Polygon of recollements and N-complexes

arXiv:1603.06056v1 [math.CT] 19 Mar 2016 POLYGON OF RECOLLEMENTS AND N -COMPLEXES OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI Abstract. We study a structure of subcategories which are called a polygon of recollements in a triangulated category. First, we study a 2n-gon of recollements in an (m/n)-Calabi-Yau triangulated category. Second, we show the homotopy category K(MorN−1 (B)) of complexes of an additive category MorN−1 (B) of N −1 sequences of split monomorphisms of an additive category B has a 2N -gon of recollments. Third, we show the homotopy category KN (B) of N -complexes of B has also a 2N -gon of recollments. Finally, we show there is a triangle equivalence between K(MorN−1 (B)) and KN (B). Contents 0. Introduction 1. Stable t-structures and recollements 2. Contravariantly finite subcategories and Stable t-structures 3. Constructions of Stable t-structures 4. Recollement of K(Morsm N −1 (B)) 5. Recollement of KN (B) 6. Triangle equivalence between homotopy categories 7. Appendix References 1 3 5 7 9 14 21 29 31 0. Introduction The notion of recollement of triangulated categories was introduced by Beilinson, Bernstein and Deligne in connection with derived categories of sheaves of topological spaces ([BBD]). One of the authors introduced the notion of stable t-structure in a triangulated category [Mi], and studied relations to recollements. Afterwards this notion was studied by many authors under a lot of names, e.g. a torsion pair, a semiorthogonal decomposition, Bousfield localization. Definition 0.1. Let D be a triangulated category with the translation functor Σ. A pair (U, V) of full subcategories of D is called a stable t-structure in D provided that (a) U = ΣU and V = ΣV. (b) HomD (U, V) = 0. (c) For every X ∈ D, there exists a triangle U → X → V → ΣU with U ∈ U and V ∈ V. Date: November 18, 2021. 1991 Mathematics Subject Classification. 18E30, 16G99. 1 2 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI In [IKM1], we introduced the notion of polygons of recollements in a triangulated category, and studied the properties of triangle of recollements in connection with the homotopy category of unbounded complexes with bounded homologies, its quotient category by the homotopy category of bounded complexes, and the stable category of Cohen-Macaulay modules over an Iwanaga-Gorenstein ring. Moreover, we studied derived categories DN (A) of N -complexes of an abelian category A, and showDN (A) is a triangle equivalent to the derived category D(MorN −1 (A)) of ordinary complexes over an abelian category MorN −1 (A) of N − 1 sequences of morphisms of A ([IKM2]). In this article, we study the properties of polygons of recollements in various categories. Definition 0.2. Let D be a triangulated category, and let U1 , · · · , Un be full triangulated subcategories of D. An n-tuple (U1 , U2 , · · · , Un ) is called an n-gon of recollements in D if (Ui , Ui+1 ) is a stable t-structure in D (1 ≤ i ≤ n), where U1 = Un+1 . In Section 1, we recall the notions of stable t-structures and recollement polygons of recollements in a triangulated category. Proposition 0.3 (Proposition 1.7). Let D1 , D2 be triangulated categories. Let (U1 , · · · , Un ) and (V1 , · · · , Vn ) be n-gons of recollements in D1 and D2 , respectively. Assume a triangle functor F : D1 → D2 sends (U1 , · · · , Un ) to (V1 , · · · , Vn ). If F |Ut and F |Ut+1 are triangle equivalences for some t, so is F . In Section 1, we study stable t-structures with relation to relations to contravariantly finite categories and Calabi-Yau triangulated categories. Proposition 0.4 (Proposition 2.5). Let D be an (m/n)-Calabi-Yau triangulated category. For any functorially finite thick subcategory U1 of D, we put Ui+1 := Ui⊥ for any i. Then we have an l-gon (U1 , · · · , Ul ) of recollements in D for some positive divisor l of 2n. In Section 3, we constructs polygons of recollements in the derived derived category of modules over an algebra, and them in the stable category of Cohen-Mcaulay modules over an Iwanaga-Gorenstein ring. Theorem 0.5 (Theorem 3.6). Let A be a finite dimensional k-algebra of finite global dimension such that A/JA is separable over a field k and Db (mod A) is (m/n)Calabi-Yau. Let R be a coherent k-algebra of finite self-injective dimension as both sides. For any functorially finite thick subcategory U1 of Db (mod A), we put Ui+1 := Ui⊥ for any i. Then there is a positive divisor l of 2n such that we have an l-gon (U1R , · · · , UlR ) of recollements in Db (mod R⊗k A) and an l-gon (Q(U1R ), · · · , Q(UlR )) of recollements in CM(R ⊗k A) sm In Section 4, we the homotopy category K(Morsm N−1 (B)) of the category MorN −1 (B) of N − 1 sequences of split monomorphisms in an additive category B. Theorem 0.6 (Theorem 4.8). Let B be an additive category. Then there is a 2N -gon of recollements in K(Morsm N −1 (B)): (F [1,N−1] , E [2,N−1] , E 1 , F [1,2] , · · · , E s , F [s,s+1] , · · · , E N−2 , F [N−2,N−1] , E N−1 , E [1,N−2] ) In Section 5, we study the homotopy category KN (B) of N -complexes of objects of an additive category B. POLYGON OF RECOLLEMENTS 3 Theorem 0.7 (Corollary 5.10). We have a recollement of KN (B): i∗ s is∗ | KN −r (B) b | / KN (B) b i!s js! js∗ js∗ / Kr+1 (B) Corollary 0.8 (Corollary 5.11). There is a 2N -gon of recollements in KN (B): N −2 N −2 N −2 1 1 , FN , Fr1 , · · · , FN (F1N −2 , F01 , F2N −2 , F11 , · · · , Fr+1 −1 ) −1 , FN −2 , F0 In Section 6, we construct a triangle functor F N : K(Morsm N −1 (B)) → KN (B) which sends the above 2N -gon of KN (B) to the above 2N -gon of KN (B). Therefore we have the result. Theorem 0.9 (Theorem 6.8). Let B be an additive category, then we have triangle equivalences: ♯ K♯ (Morsm N −1 (B)) ≃ KN (B) where ♯ = nothing, −, +, b. 1. Stable t-structures and recollements We recall the notion of recollements and study their relationship with stable t-structures. This correspondence enables us to understand (co)localizations and recollements by way of subcategories instead of quotient categories. In Proposition 1.2 we see that a recollement corresponds to two consecutive stable t-structures. First we see that a (co)localization and a stable t-structure essentially describe the same phenomenon, using the methods which are similar to ones in recollements [BBD]. Next we recall the notion of a recollement which consists of a localization and a colocalization. Definition 1.1 ([BBD]). We call a diagram i∗ i∗  ′ DZ i! j! j∗  / DZ j∗ ′′ /D of triangulated categories and functors a recollement if it satisfies the following: (1) i∗ , j! , and j∗ are fully faithful. (2) (i∗ , i∗ ), (i∗ , i! ), (j! , j ∗ ), and (j ∗ , j∗ ) are adjoint pairs. (3) there are canonical embeddings Im j! ֒→ Ker i∗ , Im i∗ ֒→ Ker j ∗ , and Im j∗ ֒→ Ker i! which are equivalences. We remember that a recollement corresponds to a pair of consecutive stable t-structures. Proposition 1.2 ([Mi]). (1) Let ′ DZ  i∗ i∗ i! / DZ  j! j∗ j∗ ′′ /D be a recollement. Then (U, V) and (V, W) are stable t-structures in D where we put U = Im j! , V = Im i∗ and W = Im j∗ . 