of Rickard by Theorem 3.7. Lemma 3.9. Suppose Q ∈ add ( A E ) , and ¯ Q ∈ add ( B ¯ E ) . Then ( 1 ) for ea ch P • in K b ( add ( A E )) , the comple x F ( P • ) is iso morphic in D b ( B ) to a comple x in K b ( add ( B ¯ E )) . ( 2 ) for ea ch ¯ P • in K b ( add ( B ¯ E )) , the comple x G ( ¯ P • ) i s isomorphic in D b ( A ) to a comple x in K b ( add ( A E )) . Pr oof. (1) It suf ﬁces to sho w t hat, for ea ch ind ecomposa ble A -module U in a dd ( A E ) , the comp lex F ( U ) is isomo rphic t o a comple x in K b ( add ( B ¯ E )) . Suppose U ∈ add ( A E ) . By Lemma 3.1, F ( U ) is isomorph ic in D b ( B ) to a radica l comple x ¯ Q • U : 0 / / ¯ Q 0 U / / ¯ Q 1 U / / · · · / / ¯ Q n U / / 0 with ¯ Q i U ∈ add ( B ¯ Q ) for all i > 0. For simplicity , we assume that F ( U ) is just ¯ Q • U . Since ¯ Q i U ∈ add ( B ¯ Q ) ⊆ add ( B ¯ E ) for i > 0, it remains to sho w tha t ¯ Q 0 U is in add ( B ¯ E ) . C learly , ¯ Q 0 U is pro jecti v e since U is proje cti ve . Note that we ha ve an isomorphism ν B F ( U ) ≃ F ( ν A U ) in D b ( B ) , that is, ν B ¯ Q • U is isomorphic to ¯ Q • ν A U . Note that ν B ¯ Q i U ∈ add ( ν B ¯ Q ) for all i > 0 a nd add ( ν B ¯ Q ) ⊆ add ( B ¯ E ) by the deﬁnition of B ¯ E . Thus ν B ¯ Q i U is projec ti ve-i njecti v e for all i > 0, and ν B ¯ Q • U and ¯ Q • ν A U are iso morphic in K b ( B ) by Lemma 2.2. S ince both ν B ¯ Q • U and ¯ Q • ν A U are radica l comp lex es, ν B ¯ Q • U and ¯ Q • ν A U are actually isomorphic in C b ( B ) , and particular ly we hav e ν B ¯ Q 0 U ≃ ¯ Q 0 ν A U . Note that if U ∈ add ( A E ) , then ν i A U ∈ add ( E ) fo r all i ≥ 0 by deﬁnitio n. Hence, fo r ea ch i nteg er m > 0, we ha ve ν m B ( ¯ Q 0 U ) ≃ ν m − 1 B ( ¯ Q 0 ν A U ) ≃ · · · ≃ ¯ Q 0 ν m A U , and there fore ν m B ( ¯ Q 0 U ) is projecti ve -injecti ve. T hus ¯ Q 0 U ∈ add ( B ¯ E ) by deﬁniti on. (2) is a dual state ment of (1). The follo wing is a conse quence of Lemma 3.9. Cor ollary 3.10. If F is an almost ν -stable derive d equivalen ce between Artin algebr as A and B, then ther e is a derive d equiva lence between the self-in jective algebr as End A ( E ) and E nd B ( ¯ E ) . Pr oof. B y Lemma 3.9, F induces an equi v alence between K b ( add ( A E )) and K b ( add ( B ¯ E )) as triangu lated cate gories. Since K b ( add ( A E )) and K b ( End ( A E ) -proj ) are equi v alent as triangulate d categorie s, we obtain an equi v alen ce between K b ( End ( A E ) -proj ) and K b ( End ( B ¯ E ) -proj ) as triang ulated cate gories . By [13, Theorem 6.4], the algeb ras E nd A ( E ) and End B ( ¯ E ) are deriv ed- equi v alen t. N ote that End A ( E ) is self-inject iv e. Let us end thi s section by the follo wing result which tells us ho w to get an almost ν -st able deri ved equiv a- lence from a tilting module. Let A be an Artin algebr a. R ecall that an A -module T is called a tilting m odule if ( 1 ) the proje cti ve dimension of T is ﬁnite, ( 2 ) Ext i A ( T , T ) = 0 for all i > 0, and ( 3 ) there is an exac t sequence 0 − → A − → T 0 − → · · · − → T m − → 0 of A -mod with each T i in add ( A T ) . It is well-k no wn that a tilting A -module A T induce s a deriv ed equi v alenc e between A and End A ( T ) (see [5 , Theorem 2.10, p. 109 ] and [4, Theorem 2.1]). 15 Pro position 3.11. Let A be an Artin alge bra. Suppo se A T is a tilting A-module with B = End A ( T ) . Let 0 − → P n d n − → P n − 1 − → · · · − → P 0 d 0 − → T − → 0 be a min imal p r ojective r esolution of A T . Set P : = L n − 1 i = 0 P i . If add ( A P ) = add ( ν A P ) , then ther e is an almost ν -stab le derived equivalenc e between A and B. Pr oof: By [4, Theorem 2.1], the functor F ′ = A T ⊗ L B − : D b ( B ) − → D b ( A ) is a deriv ed equi v alenc e. N o w we deno te F ′ [ − n ] by F . Let P • be the complex 0 − → P n d n − → P n − 1 − → · · · − → P 0 − → 0 with P n in degree zero. Then we ha v e F ( B ) = ( A T ⊗ L B B )[ − n ] = A T [ − n ] ≃ P • in D b ( A ) . Let G b e a quas i-in verse of F . Then G ( P • ) ≃ G ( F ( B )) ≃ B in D b ( B ) , and therefor e P • is a radical tilting comple x associated to G . Since add ( ν A P ) = ad d ( A P ) , the modu le A P is proj ecti ve -injecti ve. Thus A P ∈ add ( A T ) and P i ∈ add ( A T ) for all 0 ≤ i ≤ n − 1. W e den ote by T • the comple x 0 − → P n − 1 ⊕ P h d n − 1 0 i − → P n − 2 − → · · · − → P 0 d 0 − → T − → 0 with T in degree zero. Then Hom • A ( T , T • ) is a comple x in K b ( B -proj ) , and F ( Hom • A ( T , T • )) = A T ⊗ L B Hom • A ( T , T • )[ − n ] ≃ A T ⊗ • B Hom • A ( T , T • )[ − n ] ≃ T • [ − n ] ≃ P n ⊕ P in D b ( A ) . Since P • is a tilting comp lex o ve r A , we ha ve A A ∈ add ( P n ⊕ P ) . Thus ther e is a radical comple x ¯ P • in K b ( B -proj ) such that ¯ P • ∈ add ( Hom • A ( T , T • )) and F ( ¯ P • ) ≃ A in D b ( A ) . By deﬁnition, ¯ P • is a tilting complex associ ated to F . (For t he unexp lained notations appearing in this proof, we refer the reader to Section 5 belo w). W e clai m that F is almos t ν -stab le. In fact, L − n i = − 1 ¯ P i is in add ( Hom A ( T , P )) , and L n i = 1 P i = L n − 1 i = 0 P i = P . Let A E (respecti v ely , B ¯ E ) be the maximal ν -stable A -module (resp ecti ve ly , B -module) . Then A P ∈ add ( A E ) . Note that we ha ve the foll o wing isomorphisms of B -modules: ν B Hom A ( T , P ) = D Hom B ( Hom A ( T , P ) , H om A ( T , T )) ≃ D Hom A ( P , T ) ≃ D ( P ∗ ⊗ A T ) ≃ Hom A ( T , ν A P ) . Since add ( A P ) = add ( ν A P ) , w e hav e add ( ν B Hom A ( T , P )) = add ( Hom A ( T , P )) , that is, Hom A ( T , P ) ∈ add ( B ¯ E ) . It follo w s that L − n i = − 1 ¯ P i is in add ( B ¯ E ) . By P roposi tion 3.8, the functo r F is a lmost ν -stable . Remark: Let A be a self-injecti ve Artin alge bra, and let X be an A -module. By [6, Corollary 3.7], there is a deri v ed equi v alenc e between End A ( A ⊕ X ) and E nd A ( A ⊕ Ω A ( X )) induce d by the almost add ( A ) -split sequence 0 → Ω A ( X ) → P X → X → 0, where P X is a projecti v e cov er of X . By Propositio n 3.11, it is easy to check that this is an almost ν -stab le deri ved equi v ale nce. Thu s the algebras End A ( A ⊕ X ) and End A ( A ⊕ Ω A ( X )) are stably equi v alen t by Theorem 3.7. 