Derived categories of N-complexes

DERIVED CATEGORIES OF N -COMPLEXES arXiv:1309.6039v5 [math.CT] 2 Oct 2017 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI Abstract. We study the homotopy category KN (B) of N -complexes of an additive category B and the derived category DN (A) of an abelian category A. First we show that both KN (B) and DN (A) have natural structures of triangulated categories. Then we establish a theory of projective (resp., injective) resolutions and derived functors. Finally, under some conditions of an abelian category A, we show that DN (A) is triangle equivalent to the ordinary derived category D(MorN−2 (A)) where MorN−2 (A) is the category of sequential N − 2 morphisms of A. 0. Introduction The notion of N -complexes, that is, graded objects with N -differentials d (dN = 0), was introduced by Mayer [32] in his study of simplicial complexes. Recently Kapranov and Dubois-Violette gave abstract framework of homological theory of N -complexes [22, 10]. Since then the N -complexes attracted many authors, for example [4, 5, 9, 11, 12, 13, 20, 22, 33, 34]. The aim of this paper is to give a solid foundation of homological algebra of N -complexes by generalizing classical theory of derived categories due to Grothendieck-Verdier. In particular we study homological algebra of N -complexes of an abelian category A based on the modern point of view of Frobenius categories (see [17] for the definition) and their corresponding algebraic triangulated categories. In section 2, we study the category CN (B) of N -complexes over an additive category B and the homotopy category KN (B). Precisely speaking, we introduce an exact structure on CN (B) to prove the following results. Theorem 0.1 (Theorems 2.1 and 2.6). (1) The category CN (B) has a structure of a Frobenius category. (2) The category KN (B) has a structure of a triangulated category. We give an explicit description of the suspension functor Σ and triangles in KN (B). Unlike the classical case N = 2, the suspension functor Σ does not coincide with the shift functor Θ. However we have the following connection between Σ and Θ in KN (B). Theorem 0.2 (Theorem 2.7). There is a functorial isomorphism Σ2 ≃ ΘN on KN (B). In Section 3, we introduce the derived category DN (A) of N -complexes for an abelian category A. We generalize the theory of projective resolutions of complexes initiated by Verdier [42] and extended to unbounded complexes by Spaltenstein and Böckstedt-Neeman [41, 7]. Our main result is the following, where Date: September 19, 2018. 1991 Mathematics Subject Classification. 18E30, 16G99. 1 2 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI Prj A (resp., Inj A) is the subcategory of projective (resp., injective) objects in A and KaN (A) (resp., KpN (A), KiN (A)) is the homotopy category of N -acyclic (resp., K-projective, K-injective) N -complexes (see Definitions 3.3, 3.20). We denote by −,b −,a K− N (Prj A) (resp., KN (Prj A), KN (Prj A)) the subcategory of KN (Prj A) consisting of N -complexes bounded above (resp., bounded above with bounded homologies, bounded above and N -acyclic). For other unexplained notations, we refer to the paragraph before Theorem 3.16. Theorem 0.3 (Theorems 3.16 and 3.21). The following hold for ♮ =nothing, b. (1) Assume that A has enough projectives. −,a −,♮ (a) (K−,♮ N (Prj A), KN (A)) is a stable t-structure in KN (A) and we have −,b − b triangle equivalences K− N (Prj A) ≃ DN (A) and KN (Prj A) ≃ DN (A). p a (b) If A is an Ab4-category, then (KN (A), KN (A)) is a stable t-structure in KN (A) and we have a triangle equivalence KpN (A) ≃ DN (A). (2) Assume that A has enough injectives. +,♮ +,♮ (a) (K+,a N (A), KN (Inj A)) is a stable t-structure in KN (A) and we have +,b + b triangle equivalences K+ N (Inj A) ≃ DN (A) and KN (Inj A) ≃ DN (A). a i ∗ (b) If A is an Ab4 -category, then (KN (A), KN (A)) is a stable t-structure in KN (A) and we have a triangle equivalence KiN (A) ≃ DN (A). Moreover, we generalize a result of Krause [29] characterizing the compact objects in classical homotopy categories. We deal with a locally noetherian Grothendieck category, that is, a Grothendieck category with a set of generators of noetherian objects. We give the following result, where C c denotes the subcategory of compact objects in an additive category C. Theorem 0.4 (Theorem 3.27). Let A be a locally noetherian Grothendieck category with the subcategory noeth A of noetherian objects in A. (1) KN (Inj A) is compactly generated. (2) The canonical functor KN (Inj A) → DN (A) induces an equivalence between KN (Inj A)c and DbN (noeth A). We generalize the classical existence theorem of derived functors to our setting by showing that any triangle functor KN (A) → KN ′ (A′ ) has a left/right derived functor DN (A) → DN ′ (A′ ) (see Definition 3.30) under certain mild conditions on A. Our result is the following. Theorem 0.5 (Theorem 3.33). Let A, A′ be abelian categories, F : KN (A) → KN ′ (A′ ) a triangle functor. Then the following hold. (1) If A is an Ab4-category with enough projectives, then the left derived functor LF : DN (A) → DN ′ (A′ ) exists. (2) If A is an Ab4∗ -category with enough injectives, then the right derived functor RF : DN (A) → DN ′ (A′ ) exists. In section 4, we give our main result in this paper. We show that the derived category DN (A) is triangle equivalent to the ordinary derived category D(MorN −2 (A)) of MorN −2 (A), where MorN −2 (A) is the category of sequences of N − 2 morphisms of A (see Definition 4.1). Theorem 0.6 (Theorems 4.2 and 4.10). Let A be an Ab3-category with a small full subcategory of compact projective generators. Then we have a triangle equivalence DERIVED CATEGORIES OF N -COMPLEXES 3 for ♮ =nothing, +, −, b. D♮N (A) ≃ D♮ (MorN −2 (A)). As applications, we have the following triangle equivalences. Here B is an additive category, Morsm N −2 (B) is the category of sequences of N −2 split monomorphisms of B (see Definition 4.1) and TN −1 (R) is the upper triangular matrix ring of size N − 1 over a ring R. For a full subcategory C of an additive category B with arbitrary coproducts, AddB C is the category of direct summands of coproducts of objects of C in B. For a ring R, mod R (resp., prj R) is the category of finitely presented (resp., finitely generated projective) R-modules. Corollary 0.7 (Corollary 4.12, Proposition 4.15). (1) Let B be an additive category with arbitrary coproducts. If the subcategory B c of compact objects of B is skeletally small and satisfies B = Add(B c ), then we have triangle b sm − sm b equivalences K− N (B) ≃ K (MorN −2 (B)) and KN (B) ≃ K (MorN −2 (B)). (2) For a ring R, we have a triangle equivalence K♮N (prj R) ≃ K♮ (prj TN −1 (R)) for ♮ = −, b, (−, b). For a right coherent ring R, we have a triangle equivalence D♮N (mod R) ≃ D♮ (mod TN −1 (R)) for ♮ =nothing, −, b. In [16], we will study more precise relations between the homotopy categories. 1. Preliminaries In this section, we collect preliminary results on additive and triangulated categories. We will omit proofs of elementary facts. Lemma 1.1. In an abelian category, consider a pull-back (resp., push-out) diagram X f  Y and morphisms ( g′ hold. f′ g g′ ) : X ′ ⊕ Y → Y ′, / X′  f′ / Y′ g
: X → X ′ ⊕ Y . Then the following f (1) If f ′ (resp., f ) is epic (resp., monic), then the above diagram is also pushout (resp., pull-back), and f (resp., f ′ ) is also epic (resp., monic). (2) The induced morphism Ker f → Ker f ′ is an isomorphism (resp., an epimorphism). (3) The induced morphism Cok f → Cok f ′ is a monomorphism (resp., an isomorphism). (4) We have an
exact sequence 0 → Cok f → Cok f ′ → Cok ( g′ f ′ ) → 0 (resp., g 0 → Ker f → Ker f → Ker f ′ → 0. A commutative square is called exact if it is pullback and push-out [39]. Lemma 1.2. In an abelian category, consider two pull-back squares (X) and (Y) A  D a (X) /B b /E (Y ) /C c / F. Then the square (X+Y) is exact if and only if the squares (X) and (Y) are exact. 4 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI Lemma 1.3. In an abelian category, consider an exact square with a split epimorphism d. A⊕B ι=( ι1 ι2 )  D (0 1)   d= d1 d2 /B 1  (0) / B⊕C Then there exists an isomorphism a : A ⊕ B ⊕ C → D such that ι = a da = ( 00 10 01 ). 1 0 01 00 and Proof. Since d is a split epimorphism, there exists ι3 : C → D such that d1 ι3 = 0 and d2 ι3 = 1. Then a = (ι1 ι2 ι3 ) satisfies the desired conditions.  For a triangulated category T and a full subcategory C of T , we denote by tri C = triT C the smallest triangulated subcategory of T containing C, and by thick C = thickT C the smallest triangulated subcategory of T containing C and closed under direct summands, and by Loc C = LocT C the smallest triangulated subcategory of T containing C and closed under coproducts. Definition 1.4 (Triangle Functor). Let T and T ′ be triangulated categories with suspensions ΣT and ΣT ′ respectively. A triangle functor is a pair (F, α), where ∼ F : T → T ′ is an additive functor and α : F ΣT → ΣT ′ F is a functorial isomorphism such that (F X, F Y, F Z, F (u), F (v), αX F (w)) is a triangle in T ′ whenever (X, Y, Z, u, v, w) is a triangle in T . If a triangle functor F is an equivalence, then we say that T is triangle equivalent to T ′ . Let (F, α), (G, β) : T → T ′ be triangle functors. A functorial morphism of triangle functors is a functorial morphism φ : F → G satisfying (ΣT ′ φ)α = βφΣT . Let T be a triangulated category and U, V be full subcategories. The category of extensions U ∗ V is the full subcategory of T consisting of objects X such that there exists a triangle U → X → V → ΣU with U ∈ U and V ∈ V. Note that (U ∗ V) ∗ W = U ∗ (V ∗ W) holds by octahedral axiom. Definition 1.5 ([36]). Let T be a triangulated category. A pair (U, V) of full triangulated subcategories of T is called a stable t-structure (also known as semiorthogonal decomposition, torsion pair, Bousfield localization) in T provided that