4 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI (2) Let (U, V) and (V, W) be stable t-structures in D. Then for the canonical embedding i∗ : V → D, there is a recollement VY  i∗ i∗ i! / D[ j! j∗  j∗ / D/V such that Im j! = U and Im j∗ = W. In each cases, for every object X of D adjunction arrows of adjoints induce triangles i∗ i! X → X → j∗ j ∗ X → Σi∗ i! X, j! j ∗ X → X → i∗ i∗ X → Σj! j ∗ X. Thirdly, we introduce the notion of a polygon of recollements. Definition 1.3. Let D be a triangulated category, and let U1 , · · · , Un be full triangulated subcategories of D. We call (U1 , U2 , · · · , Un ) an n-gon of recollements in D if (Ui , Ui+1 ) is a stable t-structure in D (1 ≤ i ≤ n), where U1 = Un+1 . An n-gon of recollements results in strong symmetry, and it induces three recollements as the name suggests. Proposition 1.4 ([IKM1]). Let D be a triangulated category. Then (U1 , · · · , Un ) is an n-gon of recollements in D if and only if there is a recollement  UtZ i∗ t it∗ i!t / D[  jt! jt∗ jt∗ / D/Ut such that the essential image Im jt! is Ut−1 , and that the essential image Im jt∗ is Ut+1 for any t mod n. In this case all the relevant subcategories Ut and the quotient categories D/Ut are triangle equivalent. Finally we study the case that triangle functors preserve localizations, colocalizations or recollements, etc. Definition 1.5. Let D1 and D2 be triangulated categories and let F : D1 → D2 be a triangle functor. (1) Let (Un , Vn ) be a stable t-structure in Dn (n = 1, 2). We say that F sends (U1 , V1 ) to (U2 , V2 ) if F (U1 ) is contained in U2 and F (V1 ) is in V2 . (2) Let (Uin , · · · , Uin ) be an n-gon of recollements in Di (i = 1, 2). We say that F sends (U1n , · · · , U1n ) to (U2n , · · · , U2n ) if F (U1k ) is contained in U2k for any k. Lemma 1.6 ([IKM1]). If a triangle functor F : D1 → D2 sends a stable t-structure (U1 , V1 ) in D1 to a stable t-structure (U2 , V2 ) in D2 . Then we have the following: (1) If F |U1 is full (resp., faithful), then HomD1 (U, X) → HomD2 (F U, F X) is surjective (resp., injective) for U ∈ U1 and X ∈ D1 . (2) If F |V1 is full (resp., faithful), then HomD1 (X, V ) → HomD2 (F X, F V ) is surjective (resp., injective) for X ∈ D1 and V ∈ V1 . (3) If F is full and F |U1 : U1 → U2 and F |V1 : V1 → V2 are dense, then F is dense. POLYGON OF RECOLLEMENTS 5 Proposition 1.7. Let D1 , D2 be triangulated categories. Let (U1 , · · · , Un ) and (V1 , · · · , Vn ) be n-gons of recollements in D1 and D2 , respectively. Assume a triangle functor F : D1 → D2 sends (U1 , · · · , Un ) to (V1 , · · · , Vn ). Then the following hold. (1) In the case that n is odd, if F |Ut is fully faithful (equivalent) for some t, so is F . (2) In the case that n is even, if F |Ut and F |Ut+1 are fully faithful (equivalent) for some t, so is F . Proof. By Lemma 1.6.  2. Contravariantly finite subcategories and Stable t-structures In this section let k be a field and D := Homk (−, k). The concept of stable t-structures is closely related to functorially finite subcategories [AS]. If (U, V) is a stable t-structure of D, then clearly U (resp., V) is a contravariantly (resp., covariantly) finite subcategory of D. We shall show that a certain converse of this statement holds. For a full subcategory U of D, we put U⊥ := {T ∈ D | HomD (U, T ) = 0}, ⊥ := {T ∈ D | HomD (T, U) = 0}. U Recall that an additive category is called Krull-Schmidt if any object is isomorphic to a finite direct sum of objects whose endomorphism rings are local. Definition 2.1. Let C be an additive category, and U its full subcategory. For an f → X with UX ∈ U is called a right U-approximation object X of C, a morphism UX − if HomC (U, f ) : HomC (U, UX ) → HomC (U, X) is surjective for any U ∈ U. Moref → X is called minimal if g is an isomorphism over, a right U-approximation UX − whenever g : UX → UX satisfies f ◦ g = f . A full subcategory U is called a contravariantly finite subcategory if every object X of C has a right a U-approximation. A left a U-approximation and a covariantly finite subcategory are defined dually. U is called a functorially finite subcategory if it is contravariantly finite and covariantly finite. Proposition 2.2. Let D be a Krull-Schmidt triangulated category. For any contravariantly (resp., covariantly) finite thick subcategory U of D, we have a stable t-structure (U, U ⊥ ) (resp., (⊥ U, U)) in D. Proof. This is a consequence of Wakamatsu-type Lemma (see [IY, Prop. 3.6] for example). For the convenience of the reader, we give the proof here. Since D is Krull-Schmidt category and U is closed under direct summands, for any object X f f g h → X is a minimal →X − →V − → ΣUX such that UX − of D there is a triangle UX − right U-approximation. For any morphism α : U → V with U ∈ U, by octahedral 6 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI axiom we have a commutative diagram Σ−1 U Σ−1 U Σ−1 α  Σ−1 V −Σ−1 h  / UX  U /X g /V γ β  Z f h ′  /M Σβ f′ /X g′  / ΣZ  U where all columns and rows of 4 terms are triangles. Then M belongs to U. Since f is a right U-approximation, there is a morphism δ : M → UX such that f ◦ δ = f ′ . Then there is a morphism ǫ : Z → Σ−1 V such that −Σ−1 h ◦ ǫ = δ ◦ h′ . Then γ is a split monomorphism because of the minimality of f . Therefore β is a split monomorphism, and Σ−1 α and α are zero. Hence we have V ∈ U ⊥ .  Definition 2.3. Let D be a k-linear triangulated category such that dimk HomD (X, Y ) < ∞ for any X, Y ∈ D. An autofunctor S : D → D is called a Serre functor [BK, RV] if there exists a functorial isomorphism HomD (X, Y ) ≃ D HomD (Y, SX) for any X, Y ∈ D. We say that D is (m/n)-Calabi-Yau for a positive integer n and an integer m if we have an isomorphism S n ≃ Σm of functors 1. Immediately we have the following. Lemma 2.4. Let D be a Krull-Schmidt triangulated category with a Serre functor S. Then the following hold. (1) U ⊥ = ⊥ (SU) for any subcategory U of D. (2) If (U, V) is a stable t-structure and U is functorially finite in D, then (V, SU) is a stable t-structure and V is functorially finite in D. Proof. (1) Immediate from the definition of Serre functor. (2) We have ⊥ (SU) = U ⊥ = V by (1). Since SU is a covariantly finite subcategory of D, we have that (V, SU) is a stable t-structure by the dual of Proposition 2.2. Consequently V is a functorially finite subcategory of D by Proposition 2.2.  Proposition 2.5. Let D be an (m/n)-Calabi-Yau triangulated category. For any functorially finite thick subcategory U1 of D, we put Ui+1 := Ui⊥ for any i. Then we have an l-gon (U1 , · · · , Ul ) of recollements in D for some positive divisor l of 2n. Proof. By Propositions 2.2, we have a stable t-structure (U1 , U2 ) in D. Using Lemma 2.4(2) inductively, we have a stable t-structure (Ui , Ui+1 ) in D such that Ui+2 = SUi for any i. We have the statement because of S n ≃ Σm .  1We notice that we can not cancel a common divisor of m and n. POLYGON OF RECOLLEMENTS 7 3. Constructions of Stable t-structures In this section, we investigate polygons of recollements in derived categories and in stable module categories. For a ring R, we denote by Mod R (resp., mod R) the category of right (resp., finitely generated right) R-modules, and denote by Proj R (resp., Inj R, proj R) the full subcategory of Mod R consisting of projective (resp., injective, finitely generated projective) modules. For right (resp., left) R-module MR (resp., R N ), we denote by idim MR (resp., idim R N ) the injective dimension of MR (resp., R N ), and by pdim MR (resp., pdim R N ) the projective dimension of MR (resp., R N ). For A be an abelian category and its addtive subcategory B. we denote by Db (A) (resp., Kb (B) the derived category (resp., the homotopy category) of bounded complexes of objects of A (resp., B). Definition 3.1. We call a ring R Iwanaga-Gorenstein if it is Noetherian with idimR R < ∞ and idim RR < ∞ [Iw]. We define the category of Cohen-Macaulay R-modules 2 and the category of large Cohen-Macaulay R-modules by CM R LCM R := {X ∈ mod R | ExtiR (X, R) = 0 (i > 0)}, := {X ∈ Mod R | ExtiR (X, Proj R) = 0 (i > 0)}. Then CM R forms a Frobenius category with the subcategory proj R of projectiveinjective objects, and the stable category CMR forms a triangulated category [Ha]. By [IKM1] there exist triangle equivalences CMR ≃ Db(mod R)/ Kb (proj R), LCMR ≃ Db(Mod R)/ Kb (Proj R). For subcategories U and V of a triangulated category D, we put U ∗ V := {X ∈ D | U → X → V → ΣU is a triangle in D (U ∈ U, V ∈ V)}. By octahedral axiom, we have (U ∗ V) ∗ W = U ∗ (V ∗ W). Lemma 3.2. Let D be a triangulated subcategory, and U and V triangulated (resp., thick) subcategories of D satisfying HomD (U, V) = 0. Then U ∗ V is a triangulated (resp., thick) subcategory of D. Proof. We only have to show (U ∗ V) ∗ (U ∗ V) ⊂ U ∗ V. Since HomD (U, ΣV) = 0, we have V ∗ U = add{U, V}. Thus we have (U ∗ V) ∗ (U ∗ V) = U ∗ (V ∗ U) ∗ V = U ∗ add{U, V} ∗ V ⊂ (U ∗ U) ∗ (V ∗ V) = U ∗ V, where add{U, V} is the additive subcategory of consisting of finite direct sums of objects of U and V. Thus U ∗ V is a triangulated subcategory of D. If U and V are closed under direct summand, then so is U ∗ V (e.g. [IY, Prop. 2.1]). Thus the assertion for thick subcategories follows.  