4 Comparison of homological dimensions In this sec tion, we shall deduc e some basic homolog ical propert ies of the functo r ¯ F and compar e homolo gical dimensio ns of A with that of B . W e make the follo wing con v ention: From no w on, throug hout this paper , we keep our notations introduc ed in the pre viou s sections. Recall that the ﬁnitistic dimension of an Artin algebra A , denoted by ﬁn.dim ( A ) , is deﬁned to be the supre- mum of the projecti ve dimensions of ﬁnitely generated A -modules of ﬁnite projecti v e dimension. The ﬁ nitistic 16 dimensio n c onjectu re states that ﬁn.dim ( A ) should be ﬁnite for any Artin algebra A . Concerning the n e w ad- v ances on this conje cture, w e refer the reader to the recent paper [20] and the referenc es therein. For an A -module X , we denote by pd ( A X ) the projec ti ve dimension of X , and by gl.dim ( A ) the global dimensio n of A , which is by deﬁnitio n the supremum of the project iv e dimension s of all ﬁnitely generated A -modules. Clearl y , if gl.dim ( A ) < ∞ , then ﬁn.dim ( A ) = gl.di m ( A ) . Pro position 4.1. Let ¯ F b e the stable functor of F de ﬁned in Pr oposi tion 3.4. Then: ( 1 ) F or each e xact sequence 0 − → X f − → Y g − → Z − → 0 i n A -mod , ther e is an e xact sequen ce 0 − − − − → ¯ F ( X ) [ a , f ′ ] − − − − → P ⊕ ¯ F ( Y ) [ b g ′ ] − − − − → ¯ F ( Z ) − − − − → 0 in B -mod , wher e P ∈ add ( B ¯ Q ) , ¯ F ( f ) = f ′ and ¯ F ( g ) = g ′ . ( 2 ) F or ea ch A -module X , we have a B-module isomorp hism: ¯ F ( Ω A ( X )) ≃ Ω B ( ¯ F ( X )) ⊕ P, wher e P i s a pr ojective B -module , and Ω is the syzygy operat or . ( 3 ) F or each A-module X , we have pd ( B ¯ F ( X )) ≤ pd ( A X ) ≤ pd ( B ¯ F ( X )) + n. ( 4 ) If ¯ F i s an equival ence , then A and B have the same ﬁnitis tic and global dimension s. Pr oof. For eac h A -module X , we may assume that F ( X ) is the complex ¯ Q • X deﬁned in Lemma 3.1. (1) From the e xac t sequence 0 − → X f − → Y g − → Z − → 0 in A -mod we ha ve a distinguis hed triangle in D b ( A ) : X f − − − − → Y g − − − − → Z ε − − − − → X [ 1 ] . Applying the functo r F , we get a distinguish ed trian gle ¯ Q • X F ( f ) − − − − → ¯ Q • Y F ( g ) − − − − → ¯ Q • Z F ( ε ) − − − − → ¯ Q • X [ 1 ] in D b ( B ) . M oreo ver , by Lemma 2.2, the morphisms F ( f ) and F ( g ) are induced by chain maps p • and q • , respec ti vely . So, w e may assume that F ( f ) = p • and F ( g ) = q • . Let con ( q • ) be the mapping cone of the chain map q • . Then we ha ve a commut ati ve diagram in D b ( B ) ¯ Q • Z [ − 1 ] − − − − → ¯ Q • X p • − − − − → ¯ Q • Y q • − − − − → ¯ Q • Z    s   y       ¯ Q • Z [ − 1 ] − − − − → con ( q • )[ − 1 ] π • − − − − → ¯ Q • Y q • − − − − → ¯ Q • Z for some isomorphism s , where π • = ( π i ) with π i : ¯ Q i Y ⊕ ¯ Q i − 1 Y − → ¯ Q i Y the canoni cal pro jection for each integer i . By Lemma 2.2, the morphism s is ind uced by a chai n map s • . By d eﬁnition, con ( s • ) is the f ollo wing complex 0 − − − − → ¯ Q 0 X [ − d , s 0 ] − − − − → ¯ Q 1 X ⊕ ¯ Q 0 Y   − d u v 0 − d q 0   − − − − − − − − − − − − − → ¯ Q 2 X ⊕ ¯ Q 1 Y ⊕ ¯ Q 0 Z − − − − → · · · − − − − → ¯ Q n Z − − − − → 0 , where s 1 = [ u , v ] : ¯ Q 1 X − → ¯ Q 1 Y ⊕ ¯ Q 0 Z , and where the mo dules ¯ Q i X , ¯ Q i Y and ¯ Q i Z are proje cti ve for i > 0. Since s is an isomorphism in D b ( B ) , the mapping cone con ( s • ) of s • is an acyc lic comple x. Note that the map − d : ¯ Q 0 Y − → ¯ Q 1 Y is a radical map. Thus, droppin g the split direct summands of the acy clic complex con ( s • ) , we get an ex act sequenc e ( ∗ ) 0 − − − − → ¯ Q 0 X [ a , s 0 ] − − − − → P ⊕ ¯ Q 0 Y [ b , q 0 ] T − − − − → ¯ Q 0 Z − − − − → 0 17 in B -mod, where P is a direct summan d of ¯ Q 1 X , and where a and b are some homomorphisms of B -modules. It follo ws from p • − s • π • = 0 that the morphism p • − s • π • is null-h omotopic by L emma 2.2 . Therefore p 0 − s 0 fact orizes through ¯ Q 1 X , and p 0 = s 0 . By sett ing f ′ = s 0 and g ′ = q 0 , we re-write ( ∗ ) as 0 − − − − → ¯ F ( X ) [ a , f ′ ] − − − − → P ⊕ ¯ F ( Y ) [ b g ′ ] − − − − → ¯ F ( Z ) − − − − → 0 with P ∈ ad d ( B ¯ Q ) , ¯ F ( f ) = f ′ and ¯ F ( g ) = g ′ . This prov es ( 1 ) . ( 2 ) Let X be an A -module. W e hav e an exac t sequence 0 − → Ω A ( X ) − → P X − → X → 0 in A -mod w ith P X projec ti ve. By ( 1 ) , we get an exact sequen ce 0 − → ¯ F ( Ω A ( X )) − → P ⊕ ¯ F ( P X ) − → ¯ F ( X ) − → 0 in B -mod for some proj ecti ve B -module P . By the deﬁnition of ¯ F , the B -module ¯ F ( P X ) is projec ti ve. Thus (2) follo ws. ( 3 ) The inequa lity pd ( B ¯ F ( X )) ≤ pd ( A X ) follows from (2). In fact, we may assume pd ( A X ) = m < ∞ . Then Ω m A ( X ) is projecti v e. There fore Ω m B ¯ F ( X ) is projecti v e by (2), and pd ( B ¯ F ( X )) ≤ m . For the se cond inequ ality in (3 ), we may assume pd ( B ¯ F ( X )) = m < ∞ . Let Y be an A -module. W e claim that Ext i A ( X , Y ) ≃ Hom D b ( A ) ( X , Y [ i ]) ≃ Hom D b ( B ) ( F ( X ) , F ( Y )[ i ]) = 0 for all i > m + n . Indeed, by Lemma 3.1, the complex F ( X ) is isomorphic in D b ( B ) to a comple x ¯ Q • X with Q i X being proje cti ve for all i > 0. Since pd ( B ¯ Q 0 X ) = pd ( B ¯ F ( X )) = m , we see that ¯ Q • X is isomorp hic in D b ( B ) to a c om- ple x P • in K b ( B -proj ) with P k = 0 fo r all k < − m . Note th at F ( Y ) is is omorphic to t he complex ¯ Q • Y with ¯ Q k Y = 0 for all k > n . Clearly , we hav e Hom D b ( B ) ( F ( X ) , F ( Y )[ i ]) ≃ Hom D b ( B ) ( P • , ¯ Q • Y [ i ]) = Hom K b ( B ) ( P • , ¯ Q • Y [ i ]) = 0 f or all i > m + n . Then the second inequality follo ws. ( 4 ) is a conseq uence of ( 2 ) . In fact, suppose ¯ F is an equi v alenc e. Then, for an A -module X and a pos iti ve inte ger m , the A -module Ω m A ( X ) is projecti v e if and only if ¯ F ( Ω m A ( X )) is projecti v e. By (2), ¯ F ( Ω m A ( X )) is projec ti ve if and only if Ω m B ( ¯ F ( X )) is projecti ve. It follo ws that pd ( A X ) ≤ m if and only if pd ( B ¯ F ( X )) ≤ m , and conseq uently pd ( A X ) = pd ( B ¯ F ( X )) for arbitrary A -module X . Thus ( 4 ) follo ws. Remark. Proposition 4.1 (3) can be re garded as an alternati v e proof of the fact that if two Artin algebras A and B are deri v ed-equ iv alent then ﬁn.dim ( A ) < ∞ if an d only if ﬁn.dim ( B ) < ∞ . In Propo sition 3.5, w e h av e con structed a functor ¯ G : B -mod − → A -mod under the c onditio n add ( A Q ) = add ( ν A Q ) . This funct or ¯ G has many p roperti es similar to that of ¯ F . Pro position 4.2. Suppose add ( A Q ) = add ( ν A Q ) . Let ¯ G : B -mod − → A -mod be the functor de ﬁned in P r oposition 3.5. Then: ( 1 ) F or each e xact sequence 0 − → U