HomT (U, V) = 0 and T = U ∗ V. In this case, the canonical quotient T → T /U (resp., T → T /V) has a right (resp., left) adjoint, and we have a triangle equivalence T /U ≃ V (resp., T /V ≃ U). Lemma 1.6. [21] Let T be a triangulated category and U, V be full triangulated subcategories. Then the following conditions are equivalent. (1) V ∗ U ⊂ U ∗ V. (2) U ∗ V is a triangulated subcategory of T . (3) Any morphism f : U → V with U ∈ U and V ∈ V factors through an object in U ∩ V. In this case, (U/(U ∩V), V/(U ∩V)) is a stable t-structure in (U ∗ V)/(U ∩V). Hence we have triangle equivalences U/(U ∩ V) ≃ (U ∗ V)/V and V/(U ∩ V) ≃ (U ∗ V)/U. Thus the canonical functors U/(U ∩ V) → T /V and V/(U ∩ V) → T /U are fully faithful. DERIVED CATEGORIES OF N -COMPLEXES 5 2. Homotopy category of N -complexes In this section, we study the homotopy category of N -complexes. We fix a positive integer N ≥ 2. Throughout this section B is an additive category. An N -complex X = (X i , diX ) is a diagram di−1 di+1 di · · · −−X−→ X i −−X → X i+1 −−X−→ · · · with X i ∈ B and diX ∈ HomB (X i , X i+1 ) satisfying −1 i di+N · · · di+1 X X dX = 0 for any i ∈ Z. We often denote the r-th power of dX by {r} i+1 i dX = di+r X · · · dX dX {0} without mentioning grades, where dX = 1. A morphism f : X → Y between N -complexes is a commutative diagram ··· ··· di−1 X di−1 Y diX / Xi f / Yi / X i+1 i diY /Y  di+1 X f i+1 i+1 dY i+1 / ··· / ··· with f i ∈ HomB (X i , Y i ) for any i ∈ Z. We denote by CN (B) the category of N -complexes. We call an N -complex X bounded above (resp., bounded below ) if X i = 0 for all i ≫ 0 (resp., i ≪ 0), and bounded if X is both bounded above and bounded below. + b We denote by C− N (B) (resp., CN (B), CN (B)) the full subcategory of bounded above (resp., bounded below, bounded) N -complexes. Our approach to the category CN (B) of N -complexes is based on the theory of exact categories [40] (see [24] for modern account). Let SN (B) be the collection of f fi g sequences 0 → X − →Y − → Z → 0 of morphisms in CN (B) such that 0 → X i −→ gi Y i −→ Z i → 0 is split exact in B for any integer i. Then we have the following basic observation. Theorem 2.1. The category (CN (B), SN (B)) of N -complexes is a Frobenius category. For an object M of B and integers s and 1 ≤ r ≤ N , let ds−r+1 ds−2 ds−1 µsr (M ) : · · · → 0 → M s−r+1 −−−−→ · · · −−−→ M s−1 −−−→ M s → 0 → · · · be an N -complex given by M s−i = M (0 ≤ i ≤ r−1) and ds−i = 1M (0 < i ≤ r−1). One can easily check the functorial isomorphisms (2.2) HomCN (B) (X, µsN (M )) ≃ HomB (X s , M ) and HomCN (B) (µsN (M ), X) ≃ HomB (M, X s−N+1 ) 6 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI where f ∈ HomB (X s , M ) and g ∈ HomB (M, X s−N +1 ) are mapped to ρsf and λsg respectively by the following commutative diagrams. µsN (M ) : /0 ··· ρs f  X:  d ···  / X s−N λs g µsN (M ) : /M d  gd /M  /0 ··· f / X s−N+1 1 / ··· 1 d / ··· d d / Xs / ··· 1  /M /M {N −1} 1 / ··· /0 {N −1} d f  / X s+1 d / ··· g  /0 / ··· Lemma 2.3. The object µsN (M ) is projective-injective in (CN (B), SN (B)) for any object M ∈ B and any integer s. Proof. For any exact sequence 0 → X → Y → Z → 0 in SN (B), the isomorphism (2.2) gives a commutative diagram of exact sequences 0 / HomC N (B) (Z, µsN (M )) / HomC N (B) ≀ 0 (Y, µsN (M )) / HomC ≀  / HomB (Z s , M ) N (B) (X, µsN (M )) ≀  / HomB (Y s , M )  / HomB (X s , M ) where the lower sequence is exact since 0 → X s → Y s → Z s → 0 is split exact. This means that µsN (M ) is injective. Dually one can show that µsN (M ) is projective.  Let X ∈ CN (B) be given. We have morphisms ρn1X n−N +1 : µnN (X n−N +1 ) → X L and λn1X n : X → µnN (X n ), using (2.2). Set ρX = (ρn1X n−N +1 )n : n∈Z µnN (X n−N +1 ) → L n n X and λX = (λn1X n )n : X → n∈Z µN (X ). Then we have the following exact sequences in SN (B). (2.4) ǫ X → 0 → Ker ρX −− M ρ n−N+1 → X → 0, µn ) −−X N (X n∈Z λ → 0 → X −−X M η n −X → Cok λX → 0. µn N (X ) − n∈Z of Theorem 2.1. The exact sequences (2.4) with Lemma 2.3 show that (CN (B), SN (B)) has enough projectives and enough injectives. Let X be an arbitrary projective (resp., injective) object. Then, on the first (resp., second) sequence of (2.4), X is a direct summand of the middle term. By Lemma 2.3, X is injective (resp., projective).  