Let A and R be k-algebras. For a subcategory U of Db (mod A), we denote by U the thick subcategory of Db (mod R ⊗k A) generated by R {L ⊗k X | L ∈ Db (mod R), X ∈ U}. The following observation gives us a lot of examples of stable t-structures in derived categories. 2 In the representation theory of orders [CR, Au, Y], there is another notion of Cohen-Macaulay modules which generalizes the classical notion in commutative ring theory. These two concepts coincide for Gorenstein orders. 8 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI Proposition 3.3. Let R be a k-algebra and A a finite dimensional k-algebra such that A/JA is a separable k-algebra, where JA is the Jacobson radical. For any stable t-structure (U, V) in Db (mod A), we have a stable t-structure (U R , V R ) in Db (mod R ⊗k A). Proof. Let D := Db (mod R ⊗k A). Since HomD (L ⊗k U, M ⊗k V ) = HomDb (mod R) (L, M ) ⊗k HomDb (mod A) (U, V ) for any L, M ∈ Db (mod R) and any U, V ∈ Db (mod A), we have HomD (U R , V R ) = 0. Since U R ∗ V R is a thick subcategory of D by Lemma 3.2, we only have to show R U ∗ V R contains mod R ⊗k A. Any R ⊗k A-module M is filtered by R ⊗k Ai+1 i modules M JA /M JA which are semisimple A-modules. We only have to show that any R ⊗k A-module N which is a semisimple A-modules belongs to U R ∗ V R . Since the map (A/JA ) ⊗k (A/JA ) → A/JA , x ⊗ y 7→ xy is a split epimorphism of Aop ⊗k A-modules, we have that the map N ⊗k (A/JA ) → N , n ⊗ y 7→ ny is a split epimorphism of R ⊗k A-modules. Since A/JA ∈ U ∗ V, we have that N ⊗k (A/JA ) ∈ U R ∗ V R . Thus N ∈ U R ∗ V R .  The following result gives a criterion for a stable t-structure in the derived category to give a stable t-structure in the stable category. Lemma 3.4 ([IKM1]). Let D be a triangulated category, C a thick subcategory of D, and Q : D → D/C the canonical quotient [Ne]. For a stable t-structure (U, V) in D, the following are equivalent, where Q(U) is the full subcategory of D/C consisting of objects Q(X) for X ∈ E. (1) (Q(U), Q(V)) is a stable t-structure in D/C. (2) (U ∩ C, V ∩ C) is a stable t-structure in C. The following example provides us a rich source of triangulated categories with Serre functors. Proposition 3.5. Let A be a finite dimensional k-algebra of finite self-injective dimension as both sides. Then the following hold. (1) Kb (proj A) has a Serre functor νA := −⊗L A (DA). L b (2) K (proj A) is (m/n)-Calabi-Yau if and only if (DA)⊗A n ≃ Σm A in Db (mod Aop ⊗k A). We have the following main result in this section. Theorem 3.6. Let A be a finite dimensional k-algebra of finite global dimension such that A/JA is separable over k and Db (mod A) is (m/n)-Calabi-Yau. Let R be a coherent k-algebra of finite self-injective dimension as both sides. For any functorially finite thick subcategory U1 of Db (mod A), we put Ui+1 := Ui⊥ for any i. Then there is a positive divisor l of 2n such that we have an l-gon (U1R , · · · , UlR ) of recollements in Db (mod R ⊗k A) and an l-gon (Q(U1R ), · · · , Q(UlR )) of recollements in CM(R⊗k A), where Q : Db (mod R⊗k A) → CM(R⊗k A) is the canonical quotient. R Proof. According to Propositions 2.5 and 3.3, we have an n-gon (U1R , U2R , · · · , U2n ) b of recollements in D (mod R ⊗k A). Since A is of finite global dimension, there exists a triangle Ui → A → Ui+1 → ΣUi POLYGON OF RECOLLEMENTS 9 with Ui ∈ Ui ∩ Kb (proj A) and Ui+1 ∈ Ui+1 ∩ Kb (proj A). Applying R ⊗k −, we have a triangle R ⊗k Ui → R ⊗k A → R ⊗k Ui+1 → ΣR ⊗k Ui R with R ⊗k Ui ∈ UiR ∩ Kb (proj R ⊗k A) and R ⊗k Ui+1 ∈ Ui+1 ∩ Kb (proj R ⊗k A). By R R  Lemma 3.4, we have a stable t-structure (Q(Ui ), Q(Ui+1 )) in CM(R ⊗k A). We have the following example of recollements by [Mi, Cor. 5.11]. Proposition 3.7. Let A be a finite dimensional k-algebra, and e an idempotent of A. Assume that ExtiA (A/AeA, A/AeA) = 0 (i > 0), pdim A (AeA) < ∞ and pdim(AeA)A < ∞. Then we have a recollement x D (mod A/AeA) e b i∗ e ie∗ i!e x b / D (mode A) je! je∗ je∗ b / D (mod eAe) In particuler, (Im je! , Im i∗e ) and (Im i∗e , Im je∗ ) are stable t-structures in Db (mod A). 4. Recollement of K(Morsm N −1 (B)) In this section we study the properties of complexes of the category of N − 1 sequences of morphisms in an additive category B. Throughout this section B is an additive category. Then the category C(B) of complexes of objects of B is a Frobenius category such that its cconflations are short exact sequences of which each term is a split exact sequence in B. Definition 4.1. We define the category Morsm N −1 (B) (resp., MorN −1 (B)) of sequences of morphisms in B as follows. • An object is a sequence of split monomorphisms (resp., morphisms) X : α1 αN −2 X 1 −−X → · · · −−X−−→ X N −1 in B. • A morphism from X to Y is an (N − 1)-tuple f = (f 1 , · · · , f N −1 ) of i morphisms f i : X i → Y i such that f i+1 αiX = αi+1 Y f for 1 ≤ i ≤ N − 2. We give technical tools to investigate the homotopy category K(Morsm N −1 (B)). Definition 4.2. For an additive functor G : B → B ′ between additive categories, α → Y for let (G ↓ 1B′ ) be a comma category, that is the category of objects G(X) − ′ sm X ∈ A, Y ∈ B . We denote by (G ↓ 1B′ ) the subcategory of (G ↓ 1B′ ) consisting α of objects G(X) − → Y , where α are split monomorphisms. Example 4.3. For 1 ≤ r < N − 1, let G : Morr (B) → MorN −r−1 (B) (resp., sm G : Morsm r (B) → MorN −r−1 (B)) be an additive functor defined by α1 αr−1 β1 β N −r−2 G(X 1 −→ · · · −−−→ X r ) = Y 1 −→ · · · −−−−−→ Y N −r−1 where Y 1 = · · · = Y N −r−1 = X r and β 1 = · · · = β N −r−2 = 1X r . Then the category (G ↓ 1MorN −r−1 (B) ) (resp., (G ↓sm 1MorN −r−1 (B) )) is equivalent to MorN −1 (B) (resp., Morsm N −1 (B)). Lemma 4.4. For an additive functor G : B → B ′ between additive categories, the following hold. 10 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI (1) Evrey complex of (G ↓sm 1B′ ) has the following form: uf G(X) −−→ C(f ) where f : Y → G(X) is a morphism of complexes of B ′ , C(f ) is the mapping cone of f , and uf is the canonical morphism. uf (2) If a complex X of B is homotopically trivial, then a complex G(X) −−→ C(f ) of (G ↓sm 1B′ ) is isomorphic to 0 → ΣB′ Y in K(G ↓sm 1B′ ), where f : Y → G(X). uf (3) For a complex G(X) −−→ C(f ) of (G ↓sm 1B′ ), if C(f ) is homotopically uf u1 trivial, then G(X) −−→ C(f ) is isomorphic to G(X) −→ C(1G(X) ) in in K(G ↓sm 1B′ ). Proof. (1) It is trivial. f (2) For a morphism Y − → G(X) of complexes of B ′ , we have a triangle in K(G ↓sm 1B′ ): / G(X) 0 1 / G(X) uf  / C(f ) /0 uf  Y f / G(X) vf  / ΣB ′ Y 1 If G(X) is homotopically trivial, then G(X) − → G(X) is 0 in K(G ↓sm 1B′ ), and uf hence G(X) −−→ C(f ) is isomorphic to 0 → ΣB′ Y . f (3) For a morphism Y − → G(X) of complexes of B ′ , we have a morphism between sm triangles in K(G ↓ 1B′ ): / G(X) 1 / G(X) 1 / 0 1 ⑧ u ⑧⑧ f ⑧  ⑧⑧ 1 ⑧ ⑧ ⑧ ⑧ / G(X) / G(X) /0 0    f / G(X)uf / C(f )vf / ΣY Y f ⑧ 1 ⑧ ⑧  ⑧  ⑧⑧ ⑧ ⑧  ⑧⑧ ⑧  ⑧ 1 / G(X) / C(1G(X) ) / ΣG(X) G(X) 0 ⑧⑧ ⑧⑧ 1 If C(f ) is homotopically trivial, then f : Y → G(X) is an isomorphism in K(B ′ ), and therefore (0, f ) : (0 → Y ) → (0 → G(X)) is an isomorphism in K(G ↓sm uf 1B′ ). By the above morphism between triangles, G(X) −−→ C(f ) is isomorphic to u1  G(X) −→ C(1G(X) ) in in K(G ↓sm 1B′ ). Definition 4.5. We define the following functors: sm D[s,t] : Morsm N −1 (B) → Mort−s+1 (B) (1 ≤ s ≤ t ≤ N − 1) −1 ⇑N E r : Morr (B) → MorN −1 (B) (1 ≤ r ≤ N − 2) UN −1 : B → MorN −1 (B) αN −2 α1 as follows. For X 1 −−X → · · · −−X−−→ X N −1 ∈ Morsm N −1 (B), α1 αN −2 α1 αt−s Y D[s,t] (X 1 −−X → · · · −−X−−→ X N −1 ) = Y 1 −−→ · · · −−Y−→ Y t−s+1 POLYGON OF RECOLLEMENTS 11 where Y i = X i+s−1 , αiY = αi+s−1 (1 ≤ i ≤ t − s + 1). We denote D[s,s] by D[s] . X α1 αr−1 For X 1 −−X → · · · −−X−→ X r ∈ Morsm r (B), N −1 E ⇑r i Y = ( α1 αr−1 α1 αN −2 Y · · · −−Y−−→ Y N −1 (X 1 −−X → · · · −−X−→ X r ) = Y 1 −−→ 0 (1 ≤ i < N − r) X i−N+r+1 (N − r ≤ i ≤ N − 1) , αiY = ( 0 (1 ≤ i < N − r) αi−N+r+1 (N − r ≤ i ≤ N − 1) X For X ∈ B, αN −2 α1 Y UN −1 (X) = Y 1 −−→ · · · −−Y−−→ Y N −1 where Y i = X, αiY = 1X (1 ≤ i ≤ N − 1). Moreover, we use the same symbols −1 sm ⇑N r for the corresponding functors D[s,t] : K(Morsm : N −1 (B)) → K(Mort−s+1 (B)), E K(Morr (B)) → K(MorN −1 (B)) and UN −1 : K(B) → K(MorN −1 (B)). Definition 4.6. We define the following full triangulated subcategories of K(Morsm N −1 (B)): E [2,N −1] = Ker D[1] T E s = Ker D[1,s−1] Ker D[s+1,N −1] E [1,N −2] = Ker D[N −1] E N −1 = Ker D[1,N −2] E 1 = Ker D[2,N −1] For 1 ≤ s < t ≤ N − 1, F [s,t] is the full triangulated subcategory of K(Morsm N −1 (B)) α1 αN −2 consisting of objects X 1 −→ · · · −−−−→ X N −1 such that αs = · · · = αt−1 = 1. Immediately, we have the following. Proposition 4.7. The following hold. N −1 (1) A functor E ⇑N −2 : K(MorN −2 (B)) → K(MorN −1 (B)) induces a triangle equivalence between K(MorN −2 (B)) and E [2,N −1] . (2) A functor UN −1 : K(B) → K(MorN −1 (B)) induces a triangle equivalence between K(B) and F [1,N −1] . Proof. (1) By Lemma 4.4, every complex of E [2,N −1] is isomorphic to some complex of the form / ··· / X N −1 / X2 0 N −1 Then it is easy to see that a triangle functor E ⇑N −2 : K(MorN −2 (B)) → E [2,N −1] is a triangle equivalence. (2) Every complex of F [1,N −1] is of the form X1 X2 ··· X N −1 Then it is easy to see that a triangle functor U N −1 : K(B) → F [1,N −1] is a triangle equivalence.  Theorem 4.8. Let B be an additive category. Then a 2N -tuple of full subcategories (F [1,N−1] , E [2,N−1] , E 1 , F [1,2] , · · · , E s , F [s,s+1] , · · · , E N−2 , F [N−2,N−1] , E N−1 , E [1,N−2] ) is a 2N -gon of recollements in K(Morsm N −1 (B)). Proof. First, we prove HomK(Morsm (X , Y) = 0, where X , Y are two successive N −1 (B)) subcategories of the above 2N -tuple. By Example 4.3, Lemma 4.4 (2), any complex of E [2,N −1] is isomorphic to a complex 0 → X 2 → · · · → X N −1 in K(Morsm N −1 (B)). [1,N −1] [2,N −1] Then HomK(Morsm (F ,E ) = 0 is easy. By Example 4.3, Lemma N −1 (B)) 12 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI 4.4 (3), for any object of E 1 there is a complex X such that it is isomorphic to an object X → C(1X ) → · · · → C(1X ) in K(Morsm N −2 (B)). Since it is easy to see HomK(Morsm (E [2,N−1] , E 1 ) ≃ HomK(Morsm (D[2,N−1] (E [2,N−1] ), D[2,N−1] (E 1 )) N −1 (B)) N −2 (B)) and D[2,N −1] (X → C(1X ) → · · · → C(1X )) = C(1X ) → · · · → C(1X ) is homotopically 0 in K(Morsm (E [2,N −1] , E 1 ) = 0. By ExN −2 (B)), HomK(Morsm N −1 (B)) ample 4.3, Lemma 4.4 (2), (3), we may assume any morphism from E s to F [s,s+1] is of the form 0 / ··· /0 / Xs / C(1X ) ··· C(1X )  Y1 / ···  / Y s−1  / Ys  Y s+1 / ···  / Y N −1 It is easy that the above morphism is null homotopic, and then HomK(Morsm (E s , F [s,s+1] ) = 0. Similarly, we may assume any morphism from N −1 (B)) F [s,s+1] to E s+1 is of the form X1 / ··· / Xs X s+1 / X s+2 / ··· / X N −1  0 / ···  /0  /Y  / C(1Y ) ···  C(1Y ) It is easy that the above morphism is null homotopic, and then HomK(Morsm (F [s,s+1] , E s+1 ) = 0. Since any complex of E N −1 is isomorphic to N −1 (B)) 0 → · · · → 0 → X N −1 , and any complex of E [1,N −2] is isomorphic to Y 1 → · · · → Y N −2 → C(1Y N −1 ) in Morsm (E N −1 , N −1 (B), it is easy to see that HomK(Morsm N −1 (B)) E [1,N −2] ) = 0. Since we may assume any morphism from E [1,N −2] to F [1,N −1] is of the form / ··· / X N −2 / C(1X N −1 ) X1  Y1  Y N −2 ···  Y N −1 It is easy to see it is null homotopic, and then HomK(Morsm (E [1,N −2] , F [1,N −1] ) = N −1 (B) 0. Second, we prove K(Morsm N −1 (B)) = X ∗ Y, where X , Y are two successive subcateα1 α2 αN −2 gories of the above 2N -tuple. Let X 1 −→ X 2 −→ · · · −−−−→ X N −1 be a complex of K(Morsm N −1 (B)). Since we have a short exact sequence of complexes: X1 X1 α1 1  X1  0 X1 ··· α1  / X2  / Cok α1 αN −2 ···α1 α2 / ··· / ··· αN −2  / X N −1  / Cok αN −2 · · · α1 POLYGON OF RECOLLEMENTS 13 [1,N −1] we have K(Morsm ∗ E [2,N −1] . Since we have a triangle in N −1 (B)) = F sm K(MorN −1 (B)): α1 0 α2 / X2 αN −2 / ··· / X N −1 1  X1  X1 1  / X2 α1 α2 / ···  / C(1X 2 )  0 Σα1 αN −2 Σα2 / ··· / X N −1  / C(1X N −1 ) / ···  / ΣX 2   ΣX N −1 ΣαN −2 / [2,N −1] we have K(Morsm ∗ E 1 . Since we have a triangle in K(Morsm N −1 (B)) = E N −1 (B)): X1 α1 / ··· αs−2 / 1  X 1 α1 /   / Xs / ··· αs−2 / ··· X s−1 αs αs−1 /  / ··· Σα1 /  /  ΣX s−1 X / Σαs−1 /  ΣX s / ··· αN −2 / X N −1 / N −1 1 s+1 C(αs ) αs+1 X s+1   / C(1X s−1 ) Σαs−2 αs αs 1 C(1X 1 ) ΣX 1 αs−1 X s−1 X αs+1 1  αs+1 s+1 / ···  / C(1X s+1 ) Σαs  / ΣX s+1 / αN −2  / ··· Σαs+1 / ··· X  C(1X N −1 )  ΣαN −2 / ΣX N −1 s [s,s+1] we have K(Morsm . Since we have a triangle in K(Morsm N −1 (B)) = E ∗ F N −1 (B)): / X1 ··· αs−1 / Xs 1 / ··· / ··· X1  C(1X 1 )  Σα1 / ··· αs−1 / /  Xs  C(1X s−1 ) Σαs−1 /  ΣX s αs+1 αs / αs / /  X s+1  C(αs )  ΣX s αs+2 X s+2 αs 1  ΣX 1 Xs / ··· αN −2 / X N −1 1 αs+1 αs+1 / 1  αs+2 X s+2  / C(1X s+1 ) Σαs+1 αs /  ΣX s+2 Σαs+2 / ··· / / ··· ··· αN −2 / /  X N −1  C(1X N −1 ) ΣαN −2 /  ΣX N −1 14 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI [s,s+1] we have K(Morsm ∗E s+1 . Since we have a triangle in K(Morsm N−1 (B)): N −1 (B)) = F / ··· 0 / X N −1 /0 1  X1 1 α  N −3 α / ··· / X N −2 1  X1  N −2 α / X N −1 1 α1  0 αN −3/ / ···  X N −2  / C(1X N −1 )  /0  / ΣX N −1 / ··· N −1 we have K(Morsm ∗E [1,N −2] . Since we have a triangle in K(Morsm N−1 (B)): N −1 (B)) = E α1 X1 / ··· αN −3 / αN −2 X N −2 / X N −1 αN −2 αN −2 ···α1  1   X N −1 ··· X N −1 X N −1  C(αN −2 · · · α1 ) / ···  / C(αN −2 )  / C(1X N −1 )  ΣX 1 / ··· Σα1  ΣX N −2 ΣαN −3 / ΣαN −2  / ΣX N −1 N −1 we have K(Morsm ∗ F [1,N −1] . N −1 (B)) = E  5. Recollement of KN (B) We fix a positive integer N ≥ 2. Let B be an additive category. An N -complex is a diagram di−1 di di+1 → X i+1 −−X−→ · · · · · · −−X−→ X i −−X with objects X i ∈ B and morphisms diX ∈ HomB (X i , X i+1 ) satisfying −1 i di+N · · · di+1 X X dX = 0 for any i ∈ Z. A morphism between N -complexes is a commutative diagram di−1 · · · −−X−→ di−1 · · · −−Y−→ di Xi i ↓f −−X → X i+1 i+1 ↓f Yi Y −−→ di Y i+1 di+1 −−X−→ · · · di+1 −−Y−→ · · · with f i ∈ HomB (X i , Y i ) for any i ∈ Z. We denote by CN (B) the category of N -complexes. A collection SN (B) of conflations is the collection of short exact sequences of N -complexes of which each term is a split short exact sequence in B. Proposition 5.1 ([IKM2]). A category (CN (B), SN (B)) is a Frobenius category. POLYGON OF RECOLLEMENTS 15 Definition 5.2 ([IKM2]). Let (X, d), (Y, e) be objects and f : Y → X be a morphism in CN (B). Then the mapping cone C(f ) of f is given as  C(f )m = X m ⊕ m+N−1 a Y i , dm C(f ) i=m+1 (Σ−1 C(f ))m= m−1 a       =      d f 0 0 .. . . .. 0 0 0 .. . 1 .. 0 0 ··· ··· X i ⊕ Y m , dm Σ−1 C(f ) i=m−N+1 . ··· .. . .. . ··· 0 ··· ··· 1 0 ··· −e{N−1} −e{N−2}  −d 1 0   2 −d 0 1   .. ..   . . =  .. .. ..   . . .   −d{N−1} 0 ··· 0 ··· ··· .. 0 . .. .. . . ··· .. . .. . .. . ··· ··· 0 1 −e ··· .. . 1 0 0 Here d{N −1} means the (N − 1)-power of d.              0 .. . .. .        .   