f − → V g − → W − → 0 in B -mod , th er e is an e xact sequence 0 − − − − → ¯ G ( U ) [ a , f ′ ] − − − − → P ⊕ ¯ G ( V ) [ b g ′ ] − − − − → ¯ G ( W ) − − − − → 0 in A -mod , wher e P ∈ add ( A Q ) , ¯ G ( f ) = f ′ and ¯ G ( g ) = g ′ . ( 2 ) F or each B-module Y , we have ¯ G ( Ω B ( Y )) ≃ Ω A ( ¯ G ( Y )) ⊕ P in A -mod for a pr ojective A-module P. ( 3 ) F or each B-module Y , we have pd ( A ¯ G ( Y )) ≤ pd ( B Y ) . ( 4 ) If I is an injecti ve B-module, then ¯ G ( I ) is an injec tive A-module . Mor eov er , ¯ G ( D ( B )) ≃ ν A Q 0 . Pr oof: ( 1 ) , ( 2 ) and ( 3 ) are dual statement s of Propositio n 4.1, and their proofs will be omitted here. W e only prov e ( 4 ) . Let I be an inj ecti ve B -module. Then I = ν B P for a projecti v e B -module P . Since G ( B ) ≃ Q • , we know that G ( P ) is isomorp hic in K b ( A -proj ) to a r adical co mplex Q • 1 ∈ add ( Q • ) . T hus the comple x Q • I deﬁned in Lemma 3.2 is isomorphic to G ( I ) ≃ ν A G ( P ) ≃ ν A Q • 1 by Lemma 2.3. Moreov er , all the terms of ν A Q • 1 in neg ati ve de grees are in add ( ν A Q ) . Thus, by Lemma 3.2, the comple xes Q • I and ν A Q • 1 are iso morphic in C b ( A ) , and conseq uently ¯ G ( I ) = Q 0 I ≃ ν A Q 0 1 is injecti v e. In particula r , if we take P = B B and Q • 1 = Q • , then ¯ G ( D ( B )) = ¯ G ( ν B B ) ≃ ν A Q 0 . 18 Let X b e an A -module, and let 0 − → X − → I 0 − → I 1 − → · · · be a minimal injecti ve resolu tion of X with all I j injecti ve. T he domina nt dimension of X , denoted by dom . dim ( X ) , is deﬁne d to be dom . dim ( X ) : = sup { m | I i is proje cti ve for all 0 ≤ i ≤ m − 1 } . The dominant dimensi on of the alge bra A , den oted by dom . dim ( A ) , is de ﬁned to be the dominant di mension of the m odule A A . Concer ning the dominant dimension of an Artin algebra, there is a conjectu re, namely the N akayama con jecture , which states that an Artin algebr a with inﬁnite dominan t dimension should be self- injecti ve. It is well-kno w n that the ﬁnitistic dimension conjecture implies the Nakayama conjecture . Usually , a d eri ve d equi v alen ce does not pr eserv e the us ual homolog ical dimensio ns of an algebra. How- e ver , under the cond ition add ( A Q ) = add ( ν A Q ) , we hav e the followin g inequal ities about these homological dimensio ns. Cor ollary 4.3. Let F : D b ( A ) − → D b ( B ) be a derived equival ence b etween A rtin algebr as A and B. If add ( A Q ) = add ( ν A Q ) , then ( 1 ) ﬁn . dim ( A ) ≤ ﬁn . dim ( B ) , ( 2 ) gl . di m ( A ) ≤ gl . dim ( B ) , ( 3 ) dom . di m ( A ) ≥ dom . dim ( B ) . Pr oof. ( 1 ) For each A -module X , we ha ve pd ( A X ) = pd ( A ¯ G ¯ F ( X )) by Proposit ion 3.6. Accordin g to Proposit ion 4.2 (3), we hav e pd ( A ¯ G ¯ F ( X )) ≤ p d ( B ¯ F ( X )) . By Proposition 4.1(3), we hav e another inequa l- ity pd ( B ¯ F ( X )) ≤ pd ( A X ) . Thus pd ( A X ) = pd ( B ¯ F ( X )) . This impli es that gl.dim ( A ) ≤ gl.dim ( B ) . Moreov er , if pd ( A X ) < ∞ , we ha ve pd ( A X ) = pd ( B ¯ F ( X )) ≤ ﬁ n . dim ( B ) . Hence ﬁ n . dim ( A ) ≤ ﬁn . dim ( B ) . This prov es ( 1 ) and ( 2 ) . ( 3 ) Suppose dom . dim ( B ) = m . L et 0 − → B B − → I 0 − → I 1 − → · · · be a minimal injecti ve res olution of B B . Then, by de ﬁnition, the in jecti v e B -modules I 0 , · · · , I m − 1 all are projecti v e. By Prop osition 4.2(1), we get an exa ct sequence 0 − → A Q 0 − → P 0 ⊕ ¯ G ( I 0 ) − → P 1 ⊕ ¯ G ( I 1 ) − → · · · with P i ∈ add ( A Q ) . From this sequen ce we get anothe r exact seque nce 0 − → A Q 0 ⊕ A Q − → P 0 ⊕ ¯ G ( I 0 ) ⊕ A Q − → P 1 ⊕ ¯ G ( I 1 ) − → · · · . Since add ( A Q ) = add ( ν A Q ) and ¯ G ( I i ) is injecti ve for all i by Propositio n 4.2, the A -modules P i ⊕ ¯ G ( I i ) is injecti ve for all i . Thus, the abov e exact seque nce actually g i ves an injecti v e r esoluti on of A Q 0 ⊕ A Q . S et I ′ 0 : = P 0 ⊕ ¯ G ( I 0 ) ⊕ A Q and I ′ i : = P i ⊕ ¯ G ( I i ) for i > 0. Since I i is pro jecti v e fo r all i ≤ m − 1, we see tha t ¯ G ( I i ) is projecti v e for all 0 ≤ i ≤ m − 1, and consequen tly I ′ i is projec ti ve for a ll 0 ≤ i ≤ m − 1. Hence dom . dim ( A Q ⊕ A Q 0 ) ≥ m . Moreo ver , since Q • is a tilting complex, w e ha ve A A ∈ add ( L i Q i ) , th at is, A A ∈ add ( A Q ⊕ A Q 0 ) . Hence dom . dim ( A ) ≥ m = do m . dim ( B ) . 5 Stable equiv alences of Morita type induce d by deriv ed equi valences In this secti on, we sha ll prov e that an almost ν -stabl e de ri ved funct or F between two ﬁnite-dimen sional alg e- bras actu ally induces a stable equiv alence o f Morita type. Our result in this sectio n g enerali zes a well-kno wn result of Rickar d [15, C orollar y 5.5], w hich states that, for ﬁnite-d imension al s elf-inje cti ve algeb ras, a de ri ved equi v alen ce induces a stable equi v alen ce of Morita type. Through out this section, we keep the notation s introduced in Section 3 and conside r exclusi v ely ﬁnite- dimensio nal algebra s over a ﬁeld k . Let Λ be an algebra. By C + ( Λ ) (re specti vely , C − ( Λ ) ) we denote the full subcat ego ry of C ( Λ ) co nsisting of all complex es bounded belo w (respec ti vel y , bounded ab ov e). Analog ously , one has the correspond ing homotopy 19 cate gories K + ( Λ ) an d K − ( Λ ) as well as th e correspo nding deriv ed cat egor ies D + ( Λ ) an d D − ( Λ ) . Recall that the categor y D − ( Λ ) is equi v alent to the cate gory K − ( Λ -pro j ) , and the category D + ( Λ ) is equi v alent to the categor y K + ( Λ -inj ) (see [17, Theorem 10.4.8, p.388], for example ). Thus, for each complex U • in D − ( Λ ) (resp ecti ve ly , D + ( Λ ) ), we can ﬁnd a co mplex P • U ∈ K − ( Λ -pro j ) (res pecti v ely , I • U ∈ K + ( Λ -inj ) ) tha t is isomorph ic to U • in D ( Λ ) . No w , le t X • be a comple x in D − ( Λ op ) a nd Y • a c omplex in D − ( Λ ) . By X • ⊗ • Λ Y • we mea n t he total comple x of the double complex with ( i , j ) -term X i ⊗ Λ Y j , and by X • ⊗ L Λ Y • the comple x X • ⊗ • Λ P • Y . U p to isomorphisms of complex es, X • ⊗ L Λ Y • does not depend on the choice of P • Y . It is kno wn that X • ⊗ L Λ − is a funct or from D − ( Λ ) to D − ( k ) , and is called the left derive d functo r of X • ⊗ • Λ − : K − ( Λ ) − → K − ( k ) . Note that if we choos e P • X ∈ D − ( Λ op ) such that P • X ≃ X • in D ( Λ op ) then there is a natural isomorphi sm between X • ⊗ • Λ P • Y and P • X ⊗ • Λ Y • in D ( k ) (see [17, Exerci se 10.6.1, p.395]). Thus X • ⊗ L