The stable category F of a Frobenius category (F , S) has the same objects as F and the homomorphism set between X, Y ∈ F is given by HomF (X, Y ) = HomF (X, Y )/I(X, Y ) where I(X, Y ) is the subgroup of HomF (X, Y ) consisting of morphisms which factor through some projective-injective object of (F , S). By [17], F has a structure of a triangulated category, which is nowadays called an algebraic triangulated category. Now we shall describe the stable category of our Frobenius category (CN (B), SN (B)) more explicitly. Indeed, as in the classical case, it coincides with the homotopy category of N -complexes. Recall that 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 (2.5) fi = N −1 X j=1 +j i+j−1 i+j−2 di−1 · · · di−N s dX · · · diX Y Y / 0, DERIVED CATEGORIES OF N -COMPLEXES 7 for any i ∈ Z. For morphisms f, g : X → Y in CN (B), we denote f ∼ g if f − g is null-homotopic. We denote by KN (B) the homotopy category, that is, the category consisting of N -complexes such that the homomorphism set between X, Y ∈ KN (B) is given by HomKN (B) (X, Y ) = HomCN (B) (X, Y )/ ∼ . Theorem 2.6. The stable category of the Frobenius category (CN (B), SN (B)) is the homotopy category KN (B) of B. In particular, KN (B) is an algebraic triangulated category. Proof. It suffices to show that a morphism f : XL→ Y is null-homotopic if and only if f factors through the morphism λX : X → n∈Z µnN (X n ) given in (2.4). This can be easily checked by (2.2).  Now we define functors Σ, Σ−1 : CN (B) → CN (B) by Σ−1 X = Ker ρX and ΣX = Cok λX in the exact sequences (2.4). Then Σ and Σ−1 induce the suspension functor and its quasi-inverse of the triangulated category KN (B). On the other hand, we define the shift functor Θ : CN (B) → CN (B) by Θ(X)i = X i+1 and diΘ(X) = di+1 X for X = (X i , diX ) ∈ CN (B). This induces the shift functor Θ : KN (B) → KN (B) which is a triangle functor. Unlike classical case, Σ does not coincide with Θ. However we have the following observation. Theorem 2.7. There is a functorial isomorphism Σ2 ≃ ΘN on KN (B). To prove this, we give a more explicit description of Σ and Σ−1 . Let X = (X i , di ) be an object of CN (B). In (2.4), the first sequence is given by  (Σ−1 X)m = m−1 M X i, dm Σ−1 X i=m−N+1  (ǫX )m     =     1 −d 0 .. . 0 0 0 1 −d .. . 0 0 0 0 1 .. . ··· ··· ··· ··· ··· .. . −d 0 0 0 0 .. . 1 −d −d  −d{2}   ..  . =   −d{N−3}   −d{N−2} −d{N−1} 1 0 .. . 0 0 0 0 1 .. . 0 0 0 ··· ··· .. . ··· ··· ··· 0 0 .. . 1 0 0 0 0 .. . 0 1 0                 and (ρX )m =     d{N−1} d{N−2} ··· d 1
. while the second sequence by  (ΣX)m = m+N−1 M i=m+1 Xi, dm ΣX     =     0 0 . .. 0 0 1 0 . .. 0 0 0 1 . .. 0 0 −d{N−1} −d{N−2} −d{N−3} ··· ··· .. . ··· ··· ··· 0 0 . .. 1 0 −d{2} 0 0 . .. 0 1 −d      ,     8 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI  m (λX ) 1 d . ..    =   {N−2}  d d{N−1}         m  and (ηX ) =        1 −d 0 .. . 0 −d 0 0 .. . 0 0 1 −d .. . 0 ··· ··· ··· .. . ... 0 0 0 .. . −d 0 0 0 .. . 1     .    of Theorem 2.7. We shall construct a functorial isomorphism Σ → ΘN Σ−1 . Given Lm+N −1 an object X = (X i , di ) ∈ CN (B), we have (ΣX)m = i=m+1 X i = (Σ−1 X)m+N m −1 for each m by (2.4). Let φm X)m+N be a morphism given as X : (ΣX) → (Σ  φm X    =    1 d d{2} .. . 0 1 d .. . d{N−2} d{N−3} 0 0 1 .. . ··· ··· ··· ··· .. . d 0 0 0 .. . 1     .    Then it is easy to check that φX makes the following diagram commutative (ΣX)m φm  X −1 (Σ X)m+N dm ΣX / (ΣX)m+1 m+1 dm+N −1 Σ X  φX / (Σ−1 X)m+N +1 . Thus φX : ΣX → ΘN Σ−1 X is an isomorphism in CN (B). Next let f be a morphism from X to Y in CN (B). It is routine to show (ΘN Σ−1 f )φX = φY Σf holds. Thus φ gives a functorial isomorphism Σ ≃ ΘN Σ−1 .  + b We denote by K− N (B) (resp., KN (B), KN (B)) the full subcategory of KN (B) + b corresponding to C− N (B) (resp., CN (B), CN (B)). Then they are full triangulated subcategories of KN (B) by the above descriptions of Σ and Σ−1 . Definition 2.8 (Hard truncations). For an N -complex X = (X i , di ), set τ≤n X : · · · → X n−2 → X n−1 → X n → 0 → · · · , τ≥n X : · · · → 0 → X n → X n+1 → X n+2 → · · · . Then we have a triangle τ≥n X → X → τ≤n−1 X → Σ(τ≥n X) in KN (B). Later we will use the following observation. Lemma 2.9. We have the following. (1) For any C ∈ B, i, s ∈ Z and 0 < r < N , we have Σ2i+k µsr (C) ≃  −iN µr +s (C) (k = 0) −iN +s−r µN (C) (k = 1). −r b s (2) KN (B) = tri{µ1 (C) | C ∈ B, 0 < s < N }. Proof. (1) For each C ∈ B and r, i ∈ Z with 1 ≤ r ≤ N − 1, we have a term-wise +s −iN +s−r +s split exact sequence 0 → µ−iN (C) → µ−iN (C) → µN (C) → 0 in C(B). r N −r −iN +s Since µN (C) is a projective-injective object in CN (B), we have the desired isomorphisms in KN (B). (2) Using triangles in Definition 2.8, we can show KbN (B) = tri{µs1 (C) | C ∈ B, s ∈ Z} by an induction on the number of non-zero terms. Moreover, we can replace the condition s ∈ Z by 0 ≤ s < N since Σ2 ≃ ΘN holds by Theorem 2.7. We can DERIVED CATEGORIES OF N -COMPLEXES 9 −1 s further replace it by 0 < s < N since µ01 (C) = ΣµN N −1 (C) belongs to tri{µ1 (C) | C ∈ B, 0 < s < N }.  We end this section with an explicit description of the mapping cone. For a morphism f : Y = (Y i , ei ) → X = (X i , di ) in CN (B), the mapping cone C(f ) is given by the diagram λY 0 /Y 0  /X g  / C(f )  where C(f )m = X m ⊕( Y i ), dm C(f ) i=m+1 gm   1 0   = . ,  ..  0 0 1 0   0 hm =   .. . 0 .. . 0 1 .. . 0  0 f / ΣY /0 / ΣY / 0, f 0 .. . 0 0 0 1 .. . 0 0 −e{N−1} −e{N−2} ψf f m+N−1 M ηY / I(Y )     =     ··· .. . .. . ··· h d 0 .. . 0 0 0   f 0 −e  ..    .   and ψfm =  0    .. 0  . 1 0 g 0 0 ··· ··· .. . 1 0 −e{2} .. . ··· ··· ··· 0 1 0 0 −e .. . ··· 1 .. . 0 ··· ··· .. . .. . −e 0 0 .. . 0 1 −e            0 0  ..   . .   0 1 h Thus we have a triangle Y − →X− → C(f ) − → ΣY in KN (B). 3. Derived category of N -complexes In this section, we introduce the derived category of N -complexes as the Verdier quotient of the homotopy category with respect to the N -quasi-isomorphisms as in the case of 2-complexes. 3.1. Homologies of N -complexes. Let A be an abelian category, and Prj A (resp., Inj A) the subcategory of A consisting of projective (resp., injective) objects of A. Let X be an N -complex in A di−1 di · · · → X i−1 −−X−→ X i −−X → X i+1 → · · · . For 0 ≤ r ≤ N and i ∈ Z, we define i+r−1 Zi(r) (X) := Ker(dX · · · diX ), i−r Bi(r) (X) := Im(di−1 X · · · dX ), i−r Ci(r) (X) := Cok(di−1 X · · · dX ), Hi(r) (X) := Zi(r) (X)/ Bi(N −r) (X). For example, Zn(N ) (X) = Bn(0) (X) = X n and Zn(0) (X) = Bn(N ) (X) = 0 hold. With this in mind, using the notation dn(r) := dnX |Zn(r) (X) , we can understand a homology as follows (3.1) ! +r dn−N (N ) dn−2 (r+2) dn−1 (r+1) n−N +r n−1 Hn(r) (X) = Cok Z(N (X) −−−−−→ · · · −−−−→ Z(r+1) (X) −−−−→ Zn(r) (X) . ) 10 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI X n−N +1 ❄❄ ❄❄ ❄ X n−N +2 ··· ? ⑧⑧ ⑧ ⑧⑧ ③< ③③ ③③ ❄❄ ❄❄ ❄ ❄❄ ❄❄  < ③③ ③③ ❄❄ ❄❄  ❄❄ ❄❄  < ③③ ③③ n−N +2 n−N +2 n−N +3 Z(N D(N −1) Z(N −1) −1) ··· ? ⑧⑧ ⑧ ⑧⑧ n−N +3 n−N +3 n−N +4 Z(N D(N −2) Z(N −2) −2) ❄❄ ❄❄  n−N +4 Z(N −3) ❇❇ ❇❇ ❇ . .. . . n−1 ❄❄ ❄❄ ❄ Z n+1 . . ❄❄ ❄❄  n Z(1) n X X n+1 ⑧? ❄❄❄ ⑧? ⑧ ⑧ ❄❄ ⑧ ⑧ ⑧ ⑧ ⑧  ⑧ n n n−1 n+1 Z(N Z(N D n−1 −1) D(N −1) Z(N −1) ?⑧ −1)❄❄ (N −1) ⑧? ❄❄ ? ⑧ ❄❄ ⑧ ⑧ ❄❄ ⑧⑧ ⑧⑧   ⑧⑧ ⑧⑧ n n n+1 Z(N −2) D(N −2) Z ··· (N −2) ⑧? ❄❄❄ ⑧? ⑧ ⑧ ❄ ⑧⑧ ⑧⑧ X ? ⑧⑧ ⑧ ⑧⑧ . ? (N −3) ❁❁ ⑧? . ⑧ ❁❁ ⑧⑧ ⑧ ⑧ ⑧  ⑧ n . Z(2) ?⑧ ❄❄ ?⑧ . . ❄❄ ⑧ ⑧  ⑧⑧ ⑧⑧ Dn Z n+1 ❄❄ (1) ⑧? (1) ❄❄ ⑧ ❄ ⑧⑧⑧ 0 Figure 1. For 1 ≤ r ≤ N − 1, we have a pull-back diagram with the canonical inclusion ιn(r) . (3.2) / Zn(1) (X) 0 / Zn(r) (X) _  / Zn(1) (X) 0 ιn (r) / Zn(r+1) (X) dn (r) n (D(r) ) dn (r+1) / Zn+1 (X) (r−1) _  ιn+1 (r−1) / Zn+1 (X), (r) n Then (D(r) ) forms a commutative diagram in Figure 1. Definition 3.3. We call X ∈ CN (A) N -acyclic if Hi(r) (X) = 0 for any 0 < r < N and i ∈ Z. For example, the complex µiN (M ) is N -acyclic for any M ∈ A and i ∈ Z. An N -complex X is N -acyclic if and only if there exists some r with 0 < r < N such that Hi(r) (X) = 0 for each integer i [22]. ♮,a For ♮ =nothing, −, +, b, let C♮,a N (A) (resp., KN (A)) denote the full subcategory of C♮N (A) (resp., K♮N (A)) consisting of N -acyclic N -complexes. Proposition 3.4. We have the following. ♮ (1) K♮,a N (A) is a thick subcategory of KN (A) for ♮ = −, +, b. i i+r i +r (2) H(r) (ΣX) = H(N −r) (X) and H(r) (Σ−1 X) = Hi−N (N −r) (X) hold for any X ∈ CN (A). To prove this, we recall that CN (A) forms an abelian category. A sequence α β β α 0→X − →Y − → Z → 0 is exact if and only if 0 → X i − → Yi − → Z i → 0 is (not necessarily split) exact in A for each i. In this case, for any 0 ≤ r ≤ N and i ∈ Z, we have the following exact sequence [10]. (3.5) ··· ∂ β∗ α ∂ β∗ α ∗ ∗ ∗ ∗ i+r Hi(r) (Y ) −→ Hi(r) (Z) −→ Hi+r −→ Hi(r) (X) −−→ −→ Hi+r (N −r) (X) − (N −r) (Y ) −→ H(N −r) (Z) ∂ α β∗ ∂ α ∗ ∗ ∗ ∗ i+N i+r+N −→ Hi+N −→ Hi+N −→ ··· . (r) (X) − (r) (Y ) −→ H(r) (Z) −→ H(N −r) (X) − DERIVED CATEGORIES OF N -COMPLEXES 11 of Proposition 3.4. (2) It is immediate by applying (3.5) to the exact sequences (2.4). −1 (1) It follows from (2) that K♮,a . Let X → Y → N (A) is closed under Σ and Σ Z → ΣX be a triangle in KN (A). This comes from a term-wise split short exact sequence. Therefore if X and Y belong to K♮,a  N (A), then so does Z by (3.5). As in the classical case, we have the following observation. + Lemma 3.6. If X ∈ KaN (A) and P ∈ K− N (Prj A) (resp., I ∈ KN (Inj A)), then we have HomKN (A) (P, X) = 0 (resp., HomKN (A) (X, I) = 0). Proof. Let f : P → X be as follows. P : ··· / P n−2 dn−2 P / P n−1 n−2 f  X : ··· f / X n−2 dn−2 X /X dn−1 P / Pn f n−1n−1 d n−1 X f   / Xn /0 / ···  / X n+1 / ··· . n dn X Since dnX f n = 0 and Hn(1) (X) = 0, there is sn : P n → X n−N +1 such that n−1 +1 n n−1 n−1 n−2 +1 n n−1 n−1 n−1 f n = dX · · · dn−N s . Since dX (f −dX · · · dn−N s dP ) = dX f − X X n−2 n−N +1 n n−1 n n−1 n−1 n−1 n−N n−1 f dP = 0, there is s :P →X such that f = dX · · · dX s dP + n−2 n−N n−1 dX · · · dX s . Repeating similar argument, we obtain si : P i → X i−N +1 for i ≤ n satisfying (2.5).  