On iterated almost $nu$-stable derived equivalences

On iterated almost ν -stable deri ved equi v alences W ei Hu Abstract In a recent paper [6], we introduced a classes o f derived equiv alences called a lmost ν -stable der i ved equiv- alences. The crucial prope rty is that an almost ν -stable derived eq uiv alence a lw ays indu ces a stable equiv- alence o f Mo rita type, which gene ralizes a well-known result of Rickard: derived-equiv alent self-in jecti ve algebras ar e stably e quiv alent of M orita type. In this p aper, we shall con sider the compositions of almost ν - stable de riv ed equiv alences an d their quasi-in verses, which are called iterated almost ν -st able deri ved equiv a- lences. W e give a sufficient and ne cessary c ondition fo r a derived equiv alence to be an iterated almost ν -stable derived equivalence, and give an explicit co nstruction of the stable equi valence fu nctor in duced by an iter ated almost ν -stable deri ved equi valence. As a consequence, we get som e new suf ficient con ditions for a d erived equiv alence between general finite-dimensional algebras to induce a stable equi valence of Morita typ e. 1 Introd uction In [6], w e introduced a class of deri ve d equi v alences called almost ν -stab le deriv ed equi v alences . The crucial proper ty is that an almost ν -sta ble deri ve d equi v alen ce al ways induces a stable equi v alenc e of Morita typ e, which genera lizes a classical resul t of Rickard ([13, Corollar y 5.5]). This als o giv es a su fficie nt condition for a d eri ved equi v alence betwee n g eneral fini te-dimensi onal algebra s to in duce a stable equiv alen ce of Mori ta ty pe. Note that many homolog ical dimensi ons, suc h as global dimen sion, finitis tic d imension, and representat ion dimension, are not in v ari ant under deri ved equi v alences in general. But they are all preserve d by stable equi va lences of Morita type. So, this also helps us to compare the homologi cal dimensions of deriv ed-eq ui val ent algebra s. Let us first recall the definition of almost ν -stabl e deri ved equi v alence s. Let F : D b ( A ) − → D b ( B ) be a deriv ed equi v alence between two Artin algebra s A and B , where D b ( A ) and D b ( B ) stand for the deri v ed cate gories of bounded comple xes ove r A and B , respect iv ely . W e use F − 1 to denote a quasi-in ver se of F . F is called an almost ν -sta ble derived equival ence if the follo wing hold: (1) The tilting comple x T • associ ated to F has the following form: 0 − → T − n − → · · · − → T − 1 − → T 0 − → 0 In this case, the tiltin g ¯ T • associ ated to F − 1 has the follo wing form (see [6, Lemma 2.1]): 0 − → ¯ T 0 − → ¯ T 1 − → · · · − → ¯ T n − → 0 (2) 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 ) , wher e ν is the Nakayama functo r . Let us re mark t hat the co mposition of two almost ν - stable d eri ved eq ui val ences (or their quasi- in verses) is no longer a lmost ν -s table in genera l. If a deri ved equi valenc e is a comp osition F ≃ F 1 F 2 · · · F m with F i or F − 1 i being 2000 Mathematics Subject Classification: 18E30,16G10 ;18G20,16D90. Ke yword s: deri ved equi valen ce, stable equiv alence of Morita t ype, tilting complex. 1 an almost ν - stable deri ved equi v alence for a ll i , then F is ca lled an iter ated almost ν -stable deriv ed equiva lence . By definition, we see that an almost ν -stable deri ved equi v alence and its quasi-i n verse are iterated almost ν - stable deriv ed equi valen ces, and that the compositio n of two iterated almost ν -sta ble deriv ed equ iv alences is again an iterated almost ν -stable deri ved equi valen ce. Clearly , an iterated almost ν -stable deri ved equi v alence alw ays induces a stable equi valen ce of Morita type, and therefo re the in v olve d algebras ha ve many common homolog ical dimension s. B ut the probl em is: Question: Given a derive d equival ence F , how to determine whether F is iter ated almost ν -stable or not? The main purpose of this note is to gi ve a complete answer to the abov e question. For a bounded complex X • ov er an algebr a A , we use X ± to denote L i 6 = 0 X i . T he main result of this note can be stated as the follo wing theore m. Theor em 1.1. Let F : D b ( A ) − → D b ( B ) be a de rived e quivale nce between two Artin algeb ras A an d B. Suppos e that T • and ¯ T • ar e the tilting comple xes associ ated to F and F − 1 , r especti vely . Then F is an itera ted almost ν -stab le derived equiva lence if and only if add ( A T ± ) = add ( ν A T ± ) and add ( B ¯ T ± ) = add ( ν B ¯ T ± ) . The abov e theorem tells us that, by checking the terms of tilting complex es, we can det ermine whether a deri ve d equi valenc e is iterated almost ν -stable or not. If we work with finite-dimens ional algebras over a field, then we ha ve se vera l other characte rization s of iterated almost ν -stab le deri ved equi v alences. For details, see Theorem 3.6 belo w . As a consequen ce of Theorem 1.1, we hav e the follo w ing corollary , which provide s a new suf ficient conditio n for a deriv ed equiv alence to induce a stable equi valenc e of Morita type. For information on stable equ iv alences of Morita type, we refer to [3, 8, 9, 10]. Cor ollary 1.2. Let F : D b ( A ) − → D b ( B ) be a derived equivalenc e between two finite-d imensiona l algebr as A and B over a field. Suppose that T • and ¯ T • ar e the tilting complex es associated to F and F − 1 , re spective ly . If add ( A T ± ) = add ( ν A T ± ) and add ( B ¯ T ± ) = add ( ν B ¯ T ± ) , then A and B ar e stably equival ent of Morita type. This paper is or ganized as follo ws. In S ection 2, we shall recall some notations and basic fac ts. Theorem 1.1 will be prov ed in Section 3 after sev eral lemmas. Section 4 is de v oted to describing the stable equi v alence functo r induced by an iterated almost ν -stable deri ved equi v alence . Finall y , in S ection 5, w e shall gi ve se vera l methods to const ruct iterated almost ν -stab le deri ved equi v alences. 2 Pr eliminaries In this secti on, we shall recall some basic definition s and facts needed in our later proofs . Through out this paper , unless specified otherwise, all algebr as will be Artin algebras ov er a fi xed commu- tati ve Artin ring R . All modules will be finitely genera ted unitary left modules. For an algebra A , the cate gory of A -modules is denoted by A -mod; the full subcateg ory of A -mod consist ing of projecti ve m odules is denoted by A -proj . The stable module cate gory , denot ed by A -mod , is the quotient cate gory of A -mod modulo the ideal genera ted by morphisms fa ctorizin g throu gh projec tiv e modules. W e den ote by ν A the usu al Nakayama functo r . Let C be an additiv e category . The compo sition of two morphisms f : X − → Y and g : Y − → Z in C will be denote d by f g . For two functo rs F : C → D and G : D → E of cate gories, their composition is denoted by GF . For a n object X in C , add ( X ) is the full subc atego ry of C consisting of all direct summand s of finite direct sums of copie s of X . A complex X • ov er C is a sequen ce · · · − → X i − 1 d i − 1 X − → X i d i X − → X i + 1 d i + 1 X − → · · · in C such that d i X d i + 1 X = 0 for all inte gers i . T he cate gory of complex es ov er C is denoted by C ( C ) . The homotop y category of comple xes ov er C is de noted by K ( C ) . When