Λ Y • can be calcu lated by P • X ⊗ • Λ Y • . Let X • 1 and X • 2 be two c omplex es in D + ( Λ ) . B y Hom • Λ ( X • 1 , X • 2 ) we den ote the total complex of the dou- ble comple x with ( i , j ) -term Hom Λ ( X − i 1 , X j 2 ) . Choose I • X 2 ∈ K + ( Λ -inj ) with I • X 2 ≃ X • 2 in D ( Λ ) . W e deﬁne R Hom Λ ( X • 1 , X • 2 ) = Hom • Λ ( X • 1 , I • X 2 ) . It is kno wn that R Hom Λ ( X • 1 , − ) : D + ( Λ ) − → D + ( k ) is a funct or . This functo r is called the right derived functor of H om • Λ ( X • 1 , − ) : K + ( Λ ) − → K + ( k ) . Note that if we choose P • X 1 ∈ K − ( Λ -pro j ) with P • X 1 ≃ X • 1 in D ( Λ ) then the complex es Hom • Λ ( P • X 1 , X • 2 ) and H om • Λ ( X • 1 , I • X 2 ) are nat- urally isomorphic in D ( k ) (see [17 , E xerc ise 10.7.1, p.400]) . Thus R Hom Λ ( X • 1 , X • 2 ) can be calculated by Hom • Λ ( P • X 1 , X • 2 ) . Suppose Λ 1 and Λ 2 are two algebras. Let T • i be a tilting complex over Λ i with Γ i = End D b ( Λ i ) ( T • i ) for i = 1 , 2 . By a result [15, Theorem 3.1] of Rickard, T • 1 ⊗ • k T • 2 is a tilting comp lex ov er Λ 1 ⊗ k Λ 2 , and the end omorphis m algebr a of T • 1 ⊗ • k T • 2 is canonically isomorphic to Γ 1 ⊗ k Γ 2 . Thus the tensor algebra s Λ 1 ⊗ k Λ 2 and Γ 1 ⊗ k Γ 2 are deri v ed-equ iv alent. Recall that Q • is a tiltin g comple x over A with the endomorphism algebra B . By [13, Proposition 9.1], Hom • A ( Q • , A ) is a tilting comple x ov er A op with the endomorphi sm al gebra B op . Also, ¯ Q • is a tilting complex ov er B with the endomorphism A , and therefore Hom • B ( ¯ Q • , B ) is a tilting complex ove r B op with the end omorphis m algebr a A . Thus , by taking tensor pro ducts, we get fo ur deriv ed-eq ui v alent algebras A ⊗ k A op , A ⊗ k B op , B ⊗ k B op , and B ⊗ k A op . The follo w ing table, taken from [15], describ es t he corresp onding objects in va rious equi v alen t deri v ed categ ories. D b ( A ⊗ k A op ) D b ( A ⊗ k B op ) D b ( B ⊗ k B op ) D b ( B ⊗ k A op ) Q • ⊗ • k A A Q • ⊗ • k Hom • B ( ¯ Q • , B ) B B ⊗ • k Hom • B ( ¯ Q • , B ) B B ⊗ k A A Q • ⊗ • k Hom • A ( Q • , A ) Q • ⊗ • k B B B B ⊗ k B B B B ⊗ • k Hom • A ( Q • , A ) A A ⊗ • k Hom • A ( Q • , A ) A A ⊗ k B B ¯ Q • ⊗ • k B B ¯ Q • ⊗ • k Hom • A ( Q • , A ) A A ⊗ k A A A A ⊗ • k Hom • B ( ¯ Q • , B ) ¯ Q • ⊗ • k Hom B ( ¯ Q • , B ) ¯ Q • ⊗ • k A A A A A Θ • B B B ∆ • T able 1 By T able 1, on e c an easily ﬁnd the correspondi ng objects in t he abov e four equi v alent de ri ved categ ories. For instan ce, we consider the deriv ed equi v alenc e b F : D b ( A ⊗ k A op ) − → D b ( B ⊗ k A op ) induce d by the tilting c omple x Q • ⊗ • k A A . T able 1 sho ws that b F s ends Q • ⊗ • k A A to B B ⊗ k A A , Q • ⊗ • k Hom • A ( Q • , A ) to B B ⊗ • k Hom • A ( Q • , A ) , A A ⊗ • k Hom • A ( Q • , A ) to ¯ Q • ⊗ • k A A , and A A A to ∆ • . The follo wing lemma collects some proper ties of the se complex es [15]. Lemma 5.1. Let ∆ • and Θ • be the comple xes deﬁned in T able 1. W e have the following . ( 1 ) ∆ • ⊗ L A Θ • ≃ B B B in D b ( B ⊗ k B op ) . ( 2 ) Θ • ⊗ L B ∆ • ≃ A A A in D b ( A ⊗ k A op ) . 20 ( 3 ) The functor ∆ • ⊗ L A − : D b ( A ) − → D b ( B ) is a derived equ ivalenc e and ∆ • ⊗ L A X • ≃ F ( X • ) for all X • ∈ D b ( A ) . ( 4 ) The functor Θ • ⊗ L B − : D b ( B ) − → D b ( A ) is a derived equival ence with Θ • ⊗ L B U • ≃ G ( U • ) for all U • ∈ D b ( B ) . ( 5 ) Θ • ≃ R Hom B ( ∆ • , B ) in D b ( A ⊗ k B op ) . ( 6 ) ∆ • is isomorp hic to ¯ Q • when cons ider ed as an objec t in D b ( B ) and to Hom A ( Q • , A ) when con sider ed as an obje ct in D b ( A op ) . ( 7 ) Θ • is iso morphic to Q • when cons ider ed as an object in D b ( A ) and to Hom B ( ¯ Q • , B ) when c onside r ed as an obje ct in D b ( B op ) . Pr oof. The state ments (1)–( 5) follo w from [ 15, Theorem 3.3, Proposition 4.1] and th e re marks aft er [ 15, Deﬁnition 4.2]. The state ments (6) and (7) are take n from [15, Proposition 3.1]. Note that it is an open question in [15] whether the tw o functors F and ∆ • ⊗ L A − are natural ly i somorphi c, althou gh they ha ve isomorphi c images on each object by Lemma 5.1(3). Recall that a comple x T • in D b ( A ⊗ k B op ) is calle d a two-s ided tilting complex ov er A ⊗ k B op if there is a comple x ¯ T • in D b ( B ⊗ k A op ) such that T • ⊗ L B ¯ T • ≃ A A A in D b ( A ⊗ k A op ) and ¯ T • ⊗ L A T • ≃ B B B in D b ( B ⊗ k B op ) . In th is case, the comple x ¯ T • is ca lled an in verse of T • . From Lemma 5.1 we see that ∆ • and Θ • deﬁned in T able 1 are mutua lly in v erse two-si ded tilting complex es over A ⊗ k B op and B ⊗ k A op , respe cti vel y . The follo wing lemma, which is c rucial in ou r later proo fs, describes some pro perties of the ter ms of th e two-si ded tilting complex ∆ • in T able 1. Lemma 5.2. The two-sided tilting comple x ∆ • is isomorp hic in D b ( B ⊗ k A op ) to a radic al complex 0 − → ∆ 0 − → ∆ 1 − → . . . − → ∆ n − → 0 with ∆ i ∈ add ( B ¯ Q ⊗ k Q ∗ A ) for all i > 0 . Pr oof. Thanks to T able 1, there is a d eri ve d equi v alen ce b F : D b ( A ⊗ k A op ) − → D b ( B ⊗ k A op ) . Moreo ver , the comple xes Q • ⊗ • k A A and ¯ Q • ⊗ • k A A are the associated tilting complex es to b F and its quasi-in vers e, respecti ve ly . Note that the two complex es are radical and hav e the shape as assumed in S ection 3. By T able 1, the two-si ded tilting complex ∆ • ov er B ⊗ k A op is isomo rphic in D b ( B ⊗ k A op ) to b F ( A A A ) , and therefo re, by Lemma 3.1, the comple x ∆ • is isomorp hic in D b ( B ⊗ k A op ) to a radical complex R • : 0 − → R 0 − → R 1 − → · · · − → R n − → 0 with R i ∈ add ( B ¯ Q ⊗ k A A ) for all i > 0. Similarly , by T able 1, there is a deri ved equiv alence e F : D b ( B ⊗ k B op ) − → D b ( B ⊗ k A op ) induced by the tilting comple x B B ⊗ • k Hom • B ( ¯ Q • , B ) ove r B ⊗ k B op . F rom T able 1, w e kno w that the complex B B ⊗ • k Hom • A ( Q • , A ) is a tilting comple x associa ted to the quasi-in verse of e F . Moreo ver , it follo ws from T able 1 that e F ( B B B ) ≃ ∆ • in D b ( B ⊗ k A op ) . By Lemma 3.1, e F ( B B B ) is isomorphic in D b ( B ⊗ k A op ) to a radica l complex S • : 0 − → S 0 − → S 1 − → · · · − → S n − → 0 , with S i ∈ add ( B B ⊗ k Q ∗ A ) for all i > 0. Thus both R • and S • are isomorp hic to ∆ • in D b ( B ⊗ k A op ) . By L emma 2.2, the comple xes R • and S • are isomorph ic in the homotop y cate gory K b ( B ⊗ k A op ) . Since