Now let B be an additive category, pick X ∈ CN (B) and M ∈ B. Then we have N -complexes HomB (X, M ) and HomB (M, X) of abelian groups with HomB (M, X)n := HomB (M, X n ) and HomB (X, M )n := HomB (X −n , M ). One can easily check the following analogs of (2.2) for each 0 < r < n. (3.7) HomCN (B) (µsr (M ), X) ≃ Zs−r+1 (HomB (M, X)), (r) HomCN (B) (X, µsr (M )) ≃ Z−s (Hom B (X, M )), (r) s−r+1 (HomB (M, X)), HomKN (B) (µsr (M ), X) ≃ H(r) −s s HomKN (B) (X, µr (M )) ≃ H(r) (HomB (X, M )). We prepare the following observations which will be used later. Lemma 3.8. Let X ∈ KN (A), M ∈ A, and 0 < r < N be given. (1) We have a commutative diagram of exact sequences / HomA (M, X s−N +1 ) 0 /  / (X)) HomA (M, Zs−r+1 (r) d{N −r} HomA (M, Bs−r+1 (X)) (N −r) / HomA (M, Zs−r+1 (X)) (r) / / HomKN (A) (µsr (M), X)  HomA (M, Hs−r+1 (X)) (r) / 0 Ext1A (M, Bs−r+1 (X)) (N −r) (2) If M is projective in A, then HomKN (A) (µsr (M ), X) ≃ HomA (M, Hs−r+1 (X)). (r) +1 (3) If X ∈ KN (Inj A) is N -acyclic, then HomKN (A) (µsr (M ), X) ≃ Ext1A (M, Zs−N (N −r) (X)). Proof. (1) The upper sequence is exact by (3.7) and Zs−r+1 (HomA (M, X)) ≃ (r) HomA (M, Zs−r+1 (X)). The lower one is clearly exact. (r) (2) Immediate from (1). +1 s−N +1 (3) We have a short exact sequence 0 → Zs−N → Zs−r+1 (X) → (N −r) (X) → X (r) s−N +1 0. Applying HomA (M, −) and using injectivity of X , we have an exact sequence +1 HomA (M, X s−N +1 ) → HomA (M, Zs−r+1 (X)) → Ext1A (M, Zs−N (r) (N −r) (X)) → 0. 12 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI Comparing with the upper exact sequence in (1), we have the desired isomorphism.  Lemma 3.9. For a commutative diagram (3.2), the following hold. n+r−1−s ) is an exact square for any r ≤ s ≤ N − 1. (1) If Hn(r) (X) = 0, then (D(s) n−1 n−2 n−N +r In particular, (D(r) + D(r+1) + · · · + D(N −1) ) is an exact square. n (2) X is N -acyclic if and only if d(r+1) is an epimorphism for any 0 < r < N and n ∈ Z. (3) X is isomorphic to 0 in KN (A) if and only if dn(r+1) is a split epimorphism for any 0 < r < N and n ∈ Z. Proof. (1) (2) The assertions immediately follow from (3.1). (3) We prove the ‘only if’ part. Clearly dn(r+1) is a split epimorphism for X = µsN (M ). Since every projective-injective object of CN (A) is in Add{µsN (M ) | s ∈ Z, M ∈ A}, the assertion follows. Lr−1 n+i To show the converse, set Wn(r) (X) := i=0 Z(1) (X) for 1 ≤ r ≤ N . Then n n we have natural morphisms p(r) := ( 0 1 ) : Wn(r) (X) → Wn+1 (r−1) (X) and i(r) := n n ( 10 ) : W(r) (X) → W(r+1) (X). We show the existence of an isomorphism an(r) : Wn(r) (X) → Zn(r) (X) such that the following diagram commute. Wn (r) (X) ❳❳❳❳ ❳ in (r)  Wn (r+1) (X) ❳❳ ❳an / Wn+1 (X) (r−1) ❳❳❳ n+1 a pn (r) an (r) ❳❳❳❳❳, pn (r+1) (r+1) ❳❳❳, Zn (r) (X) dn (r) in+1 (r−1) (r−1) ❳❳❳, / Zn+1 (X) (r−1)  / Wn+1 (X) ❳ ιn+1 (r−1) (r) ❳❳❳ n+1 a(r) ❳❳❳  ❳ , / Zn+1 (X). dn (r+1) ιn (r)  Zn (r+1) (X) (r) an(1) For r = 1, set = 1. Suppose r > 1 and that we have defined an(i) for any n, i ∈ Z with 0 < i ≤ r. Applying Lemma 1.3 to the exact square n+1 n Wn (r) (X) = Z(1) (X) ⊕ W(r−1) (X)  n ιn (r) a(r) Zn (r+1) (X) pn (r) (an+1 )−1 dn (r+1) (r) / Wn+1 (X) (r−1)  in+1 (r−1) / Wn+1 (X) ⊕ Zn+r (X) = Wn+1 (X), (r) (1) (r−1) we get an isomorphism an(r+1) : Wn(r+1) (X) → Zn(r+1) (X) as desired. LN −1 n+i Consequently we have an isomorphism an(N ) : Wn(N ) (X) = i=0 Z(1) (X) → n n+1 n n+1 n n n holds, it is easy to check Z(N ) (X) = X . Since d = ι(N −1) d(N ) : X → X L X ≃ n∈Z µnN (Zn(1) (X)) in CN (A). Thus X is zero in KN (A).  Definition 3.10. A morphism f : X → Y of KN (A) is called an N -quasiisomorphism if Hi(r) (f ) : Hi(r) (X) → Hi(r) (Y ) is an isomorphism for any 0 < r < N and i ∈ Z, or equivalently by (3.5), the mapping cone C(f ) is N -acyclic. For ♮ =nothing, +, −, b, the derived category of N -complexes is defined as the quotient category D♮N (A) = K♮N (A)/ K♮,a N (A). By definition, a morphism in K♮N (A) is an N -quasi-isomorphism if and only if it is an isomorphism in D♮N (A). DERIVED CATEGORIES OF N -COMPLEXES f 13 g (1) If 0 → X − →Y − → Z → 0 is an exact sequence in the Proposition 3.11. f g abelian category CN (A), then it can be embedded into a triangle X − →Y − → h Z− → ΣX in DN (A). f g h (2) For any triangle X − → Y − → Z − → ΣY in DN (A), we have a long exact sequence ··· f∗ g∗ g∗ f∗ h ∗ i+r i+r i+r Hi(r) (X) −→ Hi(r) (Y ) −→ Hi(r) (Z) − (Z) (Y ) −→ H(N−r) (X) −→ H(N−r) − → H(N−r) h ∗ i+r+N i+N i+N i+N (X) −→ · · · . −→ H(N−r) (Z) − (Y ) −→ H(r) (X) −→ H(r) H(r) ∗ − − → g∗ f∗ ∗ − − → h f∗ h Proof. (1) We have the following commutative diagram of exact sequences in CN (A). 0 0   / I(X) /X 0 0  /Y f  g u Z  0 f u / ΣX /0 / ΣX /0 ψf  / C(f )  v s Z  0 v Then X − →Y − → C(f ) − → ΣX is a triangle in KN (A). Since I(X) is N -acyclic, s f su=g vs−1 is an N -quasi-isomorphism. Thus we have a triangle X − → Y −−−→ Z −−−→ ΣX in DN (A). f (2) We have only to verify the assertion for the triangle X − →Y → − C(f ) − → ΣY . Applying (3.5) to a short exact sequence 0 → X → Y ⊕ I(X) → C(f ) → 0 in CN (A), we get the desired sequence.  