C is an abeli an cate gory , the deri ved categ ory of comple xes ov er C is den oted by D ( C ) . The f ull sub categ ory of K ( C ) and D ( C ) consisti ng of bo unded comp lex es o ver C is de noted by K b ( C ) 2 and D b ( C ) , respecti vely . As usual, for a giv en algebra A , we simply w rite K b ( A ) and D b ( A ) for K b ( A -mod ) and D b ( A -mod ) , respe cti vely . It is well-kno w n that, for an algebra A , K b ( A ) and D b ( A ) are triangulate d cate gories . For basic resul ts on triangulate d categori es, we refer to Happel’ s book [4]. T hrough out this paper , w e use X • [ n ] to denote the comple x obtained by shifting X • to the left by n deg ree. Let A be an alg ebra. A homomorphi sm f : X − → Y of A -modu les is called a radical map if, for an y module Z and homomorp hisms h : Z − → X and g : Y − → Z , the comp osition h f g is not a n isomor phism. A compl ex o ver A -mod is called a radical complex if all its dif ferentia l maps are radical maps. Every complex ov er A -mod is isomorph ic in the ho motopy cate gory K ( A ) to a rad ical complex. It is easy to s ee that if tw o radical comple xe s X • and Y • are isomorph ic in K ( A ) , the n X • and Y • are isomorph ic in C ( A ) . T wo algebras A and B are said to be derive d-equiva lent if their deriv ed cate gorie s D b ( A ) and D b ( B ) are equi valent as triangulated categorie s. In [12], Rickard prov ed that two algebras are deriv ed-eq ui val ent if and only if there is a comple x T • in K b ( A -proj ) satisfyi ng (1) Hom ( T • , T • [ n ]) = 0 for all n 6 = 0, and (2) add ( T • ) genera tes K b ( A -proj ) as a triang ulated categ ory such tha t B ≃ End ( T • ) . A comple x in K b ( A -proj ) sa tisfying th e ab ov e tw o cond itions is called a tilting comple x ov er A . It is kno wn that, giv en a deri ved equi v alence F between A and B , there is a unique (up to isomorphism) tilting complex T • ov er A su ch that F ( T • ) ≃ B . If T • is a ra dical co mplex, it is called a tilti ng compl ex a ssociate d to F . Note that, by definition , a tilting complex associ ated to F is unique up to isomorphism in C b ( A ) . The follo w ing lemma is useful in our later proof. For the con veni ence of the reader , we provid e a proof. Lemma 2.1. Let C and D be two additive cate gories, and let F : K b ( C ) − → K b ( D ) be a triangle functor . Let X • be a complex in K b ( C ) . F or eac h term X i , let Y • i be a complex isomorphic to F ( X i ) . Then F ( X • ) is isomorph ic to a comple x Z • with Z m = L i + j = m Y j i for all m ∈ Z . Pr oof. W e use induct ion on the number of non-z ero terms of X • . If X • has only one non-ze ro term, then it is obv ious. Assume that X • has m ore than one non-zero terms. W ithout loss of generali ty , we suppos e that X • is the foll owing comple x 0 − → X 0 − → X 1 − → · · · − → X n − → 0 with X i 6 = 0 for all i = 0 , 1 , · · · , n . Let σ > 1 X • be the complex 0 − → X 1 − → · · · − → X n − → 0. Then there is a distin guished triangl e in K b ( C ) : X 0 [ − 1 ] − → σ > 1 X • − → X • − → X 0 . Applying F , we get a disti nguishe d triangle in K b ( D ) : F ( X 0 [ − 1 ]) − → F ( σ > 1 X • ) − → F ( X • ) − → F ( X 0 ) . By induct ion, F ( σ > 1 X • ) is isomorphic to a complex U • with U m = L 1 6 i 6 n , i + j = m Y j i . Thus, F ( X • ) is isomorph ic to the mapping cone Z • of the map from Y • 0 [ − 1 ] to U • . Thus, by definition, we ha ve Z m = M 0 6 i 6 n , i + j = m Y j i = M i + j = m Y j i . This finishes the proo f. Remark: Let F : D b ( A ) − → D b ( B ) be a deri ved equiv alence between two algebras A and B . F induce s an equi valenc e F : K b ( A -proj ) − → K b ( B -proj ) . S o, for a bounded complex of projecti ve A -modules, we can use the abo ve lemma to calcu late its image under F . 3 3 Characterizations of iterated almost ν -stable deri ved equiv alences In th is sectio n, we sha ll g iv e a proof o f our main result Theo rem 1.1, whic h ch aracteriz es iterated a lmost ν -stab le deri ve d equi v alences in ter ms of tiltin g comple xes. In case th at the alg ebras are fini te-dimensio nal algebra s ov er a field, we shall giv e sev eral other characteri zations of iterate d almost ν -stab le der iv ed equiv alences. For this purpo se, we need some lemmas. Let A be an algebra, and let A E be the direct sum of all those non-isomorp hic indecompo sable projec tiv e A -modules P w ith ν i A P being projecti ve-injec tiv e for all i > 0. The A -module A E is uniqu e up to isomorphism, and is called the maximal ν -stable A -module. If A Q is a projecti ve A -module such that add ( A Q ) = add ( ν A Q ) , then clearly A Q ∈ add ( A E ) . Throughou t this paper , we use ν A -Stp to denote the catego ry add ( A E ) . Recall that for a bou nded comple x X • ov er A , we use X ± to deno te the A -module L i 6 = 0 X i . Lemma 3.1. Let T • be a tiltin g comple x as sociated to a deriv ed equiva lence F : D b ( A ) − → D b ( B ) between two alg ebras . Then the following two conditions are e quivalen t. ( 1 ) add ( ν A T ± ) = add ( A T ± ) ; ( 2 ) A T ± ∈ ν A -Stp . Pr oof. Clearly , we ha ve ( 1 ) ⇒ ( 2 ) . It remains to sho w that ( 2 ) implies ( 1 ) . N o w we assume ( 2 ) holds. Let Q 1 = L i < 0 T i . Using the same method in the proo f of [6, Lemma 3.1], F − 1 ( B ) is isomorph ic in D b ( A ) to a comple x X • with X i ∈ add ( ν A Q 1 ) for al l i < 0. Thus, T • ≃ X • , an d there is a qua si-isomorp hism f • : T • − → X • , which ind uces a quasi-i somorphism U • : · · · / / T − 2 f − 2   d − 2 T / / T − 1 f − 1   π T / / Im d − 1 T f 0 | Im d − 1 T   / / 0 V • : · · · / / X − 2 d − 2 X / / X − 1 π X / / Im d − 1 X / / 0 . W e claim th at the canonical e pimorphism π T : T − 1 − → Im d − 1 T is still a radical m ap. Otherwise, le t h : Y − → T − 1 and g : Im d − 1 T − → Y be such that h π T g = 1 Y . T hen Y is isomorphic to a direct summand of T − 1 , and therefo re Y is an injecti ve module. Thus, g factors throug h the inclusion λ : Im d − 1 T − → T 0 , say g = λ u . Consequent ly 1 Y = h π T λ u = hd − 1 T u . This means that d − 1 T : T − 1 − → T 0 is not radical which is a contradi ction. Since T i and X i are injecti ve for all i < 0, by [6, Lemma 2.2], U • and V • are isomorphic in K b ( A ) . Thu s, T i is a direct summand of X i for all i < 0, and conseq uently Q 1 = L i < 0 T i ∈ add ( ν A Q 1 ) . Since Q 1 and ν A Q 1 ha ve the same number of non-isomorph ic indecompos able direc t summands, we hav e add ( A Q 1 ) = add ( ν A Q 1 ) . Let Q 2 : = L i > 0 T i . Similarly , we hav e add ( A Q 2 ) = add ( ν A Q 2 ) . Co nsequent ly , a dd ( A T ± ) = add ( A Q 1 ⊕ A Q 2 ) = add ( ν A Q 1 ⊕ ν A Q 2 ) = add ( ν A T ± ) . Hence ( 2 ) ⇒ ( 1 ) . In th e follo wing, we shal l use Lemma 3.1 free ly . For insta nce, in the defini tion of an almost ν - stable equi va- lence, the conditio n add ( L n i = 1 T − i ) = add ( L n i = 1 ν A T − i ) is eq ui val ent to say th at T − i ∈ ν A -Stp for all i = 1 , · · · , n . Lemma 3 .2. Let F : D b ( A ) − → D b ( B ) be a deriv ed equivalen ce between two alg ebr as A and B, an d let T • and ¯ T • be the tilting comple xes associat ed to F and F − 1 , re spectivel y . If add ( A T ± ) = add ( ν A T ± ) and add ( B ¯ T ± ) = add ( ν B ¯ T ± ) , then F induces an equivale nce between K b ( ν A -Stp ) and K b ( ν B -Stp ) . Pr oof. L et A E (respect iv ely , B ¯ E ) be