R • and S • are radical comple xes , the y a re iso morphic in C b ( B ⊗ k A op ) . In particul ar , R i ≃ S i as B - A -bimodule s for all i . Thus, for each i > 0, t he bimodule R i lies in both add ( B B ⊗ k Q ∗ A ) and add ( B ¯ Q ⊗ k A A ) . As a result , w e ha ve R i ∈ add ( B ¯ Q ⊗ k Q ∗ A ) for all i > 0. Using Lemma 5.2, we no w pro ve the follo wing main result in this secti on. 21 Theor em 5.3. Let A an d B be two ﬁnite-dimens ional algebr as over a ﬁeld k, and let F : D b ( A ) − → D b ( B ) be a derive d equivalen ce. If F is almost ν -stab le, then ther e is a stable equiva lence φ : A -mod − → B -mod of Morita type suc h that φ ( X ) ≃ ¯ F ( X ) for any A-module X , wher e ¯ F is deﬁned in Pr oposition 3.4. Pr oof. First, we sho w that A may be assumed to be indecomp osable. In fact, if A = A 1 × A 2 is a product of two algebr as A i , then the comple x Q • associ ated t o F has a decompos ition Q • = Q • 1 ⊕ Q • 2 such that Q • 1 ∈ K b ( A 1 -proj ) and Q • 2 ∈ K b ( A 2 -proj ) . C orrespo ndingl y , the algebra B , which is isomorphic to the endomor phism algebr a of Q • , is a produ ct of two algebras, say B = B 1 × B 2 , such that B i ≃ End D b ( A i ) ( Q • i ) for i = 1 , 2. Thus the deri v ed eq ui v alence F : D b ( A ) − → D b ( B ) induces two deri ved equiv alences F i : D b ( A i ) − → D b ( B i ) f or i = 1 , 2. Moreo ver , for each i , the complex Q • i is a tilting comple x associate d to F i , and the tilting complex associate d to the quasi-in vers e of F i is isomorphic to F i ( A i ) ≃ F ( A i ) w hich is a direct summand of ¯ Q • . Thus, if F is almos t ν -stab le, th en F i is almos t ν -stable for i = 1 , 2. Furthermore, if A i and B i are sta bly equiv alent of Morita type for i = 1 , 2, then A 1 × A 2 and B 1 × B 2 are sta bly equi v alent of Morita ty pe. Thus, we may as sume that A i s indeco mposable . Since deri v ed equiv alence pres erve s the semis implicity of alg ebras, we kn o w that A is semi-s imple if and only if B is semisimple. Hence we can further assume that A is non-semisimpl e. No w , let A be non-semis imple and indeco mposable . Then B is also non-semis imple and indecomp osable. Let ∆ • be the comple x in T able 1. By Lemma 5.2, the complex ∆ • is isomorphic in D b ( B ⊗ k A op ) t o a radical comple x 0 − → ∆ 0 − → ∆ 1 − → · · · − → ∆ n − → 0 , with ∆ i in add ( B ¯ Q ⊗ k Q ∗ A ) for all i > 0. For simpli city , we assume that ∆ • is the above compl ex. By Lemma 5.1 (6), the complex ¯ Q • is isomorphic to ∆ • in D b ( B ) , and therefor e in K b ( B ) by Lemma 2.2(1). Thus there is a chain map α : ¯ Q • → ∆ • such that the mapping cone con ( α ) is acycli c. S ince the terms of con ( α ) in positi ve de grees are all proj ecti ve , the acyclic c omplex c on ( α ) s plits. This mean s that B ∆ 0 is a project iv e B - module. Thus all terms of ∆ • are projecti ve as left B -modules. Similarly , by the fact that ∆ • is isomorph ic to Hom A ( Q • , A ) in D b ( A op ) , we infer that ∆ 0 A is a projecti v e right A -modu le and th at all th e terms o f ∆ • are projec ti ve as right A -modules. Conseq uently , the complex R Hom B ( ∆ • , B ) is isomorphic in D b ( A ⊗ k B op ) to the comple x Hom • B ( ∆ • , B ) : 0 / / Hom B ( ∆ n , B ) / / · · · / / Hom B ( ∆ 1 , B ) / / Hom B ( ∆ 0 , B ) / / 0 . Since ∆ i ∈ add ( B ¯ Q ⊗ k Q ∗ A ) for all positi ve i , we ﬁnd that the A - B -bimodule Hom B ( ∆ i , B ) is in add ( Hom B ( B ¯ Q ⊗ k Q ∗ A , B )) for each positi v e integer i . Recall that there is a natural isomorph ism Hom B ( B ¯ Q ⊗ k Q ∗ A , B ) ≃ ν A Q ⊗ k ¯ Q ∗ . Since add ( A Q ) = add ( ν A Q ) , the term Hom B ( ∆ i , B ) is in add ( Q ⊗ ¯ Q ∗ ) for all i > 0. As to Hom B ( B ∆ 0 A , B ) , we can use Lemma 5.1 (7) and similarl y prove that Hom B ( B ∆ 0 A , B ) is project iv e as a one-sided module on both sides . Next, we sho w that ∆ 0 and Hom B ( ∆ 0 , B ) deﬁne a sta ble equi v alence of Morita type between A and B . In- deed, by Lemma 5.1 (5), the two-sided tilting complex Θ • deﬁned in T able 1 is isomorphic in D b ( A ⊗ k B op ) to R Hom B ( ∆ • , B ) , a nd the latter is isomorphic in D b ( A ⊗ k B op ) to Hom • B ( ∆ • , B ) . For simplicity , we assume that Θ • equals H om • B ( ∆ • , B ) . S ince all the terms of ∆ • are pro jecti v e as right A -modules , t he comple x ∆ • ⊗ L A Θ • is isomorph ic in D b ( B ⊗ k B op ) to the comple x ∆ • ⊗ • A Θ • . The m -th term of ∆ • ⊗ • A Θ • is M i + j = m ( ∆ i ⊗ A Θ j ) = M i + j = m ( ∆ i ⊗ A Hom B ( ∆ − j , B )) . Let A E be the maximal ν -stable A -module, and let B ¯ E b e the maximal ν -stabl e B -module (see S ection 3). Since add ( A Q ) = add ( ν A Q ) an d add ( B ¯ Q ) = add ( ν B ¯ Q ) , we ha ve A Q ∈ add ( A E ) and B ¯ Q ∈ add ( B ¯ E ) . By Lemma 5.1 22 (3) an d Lemma 3.9, the complex ∆ • ⊗ • A E is i somorphi c in D b ( B ) to a complex in K b ( add ( B ¯ E )) . Since ∆ i ∈ add ( B ¯ Q ⊗ k Q A ) for i > 0, the i -th term ∆ i ⊗ A E of ∆ • ⊗ • A E i s in add ( B ¯ Q ) ⊆ add ( B ¯ E ) for all i > 0. It follows that ∆ 0 ⊗ A E is in add ( B ¯ E ) , and therefore E ∗ ⊗ A Θ 0 = E ∗ ⊗ A Hom B ( ∆ 0 , B ) ≃ Hom B ( ∆ 0 ⊗ A E , B ) ∈ add ( ¯ E ∗ ) . Note that, for each i > 0, w e hav e ∆ i ∈ add ( ¯ Q ⊗ k Q ∗ ) ⊆ add ( ¯ E ⊗ k E ∗ ) and Θ − i = Hom B ( ∆ i , B ) ∈ add ( Q ⊗ k ¯ Q ∗ ) ⊆ add ( E ⊗ k ¯ E ∗ ) . Thus, it is n ot hard to see t hat ∆ i ⊗ A Θ j is in add ( ¯ E ⊗ k ¯ E ∗ ) for all i and j with i + j 6 = 0. This means that a ll terms in non-zero degr ees of ∆ • ⊗ • A Θ • are in add ( ¯ E ⊗ k ¯ E ∗ ) . Fo r the term in deg ree 0 o f ∆ • ⊗ • A Θ • , e xcept ∆ 0 ⊗ A Θ 0 , all of its other direct summands are in add ( ¯ E ⊗ k E ∗ ⊗ A E ⊗ k ¯ E ∗ ) which is contained in add ( ¯ E ⊗ k ¯ E ∗ ) . Note that all the bimodul es in add ( B ¯ E ⊗ k ¯ E ∗ B ) are projecti ve-injec ti ve. No w , we ha ve ∆ • ⊗ • A Θ • ≃ B B B in D b ( B ⊗ k B op ) by Lemma 5.1 (1). T hus the comple x ∆ • ⊗ • A Θ • has zero homolog y and projec ti ve-i njecti v e terms in all non-zero degrees, and therefor e it splits and is isomorphic to B B B in the homotop y c ateg ory K b ( B ⊗ k B op ) . Since B is indecomposab le and non-semis imple, the bimodule B B B is i ndecomp osable and n on-pro jecti v e, and t herefor e it is a direct summand of ∆ 0 ⊗ A Θ 0 . It fo llo ws t hat ∆ 0 ⊗ A Θ 0 ≃ B B B ⊕ U for a projecti ve -injecti ve B - B -bimodu le U . Similarly , we hav e Θ 0 ⊗ B ∆ 0 ≃ A A A ⊕ V for a projec ti ve-i