Definition 3.12 (Truncations). For an N -complex X = (X i , di ), set dn−N +1 dn−N (N ) +2 dn−N (N −1) dn+1 (2) n−N +2 σ≤n X : · · · −−−−→ X n−N +1 −−−−−→ Z(N → · · · −−−→ Zn(1) (X) → 0 → · · · . −1) (X) −−−−− Lemma 3.13. For an N -complex X = (X i , di ) and an integer n, the following hold. (1) Hi(r) (σ≤n (X)) ≃ Hi(r) (X) for any 0 < r < N and i + r ≤ n + 1. (2) If Hi(r) (X) = 0 holds for any 0 < r < N and i ≥ n + 1, then the canonical injection σ≤n X → X is an N -quasi-isomorphism. Proof. (1) If i + r ≤ n + 1, then Zi(r) (X) is the kernel of d{r} : Zi(n−i+1) (X) → i i X i+r which maps into Zi+r (n−i−r+1) (X). Hence Z(r) (X) = Z(r) (σ≤n X). Clearly Bi(N −r) (σ≤n X) = Bi(N −r) (X). (2) It remains to show Hi(r) (σ≤n (X)) ≃ Hi(r) (X) for i ≤ n and i + r > n + 1. Since Zi(r) (σ≤n X) = Zi(n−i+1) (X) holds, we have a commutative diagram {N −r} d +r / Zi Zi−N (X) (n−i+1) (n−i+N −r+1)  _ (X) _   d{N −r} / Zi(r) (X) X i−N +r / Hi (σ≤n X) (r) /0 / Hi(r) (X) /0 14 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI of exact sequences. The left square is exact. Indeed it follows from Lemmas 1.1 j and 1.2 since (D(s) ) is an exact square for j + s ≥ n + 1 by Lemma 3.9(1). Thus we have the desired isomorphism.  Proposition 3.14. Let ♮ = +, −, b. The canonical functors DbN (A) → D♮N (A) → DN (A) are fully faithful. Therefore D♮N (A) is equivalent to the full subcategory of DN (A) consisting of objects in K♮N (A). Proof. We only show that D− N (A) → DN (A) is fully faithful. Let f : X → Y be any morphism with X ∈ K− (A) and Y ∈ KaN (A). For sufficiently large n, f factors N through the natural morphism σ≤n (Y ) → Y . Since σ≤n (Y ) belongs to K−,a N (A) by Lemma 3.13(2), we get the conclusion from Lemma 1.6.  3.2. Elementary morphisms. In this subsection, we introduce the N -complex version of an elementary map of degree i in the sense of Verdier [42]. We start with the following observation. di−1 di Definition-Proposition 3.15. For an object X : · · · → X i−1 −−X−→ X i −−X → X i+1 → · · · in CN (A) and a morphism u : M → X i in A, we take successive pull-backs Y i−r−1 i−r−1 u  X i−r−1 d′i−r−1 (E i−r−1 / Y i−r  ui−r / X i−r ) di−r−1 X for 0 ≤ r < N − 1, where Y i = M and ui = u. Then there are a morphism d′i−N : X i−N → Y i−N +1 in A and a morphism Vi (X, u) :  ··· / X i−N ··· i−N / Y i−N+1 d ui−N +1 pi (u) X: d′i−N /X di−N X /X  ′i−N +1 (E i−N +1 ) i−N+1 d′i−1 / Y i−1 / ··· ui−1 /X / ··· +1 di−N X  (E i−1 ) i−1 di−1 X /M diX u u / Xi diX / X i+1 / ··· / X i+1 / ··· in CN (A). Moreover the following conditions are equivalent. (1) pi (u) is an N -quasi-isomorphism. (2) The commutative diagram (E i−N +1 + · · · + E i−1 ) is an exact square. (3) The commutative diagrams (E i−N +1 ), · · · , (E i−1 ) are exact squares. (4) (u d{N −1} ) : M ⊕ X i−N +1 → X i is an epimorphism. Proof. Set Y = Vi (X, u) and ũ = pi (u). (2) ⇔ (3)⇔ (4). These are clear from Lemmas 1.2 and 1.1. (1) ⇒ (4). The morphism ũ induces a morphism u : Y → X of 2-complexes as follows: {N −1} Y : dY u X: / Y i−N u / X i−N {N −1} dX {N −1} dY / Y i−N+1 i−N +1 dX  / X i−N+1 dY (E) /M u / Xi {N −1} dX dY dX {N −1} / Y i+1 / X i+1 dY / Y i+N / X i+N {N −1} dX dY dX / / The assumption forces u to be a 2-quasi-isomorphism. Then [42, III. 2.1.2(c] implies that (u d{N −1} ) : M ⊕ X i−N +1 → X i is an epimorphism. DERIVED CATEGORIES OF N -COMPLEXES 15 i−s i−s (3) ⇒ (1). We shall show that Hi−s (r) (ũ) : H(r) (Y ) → H(r) (X) is an isomorphism for each s ∈ Z, 0 < r < N . Set the commutative squares (A), (B), (C), (D) as follows: {r} Y i−N −s X  dY {N −r} / Y i−N −s+r dY {r} / Y i−s (B)   / X i−N −s+r / X i−s {r} {N −r} (A) i−N −s dX dX dY {N −r} / Y i−s+r (C)  / X i−s+r {r} dX dY / Y i+N −s (D)  / X i+N −s {N −r} dX Assume that (A) and (C) are exact. Consider the diagram with exact rows i−N −s+r C(r) (Y ) i−N −s+r C(r) (ũ)  i−N −s+r C(r) (X) / Zi−s (Y ) (r) Zi−s (ũ) (r)  / Zi−s (X) (r) / Hi−s (Y ) (r) /0 Hi−s (ũ) (r)  / Hi−s (X) (r) / 0. i−N −s+r Lemma 1.1 implies that C(r) (ũ) and Zi−s (r) (ũ) are isomorphisms. Hence so is i−s Hi−s (r) (ũ). Similarly H(r) (ũ) is an isomorphism provided that (B) and (D) are exact. Therefore it is enough to show that either (A), (C) or (B), (D) are exact. To prove this, notice that for any integer j other than i − N or i, the following square is exact. djY / Y j+1 Yj uj   uj+1 djX / X j+1 Xj Lemma 1.2 (1)⇒(2) implies that (B) and (D) are exact if s ∈ {0, 1, . . . , r − 1}, otherwise (A) and (C) are exact. Therefore one of the above two conditions holds.  3.3. Resolutions of N -complexes. The aim of this subsection is to establish Theorems 3.16, 3.21 which are well-known for the classical case N = 2. For a full additive subcategory B of an abelian category A and ♮ =nothing, −, +, b, ♮,b ♮,− ♮,+ we denote by C♮,a N (B) (resp., CN (B), CN (B), CN (B)) the full subcategory of C♮N (B) consisting of N -complexes X satisfying that Hi(r) (X) = 0 for any 0 < r < N and for all (resp. all but finitely many, sufficiently large, sufficiently small) i ∈ Z. ♮,b The corresponding subcategory of K♮N (B) is denoted by K♮,a N (B) (resp., KN (B), ♮,+ K♮,− N (B), KN (B)). Theorem 3.16. The following hold for ♮=nothing, b. −,a (1) If A has enough projectives, then (K−,♮ N (Prj A), KN (A)) is a stable t-structure −,♮ − in KN (A) and we have triangle equivalences K− N (Prj A) ≃ DN (A) and b K−,b N (Prj A) ≃ DN (A). +,♮ (2) If A has enough injectives, then (K+,a N (A), KN (Inj A)) is a stable t-structure +,♮ + in KN (A) and we have triangle equivalences K+ N (Inj A) ≃ DN (A) and b K+,b N (Inj A) ≃ DN (A). Our proof of Theorem 3.16 is based on Verdier’s method [42, III, Section 2.2]. Definition 3.17. Let M be an additive full subcategory of A satisfying the following. 