the maximal ν -stable A -module (respecti vely , B -module). Then by defini- tion, we hav e ν A -Stp = add ( A E ) and ν B -Stp = add ( B ¯ E ) . The complex F ( A E ) is isomorph ic to a complex ¯ T • 1 in add ( ¯ T • ) . Since ν A E ≃ A E , we hav e ν B ¯ T • 1 ≃ ¯ T • 1 in D b ( B ) . Hence there is a chain map η from ¯ T • 1 to ν B ¯ T • 1 such that the m apping cone con ( η ) is acyc lic. By our assumption, all ¯ T i 1 and ν B ¯ T i 1 with i 6 = 0 are project iv e-injec tiv e since th ey are al l in ν B -Stp. Hence co n ( η ) splits, and th erefore ν B ¯ T 0 1 ⊕ ¯ Q 1 ≃ ¯ T 0 1 ⊕ ¯ Q 2 for s ome ¯ Q 1 , ¯ Q 2 ∈ ν B -Stp. Hence, ν B ¯ T 0 1 ∈ add ( ¯ T 0 1 ⊕ B ¯ E ) . It follo ws that ν i B ¯ T 0 1 ∈ add ( ¯ T 0 1 ⊕ B ¯ E ) is projecti ve-injec tiv e for all i > 0. Hence 4 ¯ T 0 1 ∈ ν B -Stp, and conseq uently ¯ T • 1 is in K b ( ν B -Stp ) . Similarly , we can sho w that F − 1 ( B ¯ E ) is isomorphi c to a comple x in K b ( ν A -Stp ) and the lemma is pro ved. The follo w ing lemma is useful in the proof of Theorem 1.1. Lemma 3.3. Let F : D b ( A ) − → D b ( B ) and G : D b ( B ) − → D b ( C ) be derived equivalence s, and let P • , ¯ P • , Q • , ¯ Q • , T • , and ¯ T • be the tilting complexe s associa ted to F , F − 1 , G , G − 1 , GF , and F − 1 G − 1 r espec tively . If the followin g hold: ( 1 ) A P ± ∈ ν A -Stp and B ¯ P ± ∈ ν B -Stp ; ( 2 ) B Q ± ∈ ν B -Stp and C ¯ Q ± ∈ ν C -Stp , then we have A T ± ∈ ν A -Stp and C ¯ T ± ∈ ν C -Stp . Pr oof. W e only need to sho w that ¯ T ± ∈ ν C -Stp, the other stateme nt follo ws by symmetry . By definitio n, ¯ T • is isomorph ic to GF ( A ) ≃ G ( ¯ P • ) . Since ¯ P i ∈ ν B -Stp for all i 6 = 0, by Lemma 3.2, G ( ¯ P i ) is iso morphic to a co mplex Y • i in K b ( ν C -Stp ) for all i 6 = 0. For i = 0, the complex G ( ¯ P 0 ) is isomorphic to a complex Y • 0 in add ( ¯ Q • ) . By Lemma 2.1, the complex G ( ¯ P • ) is isomorph ic to a comple x Z • with Z m = L i + j = m Y j i . Since all Y j i , exce pt Y 0 0 , are in ν C -Stp, we hav e Z ± ∈ ν C -Stp. Note that both ¯ T • and Z • are in K b ( C -proj ) . T he complex es ¯ T • and Z • are isomorph ic in K b ( C -proj ) . Furthermore, since the complex ¯ T • is a radical complex, it follo ws that ¯ T i is a direct summand of Z i for inte gers i , and consequentl y ¯ T ± ∈ ν C -Stp. Finally , w e ha ve the follo wing lemma which is crucial in the proof of our main result. Lemma 3.4. Let F : D b ( A ) − → D b ( B ) be a derived equivale nce between two Artin alg ebr as A and B, and let T • be the associated tilting comple x of F . If A T ± ∈ ν A -Stp , then ther e is an almost ν -stable equi valence G : D b ( C ) − → D b ( A ) suc h that associ ated tiltin g comple x P • of F G satisfies that P i ∈ ν C -Stp for all i < 0 and P i = 0 for all i > 0 . Pr oof. L et A E be the maximal ν -stable A -module. Then ν A -Stp = add ( A E ) . S uppos e m is the m aximal integer such that T m 6 = 0. By a dual statement of [5, Proposition 3.2], the re is a tilting comple x Q • : = R • ⊕ A E [ − m ] ov er A , where R • is of the form: R • : 0 − → A − → R 1 − → · · · − → R m − → 0 with R i ∈ ν A -Stp for all i > 0. Let C be the endo morphism algebra of Q • , and let H : D b ( A ) − → D b ( C ) be a deri ved equi v alence giv en by the tilting comple x Q • . It is easy to see that H ( A E ) ≃ C P [ m ] for some C P ∈ ν C -Stp, and H ( A ) is isomorphic to a complex S • : 0 − → S − m − → · · · − → S − 1 − → S 0 − → 0 with S i ∈ ν C -Stp for all i < 0. Let G is a quasi-in verse of H . Then S • is a tiltin g comple x associa ted to G . By Lemm a 3.1, we see that G is almost ν -st able. No w let Y • i : = H ( T i ) for each integer i . Since T ± ∈ ν A -Stp, for each integer i 6 = 0, w e hav e Y • i ≃ P i [ m ] for some P i ∈ ν C -Stp. Moreove r , Y • i = 0 for all i > m since T i = 0 for all i > m . The comple x Y • 0 has the prope rty that Y i 0 = 0 for all i > 0 and Y i 0 ∈ ν C -Stp for all i < 0. By L emma 2.1, the compl ex H ( T • ) is isomorp hic to a complex Z • with Z t = L i + j = t Y j i . It follo ws that Z t = 0 for all t > 0 and Z t ∈ ν C -Stp for all t < 0. Since F G ( H ( T • )) ≃ F ( T • ) ≃ B ≃ F G ( P • ) in D b ( B ) , the comple x Z • is isomorphic in D b ( C ) to the tilting comple x P • associ ated to F G . Since both Z • and P • are in K b ( C -proj ) , the y are isomor phic in K b ( C -proj ) . Since P • is a radical complex , the term P i is a direct summand of Z i for all i , and consequen tly P • has the desired proper ty . W e are no w in the position to giv e a proof of our main result. Pro of of Theor em 1.1. Assume that F is an iterated almost ν -stable deriv ed equi va lence. Let F ≃ F 1 F 2 · · · F m be a composition such that F i or F − 1 i is an almost ν -stable deri ved equi va lence. The n by Lemma 3.3, we ha ve add ( A T ± ) = add ( ν A T ± ) and add ( B ¯ T ± ) = add ( ν B ¯ T ± ) . Con versely , assume that add ( A T ± ) = add ( ν A T ± ) and add ( B ¯ T ± ) = add ( ν B ¯ T ± ) . By Lemma 3.4, there is an almost ν -stable deri ved equi v alence G : D b ( C ) − → D b ( A ) such that the tilting complex P • associ ated to F G has the property that P i = 0 for all i > 0 and P i ∈ ν C -Stp for all i < 0. B y Lemma 3.1, we hav e add ( L i < 0 P i ) = add ( L i < 0 ν C P i ) . Let ¯ P • be the tilting comple x associat ed 5 to G − 1 F − 1 . By Lemma 3.3 , we hav e add ( B ¯ P ± ) = add ( ν B ¯ P ± ) . Since P i = 0 for all i > 0, by [6, Lemma 2.1], we ha ve ¯ P i = 0 for all i < 0. Hence F G is an almost ν -stabl e deriv ed equi v alence. Thus, F ≃ ( F G ) G − 1 is an iterate d almost ν -stabl e deriv ed equi v alence . Remark: (1) Theorem 1.1 gi ves us a method to determin e whether a deriv ed equiv alence is iterate d almost ν -stab le or not by checkin g the terms of the in v olv ed tilting complex es. (2) Let P be a projecti ve A -module. The conditio n add ( A P ) = add ( ν A P ) is equiv alent to say that P is projec tiv e-inje cti ve and add ( top ( P )) = ad d ( soc ( P )) . (3) The proo f of Cor ollary 1.2 fol lo ws immediately from Theorem 1.1 and [1 0 , Theorem 5.3]. But Corollar y 1.2 generalizes [10, T heorem 5.3]. This giv es a ne w suf ficient condit ion for a deriv ed equi v alence to induce a stable equ iv alence of Morita type. As an appli cation of Theorem 1.1, we ha ve the follo wing corollary . Cor ollary 3.5. Let F : D b ( A ) − → D b ( B ) be a derived equivale nce between two Artin algebr as A and B, and let T • and ¯ T • be the tilting comple xes associate d to F and F − 1 , r especti vely . If add ( A T ± ) = add ( ν A T ± ) and add ( B ¯ T ± ) = add ( ν B ¯ T ± ) , then the following hold: ( 1 ) fin . dim ( A ) = fin . dim ( B ) , and gl . dim ( A ) = gl . dim ( B ) ; ( 2 ) rep . dim ( A ) = rep . dim ( B ) ; ( 3 ) dom . dim ( A ) = do m . dim ( B ) , wher e fin . dim , gl . dim , rep . dim and dom . dim stand for finitis tic dimension, global dimens ion, r epr esent ation dimensio n and dominant dimensio n, r especti vely . Pr oof. T he coro llary follo w s from [6, Corollary 1.2] and Theorem 1.1. No w we work with finite-di mensional algebras over a field. In this case, we get sev eral other characte riza- tions of iterate d almost ν -stabl e deriv ed equi v alence s, which is the follo w ing theorem. Theor em 3.6. Let F : D b ( A ) − → D b ( B ) be a derived equivale nce between two finite-dimensio nal basic algeb