njecti v e A - A -bimodu le V . Hence A and B are stab ly equiv alent of Morita type. Let φ : A -mod − → B -mod b e the stable equi valenc e induced by ∆ 0 ⊗ A − . It follo ws from Lemma 5.1 ( 3 ) and Lemma 3.1 that φ ( X ) ≃ ¯ F ( X ) in B -mod for all A -modules X . Let A be an algebra . An A -module M is cal led a gener ator -co ge nera tor for A -mod if A ⊕ D ( A ) ∈ a dd ( M ) . The r epr esentatio n dimension of A , deno ted by rep . dim ( A ) , is deﬁned to be rep . dim ( A ) : = inf { gl . dim ( E nd A ( M )) | M is a genera tor -coge nerator for A -mod } . This not ion was introd uced by Auslander in [1] to m easure homolo gically how f ar an alge bra is from being repres entation -ﬁnite, and has been stu died by man y auth ors in recent y ears (see [16] and the referen ces therein). The follo wing result is a con sequen ce of Theorem 5.3 since stable equi v ale nces of Morita type prese rve repres entation dimensions [18]. Cor ollary 5.4. If F is an almost ν -stable derived functor between A and B, then A and B have the same r epr e- sentat ion and dominant dimensio ns. As anoth er consequ ence of Theorem 5.3, we re-obtain the followin g result of Rickard [14] since ev ery deri v ed equi v alenc e between self-inject iv e algebras is almost ν -stable by Propositio n 3.8. Cor ollary 5 .5. Let A and B be ﬁnite-di mensiona l self-inject ive alge bras . I f A and B ar e deri ved-eq uivalen t, the n the y ar e stably equi valent of Morita type. Remark: (1) Let A be a ﬁnite dimens ional s elf-inj ecti ve algebra and X be an A -module. By the remark at the end of S ection 3, there is a deri ved equ iv alence between the algebras End A ( A ⊕ X ) a nd End A ( A ⊕ Ω ( X )) satisfy ing the almost ν -st able condition. T hus, we hav e an alte rnati v e proof of the resu lt [10, Corolla ry 1.2] of Liu and Xi by apply ing Theorem 5.3. (2) Theorem 5.3 may be fals e if only one of the two equalities of the almost ν -stable con dition is satisﬁed. For a c ountere xampl e, we refer the read er to Example 2 in S ection 7. 6 Inductiv e constructions of almost ν -stable deriv ed equiv alences In this section , we shall gi ve se vera l inducti v e constructio ns of almost ν -stable deri v ed equi v alenc es. As a con- sequen ce, one can produce a lot of (usually not self-inject i ve) ﬁnite-dimensio nal alg ebras that are both deriv ed- equi v alen t and stably equi v alen t of Morita type. In this section, we keep the notation s int roduced in Section 3. Our ﬁ rst inducti ve constru ction is the follo wing propo sition. 23 Pro position 6.1. Suppose tha t F is an al most ν -s table deriv ed equivale nce between ﬁn ite-dimen sional algebr as A and B over a ﬁeld k. Let ¯ F be the stable funct or of F deﬁned in Pr opos ition 3.4, and let X be an A -module . Then the r e is an almost ν -sta ble derived equiv alence between the en domorph ism algebr as End A ( A ⊕ X ) and End B ( B ⊕ ¯ F ( X )) . Pr oof. W e keep the notations in the proof of Theorem 5.3. By the last part of the proof of T heorem 5.3, the two-si ded tilting complex es ∆ • and Θ • ha ve the properties : ∆ • ⊗ • A Θ • ≃ B B B in K b ( B ⊗ k B op ) and Θ • ⊗ • B ∆ • ≃ A A A in K b ( A ⊗ k A op ) . It follows th at the functor ( ∆ • ⊗ • A Θ • ) ⊗ • B − is naturally iso morphic to t he ide ntity funct or 1 K b ( B ) , an d ( Θ • ⊗ • B ∆ • ) ⊗ • A − is na turally isomo rphic to the id entity functor 1 K b ( A ) . Thus ∆ • ⊗ • A − and Θ • ⊗ • B − induce mutually in v erse equiv alences between K b ( A ) and K b ( B ) . N o w we prov e that the restrictio ns of these two functors to K b ( add ( A ⊕ X )) and to K b ( add ( B ⊕ ¯ F ( X ))) are also mutually in vers e equiv alences for each A -module X . In fa ct, the complex ∆ • ⊗ • A X is of th e follo wing form 0 / / ∆ 0 ⊗ A X / / ∆ 1 ⊗ A X / / · · · / / ∆ n ⊗ A X / / 0 . Since ∆ i is a projecti ve bimod ule for all i > 0, t he ter m ∆ i ⊗ A X i s a p rojecti ve B -module for all i > 0. Moreo v er , by Theorem 5.3, we hav e ∆ 0 ⊗ A X ≃ ¯ F ( X ) in B -mod , and therefore ∆ 0 ⊗ A X is a direct summand of ¯ F ( X ) ⊕ P for some proje cti ve B -module P . Hence the complex ∆ • ⊗ • A X is in K b ( add ( B ⊕ ¯ F ( X ))) . Note that, for each projec ti ve A -module P 1 , the c omple x ∆ • ⊗ • A P 1 is in add ( ¯ Q • ) . Thus , f or each complex X • in K b ( add ( A ⊕ X )) , the comple x ∆ • ⊗ • A X • is in K b ( add ( B ⊕ ¯ F ( X ))) . Similarly , the functo r Θ • ⊗ • B − takes comple xes in K b ( add ( B ⊕ ¯ F ( X ))) to complex es in K b ( add ( A ⊕ X )) . Thus ∆ • ⊗ • A − and Θ • ⊗ • B − induce mutually in v erse equi v alence s between the trian gulated categorie s K b ( add ( A ⊕ X )) and K b ( add ( B ⊕ ¯ F ( X ))) . Let Λ = End A ( A ⊕ X ) and Γ = End B ( B ⊕ ¯ F ( X )) . Then K b ( Λ -pro j ) and K b ( Γ -proj ) are ca nonical ly equi v- alent to K b ( add ( A ⊕ X )) and K b ( add ( B ⊕ ¯ F ( X ))) , respecti vely . B y [13, Theorem 6.4], there is a deri ve d equi v alen ce b F between Λ and Γ . M oreo ver , the tilting comple xes associated to b F and its quasi-in ve rse are Hom • A ( A ⊕ X , Q • ⊕ ( Θ • ⊗ • B ¯ F ( X ))) and Hom • B ( B ⊕ ¯ F ( X ) , ¯ Q • ⊕ ( ∆ • ⊗ • A X )) , resp ecti ve ly . By the proof of Theo- rem 5.3, the i -th term Θ i of Θ • is in add ( Q ⊗ k ¯ Q ∗ ) for al l i < 0. Hence Θ i ⊗ B ¯ F ( X ) is i n add ( Q ) for all i < 0 , and all the terms in negati ve degree s of Hom • A ( A ⊕ X , Q • ⊕ ( Θ • ⊗ • B ¯ F ( X ))) are in add ( Hom A ( A ⊕ X , Q )) . Similarly , all th e terms in p ositi v e degrees of Hom • B ( B ⊕ ¯ F ( X ) , ¯ Q • ⊕ ( ∆ • ⊗ • A X )) are in ad d ( Hom B ( B ⊕ ¯ F ( X ) , ¯ Q ) . Note th at we ha v e the follo wing isomorphisms ν Λ ( Hom A ( A ⊕ X , Q )) = D Hom Λ ( Hom A ( A ⊕ X , Q ) , H om A ( A ⊕ X , A ⊕ X )) ≃ D Hom A ( Q , A ⊕ X ) ≃ D ( Hom A ( Q , A ) ⊗ A ( A ⊕ X )) ≃ H om A ( A ⊕ X , D ( Hom A ( Q , A ))) = H om A ( A ⊕ X , ν A Q ) . Since add ( A Q ) = ad d ( ν A Q ) , we ha ve add ( Hom A ( A ⊕ X , Q )) = add ( ν Λ ( Hom A ( A ⊕ X , Q ))) . Similarly , w e hav e add ( Hom B ( B ⊕ ¯ F ( X ) , ¯ Q )) = add ( ν Γ ( Hom B ( B ⊕ ¯ F ( X ) , ¯ Q ))) . T his sho ws t hat the deriv ed equi v alenc e b etween Λ and Γ indu ced by the tiltin g comple x H om • A ( A ⊕ X , Q • ⊕ ( Θ • ⊗ • B ¯ F ( X ))) is almost ν -stable . Our nex t construct ion uses tensor products. Pro position 6.2. Let k be a ﬁeld. Suppose F is an almost ν -stable derive d equivalen ce between ﬁnite-dimensi onal k-algebr as A and B. Then, for each ﬁnite-di mensiona l self-in jective k -algebr a C , ther e is an almost ν -stable de- rived equi valence between the tenso r algebr as A ⊗ k C op and B ⊗ k C op . 