0   f  e The above mapping cone induces a morpism between conflations: 0 /X 0  /Y uX / C(1X ) vX / ΣX /0 / ΣX /0 ψf f  / C(f ) uf vf Let I(X) = C(1X ), then I(X) is a projective-injective object in CN (B). We call a uf f vf sequence Y − → X −−→ C(1X ) −→ ΣX a (distinguished) triangle. A morphism f : X → Y of N -complexes is called null-homotopic if there exists si ∈ HomB (X i , Y i−N +1 ) such that i f = N −1 X +j i+j−1 i+j−2 di−1 · · · di−N s dX · · · diX Y Y j=1 for any i ∈ Z. We denote by KN (B) the homotopy category of N -complexes. Theorem 5.3 ([IKM2]). A category KN (B) is a triangulated category. Definition 5.4. Let N be an integer greater than 2. For any integer s, we define (N −1) (N ) functions ιs : Z → Z, ρs : Z → Z as follows.  s + k(N − 1) (i = 0) −1) ι(N (s + i + kN ) = s s + i − 1 + k(N − 1) (0 < i < N )  s + kN (i = 0) ) ρ(N (s + i + k(N − 1)) = s s + i + 1 + kN (0 < i < N − 1) (N −1) For an (N − 1)-complex X = (X i , diX ), we define a complex Is Is(N −1) (X)i =X ( diI (N −1) (X) = s −1) ι(N (i) s (N −1) dιs 1 , (i) −1) −1) (ι(N (i) < ι(N (i + 1)) s s (N −1) (N −1) (ιs (i) = ιs (i + 1)). (X) by 16 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI (N ) For an N -complex Y = (Y i , diY ), we define a complex Js (N ) Js(N ) (X)i = X ρs (i) , ) ρ(N (i+1)−1 s diJ (N ) (X) = d s (N −1) Then Is (N ) : CN −1 (B) → CN (B) and Js (N −1) Lemma 5.5. A functor Is −1) I (N : KN −1 (B) → KN (B). s (Y ) by (N ) · · · dρs (i) . : CN (B) → CN −1 (B) are functors. : CN −1 (B) → CN (B) induces the triangle functor f Proof. Let X − → Y be a morphism in CN −1 (B). Consider the mapping cone (N −1) (N −1) I (N −1) (f ) (N −1) 0 (Y ), then we have an exact sequence (X) −− −−−−→ I0 (f )) of I0 C(I0 (N −1) (N −1) (N −1) 0 → I0 (Y ) → C(I0 (f )) → Σ(I0 (X)) → 0 in CN −1 (B). Let Z be an N -complex defined by  (N −1) (N −1) −1) ` Y ι(N (k) (i) (k) ι0 0 (k ≡ 0 mod N ) ⊕ X ι0 ⊕ k+N−2 i=k+1 X Z = (N −1) (N −1) `k+N−1 ι (k) (i) Y ι0 ⊕ i=k+1 X 0 (k ≡ 6 0 mod N )   1 ··· 0 0     .  .. ..  ..   ..   . . .    (k ≡ 0 mod N )     0 · · · 1 0          0 ··· 0 0        f 0 ··· 0 0   e  .. ..  .. dkZ =     . 0 0 1 . .       .   . . . .  .    . . . . . .    .  . . . . .    (k 6≡ 0 mod N )      .    ..     0 0 0 · · · 1         N−1 N−2  0 −d d · · · −d 0       0 0 0 ··· 0 0 k and W an N -complex defined by  −1) (N −1) `k+N−2 X ι(N (i) (k) 0 ⊕ X ι0 (k ≡ 0 mod N ) W = `i=k+1 (N −1) (i)  k+N−1 X ι0 (k ≡ 6 0 mod N ) i=k+1   1 ··· 0 0     .   .. ..  ..   .    .  . . .    (k ≡ 0 mod N )        0 0 · · · 1        0 · · · 0 0     0 0 1 ··· 0 dkW =   . . . .   ..   .. .. .. ..     .          .    (k 6≡ 0 mod N )  .    . 0 0 · · · 1       −dN−1 dN−2 · · · −d 0         0 0 ··· 0 0 k POLYGON OF RECOLLEMENTS (N −1) Let g : C(I0 1 0 . . . 0 0     gk =      1  0   .  .  .   0 0  1  .  .  .   0   0   0 0  1  0   0    ..  .   0 0 . . ··· ··· . . . ··· ··· 0 . . . 1 ··· ··· ··· . ··· 0 0 0 0 0 1 1 0 0 . . . . . . 0 0 0 −1 0 . . . 0 1 0 0 0 0 1 . . . 0 0 0 0 . . . 1 0 .. ··· ··· −d{N −1} . ··· ··· 0 0 0 0 1 . . . 0 ··· (f )) → Z be a isomorphism defined by 0 1 f 0 . . . 0 −1 17 0 0 . . . 1 −d      (k ≡ 0 mod N),    0 0 . . . 0 1        (k ≡ 1 mod N),             1 0 . . . 0 0 0 0 1 . . . 0 0 0 0 0 . .  . .  . .   0 0   (k ≡ 3 mod N), 1 0   0 1  −1 0  ··· 0 ··· 0   ··· 0    (k ≡ N − 1 mod N) .  . . .  . .  ··· 1  ··· 0 ··· ··· 0 0 . . . 1 0 0 . . . ··· ··· ··· 0 0 . . . 0 1 0 0 0 0 1 −1       (k ≡ 2 mod N),     ··· , Then we have the following isomorphism between short exact sequences: u / I (N −1) (Y ) 0 0 / C(I (N −1) (f )) 0 v / Σ(I (N −1) (X)) 0 ≀ g′ ≀ g where    1 0 0  .   u′ =  .  , v ′ =   ..  ..  0 0 (N −1) and Σ(I0 1 . .. 0 (N −1) (X)) ≃ I0  /Z u′ / I (N −1) (Y ) 0 0 ··· .. . ···  0 .  ..  . 1 (N ) (N )  /W v′ (N −1) Therefore, C(I0 /0 (N −1) (f )) ≃ I0 (C(f )) (Σ(X)) in KN (B). Proposition 5.6. For a functor Js (1) (2) (3) (4) (5) /0 : CN (B) → CN −1 (B), the following hold. ) Js induces the triangle functor J (N : KN (B) → KN −1 (B). s (N ) (N −1) . J s is a right adjoint of I s (N −1) ) I s+1 is a right adjoint of J (N . s ) (N −1) is an isomorphism. The adjunction arrow 1 → J (N s Is (N ) (N −1) The adjunction arrow J s I s+1 → 1 is an isomorphism.  18 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI Proof. For X, Z ∈ CN −1 (B), Y ∈ CN (B), consider the following morphisms in CN (B): (N −1) Is (X) : / Xs ··· fs  Y :  ··· / Ys ···  / Zs (N −1) Is+1 (Z) : / X s+1 ds f s d s  / Y s+1 gs  ds Xs f s+1 d s+1  / Y s+2 gs+1 ds / ··· / ··· gs+2   / Z s+1 Z s+1 / ··· Then these morphisms correspond to the following morphisms in CN −1 (B): / Xs ··· X: ds / X s+1 fs  (N ) Js (Y ) :  / Ys ··· f s+1  ds+1 ds/ Y s+2 ds  gs  Z:  / Zs ··· / ··· / ··· gs+2 (N ) / Z s+1 / ··· (N −1) (N −1) Then it is easy to see that Js is a right adjoint of Is , that Is+1 is a right (N ) (N ) (N −1) (N ) (N −1) adjoint of Js , and that the adjunction arrows 1 → Js Is and Js Is+1 → 1 are isomorphisms. By Lemmas 5.5 and Proposition 7.4, we have statements.  ⇑N −1) −r) · · · I (N : KN −r (B) → KN (B) and Definition 5.7. We denote I s N −r = I (N s s ⇓N −r+1) ) J s N −r = J (N · · · J (N : KN (B) → KN −r (B). For 1 ≤ r < N , we define the full s s subcategory of KN (B) Fsr = {(X i , di ) ∈ KN (B) | di+k = 1X i+k (0 ≤ k < r)for i ≡ s ⇑N ⇓N mod N } ⇑N Corollary 5.8. I s N −r , J s N −r are triangle functors such that I s N −r is a left (reps., ⇓N ⇓N ⇑N N −r right) adjoint of J s N −r (resp., J s+1 ), and that Fsr = Im I s N −r is a full triangulated subcategory of KN (B). Proof. By Lemma 5.5, Proposition 5.6.  N −r−1 Theorem 5.9. For 1 ≤ r < N , (Fsr , Fr+s+1 ) is a stable t-structure in KN (B). N −r−1 Proof. We may assume s = 0. It is easy to see that HomKN (B) (F0r , Fr+1 ) = 0. Let X be an N -complex. By Proposition 5.6, the adjunction arrow induce a triangle ⇑N ⇓N ε u v ⇑N ⇓N →X − → C(εX ) − → ΣI 0 N −r J 0 N −r (X). I 0 N −r J 0 N −r (X) −−X POLYGON OF RECOLLEMENTS (N −1) Let σ := ι0 Vk = dkV = (N −r) (N −r+1) (N ) ρ0 · · · ρ0 , · · · ι0 and V be an N -complex defined by (` k+N−1 σ(i) (k ≡ 0, · · · , r mod N ) i=k+1 X `k+N−1 k X ⊕ i=k+1,i6≡0 mod N X σ(i) (k ≡ r                   + 1, · · · , N − 1 mod N )  1 .. . ··· .. . 0 0 . .. ··· 1 1 ··· . .. .. .               0       0 19 0 ..   .  (k ≡ r mod N ) 0 ··· 0 ··· 0 . ..      (othewise)  1  0 Let h : V → C(ǫX ) be a monomorphism defined by 0  1    hk =  −d   .  . . 0 ··· ··· .. .   N −r  N −r    −d   .  .  .   0   0     0  0 0 1    −d   .  .  .   0   0     0   0  0 0 ··· .. . . ··· 0 . . . .. . ··· ··· 1 −d 0 ··· ··· ··· 0 0 0 ··· .. . 0 . . . .. . ··· ··· 1 −d 0 ··· ··· ··· ··· 0 0 0 0 1  −1    0    .  .  .  0   0      0   .  . . 0  r+1 1 .. 0 1 .. . .. 1 −d 0 1 0 1 −1 . . . 0 0 0 1 −1 . . . 0 0 0 .. 0 0 . . . N −r 0 . . . N −r 0 . . . ··· ··· .. . 0 0 . . .         (k ≡ r mod N ), 0 . . . ··· ··· 0 . . . 0 0 1 ··· ··· ··· .. . −1 0 0 0 0 . . . 1 −1 .. . ··· ··· 0 . . . ··· 0 0 1 .. . ··· ··· ··· 0 0 . . .  ··· 0 . . . 0 . . . ··· ··· ··· .. . −1 0 0 0 0 0 . . . 1 −1 0 0 0 0 . . . 0 0 1 r+1 0 ··· 0 ··· . .. . . . . ··· ··· . 0 ··· 1 −1 0 0 0 1 ··· . . . ··· ··· . . . ··· 0 . . . 0 −d . . . 0                  (k ≡ r + 1 mod N )                    0 0 . . . 0 0 . . . ··· ··· ··· .. . 0 0 0 . . . 0 0 0 . . . .. . ··· 1 −d 0 1 (k ≡ r + 2 mod N ), · · ·                      (k ≡ 0 mod N ) 20 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI d  −1    0    .  .  .   0  0      0   .  . . 0 0 1  r            ··· ··· .. . 0 0 . . . r 0 0 . . . −1 .. . ··· ··· . 0 ··· 1 −1 0 0 0 1 ··· . . . ··· ··· . . . ··· 0 . . . 0 −d . . . 0 .. dr−1 −1 0 0 0 1 0 . . . 0 −d . . . 0 ⇑N ··· ··· .. . 0 0 . . . 0 0 . . . ··· ··· ··· .. . 0 0 0 . . . 0 0 0 . . . 1 −d 0 1 .. . ··· ··· ··· ··· .. . 0 0 0 . . . 0 0 0 . . . .. . ··· 1 −d 0 1                      (k ≡ 1 mod N ), · · ·            (k ≡ r − 1 mod N ) ⇓N r+1 Let p : C(ǫX ) → I r+1 J r r+1 (X) be a epimorphism defined by pk =
1 0 ··· 0 (k ≡ r mod N ),
N −r−1 N −r−2 d d ··· d 1 ··· 1 1
(k ≡ r + 1 mod N ), N −r−2 N −r−3 (k ≡ r + 2 mod N ), d d ··· d 1 ··· 1 0 . . . r+1
1 1 ··· 1 0 ··· 0 (k ≡ 0 mod N ), r
1 d ··· d 0 ··· 0 (k ≡ 1 mod N ), r−1
(k ≡ 2 mod N ), 1 d2 · · · d2 0 · · · 0 . . .