ras A an d B over a fiel d, and let T • and ¯ T • be the til ting comple xes associate d to F and F − 1 , r espectively . Then the followin g ar e equiv alent: ( 1 ) The functo r F is an iterated almost ν -sta ble derived equival ence. ( 2 ) add ( ν A T ± ) = add ( A T ± ) and add ( ν B ¯ T ± ) = add ( B ¯ T ± ) . ( 3 ) T ± ∈ ν A -Stp and ¯ T ± ∈ ν B -Stp . ( 4 ) F or ea ch ind ecomposab le pr ojective A-module P 6∈ ν A -Stp , the i mage F ( top ( P )) is isomorph ic in D b ( B ) to a simple B-module . ( 5 ) F or each ind ecomposab le pr ojective A -module P 6∈ ν A -Stp , the following conditio ns ar e satisfied: ( a ) P 6∈ add ( A T ± ) ; ( b ) the multiplicity of P as a dir ect summand of A T 0 is 1 . Pr oof. It follo ws from Theorem 1.1 and Lemma 3.1 that the statements ( 1 ) , ( 2 ) and ( 3 ) are equi v alent. N ote that for an y simple module S ov er a basic algebra Λ , the dimensio n of S as an E nd Λ ( S ) -space is 1. In this proof, let A E and B ¯ E be the maximal ν -stable A -module and B -module, respecti vely . ( 4 ) ⇒ ( 5 ) For each indecomposab le projecti ve A -module P not in ν A -Stp, since F ( top ( P )) is isomorphic in D b ( B ) to a simple B -module , we ha ve Hom D b ( A ) ( T • , top ( P )[ i ]) = 0 for all i 6 = 0, and Hom D b ( A ) ( T • , top ( P )) ≃ Hom B ( B , F ( top ( P ))) ≃ F ( top ( P )) is one-dimen sional ove r the divis ion ring End A ( top ( P )) . Note that T • is a radical complex . It follows that P is not a direct summand of T ± and the multip licity of P as a direct summand of T 0 is 1. 6 ( 5 ) ⇒ ( 4 ) By condition ( a ) , we see that Hom D b ( A ) ( T • , top ( P )[ i ]) = 0 for all i 6 = 0. Hence F ( top ( P )) is isomorph ic to an indecomposab le B -module X . By conditi on ( b ) , we can assume that T • P is the only indecom- posab le direct summand of T • such that P is a direct summand of its degr ee zero term. Suppose that ¯ P is the indeco mposable project iv e B -module corresp onding to the direct summand T • P . Then Hom B ( B , X ) ≃ Hom D b ( A ) ( T • , top ( P )) ≃ Hom D b ( A ) ( T • P , top ( P )) ≃ Hom B ( ¯ P , X ) . This implies that X only contains top ( ¯ P ) as compositio n factors. If X is not a simple B -module, then there is a nonze ro map X − → soc ( X ) − → X in End B ( X ) which is not an isomorphis m. This contradict s to the fact that End B ( X ) ≃ End A ( top ( P )) is a di vision ring. Hence X ≃ F ( top ( P )) is a simple B -module . ( 3 ) ⇒ ( 4 ) By definition, we hav e add ( A E ) = ν A -Stp and add ( B ¯ E ) = ν B -Stp. L et P be an indecompos able projec tiv e A -module not in ν A -Stp. Then it is clear that Hom D b ( A ) ( T • , top ( P )[ i ]) = 0 for all i 6 = 0 since T ± ∈ ν A -Stp, and conseque ntly F ( top ( P )) is isomorph ic in D b ( B ) to a B -module X . By Lemma 3.2, the comple x F − 1 ( B ¯ E ) is isomorphi c in D b ( A ) to a comple x E • in K b ( ν A -Stp ) . Hence Hom B ( B ¯ E , X ) ≃ Hom D b ( A ) ( F − 1 ( B ¯ E ) , top ( P )) ≃ Hom K b ( A ) ( E • , top ( P )) = 0 . If B X is not simple, then th ere is a sh ort exact sequence 0 − → ¯ U − → X − → ¯ V − → 0 in B -module w ith ¯ U , ¯ V non-zero . Applying Hom B ( B ¯ E , − ) , we get that Hom B ( B ¯ E , ¯ U ) = 0 = Hom B ( B ¯ E , ¯ V ) , an d cons equently Hom D b ( B ) ( ¯ T • , ¯ U [ i ]) = 0 = Hom D b ( B ) ( ¯ T • , ¯ V [ i ]) for all i 6 = 0 since ¯ T ± ∈ ν B -Stp. Hence F − 1 ( ¯ U ) and F − 1 ( ¯ V ) are isomor phic to A -modules U and V , respecti vel y . Thus, we get a dist inguishe d triangle U − → top ( P ) − → V − → U [ 1 ] in D b ( A ) by applyin g F − 1 to the distin guished triangle ¯ U − → X − → ¯ V − → ¯ U [ 1 ] . Applying Hom D b ( A ) ( A , − ) to the abo ve triangle, we get an exact sequence 0 − → U − → top ( P ) − → V − → 0 with non-zero A -modules U and V . This contradic ts to the fa ct that top ( P ) is a simple A -module. Hence F ( top ( P )) ≃ X is a simple B -module. ( 4 ) ⇒ ( 3 ) For each indecomposab le projecti ve A -module P not in ν A -Stp, since F ( top ( P )) is isomorphic D b ( B ) to a simpl e B -module, we hav e Hom D b ( A ) ( T • , top ( P )[ i ]) = 0 fo r all i 6 = 0. T ogethe r with th e isomor phism Hom D b ( A ) ( T • , top ( P )[ i ]) ≃ Hom K b ( A ) ( T • , top ( P )[ i ]) ≃ Hom A ( T i , top ( P )) , we get Hom A ( T i , top ( P )) = 0 for all i 6 = 0 and for all indec omposable projecti ve A -module P not in ν A -Stp. Hence T i ∈ ν A -Stp for all i 6 = 0, that is, T ± ∈ ν A -Stp. Now let A Q be a projecti ve A -module such that A A ≃ A E ⊕ A Q . It follo ws by assumptio n that F ( top ( Q )) is a semi-simple B -module. S uppos e that ¯ Q is a projecti ve cov er of F ( top ( Q )) , and sup pose that B B ≃ ¯ Q ⊕ W . S ince ¯ T • is a radical comple x in B -proj, we ha ve Hom B ( ¯ T i , top ( ¯ Q )) ≃ Hom K b ( B ) ( ¯ T • , top ( ¯ Q )[ i ]) ≃ Hom D b ( B ) ( ¯ T • , top ( ¯ Q )[ i ]) ≃ Hom D b ( A ) ( A , top ( Q )[ i ]) = 0 for a ll i 6 = 0. Hence ¯ T ± ∈ add ( B W ) . It remai ns to sh ow B W ∈ ν B -Stp. Note tha t Hom B ( B W , top ( ¯ Q )[ i ]) = 0 for all inte gers i . It follo ws that H om D b ( A ) ( F − 1 ( B W ) , top ( Q )[ i ]) = 0 for all intege rs i . Let L • be a radical complex in K b ( A -proj ) such that F − 1 ( B W ) ≃ L • . The n Hom A ( L i , top ( Q )) ≃ Hom D b ( A ) ( L • , top ( Q )[ i ]) = 0 for all inte gers i . Hence L i ∈ add ( A E ) for all integers i . U sing the same proof as the proof of [1, Theorem 2.1], we can show that ν i B W is a proj ecti ve B -module for all i > 0. It follows that B W ≃ ν k B W for some k > 0. Hence B W is projec tiv e-inje cti ve and ν i B W is projecti ve-i njecti ve for i > 0, and consequen tly B W ∈ ν B -Stp. This finishes the proof. 7 Remark: (1) By T heore m 3.6 (5), we see that if we consider finite-dimens ional alg ebras over a field, then we can determin e whether a deri ved equi v alence F is iterated almost ν -stable or not by checkin g the terms of the tilting complex associated to F , and we do not need to check the terms of the tilting complex associated to F − 1 , which is need ed in Theorem 1.1. (2) It is interest ing to know whether Theorem 3.6 holds for general Artin alge bras. Note that the only proble m is the step “ ( 4 ) ⇒ ( 3 ) ”, where the method in the proof of [1, Theorem 2.1] does not work for general Artin alge bras. As a conseq uence, we ha ve the follo w ing corollary . Cor ollary 3.7. Let F : D b ( A ) − → D b ( B ) be a derive d equivale nce between two finite-d imensiona l basic alge- bra s over a field. If one of the equivalen t condition s in Theorem 3.6 is satisfi ed, then the algebr as A and B ar e stably equ ivalent of Morita type. Pr oof. T his follo w s from Theorem 3.6, and [6, Theorem 5.3]. W e end this section by using a simple examp le to illust rate Theorem 3.6 and Corollary 3.7. Example : Let k be a field, and let A and B be finite-dimensio nal k -algebras gi ven by quiv ers with relatio ns in Fig. 1 and Fig. 2, respecti vely . • α 1 2 / / • β        • 3 γ Y Y 3 3 3 3 3 • α / / • 1 2 β o o γ / / • 3 δ o o αγ = δβ = 0 αβγ = βγαβ = γαβγ = 0 αβ = δγδ = βα − γδ = 0 . Fig. 1 Fig. 2 Let P ( i ) denot e the indecompos able projecti ve A -module correspo nding to the vert ex i . Then there is a tilting comple x of A -modules T • : 0 − → P ( 2 ) ⊕ P ( 2 ) ⊕ P ( 3 ) [ f , 0 , 0 ] T − → P ( 1 ) − → 0 with P ( 1 ) in degr ee zero . One can check that End K b ( A -proj ) ( T • ) is isomorph ic to B , and that ν A -Stp = add ( P ( 2 ) ⊕ P ( 3 )) . Hence the tilting comple x satisfies the conditio n (5) in Theorem 3.6. Therefore , the complex T • induce s an iterate d almost ν -stable der iv ed eq ui va lence (actu ally e ven an almost ν -sta ble der iv ed eq ui va lence) between A and B . By Corollary 3.7, the algebras A and B are stably equi v alent of Morita type. 4 The stable equi valence functor In th is secti on, we will gi ve a des cription of the sta ble equ iv alence functor indu ced by an ite rated al most ν -st able deri ve d equiv alence. Let A be an Artin alge bra, a nd l et A E be a maximal ν -st able A -modu le. Then by definition ν A -Stp = add ( A E ) . W e use A -mod ν to denote the quotie nt cate gory of A -mod modulo m orphis ms factor izing thro ugh modules in ν A -Stp. T he Hom-space in A -mod ν is denoted by Hom ν A ( − , − ) . For a morphism f in A -mod, its image in A -mod ν under the cano nical functo r from A -mod to A -mod ν is denoted by f . The cate gory K b ( ν A -Stp ) is a clearly thick subcate gory (that is, a triangu lated full subcatego ry closed under tak ing direct summands) of D b ( A ) . Let D b ( A ) / K b ( ν A -Stp ) be the V er dier quotien t category , then we ha ve a cano nical additi ve functo r Σ ′ : A -mod − → D b ( A ) / K b ( ν A -Stp ) obtain ed by composin g the natura l embedding from A -mod to D b ( A ) and the quotient functor from D b ( A ) − → D b ( A ) / K b ( ν A -Stp ) . For the definit ion and basic prop erties of V erdier quotient, we refer to [11, Chapte r 2]. 8 Since Σ ′ ( A E ) is clea rly isomorphic to zero obje ct in D b ( A ) / K b ( ν A -Stp ) , the functor Σ ′ induce s an additi ve functo r Σ : A -mod ν − → D b ( A ) / K b ( ν A -Stp ) . Kee ping this not ation, we ha ve a propositi on, which can be vie wed as a general ization of a well-kno wn resul t of Rickard [14, Theore m 2.1] Pro position 4.1. The func tor Σ : A -mod ν − → D b ( A ) / K b ( ν A -Stp ) is fully faith ful. Mor eover , the functor Σ is an equiva lence if and only if A is self-injec tive. Pr oof. A morphism X • − → Y • in D b ( A ) / K b ( ν A -Stp ) is denote d by a fra ction s − 1 a : X • s ⇐ = Z • a − → Y • , where a and s are morphisms in D b ( A ) , and if Z • s = ⇒ X • − → U • − → Z • [ 1 ] is a distingu ished triang le in D b ( A ) , then U • ∈ K b ( ν A -Stp ) . A morphism s ′ in D b ( A ) with this proper ty will be denoted by s ′ = ⇒ . T wo morphisms X • s ⇐ = U • a − → Y • and X • r ⇐ = V • b − → Y • are equal if and only if the re are morphisms W • t = ⇒ U • and W • h = ⇒ V • such that t s = hr and t a = hb . An isomorphi sm from X to Y is of the form X s ⇐ = U • t = ⇒ Y . First, we sho w that Σ is a full functor . Fo r this purpose, it suffices to sho w that Σ ′ is a full functo r . No w let f : X − → Y be a morphis m in A -mod. Then Σ ′ ( f ) is the morphis m X 1 X ⇐ = X f − → Y . W e need to sho w that each morphism from X to Y in D b ( A ) / K b ( ν A -Stp ) is of this form. Let X s ⇐ = U • a − → Y be a morphis m in D b ( A ) / K b ( ν A -Stp ) . By definit ion, there is a d istingui shed triangle U • s − → X g − → E • − → U • [ 1 ] in D b ( A ) with E • ∈ K b ( ν A -Stp ) . Consider the distinguis hed triangle in D b ( A ) σ > 0 E • α − → E • β − → σ < 0 E • − → ( σ > 0 E • )[ 1 ] . Since E • is clearly in K b ( A -inj ) , we hav e Hom D b ( A ) ( X , σ < 0 E • ) ≃ Hom K b ( A ) ( X , σ < 0 E • ) = 0. It follo ws that g β = 0, and the refore g facto rizes throug h α . H ence we can form the follo wing commutati ve diagram in D b ( A ) with ro ws being disting uished triangle s. V • h + 3 r   X / / σ > 0 E • w / / α   V • [ 1 ] r [ 1 ]   U • s + 3 X g / / E • / / U • [ 1 ] . Since Hom D b ( A ) (( σ > 0 E • )[ − 1 ] , Y ) ≃ Hom K b ( A ) (( σ > 0 E • )[ − 1 ] , Y ) = 0, the morp hism ( w [ − 1 ]) ra = 0, and hence there is some morphism f : X − → Y in D b ( A ) such that r a = h f . Then we ha ve the follo wing commuta tiv e diagra m in D b ( A ) V • h  % B B B B B B B B r x  z z z z z z z z U • s   a ( ( Q Q Q Q Q Q Q Q Q X 1 X r z m m m m m m m m m m m m m m m m m m f   X Y , which means that the m orphis ms X s ⇐ = U • a − → Y and X 1 X ⇐ = X f − → Y in D b ( A ) / K b ( ν A -Stp ) are equal. Since the embeddin g of A -mod into D b ( A ) is fully faith ful, the morphism f is gi ven by a morphis m in A -mod. Hence the func tor Σ ′ is full, and theref ore Σ is a full functor . Suppose that f : X − → Y is a morphism in A -mod such that Σ ′ ( f ) = 0. That is, th e morp hisms X 1 X ⇐ = X 0 − → Y and X 1 X ⇐ = X f − → Y are equal in D b ( A ) / K b ( ν A -Stp ) . T hen there is a m orphis m W • s = ⇒ X such that s f = 0 in D b ( A ) . Embedding s into a disting uished triangle in D b ( A ) , we see that f factor izes in D b ( A ) through a 9 comple x in K b ( ν A -Stp ) , an d therefore it follo ws easily that f fac torizes in A -mod thro ugh an A -modu le ν A -Stp. Hence the func tor Σ is fait hful. If A is self -injecti ve, then ν A -Stp = A -proj and the equi valen ce was pro ve d by Rickard [14, Theo rem 2.1]. If A is not se lf-inject iv e, then there is a projecti ve A -modul e P not in ν A -Stp. S uppos e that Σ is an equi valenc e. Then there is some A -module X such that X ≃ P [ − 1 ] in D b ( A ) / K b ( ν A -Stp ) . That is, there is an isomorph ism X s ⇐ = U • t = ⇒ P [ − 1 ] in D b ( A ) / K b ( ν A -Stp ) . Then by Octahedral Axiom, we can form the followin g commutati ve diagra m in D b ( A ) E • 1 / / U • t + 3 s   P [ − 1 ] / / h   E • 1 [ 1 ] E • 1 g / / X t / /   con ( g ) / /   E • 1 [ 1 ] , E • 2 E • 2 where E • 1 and E • 2 are in K b ( ν A -Stp ) , and con ( g ) is the mapping cone of g . From the vertic al disting uished triang le on the right side, we see that the mapping cone con ( h ) of h is isomorphic in D b ( A ) to a complex E • 2 in K b ( ν A -Stp ) . All the terms of con ( h ) in non-zero degre es are in ν A -Stp and P ⊕ X is a direct summand of the 0-de gree term of con ( h ) . H ence P is isomorphi c to a complex in K b ( ν A -Stp ) which is impossible since P is projec tiv e and is not in ν A -Stp. This finishes the proof. Remark: In the above proposit ion, suppos e that P is a project iv e-injec tiv e A -module, if we replace A -mod ν by the quot ient cate gory of A -mod modulo morphis ms fac torizing through modul es in add ( P ) , and replace D b ( A ) / K b ( ν A -Stp ) by D b ( A ) / K b ( add ( P )) , then the proof of P roposi tion 4.1 actuall y can be used to sho w that in this case the func tor Σ is also fully faith ful. No w fo r each iterated a lmost ν -st able de riv ed equiv alence F : D b ( A ) − → D b ( B ) . By L emma 3.2, we see th at F induces an eq ui val ence between the triangulated cat egorie s D b ( A ) / K b ( ν A -Stp ) and D b ( B ) / K b ( ν B -Stp ) . W e also denote this equi valenc e by F . In the follo wing, we will see that there is an equi v alence φ F : A -mod ν − → B -mod ν such that the diagram A -mod ν Σ / / φ F   D b ( A ) / K b ( ν A -Stp ) F   B -mod ν Σ / / D b ( B ) / K b ( ν B -Stp ) of additi ve functo rs is commutati ve up to isomorph ism. Moreo ver , the functor φ F also induces an equi v alence between the stabl e module cate gories A -mod and B -mod. Before we g iv e the con structio n of φ F , we gi ve th e follo wing lemma, which gene ralizes [6, Lemm a 2.2] and will be used in the const ruction of φ F . Lemma 4.2. Let A be an arbitrar y ring , and let A -Mod be the cate gory of all left (not neces sarily finitely gen erate d) A-modules. Sup pose X • is a comple x over A -Mod bounde d above and Y • is a comple x over A -Mod bound ed below . If ther e is an inte ger m such that X i is pr ojective for all i > m and Y j is injec tive for all j < m, then θ X • , Y • : Hom K ( A -Mod ) ( X • , Y • ) → Hom D ( A -Mod ) ( X • , Y • ) induce d by the localiz ation functo r θ : K ( A -Mod ) → D ( A -Mod ) is an isomorphism. Pr oof. W ithout loss of general ity , we can assume that m = 0. For simplicit y , w e write K for K ( A -Mod ) and D for D ( A -Mod ) . A lso, the Hom-spaces Hom K ( − , − ) and Hom D ( − , − ) will be deno ted by K ( − , − ) and D ( − , − ) , respec tiv ely . 