24 Pr oof. By [15, Theor em 2.1], F indu ces a deri ved equi v alence b F between A ⊗ k C op and B ⊗ k C op . Suppose Q • and ¯ Q • be the ass ociated tiltin g comple x es of F and its quasi-in verse, resp ecti v ely . From T able 1 we know that Q • ⊗ • k C C and ¯ Q • ⊗ • k C C are the associate d ti lting compl ex es of b F and its quas i-in verse, respecti ve ly . Now we ha v e the follo wing isomorphisms: ν A ⊗ k C op ( A Q ⊗ k C C ) = D Hom A ⊗ k C op ( A Q ⊗ k C C , A A ⊗ k C C ) ≃ D  Hom A ( A Q , A A ) ⊗ k Hom C op ( C C , C C )  ≃ D Hom A ( A Q , A A ) ⊗ k D Hom C op ( C C , C C ) ≃ ν A Q ⊗ k ν C op C C ≃ ν A Q ⊗ k C C ( becau se C is self-injec ti ve ) . Since F is almost ν -stable, we hav e add ( Q ) = ad d ( ν A Q ) and add ( ν A ⊗ k C op ( A Q ⊗ k C C )) = add ( A Q ⊗ k C C ) . Simi- larly , we hav e add ( ν B ⊗ k C op ( B ¯ Q ⊗ k C C )) = add ( B ¯ Q ⊗ k C C ) . Henc e b F is almost ν -stable and the proof is co mpleted. Let A be a ﬁnite-dimensio nal algebra o ver a ﬁeld k , and let X be an A -module . The one-point e xten sion of A by X , denoted by A [ X ] , is the triangular matrix algebra h k X 0 A i . T he natural projection A [ X ] − → A shows that A is a quotie nt algebra of A [ X ] , and that A -mod can be vie wed as a full subcateg ory of A [ X ] -mod. Let e X denote the A [ X ] -module h k X i . Then, for each A -module M , we hav e Hom A [ X ] ( M , e X ) ≃ Hom A ( M , X ) . Our third cons truction of an almost ν -stable deri v ed equiv alence is gi ven by one- point extensio ns. Pro position 6 .3. Let k be ﬁel d. Suppose F is an almost ν -stable deriv ed equiv alence between ﬁnite-dimensi onal k-algebr as A and B. If X is an A -module such that F ( X ) is isomorphic in D b (() B ) to a B-module Y , then ther e is an almost ν -stab le derived equiva lence between the one-po int ex tension s A [ X ] and B [ Y ] . Pr oof. Let G be a quasi-in verse of F . Recall that Q • and ¯ Q • denote the radical tilting complexe s a ssociat ed to F and G , respecti ve ly . Then Q • can be viewed as a comple x in K b ( A [ X ] -proj ) . By a r esult o f Bar ot and Len zing [2], the comple x Q • ⊕ e X is a t ilting comp lex o ver A [ X ] suc h that it s endo morphism algebra is isomorphic to B [ Y ] , w here e X is re garde d as a comple x conc entrated only on deg ree ze ro. Moreov er , ¯ Q • ⊕ e Y is a tilting c omple x associ ated to the quasi-in verse of the deriv ed equiv alence induced by Q • ⊕ e X . Recall that Q is the direct sum of all the terms of Q • in negati v e degrees . Then add ( ν A Q ) = ad d ( A Q ) by assumption. Since the direct sum of all terms in neg ati ve de grees of Q • ⊕ e X e quals Q , we ha ve to sho w that add ( ν A [ X ] Q ) = add ( A [ X ] Q ) . Since F ( X ) is iso morphic in D b ( B ) to the B -modu le Y , we ha ve Hom ( Q • , X [ i ]) = 0 f or all i 6 = 0. Then there is a uniq ue maximal s ubmodule L of X with r espect to the proper ty Hom A ( Q , L ) = 0. This sho w s that 0 = Hom D b ( A ) ( Q • , X [ i ]) ≃ Hom D b ( A ) ( Q • , ( X / L )[ i ]) for all integer s i > 0. If X / L 6 = 0, then Hom A ( Q , soc ( X / L )) 6 = 0 by the deﬁnition of L . This implies that Hom D b ( A ) ( Q • , ( X / L )[ i ]) 6 = 0 for some i > 0, a contrad iction. Thus X / L = 0, Hom A ( Q , X ) = 0, and Hom A [ X ] ( Q , e X ) = 0. Conseque ntly , ν A [ X ] Q = D Hom A [ X ] ( Q , A [ X ]) ≃ D Hom A [ X ] ( Q , A ⊕ e X ) = D Hom A [ X ] ( Q , A ) ≃ ν A Q . Hence add ( ν A [ X ] ( Q )) = add ( A [ X ] Q ) . Similarly , we hav e add ( ν B [ Y ] ¯ Q ) = add ( B [ Y ] ¯ Q ) . This ﬁnishes the proof. 7 Examples and questions In the follo w ing, we shall illustrate our results with examples . Example 1 : L et A and B be ﬁ nite-di mensiona l k algebr as g i ven by qui ver with relation s in Fig. 1 and F ig. 2, respec ti vely . 25 • α 1 2 / / • β        • 3 γ Z Z 4 4 4 4 4 • α / / • 1 2 β o o γ / / • 3 δ o o αγ = δβ = 0 αβγα = βγαβ = γ αβγ = 0 αβα = δγδ = βα − γδ = 0 . Fig. 1 Fig. 2 Let P A ( i ) , I A ( i ) and S A ( i ) denote the indecomposa ble projec ti ve, injecti v e and simple A -modules correspondi ng to the verte x i , respec ti vely . W e take a non-z ero homomorph ism f : P A ( 2 ) → P A ( 1 ) . Then there is a tilting comple x of A -modules Q • : 0 / / P A ( 2 ) ⊕ P A ( 2 ) ⊕ P A ( 3 ) [ f , 0 , 0 ] T / / P A ( 1 ) / / 0 . The endo morphism algebra of Q • is isomo rphic to B . Let F : D b ( A ) → D b ( B ) be a der iv ed equi v alen ce with Q • as its associate d tilting complex . Clearly , F is almost ν -stabl e since A and B are symmetric algebras. By [6, Proposit ion 7.3], there is a deri v ed equi v alenc e F 1 between ¯ A = A / ( αβγ ) and ¯ B = B / ( αβ ) with in v erse G 1 such that the asso ciated tilting comple xes ov er ¯ A and ov er ¯ B are Q • 1 : 0 / / P ¯ A ( 2 ) ⊕ P ¯ A ( 2 ) ⊕ P ¯ A ( 3 ) / / P A ( 1 ) / so c P A ( 1 ) / / 0 , ¯ Q • 1 : 0 / / P B ( 1 ) / so c P B ( 1 ) / / P ¯ B ( 2 ) ⊕ P ¯ B ( 2 ) ⊕ P ¯ B ( 3 ) / / 0 , respec ti vely . Clearly , the two complex es satisfy the condition s: add ( ¯ A Q 1 ) = add ( ν ¯ A Q 1 ) and add ( ¯ Q 1 ) = add ( ν ¯ B ¯ Q 1 ) . Hence the algebras ¯ A and ¯ B are both deri ved-equi valen t and stab ly equi v alen t of Morita typ e by Theorem 5.3. W e kno w that F 1 ( S ¯ A ( 1 )) i s isomor phic to th e simple ¯ B -modul e S ¯ B ( 1 ) . The o ne-poi nt extens ion ¯ A [ S ¯ A ( 1 )] is giv en by the qui v er Fig. 3. • 4 η } } { { { • α 1 2 / / • β        • 3 γ Z Z 4 4 4 4 4 • 4 η   • α / / • 1 2 β o o γ / / • 3 δ o o Fig. 3 Fig. 4 with relation s αβγ = βγαβ = γαβγ = ηα = 0, and the one -point e xtensi on ¯ B [ S ¯ B ( 1 )] is gi v en by the qui ver F ig. 4. with relations η α = αβ = δγδ = βα − γδ = αγ = δβ = 0 . By Propositio n 6.3, there is a deri ved equiv alence between ¯ A [ S ¯ A ( 1 )] and ¯ B [ S ¯ B ( 1 )] , which indu ces a stable equi v alenc e of Morita type. An c alculati on shows that F 1 ( I ¯ A ( 1 )) i s isomor phic to the ¯ B -modul e I ¯ B ( 1 ) . The a lgebras End ¯ A ( ¯ A ⊕ I ¯ A ( 1 )) a nd End ¯ B ( ¯ B ⊕ I ¯ B ( 1 )) are gi ven by Fig. 5 and Fig. 6, resp ecti v ely . • 1 2 α / / • β   • δ O O • γ 3 4 o o • η   @ @ @ @ @ @ @ • α / / • 1 2 β o o 4 δ O O • γ 3 o o αβγδ = βγδα = γδαβγ = 0 αδ = γβ = βα − δηγ = γδη = ηγδ = 0 Fig. 5 Fig. 6 Thus End ¯ A ( ¯ A ⊕ I ¯ A ( 1 )) and End ¯ B ( ¯ B ⊕ I ¯ B ( 1 )) are deri ve d-equi valent a nd stably equiv alent o f Morita type by Propo- sition 6.1. 