(k ≡ r − 1 mod N ). 1 dr−1 0 · · · 0 Then we have the following conflations. ⇑N p h ⇓N r+1 0→V − → C(εX ) − → I r+1 J r r+1 (X) → 0 Since V is a projective object in CN (B), we have an isomorphism in KN (B) ⇑N ⇓N r+1 C(εX ) ≃ I r+1 J r r+1 (X) Therefore, we have a triangle U → X → V → ΣX such that U ∈ Fsr , V ∈ N −r−1 Fr+s+1 .  Corollary 5.10. We have a recollement of KN (B): | KN −r (B) b ⇓N ⇑N i∗ s is∗ i!s | / KN (B) b ⇓N js! js∗ js∗ / Kr+1 (B) ⇑N ⇓N N −r r+1 r+1 where i∗s = J s−1 , is∗ = I s N −r , i!s = J s N −r , js! = I r+s , js∗ = J r+s and js! = ⇓N r+1 . I r+s+1 N −r−1 N −r−1 Proof. By Theorem 5.9, (Fr+s−N , Fsr ) and (Fsr , Fr+s+1 ) are stable t-structures in KN (B). By the proof of Theorem 5.9, the adjunction arrows induce triangles in K(B). Hence we have the statement.  POLYGON OF RECOLLEMENTS 21 Corollary 5.11. For any integer s, N −2 N −2 N −2 N −2 1 1 1 1 N −2 , Fs+N (Fs+1 , Fs1 , Fs+2 , Fs+1 , · · · , Fs+r+1 , Fs+r , · · · , Fs+N −1 ) −1 , Fs+N −2 , Fs is a 2N -gon of recollements in KN (B). 6. Triangle equivalence between homotopy categories ds−r+1 ds−1 ds−2 µsr C : X s−r+1 −−−−→ · · · −−−→ X s−1 −−−→ X s be an N -complex satisfying that X s−i = C (0 ≤ i ≤ r − 1), r − 1). ds−i = 1C (0 < i ≤ Lemma 6.1. Let B be an additive category. Consider Morsm N −1 (B) as a subcategory of CN (B), then the following hold. (1) For every object X of Morsm N −1 (B), there are objects C1 , · · · , CN −1 of B such `N −1 N −1 that X ≃ i=1 µi Ci . (2) For any X, Y ∈ Morsm N −1 (B), we have isomorphisms HomMorsm (X, Y ) = HomCN (B) (X, Y ) N −1 (B) = HomKN (B) (X, Y ). i (3) For any X, Y ∈ Morsm N −1 (B), we have HomKN (B) (X, Σ Y ) = 0 (i 6= 0). Proof. (1) It is trivial. (2) It is easy because the term-length of objects of Morsm N −1 (B) is less than N . (3) For every C ∈ B, 1 ≤ r ≤ N − 1 and any integer i, the canonical injection and projection induce an conflation in C(B) (i+1)N −r−1 iN +N −1 0 → µriN +N −1 C → µN C → µN −r C → 0. iN +N −1 Since µN C is a projective-injective objet in CN (B), we have isomorphisms in KN (B): ( (1−j)N/2−r−1 µN −r C (j ≡ 1 mod 2) j N −1 Σ µr C ≃ (2−j)N/2−1 µr C (j ≡ 0 mod 2) For every C, C ′ ∈ B and any 1 ≤ r, r′ ≤ N − 1, we have −1 ′ −1 HomK(B) (µN C, Σj µN C ) = 0 (j 6= 0). r r′ By (1), we have the statement.  Definition 6.2. For every C ∈ B, let ( (1−j)N/2+N −r−1 µN −r C (j ≡ 1 mod 2) −1 Ξj µN C= r (2−j)N/2−1 C (j ≡ 0 mod 2) µr By the proof of Lemma 6.1, for every M ∈ Morsm N −1 (B) and any i ∈ Z, there exist the projective-injective object I ′ (Ξi M ) and a functorial conflation in CN (B) u′ i v′ i Ξ M Ξ M −→ Ξi+1 M → 0 −→ I ′ (Ξi M ) −− 0 → Ξi M −− 22 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI that is, every morphism f : M → N in Morsm N −1 (B) uniquely determines morphisms I ′ (f ) : I ′ (M ) → I ′ (N ) and Ξi f : Ξi M → Ξi N which have the following commutative diagram: / Ξi M 0 / I ′ (Ξi M ) I ′ (Ξi f ) Ξi f  / I ′ (Ξi N )  / Ξi N 0 / Ξi+1 M /0 Ξi+1 f  / Ξi+1 N /0 For any (ordinary and N ) complex X = (X i , di ), we define the following truncations: τ≤n X : · · · → X n−2 → X n−1 → X n → 0 → · · · , τ≥n X : · · · → 0 → X n → X n+1 → X n+2 → · · · , τ[m,n] X : · · · → X m → · · · → X n → 0 → · · · , τn X : · · · → 0 → X n → 0 → · · · . Lemma 6.3. Let B be an additive category. There exists an exact functor FN : sm sm C(Morsm N −1 (B)) → CN (B) which sends MorN −1 (B) to MorN −1 (B) as a subcategory of KN (B) such that FN induces a triangle functor F N : K(Morsm N −1 (B)) → KN (B). sm Proof. Let T : C(Morsm N −1 (B)) → C(MorN −1 (B)) be a translation functor. We will prove the statement by the following `steps: Step 1. We have a functor FN : i T i B → CN (B) which preserves split exact sequences. ` i sm i i For T i X ∈ T i Morsm N −1 (B), let FN (T X) = Ξ X, then FN : i T MorN −1 (B) → KN (B) which preserves split exact sequences. Step 2. We have a functor FN : Cb (Morsm N −1 (B)) → CN (B) which preserves conflations. di For i < j, let X : X i −→ X i+1 → · · · → X j ∈ Cb (Morsm N −1 (B)). By induction on j − i, we comstruct a functor FN . For X, we have a commutative diagram in Cb (B): 0 / T −i X i / I(T −i X i ) / T −i−1 X i /0  / τ≥i X / T −i−1 X i /0 di 0  / τ≥i+1 X where all rows are exact. Then we have a commutative diagram in CN (B): (6.4) 0 / FN (T −i X i ) FN (di ) 0  / FN (τ≥i+1 X) / I ′ (FN (T −i X i )) / FN (T −i−1 X i ) /0 / FN (T −i−1 X i ) /0 γi (A)  / C(FN (di )) where all rows are conflations. We take FN (τ≥i X) = C(FN (di )). By successive step, we have a functor FN : Cb (Morsm N −1 (B)) → CN (B). For any inflation (resp., b i deflation) f : X → Y in C (B), FN (T f ) : FN (T i X) → FN (T i Y ) and I ′ (FN (T i f )) : I ′ (FN (T i X)) → I ′ (FN (T i Y )) are split monomorphism (resp., epimorphism) by Definition 6.2. Then 0 → FN (T i X) → FN (τ≥i+1 X) ⊕ I ′ (FN (T −i X i )) → FN (τ≥i X) → 0 POLYGON OF RECOLLEMENTS 23 is an inflation. Let 0 → X → Y → Z → 0 be an conflation in Cb (Morsm N −1 (B)). Consider a small Frobenius subcategory C of CN (Morsm (B)) which contains FN (T i X), N −1 i i ′ −i ′ −i ′ −i FN (T Y ), FN (T Z), I (FN (T X), I (FN (T Y ), I (FN (T Z), FN (τ≥i X), FN (τ≥i Y ) and FN (τ≥i Z) (i ∈ Z). Then by the induction on j − i, 9-lemma implies that 0 → FN (τ≥i X) → FN (τ≥i Y ) → FN (τ≥i Z) → 0 is a short exact sequence in some abelian category. According to Proposition 7.2, it is a conflation in C. Therefore, FN preserves conflations. Step 3. We have a functor FN : C− (Morsm N −1 (B)) → CN (B) which preserves conflations. For X ∈ C− (B), we have X = lim τ≥−i X. By the diagram 6.4, we have i→∞ τ≥([ i+1 ])N −1 FN (τ≥−i X) = τ≥([ i+1 ])N −1 FN (τ≥−i−1 X) 2 2 Then there exists lim FN (τ≥−i X), and we take FN (X) = lim FN (τ≥−i X). It is i→∞ i→∞ not hard to see that FN becomes a functor and preserves conflations. Step 4. We have a functor FN : C(Morsm N −1 (B)) → CN (B) which preserves conflations. di Let X : · · · → X i −→ X i+1 → · · · ∈ C(B). By induction on i, we comstruct a functor FN . For X, we have a commutative diagram in C− (Morsm N −1 (B)): 0 / T −i−1 X i+1 / τ≤i+1 X / τ≤i X /0   / T −i X i+1 di 0 / T −i−1 X i+1 / I(T −i−1 X i+1 ) /0 where all rows are exact. Then we have a commutative diagram in CN (B): (6.5) / Σ−1 C(FN (di )) /0 / FN (τ≤i X) / FN (T −i−1 X i+1 ) 0 FN (di ) (B) 0 / FN (T −i−1 X i+1 )  / I ′ (T −i−1 X i+1 )  / FN (T −i X i+1 ) /0 where all rows are conflations. We take FN (τ≤i+1 X) = C(FN (di )). By the diagram 6.5, we have τ≤[ i+2 ]N −1 FN (τ≤i X) = τ≤[ i+2 ]N −1 FN (τ≤i+1 X) 2 2 Then there exists lim FN (τ≤i X), and we take FN (X) = lim FN (τ≤i X). Since the ∞←i ∞←i commutative diagram (B) is a exact square, it is not hard to see that FN becomes a functor and preserves conflations. Step 5. FN sends projective-injective objects in C(Morsm N −1 (B)) to projectiveinjective objects in CN (B). Every projective-injective object in C(Morsm N −1 (B)) is a direct summand of some biproduct ⊕i∈Z T i C(1Mi ) with Mi ∈ Morsm N −1 (B). Since FN (C(1Mi )) ≃ C(1FN (Mi ) ) is projective-injective in CN (B), we have the statement. According to Proposition 7.3, FN : C(Morsm N −1 (B)) → CN (B) induces a triangle (B)) → K (B).  functor F N : K(Morsm N N −1 Lemma 6.6. The following hold. 24 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI (1) Every complex X of C(Morsm N −1 (B)) is of the form α1 X1 where αi = "1 ··· 0 # .. . 0 ··· 1 0 ··· 0 / ⊕ 2 Xi i=1 α2 / ··· αN −2 / ⊕N −1 Xi i=1  di di12 ··· di1r 0 di22 ··· di2r 11  and (⊕ri=1 Xi , diX =  . .. 0   .. . . .. ) is a complex of . . . i 0 ··· drr B. (2) For a complex X of C(Morsm N −1 (B)), the N -complex FN (X) is equal to (Y j , djY ) where