10 First, we sho w that, for each A -module Z , the ind uced map θ X • , Z [ 1 ] : K ( X • , Z [ 1 ]) − → D ( X • , Z [ 1 ]) is monic. Indeed , applying K ( − , Z [ 1 ]) and D ( − , Z [ 1 ]) to the distingu ished triangle σ > 0 X • − → X • − → σ < 0 X • − → ( σ > 0 X • )[ 1 ] , we get a commutat iv e diagram with ex act rows. K ( σ > 0 X • , Z ) / / θ σ > 0 X • , Z   K ( σ < 0 X • , Z [ 1 ]) / / θ σ < 0 X • , Z [ 1 ]   K ( X • , Z [ 1 ]) / / θ X • , Z [ 1 ]   K ( σ > 0 X • , Z [ 1 ]) θ σ > 0 X • , Z [ 1 ]   D ( σ > 0 X • , Z ) / / D ( σ < 0 X • , Z [ 1 ]) / / D ( X • , Z [ 1 ]) / / D ( σ > 0 X • , Z [ 1 ]) By [6, Lemma 2.2], the maps θ σ > 0 X • , Z and θ σ < 0 X • , Z [ 1 ] are isomorphi sms. Since K ( σ > 0 X • , Z [ 1 ]) = 0, the map θ σ > 0 X • , Z [ 1 ] is clearl y monic. T hus, by th e F i ve Lemma (s ee, for example [15, p.13]), the map θ X • , Z [ 1 ] is monic. Next, we s ho w that the map θ X • , ( σ > 0 Y • )[ 1 ] : K ( X • , ( σ > 0 Y • )[ 1 ]) − → D ( X • , ( σ > 0 Y • )[ 1 ]) is monic. Indeed , applying K ( X • , − ) and D ( X • , − ) to the distin guished triangle Y 0 − → ( σ > 0 Y • )[ 1 ] − → ( σ > 0 Y • )[ 1 ] − → Y 0 [ 1 ] , we get a commutat iv e diagram with ex act rows. K ( X • , Y 0 ) / / θ σ X • , Y 0   K ( X • , ( σ > 0 Y • )[ 1 ]) / / θ X • , ( σ > 0 Y • )[ 1 ]   K ( X • , ( σ > 0 Y • )[ 1 ]) / / θ X • , ( σ > 0 Y • )[ 1 ]   K ( X • , Y 0 [ 1 ]) θ X • , Y 0 [ 1 ]   D ( X • , Y 0 ) / / D ( X • , ( σ < 0 Y • )[ 1 ]) / / D ( X • , ( σ > 0 Y • )[ 1 ]) / / D ( X • , Y 0 [ 1 ]) Again by [6, Lemma 2.2], the left two vertical maps are isomorphi sms. By the above discuss ion, we see that θ X • , Y 0 [ 1 ] is monic. So, by the Five Lemma again , the map θ X • , ( σ > 0 Y • )[ 1 ] is monic. Finally , applying K ( X • , − ) and D ( X • , − ) to the distinguish ed triangle ( σ < 0 Y • )[ − 1 ] − → σ > 0 Y • − → Y • − → σ < 0 Y • , we get a commutat iv e diagram K ( X • , ( σ < 0 Y • )[ − 1 ]) / / θ X • , ( σ < 0 Y • )[ − 1 ]   K ( X • , σ > 0 Y • ) / / θ X • , σ > 0 Y •   K ( X • , Y • ) / / θ X • , Y •   K ( X • , σ < 0 Y • ) θ X • , σ < 0 Y •   / / K ( X • , ( σ > 0 Y • )[ 1 ]) θ X • , ( σ > 0 Y • )[ 1 ]   D ( X • , ( σ < 0 Y • )[ − 1 ]) / / D ( X • , σ > 0 Y • ) / / D ( X • , Y • ) / / D ( X • , σ < 0 Y • ) / / D ( X • , ( σ > 0 Y • )[ 1 ]) By assumpti on, the complex σ < 0 Y • is a bounded comple x of injecti ve A -modules. So, the maps θ X • , ( σ < 0 Y • )[ − 1 ] and θ X • , σ < 0 Y • are isomorphisms . By [6, L emmma 2.2], the map θ X • , σ > 0 Y • is an isomorphis m. W e ha ve already pro ved tha t the map θ X • , ( σ > 0 Y • )[ 1 ] is monic. T hen by app lying the Five Lemma again , the proof is completed. No w we fix some nota tions for the rest of this sect ion. Let F : D b ( A ) − → D b ( B ) be an iterated almost ν -stab le deri ved equi val ences between two A rtin algebra s A and B , and let G be a quasi-in verse of F . Let T • and ¯ T • be the tilting comple xes associated to F and G , respecti vely . T hen by Theorem 1.1 and L emma 3.1, the terms of T • in non- zero degre es are all in ν A -Stp, and the terms of ¯ T • in non-zer o degrees are all in ν B -Stp. Kee ping these notation s, we hav e the follo wing lemma. 11 Lemma 4.3. F or each A -module X , the comple x F ( X ) is isomorphic in D b ( B ) to a radica l comple x ¯ T • X with ¯ T ± X ∈ ν B -Stp . More over , the comple x ¯ T • X of this form is unique up to isomorphis m in C b ( B ) . In particular , if X is a pr ojecti ve (r espectively , injective ) module, then ¯ T • X is isomorphic in C b ( B ) to a comple x in add ( ¯ T • ) (r espectively , add ( ν B ¯ T • ) ). Pr oof. By the proof of Theorem 1.1, we see that F ≃ F 2 F − 1 1 for two almost ν -stable deri ved equi valen ces F 1 : D b ( C ) − → D b ( A ) and F 2 : D b ( C ) − → D b ( B ) . For each A -module X , by [6, Lemma 3.2] and the definition of almost ν -st able deriv ed equi v alences, we see that F − 1 1 ( X ) is isomorphic in D b ( C ) to a comple x Q • X with Q i X = 0 for all i > 0 and Q i X ∈ ν C -Stp for all i < 0. Applying F 2 to the distingu ished triangle ( σ < 0 Q • X )[ − 1 ] − → Q 0 X − → Q • X − → σ < 0 Q • X , we get a distin guished triangl e in D b ( B ) F 2 ( σ < 0 Q • X )[ − 1 ] − → F 2 ( Q 0 X ) − → F 2 ( Q • X ) − → F 2 ( σ < 0 Q • X ) . Since ( σ < 0 Q • X )[ − 1 ] is a complex in K b ( ν C -Stp ) , by Lemm a 3.2, the complex F 2 ( σ < 0 Q • X )[ − 1 ] is isomorphic in D b ( B ) to a complex U • in K b ( ν B -Stp ) . By [6 , Lemm a 3.1] and the definition of almost ν -st able deriv ed equi valenc es, the complex F 2 ( Q 0 X ) is isomorphic in D b ( B ) to a comple x V • with V i ∈ ν B -Stp for all i > 0 and V i = 0 fo r all i < 0. T hus, the co mplex F 2 ( Q • X ) , which is isomorph ic in D b ( B ) to F ( X ) , is iso morphic in D b ( B ) to the mapping cone con ( α ) of a chain m ap α from U • to V • . Now it is clear that all the terms of con ( α ) in non-ze ro degrees are in ν B -Stp. T aking a radical comple x ¯ T • X which is isomorphi c to con ( α ) in K b ( B ) , we see that F ( X ) is isomorphic to ¯ T • X and ¯ T ± X ∈ ν B -Stp. Suppose that W • is another radical comple x w ith W ± ∈ ν B -Stp, and F ( X ) ≃ W • . Then W • and ¯ T • X are isomorph ic in D b ( B ) . By Lemma 4.2, the y are isomorphi c in K b ( B ) . Since both W • and ¯ T • X are radical comple xes, the y are also isomorphic in C b ( B ) . Since all the complex es in add ( ¯ T • ) and add ( ν B ¯ T • ) hav e the desired form, the last statement follows by the uniqu eness of ¯ T • X . In the follo w ing, withou t loss o f gener ality , we fix for each A -module X a compl ex ¯ T • X defined in L emma 4 .3 and assume th at F ( X ) = ¯ T • X for al l A -modules X . Let X and Y be two A -modules . There is a n atural i somorphis m Hom A ( X , Y ) ≃ Hom D b ( B ) ( ¯ T • X , ¯ T • Y ) sendin g f to F ( f ) . B y Lemma 4.2, there is a natural isomorph ism Hom K b ( B ) ( ¯ T • X , ¯ T • Y ) ≃ Hom D b ( B ) ( ¯ T • X , ¯ T • Y ) induce d by the localizat ion functor from K b ( B ) to D b ( B ) . It is easy to see that there is a natural map Hom K b ( B ) ( ¯ T • X , ¯ T • Y ) − → Hom ν B ( ¯ T 0 X , ¯ T 0 Y ) sendin g u • to u 0 . Indeed, if u • = v • in Hom K b ( B ) ( ¯ T • X , ¯ T • Y ) , then u 0 − v 0 fact orizes through ¯ T 1 X ⊕ ¯ T − 1 Y which is in ν B -Stp by definition . This means u 0 − v 0 = 0 in Hom ν B ( ¯ T 0 X , ¯ T 0 Y ) . Altogether , we hav e a natural morphism φ : Hom A ( X , Y ) − → Hom ν B ( ¯ T 0 X , ¯ T 0 Y ) sendin g f to u 0 , where u • is a chain map such that u • = F ( f ) . N o w if f factor izes through an