26 The follo wing ex ample, tak en from [14], shows that Theor em 5.3 may fail if only one of the conditi ons of an almost ν -sta ble functo r is satisﬁed . Example 2: Let A be the 17-d imension al alge bra giv en by the qui v er • ε 1 2 / / • δ o o α   • γ ? ? ~ ~ ~ ~ ~ ~ ~ • β 3 4 o o with relations γα β = γδ = εαβ = 0 , δε = αβγ . As before, we deno te by P A ( i ) the inde composab le projecti v e A -module corres pondin g to the v erte x i . Let Q • be the direc t sum of the follo wing two comple xe s 0 − → P A ( 1 ) − → P A ( 2 ) − → 0 , 0 − → 0 − → P A ( 2 ) ⊕ P A ( 3 ) ⊕ P A ( 4 ) − → 0 , where P A ( 1 ) is in de gree − 1. One c an check that Q • is a tiltin g complex o ver A . Let B = E nd D b ( A ) ( Q • ) . Then B is a 20-d imensiona l algebra gi ven by the qui ver • α 1 2 / / • β   • δ O O • γ 3 4 o o with relation s αβγδα = 0 = δαβγ , where the indecompos able projecti v e B -modules at the vertices 1, 2, 3, and 4 corr espond respecti vely to the direct summands P A ( 1 ) → P A ( 2 ) , P A ( 2 ) , P A ( 3 ) and P A ( 4 ) of the complex Q • . Let F : D b ( A ) − → D b ( B ) be the d eri ve d equi v alence induce d by the tilting complex Q • . Then F ( P A ( i )) = P B ( i ) for i = 2 , 3 , 4. L et Q • 1 be the direct summand P A ( 1 ) → P A ( 2 ) of Q • . Applying F to the follo wing distinguis hed triang le in D b ( A ) P A ( 2 )[ − 1 ] − → Q • 1 [ − 1 ] − → P A ( 1 ) − → P A ( 2 ) , we see that F ( P A ( 1 )) is of the follo w ing form 0 − → P B ( 2 ) − → P B ( 1 ) − → 0 , where P B ( 2 ) is in degre e zero. Thus F ( A ) is isomorph ic in D b ( B ) to a complex ¯ Q • which is the direct sum of the foll owin g two complex es 0 − → P B ( 2 ) − → P B ( 1 ) − → 0 , 0 − → P B ( 2 ) ⊕ P B ( 3 ) ⊕ P B ( 4 ) − → 0 − → 0 . Let G be a quas i-in verse of F . T hen ¯ Q • is a tilting complex ass ociated to G . Clearly , A Q = Q − 1 = P A ( 1 ) an d B ¯ Q = ¯ Q 1 = P B ( 1 ) . It is easy to see that F satisﬁes the c onditio n add ( B ¯ Q ) = add ( ν B ¯ Q ) , b ut not th e co nditio n add ( A Q ) = add ( ν A Q ) . Note that B is a Nakayama algebra and has 16 non-projec ti ve ind ecomposa ble modules, while A has more than 16 non-p rojecti ve indecomposab le module s. T hus A and B canno t be stably equiv alent. This example also sho ws that Coroll ary 3.10 m ay be false for deri ved equiv alences in gener al. In fact , we ha ve A E = P A ( 1 ) and B ¯ E = 0 in this exampl e. Finally , we mention the follo wing questio ns. (1) Find ne w condit ions for a deri v ed equiv alence to induce a stable equi v alence of Morita type. 27 (2) Does Theorem 5.3 hold true for Artin R -algebras A and B such that A and B both are projecti v e ov er R ? (For the deﬁnition of stable equi v alenc e of Morita type between Artin algebras see [19]). (3) Let F : D b ( A ) − → D b ( B ) be a deriv ed equi valen ce between Artin algeb ras A and B . If add ( A Q ) = add ( ν A Q ) , is it true that rep . di m ( A ) ≤ rep . di m ( B ) ? Acknowledgements. T he resear ch work of C. C. Xi is parti ally suppo rted by NSFC (No.107310 70). Refer ences [1] M . A U S L A N D E R , R epr ese ntation dimension of artin alg ebra s . Queen Mary College Mathematic s Notes, Queen Mary Colleg e, London, 1971. [2] M . B A RO T and H . L E N Z I N G , One-point extens ions and deriv ed equi v alences . J. Algeb ra 264 ( 2003) 1-5. [3] M . B RO U ´ E , Equi v alenc es of blocks of group algebras . In: “ F inite dimension al algebr as and rel ated topics ”, edited by V . D lab and L. L. Scott, Kluwer , 1994, 1-26. [4] E . C L I N E , B . P A R S H A L L and L . S C OT T , Deriv ed categori es and Mor ita t heory . J. Algebr a 1 04 (198 6) 397-4 09. [5] D . H A P P E L , T riang ulated cate gories in the r epr esentation theor y of ﬁnite dimension al alg ebra s. Cam- bridge Uni v . Press, Cambridge , 1988. [6] W . H U and C . C . X I , Almost D -split sequence s and deri ve d equi v alence s. Preprint, 2007, av ail able at : http:// math.bnu.ed u.cn/ ∼ ccxi/P apers/Arti cles/sh ort-huxi-2.pdf . [7] H . K R A U S E , Stable eq ui v alence preserv es repre sentati on type. Comment. Math. Helv . 72 (1997 ) 266-28 4. [8] Y . M . L I U and C . C . X I , Constructi ons of stable equi v alences of Morita type for ﬁnite dimensio nal algebr as, I. T rans. Amer . Math. Soc . 358 (200 6), no.6, 2537-25 60. [9] Y . M . L I U and C . C . X I , Constructi ons of stable equi v alences of Morita type for ﬁnite dimensio nal algebr as, II. Math. Z . 251 (200 5) 21-39. [10] Y . M . L I U and C . C . X I , Construct ions of stabl e equi v alence s of Morita type for ﬁnite dimensio nal algebr as, III. J. Lon don Math. Soc . 76 (2007) 567-58 5. [11] A . N E E M A N , T riangu lated cate go ries . Princeton Univ ersity Press, Princeto n and Oxford, 2001. [12] J . R I C K A R D , The abelian defect group conjec ture. Pr oceeding s of the Internatio nal Congr ess of Mathe- matician s , V ol. II (Berlin, 1998). D oc. Math. 1998 , Extra V ol. II, 121-128 . [13] J . R I C K A R D , Morita theo ry for deriv ed catego ries. J. London Math. Soc . 39 (198 9) 436-456 . [14] J . R I C K A R D , Deri v ed catego ries and stable equi v alen ces. J. Pur e Appl. Algebr a 64 (1989 ) 303-317. [15] J . R I C K A R D , Deri v ed equi v alence as deriv ed functors . J. Lond on Math. Soc . 43 (1991 ) 37-48. [16] R . R O U Q U I E R , Representatio n dimension of exterior algebras . In vent. Math. 165 (2)(20 06) 357-367. [17] C . A . W E I B E L , An intr oductio n to homolo gical algebr a. Cambridge U ni v . Press, 1994. [18] C . C . X I , Represen tation dimensio n and quasi- heredita ry algebr as. Adv . Math . 168 (20 02) 193-21 2. 28 [19] C . C . X I , Stable equi v alen ces of adjoint type. F orum M ath. 20 (2 008), no.1, 81-9 7. [20] C . C . X I A N D D . M . X U , O n the ﬁnitistic dimensio n conject ure, IV : related to relati vely projecti v e modules . P reprint , 2007, av ailable at: http://ma th.bnu.edu .cn/ ∼ ccxi/P apers/Arti cles/xi xu.pdf . June 12, 200 8. R e vised: April 15, 2009. Current address of W . Hu: School of Mathematical Scienc es, Peking Uni versity , Beijing 1008 71, P .R. China; email: huwei@math .pku.edu.cn . 29

Derived equivalences and stable equivalences of Morita type, I

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