−1 2i−1 Y j = ⊕kr=1 Xr2i ⊕ ⊕N r=k Xr   i i d d ··· di1N −1   011 d12 i i  ··· d2N −1       . 22   .. . .    . . .  .  . . .  i     0 0 ··· dN −1N −1 i−1    1 0 ··· 0 d1k 0 ···     . ..  . 0 1  .. ..  di−1 .   2k    .. . . . .. .  .. .. . 0 . . djY =    ..  ..  ..  i−1 .  . 1 dk−1k .       . .  . . di−1  ..  0    kk    .  .. .  ..  ..  . 1    .   . .   .. ..    ..    0 ··· ··· ··· ··· ··· 0   where i = 2 Nj , 0 ≤ k ≤ N − 1 and k ≡ j 1 (3) F N (E [1,N −2] ) ⊂ FN −1 (j ≡ −1 mod N ),  0 ..  .  ..   .  ..  .  ..  .  ..  .    0 1 otherwise mod N . Proof. (1) It is trivial. (2) For a complex X ∈ Cb (Morsm N −1 (B)), we assume FN (τ≥2j X) satisfies the statement. For 2j = i + 1, we have the following equations in the diagram 6.4 −1 N −1+jN 2j−1 FN (T −i X i ) = ⊕N Xr r=1 µN −r −1 N −1+jN I ′ (FN (T −i X i )) = ⊕N Xr2j−1 r=1 µN −1 −1 r−1+jN 2j−1 FN (T −i−1 X i ) = ⊕N Xr r=1 µr By easy calculations, γjk+jN : I ′ (FN (T −i X i ))k → Y k+jN is equal to dk+N in the Y statement for 0 ≤ k < N . Therefore, we have the statement for FN (τ≥i X). For 2j = i, we have the following equations in the Diagram 6.4 −1 r−1+(j+1)N FN (T −i X i ) = ⊕N Xr2j r=1 µr −1 −1+jN 2j FN (T −i X i ) = ⊕N r=1 µN −r+1 Xr r−1+(j+1)N −1 I ′ (FN (T −i X i )) = ⊕N r=1 µN −1 Xr2j POLYGON OF RECOLLEMENTS 25 By easy calculations, γjk+jN : I ′ (FN (T −i X i ))k → Y k+jN is a (k + 1) × k matrix " 1 ··· 0 # .. for 1 ≤ k < N − 1, and γjN −1+jN : I ′ (FN (T −i X i ))k → Y k+jN is equal to . 0 ··· 1 0 ··· 0 N −1+jN dY in the statement. Therefore, we have the statement for FN (τ≥i X). Similarly, we have the same result for the diagram 6.5. (3) By Lemma 4.4, any complex of E [1,N −2] is isomorphic to the mapping cone of the following morphism between complexes: X′ : /0 0 / ··· /0 / ⊕N −1 Xi i=1   f 1  X1  X: 1 α  / ⊕ 2 Xi i=1 2 α / ··· N −2 α / ⊕N −2 Xi i=1 / ⊕N −1 Xi i=1 Consider a morphism between the Diagram 6.4 for FN (τ≥i X ′ ) and the Diagram 6.4 for FN (τ≥i X). Then FN (X ′ ) = (Y ′j , diY ′ ), where ( −1 2i ⊕N (j ≡ −1 mod N ) j r=1 Xr Y = N −1 2i−1 ⊕r=1 Xr otherwise  di−1 di−1 ··· di−1  11 12 1N −1    0 di−1 ··· di−1   22 2N −1    (j ≡ −2 mod N ), j . . . .   .. .. . . .. dY =  i−1  0 0 ··· dN −1N −1    identity otherwise j where i = 2 N , 0 ≤ k ≤ N − 1 and k ≡ j mod N . Moreover, FN (f ) : FN (X ′ ) → FN (X) is equal to g : Y ′ → Y where   identity (j ≡ 0, −1 mod N )     i−1  i−1   d11 · · · d1k 0 ··· 0     ..  .. .  .  . .  0  . . . .      .  . .  j i−1 . . . . dkk g =  . . 0   otherwise  .  ..   .. ..    .  . . .  1 .         .. .. ..     . . 0 .     0 ··· ··· ··· 0 1 where i = 2 j N , 0 < k < N − 1 and k ≡ j mod N . By the proof of Theorem 5.9, ⇑N ⇓N 1 we have Y ′ = I 0 1 J 0 1 (Y ) and ǫY = g. Therefore C(FN (f )) ≃ C(g) ∈ FN −1 .  Lemma 6.7. The following hold for the above triangle functor F N : K(Morsm N −1 (B)) → KN (B). (1) F N (E [s,N −1] ∩ F [s,N −1] ) ⊂ F0s−1 ∩ FsN −s−1 . N −2 . (2) F N (E s ) ⊂ Fs+1 Proof. (1) According to Lemma 4.4, we may assume any complex Y of E [s,N −1] ∩ F [s,N −1] is of the form αs αN −2 0 → · · · → 0 → Y s −→ · · · −−−−→ Y N −1 26 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI where X = (X i , di ) = Y s = · · · = Y N −1 is a complex of B and αs = · · · = αN −2 = 1X . Then by Diagrams 6.4, 6.5, F N (Y ) is isomorphic to a complex (X ′i , d′i ) where ( i X 2[ N ]−1 (i ≡ 0, 1, · · · , s − 1 mod N ) ′i X = i 2[ N ] (i ≡ s, s + 1, · · · , N − 1 mod N ) X  1 2[ Ni ] (i ≡ 0, 1, · · · , s − 2 mod N )  X   d2[ Ni ]−1 (i ≡ s − 1 mod N ) d′i =  1 2[ i ] (i ≡ s, s + 1, · · · , N − 2 mod N )   X N   d2[ Ni ] (i ≡ N − 1 mod N ) [0,s−1] [0,N −s−1] Therefore F N (E [s,N −1] ∩ F [s,N −1] ) ⊂ F0 ∩ Fs . (2) Any complex Y of E s is isomorphic to the mapping cone of a morphism ι : Y1 → Y2 in C(Morsm N −1 (B)): Y1 : 0 / ··· /0 /0 / ···  /0  / Y s+1 ι  Y2 : αs+1 / ··· αN −2/ / ··· αN −2/ Y N −1 1  0 / Ys αs /Y  1 s+1 s+1 α  Y N −1 where X = (X i , di ) = Y s = · · · = Y N −1 is a complex of B and αs = · · · = αN −2 = ⇑N ⇓N 1X . By the construction of FN in Theorem 6.3, F N (Y2 ) = I 0 N −r J 0 N −r (F N (Y1 )) ∈ ⇑N ⇓N εF (Y1 ) N −−→ F N (Y1 ). By the Proof of Theorem F0s and F N (ι) = I 0 N −r J 0 N −r (F N (Y1 )) −−− 5.9, the mapping cone C(F N (ι)) is isomorphic to a complex (X ′i , d′i ) where  2 i −1 [ ] (i ≡ 0, 1, · · · , s − 1 mod N )  X N i X ′i = X 2[ N ] (i ≡ s mod N )   2[ Ni ]+1 (i ≡ s + 1, · · · , N − 1 mod N ) X  1 (i i  2[ N ]−1 ≡ 0, 1, · · · , s − 2 mod N )  X   d2[ Ni ]−1 (i ≡ s − 1 mod N ) d′i = i  d2[ N ] (i ≡ s mod N )     1 2 i (i ≡ s + 1, · · · , N − 1 mod N ) X [N ] N −2 Therefore F N (E s ) ⊂ Fs+1 .  Theorem 6.8. Let B be an additive category, then we have triangle equivalences: ♯ K♯ (Morsm N −1 (B)) ≃ KN (B) where ♯ = nothing, −, +, b. Proof. We prove the statement by the following steps: Step 1. F N : K(Morsm N −1 (B)) → KN (B) sends (F [1,N−1] , E [2,N−1] , E 1 , F [1,2] , · · · , E s , F [s,s+1] , · · · , E N−2 , F [N−2,N−1] , E N−1 , E [1,N−2] ) to N −2 N −2 N −2 1 1 (F1N −2 , F01 , F2N −2 , F11 , · · · , Fs+1 , Fs1 , · · · , FN , FN −1 ). −1 , FN −2 , F0 According to Lemma 6.6 (2), it is easy to see that F N (F [1,N −1] ) ⊂ F1N −2 , F N (F [s,s+1] ) ⊂ Fs1 F N (F [2,N −1] ) ⊂ F01 , (1 ≤ s < N − 1) F N (E N −1 ) ⊂ F0N −2 , POLYGON OF RECOLLEMENTS 27 N −2 (1 ≤ s < N − 1). By Lemma 6.6 (3), we By Lemma 6.7, we have F N (E s ) ⊂ Fs+1 [1,N −2] 1 have F N (E ) ⊂ FN −1 . Step 2. F N induces a triangle equivalence between F [1,N −1] and F1N −2 . Consider the following diagram: id K(B) / K(B) ⇑N UN −1  FN K(MorN −1 (B))  I1 2 / KN (B) By Proposition 4.7, Lemma 6.6, it is not hard to see that we have a functorial ⇑N isomorphism F N ◦ UN −1 ≃ I 1 2 . By Proposition 4.7, UN −1 induces a triangle ⇑N equivalence between K(B) and F [1,N −1] . On the other hand, by Corollary 5.10 I 1 2 induces a triangle equivalence between KN −1 (B) and F1N −2 . Therefore F N induces a triangle equivalence between F [1,N −1] and F1N −2 . Step 3. F N induces a triangle equivalence between E [2,N −1] and F01 . Consider the following diagram: K(MorN −2 (B)) E F N −1 / KN −1 (B) ⇑N N −1 ⇑ N −2  K(MorN −1 (B)) I 0 N −1 FN  / KN (B) N −1 ⇑N Similarly, we have a functorial isomorphism F N ◦ E ⇑N −2 ≃ I 0 N −1 ◦ F N −1 . By N −1 Proposition 4.7, E ⇑N −2 induces a triangle equivalence between K(B) and F [1,N −1] . ⇑N On the other hand, by Corollary 5.10 I 1 2 induces a triangle equivalence between KN −1 (B) and F01 . By the assumption of induction on N , F N −1 is a triangle equivalence. Therefore F N induces a triangle equivalence between E [2,N −1] and F01 . According to Proposition 1.7, F N : K(Morsm N −1 (B)) → KN (B) is a triangle equivalence. Moreover, it is easy to see the above proof is available for the case F ♯N : ♯ K♯ (Morsm  N −1 (B)) → KN (B), where ♯ = −, +, b. In [IKM2], we studied the derived category of N -complexes. We have results of [IKM2] Corollaries 4.11, 4.12 under the weaker condition. Lemma 6.9. Let D be a triangulated category, C, U full triangulated subcategories of C. We assume that for any X ∈ D there is a triangle CX → X → UX → ΣCX such that CX ∈ C and UX ∈ U. Then we have a triangle equivalence C/(C ∩ U) ≃ D/U. Proof. Let E : C → D be the canonical embedding, Q′ : C → C/(C ∩ U), Q : D → D/U the canonical quotients. Then there is a triangle functor F : C/(C ∩U) → D/U such that Q ◦ E = F ◦ Q′ . By the assumption, F is obviously dense. Given 28 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI X1 , X2 ∈ C, any morphism in D/U is represented by the following in D: (6.10) X1 ■ ■■ ■■g ■■ s1 ■■  $ F (C1 ) F (C2 ) s 1 F (C1 ) → U1 → ΣX1 is a triangle in D with U1 ∈ U. By the where X1 −→ s2 X1 → U2 → ΣC2 with U2 ∈ U. Therefore, assumption, there is a triangle C2 −→ F is a full dense functor. Let f : C1 → C2 be a morphism in C/(C ∩ U) such that F (f ) = 0 in DU . Then F (f ) is represented by the diagram 6.10 where g = 0. In the above, gs2 = 0, and then f = 0 in C/(C ∩ U). Hence F is an equivalence.  