A -module in ν A -Stp, the n u • fact orizes through a complex P • in K b ( ν B -Stp ) by Lemma 3.2. By Lemma 4.2, we can assume that u • = g • h • in K b ( B ) for chain maps g • : ¯ T • X − → P • and h • : P • − → ¯ T • Y . Thus, it follo ws that u 0 − g 0 h 0 fact orizes through ¯ T 1 X ⊕ ¯ T − 1 Y , and conseq uently u 0 fact orizes through P 0 ⊕ ¯ T 1 X ⊕ ¯ T − 1 Y which is in ν B -Stp. Hence u 0 = 0. Hence we get a natur al morphism ¯ φ : Hom ν A ( X , Y ) − → Hom ν B ( ¯ T 0 X , ¯ T 0 Y ) 12 No w we define a functor φ F : A -mod ν − → B -mod ν . For each A -module X , we set φ F ( X ) : = ¯ T 0 X , and for each morphism f ∈ Hom ν A ( X , Y ) , we define φ F ( f ) : = u 0 , where u • = F ( f ) . Now i t is easy to see that the diagram A -mod ν Σ / / φ F   D b ( A ) / K b ( ν A -Stp ) F   B -mod ν Σ / / D b ( B ) / K b ( ν B -Stp ) ( ♣ ) is commutati ve up to isomorp hism. Indeed, o ne can check that the isomorph ism ¯ T • X s ⇐ = σ > 0 ¯ T • X t = ⇒ X in D b ( B ) / K b ( ν B -Stp ) with s and t the canonic al maps is a natural map, and this gi ves rise to an isomorp hism from the funct or F Σ to the functo r Σφ F . For an Artin alg ebra, in t he follo wing theorem, we deno te by A - mod the quotie nt categor y of A -mod modulo morphisms f actorizin g throu gh injecti ve modules. Theor em 4.4. Let F : D b ( A ) − → D b ( B ) be an iterat ed almost ν -stable derived equivalence . Then we have the followin g: ( 1 ) The functo r φ F : A -mod ν − → B -mod ν is an equiva lence; ( 2 ) The functo r φ F induce s an equivale nce between A - mod and B- mod ; ( 3 ) The functo r φ F induce s an equivale nce between A -mod and B -mod ; ( 4 ) The func tor φ F is un iquely (up to isomorp hism) determined by th e commutati ve diagr am ( ♣ ) . Mor eove r , if F ′ : D b ( B ) − → D b ( C ) is anoth er iterate d almost ν -stabl e derived equivalen ce, then φ F ′ F ≃ φ F ′ φ F . Pr oof. L et G be a quasi-in ver se of F . Then G also induces an equi va lence between D b ( B ) / K b ( ν B -Stp ) and D b ( A ) / K b ( ν A -Stp ) . W e also denote it by G . Then by the abov e commutati ve diagram of addi tiv e functors, the functo r Σ φ G φ F is isomorphic to the functor GF Σ , which is isomorphi c to Σ . By Proposition 4.1 , the functor Σ is a fully faithful embedding. H ence φ G φ F is isomorphic to 1 A -mod ν . By symmetry , the functor φ F φ G is also isomorph ic to 1 B -mod ν . Hence φ F is an equi v alence , and ( 1 ) is pro ved. By the construct ion of φ F , it follows from Lemma 4.3 that φ F sends projec tiv e modules to projecti ve mod- ules, and sends injecti ve modules to injecti ve modu les. Moreov er , the modules in ν A -Stp and ν B -Stp are all projec tiv e-inje cti ve. Thus, the stat ements ( 2 ) and ( 3 ) follow . (4) If φ : A -mod ν − → B -mod ν is a functor such that Σφ ≃ F Σ , then the fun ctor Σφ is isomor phic to Σφ F . Hence φ ≃ φ F since Σ is fully fait hful. The rest of (4) follo ws similarly . Remark: (1) It f ollo ws from the d efinition of itera ted almost ν -stab le deri ved equi valen ces and [6, Theorem 3.7] that ev ery iterate d almost ν -stab le deri ved equiv alence induces an equi val ence between the stable m odule cate gories, ho wev er , the proof of Theor em 4.4 prese nted here is not b ased on the ear lier result [6, Thoer em 3.7], and is complet ely dif ferent from the proof there. M oreo ver , Theorem 4.4 is more general than [6, Theorem 3.7] since we get an equi valen ce between A -mod ν and B -mod ν which is not obtain ed in [6, Theorem 3.7]. (2) In case that F is an almost ν -st able de riv ed equiv alence, it follo ws by definiti on that the stable equi vale nce from A -mod to B -mod induced by the functor φ F coinci des with the stable functor ¯ F consider ed in [10]. 5 Constructions of iterated almost ν -stable deri ved equiv alences In this secti on, we shall gi ve some constr uctions of iterated almost ν -stabl e deriv ed equi v alence s. Let us recall from [2] the definition of approximati ons. Let C be a catego ry , and let D be a full subcate gory of C , and X an object in C . A morphism f : D − → X in C is called a right D - appr oximation of X if D ∈ D and the induced map Hom C ( D ′ , f ) : Hom C ( D ′ , D ) − → Hom C ( D ′ , X ) is surjecti ve for ev ery object D ′ ∈ D . D ually , one can define left D - appr oximation s . 13 By Theorem 1.1, to get an iterated almost ν -stable deri ved equi v alence, we only need to const ruct a deriv ed equi valenc e w ith the in vol ved tilting complex es satisfyin g the conditi ons in Theorem 1.1. Let A be an algebra, and let P , Q be two pr ojecti ve A -modules satisfyi ng the follo wing two conditio ns: (1) add ( A P ) = add ( ν A P ) , add ( A Q ) = add ( ν A Q ) ; (2) Hom A ( P , Q ) = 0 . For e ach positi ve inte ger r , we can form the follo wing comple x: 0 − → P − r f r − →− → P − r + 1 − → · · · − → P − 1 f 1 − → A − → 0 , where f 1 : P − 1 − → A is a right add ( A P ) -appro ximation of A , and f i + 1 : P − i − 1 − → P − i is a righ t add ( A P ) - approx imation of Ker ( f i ) for i = 1 , · · · , r − 1. Similarly , we can form a complex 0 − → A g 1 − → Q 1 − → · · · − → Q s − 1 g s − → Q s − → 0 , where g 1 is a left add ( A Q ) -appro ximation of A , and g i + 1 is a left add ( A Q ) -appro ximation of Coker ( g i ) for i = 1 , 2 , · · · , s − 1. Since Hom A ( P , Q ) = 0, con necting the two comple xe s togethe r , we get a comple x 0 − → P − r − → · · · − → P − 1 f 1 − → A g 1 − → Q 1 − → · · · − → Q s − → 0 , where A is in de gree zero. W e denote this comple x by T • P , Q , and let T • : = T • P , Q ⊕ P [ r ] ⊕ Q [ − s ] . Pro position 5.1. K eepin g the nota tions above , we have the following: ( 1 ) The comple x T • is a tiltin g comple x. ( 2 ) Let B : = End D b ( A ) ( T • ) . Then T • induce s an iterate d almost ν -stable derived equivalenc e between the alg ebras A and B. Pr oof. (1) B y the constr uction of T • , we ha ve T i =                P − r ⊕ P , i = − r ; P i , − r < i < 0 ; A , i = 0 ; Q i , 0 < i < s ; Q s ⊕ Q , i = s ; 0 otherwis e. , and d i T =             f r 0  , i = − r ; f − i , − r < i < 0 ; g i + 1 , 0 6 i < s − 1 ;  g s 0  , i = s − 1 ; 0 otherwis e. W e first sho w that Hom K b ( A -proj ) ( T • , T • [ i ]) = 0 for all i 6 = 0. Assume that i is a positi ve intege r . Let u • be a morphis m in Hom K b ( A -proj ) ( T • , T • [ i ]) . Then we ha ve the follo wing commutati ve diagram · · · / / T − i − 1 d − i − 1 T / / u − i − 1   T − i d − i T / / u − i   T − i + 1 d − i + 1 T / / u − i + 1   · · · / / T − 1 d − 1 / / u − 1   T 0 d 0 T / / u 0   T 1 d 1 T / / u 1   · · · · · · / / T − 1 d − 1 T / / T 0 d 0 T / / T 1 d 1 T / / · · · / / T i − 1 d i − 1 T / / T i d i T / / T i + 1 d i + 1 T / / · · · Since Hom A ( P , Q ) = 0, we hav e u k = 0 for all − i < k < 0. By definitio n, T − i ∈ add ( A P ) . Since d − 1 T = f 1 is a right add ( A P ) -appro ximation, there is a map h − i : T − i − → T − 1 such that u − i = h − i d − 1 T . Thus, ( u − i − 1 − d − i − 1 T h − i ) d − 1 T = d − i − 1 T u − i − d − i − 1 T h − i d − 1 T = d − i − 1 T u − i − d − i − 1 T u − i = 0. Since d − 2 T is a right ad d ( A P ) -appro ximation of K er ( d − 1 T ) , ther e is a map h − i − 1 : T − i − 1 − → T − 2 such that u − i − 1 − d − i − 1 T h − i = h − i − 1 d − 2 T , that is u − i − 1 = d − i − 1 T h − i + h − i − 1 d − 2 T . Similarly , for each inte ger k < − i − 1, there are map s h k + 1 : T k + 1 − → T k + i and h k : T k − → T k + i − 1 such that u k = d k T h k + 1 + h k d k + i − 1 T . Defining h k = 0 for all − i < k 6 0, we hav e u k = d k T h k + 1 + h k d k + i − 1 T 14 for all k < 0. Similarly , we can prov e that u k = d k T h k + 1 + h k d k + i − 1 T for k > 0. Altogeth er , we ha ve shown that u • = 0 in K b ( A -proj ) . Hence Hom K b ( A -proj ) ( T • , T • [ i ]) = 0 for all i > 0. By an analog ous proof, w e hav e Hom K b ( A -proj ) ( T • , T • [ i ]) = 0 for all i < 0. F inally , since P [ r ] and Q [ − s ] are in add ( T • ) , we deduc e that A A is in the tri angulate d subca tegory of K b ( A -proj ) generated by add ( T • ) . Hence add ( T • ) genera tes K b ( A -proj ) as a triang ulated categ ory , and consequentl y T • is a tilting comple x over A . (2) Let F : D b ( A ) − → D b ( B ) be a deri ved equi va lence induce d by T • . Also, we use F to denote the equi valenc e between K b ( A -proj ) and K b ( B -proj ) induced b y F . S et ¯ P : = Hom K b ( A -proj ) ( T • , P [ r ]) ≃ F ( P [ r ]) , ¯ Q : = Hom K b ( A -proj ) ( T • , Q [ − s ]) ≃ F ( Q [ − s ]) , and ¯ U : = H om K b ( A -proj ) ( T • , T • P , Q ) ≃ F ( T • P , Q ) . T hen F ( P ) ≃ ¯ P [ − r ] and F ( Q ) ≃ ¯ Q [ s ] . For simplic ity , we list some subc omplex es of T • : P • : 0 − → P − r − → · · · − → P − 1 − → 0 , Q • : 0 − → Q 1 − → · · · − → Q s − → 0 , R • : 0 − → A − → Q 1 − → · · · − → Q s − → 0 . By Lemma 2.1, F ( P • ) is iso morphic to a comple x ¯ P • in K b ( add ( ¯ P )) such that ¯ P i = 0 for all i < 0 and all i > r . The complex F ( Q • ) is isomorph ic to a complex ¯ Q • in K b ( add ( ¯ Q )) with ¯ Q i = 0 for all i > 0 and all i 6 − s . Note that there is a disti nguished triangle in K b ( A -proj ) P • [ − 1 ] − → R • − → T • P , Q − → P • . Applying F , we get a distinguis hed triangl e in K b ( B -proj ) : F ( P • )[ − 1 ] − → F ( R • ) − → F ( T • P , Q ) − → F ( P • ) . Hence F ( R • ) is isomorp hic to a complex of the foll owin g form: 0 − → ¯ U − → ¯ P 0 − → ¯ P 1 − → · · · with ¯ U in deg ree 0. Next we h av e a distinguis hed triangle Q • − → R • − → A − → Q • [ 1 ] in K b ( A -proj ) . Applying F , we see that F ( A ) is isomorp hic to a complex ¯ T • of the form · · · − → ¯ Q − 1 − → ¯ Q 0 − → ¯ U − → ¯ P 0 − → ¯ P 1 − → · · · , where ¯ U is in degree zero. No te that ¯ T • is a tilting comple x associated to F − 1 since F − 1 ( ¯ T • ) ≃ A . S ince add ( A P ) = add ( ν A P ) and add ( A Q ) = add ( ν A Q ) , we ha ve ad d ( B ¯ P ) = add ( ν B ¯ P ) and add ( B ¯ Q ) = add ( ν B ¯ Q ) . Thus, we ha ve add ( A T ± ) = add ( ν A T ± ) and add ( B ¯ T ± ) = add ( ν B ¯ T ± ) . By Theorem 1.1, the stat ement ( 2 ) follo ws. T o illustra te Propositio n 5.1, we giv e an example . Let A be the finite-dimens ional k -algebr a gi ven by the qui ver • α / / • 1 2 α ′ o o β / / • 3 β ′ o o γ / / • 4 γ ′ o o with relations α ′ α = ββ ′ = αβ = βγ = β ′ α ′ = γ ′ β ′ = β ′ β − γγ ′ = 0. W e use P i to denote the indecomposab le projec tiv e A -module corresp onding to the v ertex i for i = 1 , 2 , 3 , 4. The Loewy structur e of the proj ecti ve A - modules can be listed as follo ws. P 1 : 1 2 1 P 2 : 2 1 3 P 3 : 3 2 4 3 P 4 : 4 3 4 15 Let P : = P 1 and Q : = P 3 ⊕ P 4 . Then we ha ve add ( A P ) = add ( ν A P ) , add ( A Q ) = add ( ν A Q ) , and Hom A ( P , Q ) = 0. Using Propo sition 5.1 , we ha ve a tilting comple x T • ov er A . The indecompo sable direct summands of T • are: T • 1 : 0 − → P 1 − → 0 T • 2 : 0 − → P 1 − → P 2 − → P 3 − → 0 T • 3 : 0 − → P 3 − → 0 T • 4 : 0 − → P 4 − → 0 A calcu lation sho ws that the algebra B : = End D b ( A ) ( T • ) is gi ven by the qui ver • δ y y s s s s s s • α / / • 1 2 α ′ o o β % % K K K K K K • 4 3 γ O O with relatio ns α ′ α = αβ = δα ′ = βγδ = γδβγ = 0. By Propositio n 5.1, T • induce s an iterated almost ν -stable deri ve d equiv alence between A and B . Therefore, A and B are also stably equi va lent of Morita type. The fol lowin g propos ition sho ws ho w we can con struct iterated almost ν -stable deri ve d equiv alences induc- ti vely . Pro position 5.2. L et F : D b ( A ) − → D b ( B ) be an iterat ed almost ν -stable deri ved equivalenc e between two finite- dimensiona l alg ebra s A and B over a field k, and let φ F be the stable equival ence induced by F (see, Theorem 4.4 ). Then we have the following: ( 1 ) F or each A -module X , ther e is an iterate d almost ν -stabl e deriv ed equ ivalence between the endomor - phism alg ebr as End A ( A ⊕ X ) and End B ( B ⊕ φ F ( X )) ; ( 2 ) F or a finite-dimens ion self-injecti ve k -alge bra C , ther e is an iter ated almost ν -s table der ived eq uivalenc e between A ⊗ k C and B ⊗ k C . Pr oof. S uppos e that F ≃ F 1 F 2 · · · F n such that F i or F − 1 i is almost ν -stabl e for all i . By T heorem 4.4, we ha ve φ F ≃ φ F 1 φ F 2 · · · φ F n . By the remark after Theorem 4.4, we kno w that φ F i coinci des with the ¯ F i consid ered in [6 ] for all i . Thus, the stat ements (1) follo ws from [6, Corollary 1.3]. The proof of (2) is similar to that of [6, Proposit ion 6.2]. Let us recal l from [7 ] the definitio n of Φ -Auslan der -Y oneda algebras. A subse t Φ of the set of natur al numbers N is called admissibl e provi ded that: (1) 0 ∈ Φ ; (2) If i + j + k ∈ Φ for i , j , k ∈ Φ , then i + j ∈ Φ implies that j + k ∈ Φ . For instance, the sets N , { 0 , 1 , · · · , n } are admissible subsets of N . S uppos e that Φ be an admissible subset of N . Let A be an Artin algebra, and let X be an A -module. Now we consider the Y oneda algebr a Ext ∗ A ( X , X ) = L i > 0 Hom D b ( A ) ( X , X [ i ]) of X , and define E Φ A ( X ) : = L i ∈ Φ Hom D b ( A ) ( X , X [ i ]) with multiplic ation: for a i ∈ Hom D b ( A ) ( X , X [ i ]) and a j ∈ Hom D b ( A ) ( X , X [ j ]) , we d efine a i · a j = a i a j if i + j ∈ Φ , and zero otherwis e. T hen one can chec k that E Φ A ( X ) is an associat ed algebra. If Φ = { 0 } , then E Φ A ( X ) is isomorphic to End A ( X ) . If Φ = N , then E Φ A ( X ) is just the Y oneda algebra of X . Pro position 5.3. L et F : D b ( A ) − → D b ( B ) be an iterat ed almost ν -stable deri ved equivalenc e between two Artin alg ebra s A and B. Suppose that Φ is an admissib le subset of N . Then we have the following: ( 1 ) F or any A-module X , ther e is a derived equivalence between the Φ -Ausla nder -Y oneda alg ebra s E Φ A ( A ⊕ X ) and E Φ B ( B ⊕ φ F ( X )) ; ( 2 ) If Φ is a finite set, then for any A-module X , ther e is an itera ted almost ν -stable derived equivalen ce between E Φ A ( A ⊕ X ) and E Φ B ( B ⊕ φ F ( X )) . Pr oof. Using the result [7, Theorem 3.4], the proof is similar to that of Proposi tion 5.2 (1). 16 Ackno wledgement This wo rk is partia lly supported by China Pos tdoctora l Science Fou ndation (No. 2008044000 3). The auth or also thank s the Alexand er v on Humboldt Founda tion for support . Refer ences [1] S . A L - N O F A Y E E and J . R I C K A R D , Rigidity of tilting comp lex es and deri ve d equiv alence for self-inje cti ve algebr as. preprint, 2004. [2] M . A U S L A N D E R and S . O . S M A L Ø , Preprojecti ve modules over Artin algebras . J . Algebr a 66 (1980) 61-12 2. [3] M . 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W E I H U , School of Mathematical Sciences, Beijing Normal Univ ersity , Beijing, 100875 , China 17