We say that A is an Ab3 category provided that it has any coproduct of objects. Moreover, A is an Ab4 category provided that it has any coproduct of objects, and that the coproduct of monics is monic. Proposition 6.11. Let A be an AB4 category with enough projectives, and P the category of projective objects, P the full subcategory of A consisting of projective objects. Then the following hold. (1) We have a triangle equivalence φ sm K(Morsm N −1 (P))/ K (MorN −1 (P)) ≃ D(MorN −1 (A)). (2) We have a triangle equivalence KN (P)/ KφN (P) ≃ DN (A). φ sm Here Kφ (Morsm N −1 (P)) (resp., KN (P)) is the full subcategory of Here K(MorN −1 (P)) (resp., KN (P)) consisting of complexes (resp., N -complexes) of which all hmologies are null. Proof. (1) It is easy to see that Morsm N −1 (P) is the full subcategory of MorN −1 (A) consisting of projective objects, and that MorN −1 (A) is an Ab4 category with enough projectives. According to [BN], for any complex X ∈ K(MorN −1 (A)), there is a quasi-isomorphism P → X with P ∈ K(Morsm N −1 (P)). By Lemma 6.9, we have φ sm (P)) ≃ D(Mor (A)). (P))/ K (Mor K(Morsm N −1 N −1 N −1 (2) According to [IKM2] Theorem 2.23, for any complex X ∈ KN (A), there is a quasi-isomorphism P → X with P ∈ KN (P). By Lemma 6.9, we have KN (P)/ KφN (P) ≃ DN (A).  Corollary 6.12. Let A be an abelian category with enough projectives, and P the category of projective objects. Then we have triangle equivalences − b sm b K− (Morsm N −1 (P)) ≃ KN (P), K (MorN −1 (P)) ≃ KN (P), D− (MorN −1 (A)) ≃ D− N (A). Moreover if A is an Ab4 category, then Then we have triangle equivalences: D(MorN −1 (A)) ≃ DN (A). Proof. According to [IKM2] Lemma 4.8, we have a triangle equivalence − K− (Morsm N −1 (P)) ≃ D (MorN −1 (A)). By [IKM2] Theorem 2.18, we have a triangle − equivalence KN (P) ≃ D− N (A). By Theorem 6.8, we have the statement.  POLYGON OF RECOLLEMENTS 29 7. Appendix In this section, we give results concerning Frobenius categories. Let C be an exact category with a collection E of exact sequences in the sense of Quillen [Qu]. f g An exact sequence 0 → X − →Y − → Z → 0 in E is called a conflation, and f and g are called an inflation and a deflation, respectively. An additive functor F : C → C ′ is called exact if it sends conflations in C to conflations in C ′ . An exact category C is called a Frobenius category provided that it has enough projectives and enough injectives, and that any object of C is projective if and only if it is injective. In this case, the stable category C of C by projective objects is a triangulated category (see [Ha]). Remark 7.1. For a Frobenius category C, we treat the only case that for any object X of C we can choose conflations v u → ΣC X → 0 → IC (X) −−X 0 → X −−X u v −1 −1 Σ X Σ X 0 → Σ−1 C X −−−−→ PC (X) −−−−→ X → 0 where IC (X) and PC (X) are projective-injective objects in C. For a morphism f : X → Y in C, we have a commutative diagram 0 /X 0  /Y uX / IC (X) vX / ΣC X If f uX  / IC (Y ) /0 Σf vY  / ΣC Y /0 It is easy to see that Σf is uniquely determined in the stable category C. Therefore C has a suspension functor ΣC : C → C. x Proposition 7.2 ([Ke] A.2 Proposition). If (C, E) is a small exact category, there is an equivalence G : C → M onto a full subcategory M of an abelian category A such that M is closed under extensions and that E is formed by the collection of f g sequences 0 → X − →Y − → Z → 0 inducing exact sequences in A: G(f ) G(g) 0 → G(X) −−−→ G(Y ) −−−→ G(Z) → 0 Proposition 7.3. Let C, C ′ be Frobenius categories, F : C → C ′ an exact functor. If F sends projective objects in C to projective objents in T ′ , then it induces the triangle functor F : C → C ′ . Proof. For X ∈ C, since F (IC (X)), IC ′ (F (X)) are projective-injective objects in C ′ , we have a commutative diagram 0 / F (X) F (uX ) / F (IC (X)) F (vX ) 0 / F (X) vF (X)  / ΣC ′ F (X) F (vX )  / F (ΣC X) ′ θX 0 / F (X) F (uX )  / F (IC (X)) /0 ηX θX  / IC ′ (F (X)) uF (X) / F (ΣC X) /0 ′ ηX /0 30 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI ′ There are γX : F (ΣC X) → F (IC (X)) and γX : ΣC ′ F (X) → IC ′ (F (X)) such that ′ ′ ′ in C ′ . Then η X F (vX )γX = 1F (ΣC X) − ηX ηX and vF (X) γX = 1ΣC′ F (X) − ηX ηX ′ is an isomorphism in C . For a morphism f : X → Y in C, we have the following diagram by the diagrams of the above and Definition 7.1: F (uX ) F (vX ) / F (I(X)) / F (ΣX) / F (X) F (f ) ⑧  ⑧⑧  ⑧⑧ F (If ) ⑧⑧⑧ F (Σf ) ⑧ /0 / F (Y ) / F (I(Y )) / F (ΣY ) η Y θY  θX  ηX / F (X) / I(F (X)) / ΣF (X) 0 ⑧⑧⑧ F (f )  ⑧  ⑧⑧ IF (f )  ⑧⑧⑧ ΣF (f ) / F (Y ) / I(F (Y )) / ΣF (Y ) /0 vF (Y ) uF (Y ) 0 0 0 /0 /0 where the diagrams on the top, the bottom, the front and the back surfaces are commutative. Since (θY F (If ) − IF (f ) θX )F (uX ) = uF (Y ) F (f ) − IF (f ) uF (X) = 0 there is a morphism δ : F (ΣX) → F (I(Y )) such that θY F (If )−IF (f ) θX = δF (vX ). Then we have ηY F (Σf ) − ΣF (f ) ηX = vF (Y ) δ, and then η Y F (Σ(f )) = Σ(F (f ))η X ∼ in C ′ . Therefore we have a functorial isomorphism η : F ◦ ΣC → ΣC ′ ◦ F . For a f g h triangle X − →Y − →Z− → ΣX in C, we may have a morphism between conflations: 0 /X 0  /Y uX / I(X) vX / ΣX /0 / ΣX /0 ψf f g  /Z h There are morphisms ψF (f ) : IC ′ (F (X)) → Z ′ and z : F (Z) ❴ ❴ ❴/ Z ′ such that we have a commutative diagram F (uX ) F (vX ) / F (X) / F (I(X)) / F (ΣX) ⑧ ⑧ F (ψ ) F (f ) ⑧ ⑧ ⑧ f ⑧⑧⑧⑧⑧⑧ ⑧  /0 / F (Z) / F (ΣX) / F (Y ) ✤z ηX ηX θ X   ✤ / F (X) ✤ / I(F (X)) / ΣF (X) 0 ✤ ⑧⑧ ψF (f ) ⑧⑧⑧⑧⑧ ⑧⑧⑧ F (f )  ⑧   ⑧ /0 / ΣF (X) / Z′ / F (Y ) 0 0 0 /0 /0 Similarly, there is a morphism z ′ : Z ′ → F (Z) such that we have the above commutative diagram of which all vertical arrows are reversed. Put ζ = z ′ z + F (ψf )γX F (h), we have a commutative diagram 0 / F (Y ) F (g) / F (Z) F (h) / F (ΣX) /0 / F (ΣX) /0 ζ 0 / F (Y ) F (g)  / F (Z) F (h) POLYGON OF RECOLLEMENTS 31 Then z ′ z is an isomorphism in C ′ . Similarly zz ′ is also an isomorphism in C ′ , and then z is an isomorphism in C ′ . Therefore we have a commutative diagram in C ′ : F (X) F (f ) / F (Y ) F (g) / F (Z) F (h) / F (ΣX) ηX / ΣF (X) z F (X) F (f ) / F (Y ) g′  / Z′ h′ / ΣF (X) where all vertical arrows are isomorphisms. Hence F induces a triangle functor F : C → C ′.  Proposition 7.4. Let C, C ′ be Frobenius categories, F : C → C ′ , G : C ′ → C exact functors such that F is a right adjoint of G. Then F and G induce triangle functors F : C → C ′ , G : C ′ → C such that F is a right adjoint of G. Proof. Let 0 → A → B → C → 0 be a conflation in C, then 0 → F (A) → F (B) → F (C) → 0 is a conflation in C ′ . For a projective-injective object P of C ′ , we have an isomorphism between exact sequences: 0 / HomC (P, F (A)) 0  / HomC (G(P ), A) / HomC (P, F (B)) ≀ / HomC (P, F (C)) ≀ /0 ≀  / HomC (G(P ), B)  / HomC (G(P ), C) Then G(P ) is a projective-injective object in C. Similarly, if Q is a projectiveinjective object of C, then F (Q) is also a projective-injective object of C ′ . 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Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, 146, Cambridge University Press, Cambridge, 1990. O. Iyama: Graduate School of Mathematics, Nagoya University Chikusa-ku, Nagoya, 464-8602 Japan E-mail address: [email protected] K. Kato: Graduate School of Science, Osaka Prefecture University, 1-1 Gakuencho, Nakaku, Sakai, Osaka 599-8531, JAPAN E-mail address: [email protected] J. Miyachi: Department of Mathematics, Tokyo Gakugei University, Koganei-shi, Tokyo, 184-8501, Japan E-mail address: [email protected]