16 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI V1 For any epimorphism u : X → L with X ∈ A and L ∈ M, there is an epimorphism v : L′ → L with L′ ∈ M which factors through u. V2 For any exact sequence 0 → X → Ln → · · · → L0 → 0 with L0 , · · · , Ln ∈ M, there is an epimorphism L′ → X with L′ ∈ M. c be the full subcategory of A consisting of objects X satisfying the following Let M conditions. (1) X has an ∞-M-presentation, that is, an exact sequence · · · → Ln → · · · → L1 → L0 → X → 0 with Li ∈ M for any i ≥ 0. (2) For any exact sequence 0 → X ′ → Ln → · · · → L0 → X → 0 with L0 , · · · , Ln ∈ M, X ′ has an ∞-M-presentation. c Obviously we have M ⊂ M. For example, M = Prj A satisfies (V1 ) and (V2 ). If A has enough projectives, c = A. then M Lemma 3.18. Let M be an additive full subcategory of A satisfying (V1 ) and (V2 ). (1) [42, III 2.2.4] For an exact sequence 0 → X → Y → Z → 0, if two out of c then so does the other. three terms belong to M, c there exists a morphism (2) For an epimorphism ρ : X → L with L ∈ M, µ : M → X with M ∈ M such that ρµ is an epimorphism. Proof. (2) Take an epimorphism π : M0 → L with M0 ∈ M and a pull-back diagram ρ′ K π′  X ρ / M0 π / L. Then ρ′ and π ′ are epimorphisms. The condition (V1 ) gives a morphism k : M → K with M ∈ M such that ρ′ k is an epimorphism. Set µ = π ′ k, then ρµ = πρ′ k is an epimorphism.  Proposition 3.19. Under the conditions (V1 ) and (V2 ), we have the following. c there exists an N -quasi-isomorphism s : L → X with (1) Given X ∈ C− (M), N L ∈ C− N (M). −,♮ c c for ♮ =nothing, b. (2) We have K−,♮ ∗ K−,a (M) N (M) = KN (M)  −,♮ N  c c KN (M) K−,a K−,♮ N (M) N (M) (3) We have a stable t-structure K−,a , in K−,a and a triangle −,a (M) K (M) (M) equivalence K−,♮ N (M) K−,a N (M) ≃ c K−,♮ N (M) c K−,a (M) N N N for ♮ =nothing, b. N Proof. (1) We shall construct a series of N -quasi-isomorphisms vn+1 : Ln → Ln+1 i i c satisfying Ln ∈ C− N (M), Ln ∈ M (i > n) and vn+1 = id (i > n + 1) by an induction on n. We set Lm = X and vm = idX for m large enough. Suppose we get Ln and vn+1 . c there exists an epimorphism f : M → Lnn with M ∈ M. Then Since Lnn ∈ M, Ln−1 = Vn (Ln , f ) and vn = pn (f ) satisfy the conditions above. Indeed, vn is an N quasi-isomorphism by Definition-Proposition 3.15(4)⇒(1), Lin−1 ∈ M (i > n − 1) c (i ≤ n − 1) by Lemma and vni = id (i > n) by the construction, and Lin−1 ∈ M 3.18(1). DERIVED CATEGORIES OF N -COMPLEXES 17 i Since vn+1 : Lin → Lin+1 (i > n + 1) is an identity, the canonical morphism L := limLn → X gives a desired N -quasi-isomorphism. ← c (2) It suffices to prove ”⊂”. Given an object X ∈ K− N (M), there exists an N -quasis − c isomorphism L → X with L ∈ KN (M) by (1). Then C(s) ∈ K− N (M) is N -acyclic, −,a c −,b c − c − and we have KN (M) ⊂ KN (M) ∗ KN (M). If X ∈ KN (M), then L ∈ K−,b N (M) −,b c −,b −,a c holds, and hence KN (M) ⊂ KN (M) ∗ KN (M). −,a c −,♮ c (3) Set U = K−,♮ holds by (2). ApN (M) and V = KN (M). Then U ∗ V = KN (M)  −,♮  c KN (M) K−,a (M) V U plying Lemma 1.6, we have a stable t-structure ( U ∩V , U ∩V ) = K−,a (M) , KN −,a (M) in U ∗V U ∩V = c K−,♮ N (M) K−,a N (M) and triangle equivalences K−,♮ N (M) K−,a N (M) N ≃ U U ∩V ≃ U ∗V V N = c K−,♮ N (M) c . K−,a N (M)  of Theorem 3.16. We only prove (1) since (2) is the dual. Set M = Prj A, then c = A. By Lemma 3.6, we have K−,a (M) = 0. By Proposition 3.19(3), we M N −,a −,♮ have a stable t-structure (K−,♮ (Prj A), K N N (A)) in KN (A) and a triangle equivalence K−,♮ N (Prj A) ≃ Proposition 3.14. K−,♮ N (A) . K−,a N (A) b This is D− N (A) if ♮=nothing, and DN (A) if ♮ = b by  Recall that an abelian category A is an Ab3-category (resp., Ab3∗ -category) provided that it has an arbitrary coproduct (resp., product) of objects. It is clear that coproducts (resp., products) preserve cokernels (resp., kernels). Moreover A is an Ab4-category (resp., Ab4∗ -category) provided that it is an Ab3-category (resp., Ab3∗ -category), and that the coproduct (resp., product) of monomorphisms (resp., epimorphisms) is monic (resp., epic) (see e.g. [39]). Definition 3.20 (cf. [7, 41]). We say that X ∈ KN (A) is K-projective if HomKN (A) (X, KaN (A)) = 0. We say that X ∈ KN (A) is K-injective if HomKN (A) (KaN (A), X) = 0. We denote by KpN (A) (resp., KiN (A)) the full triangulated subcategory of KN (A) consisting of K-projective (resp., K-injective) N -complexes. A projective N resolution (resp., injective N -resolution) of X ∈ KN (A) is an N -quasi-isomorphism PX → X (resp., X → IX ) with PX ∈ KpN (A) ∩ KN (Prj A) (resp., IX ∈ KiN (A) ∩ KN (Inj A)). Clearly KpN (A) (resp., KiN (A)) is a triangulated subcategory closed under coproducts (resp., products) in KN (A). The canonical functor KN (A) → DN (A) restricts to fully faithful functors KpN (A) → DN (A) and KiN (A) → DN (A) by Lemma 1.6. p + i By Lemma 3.6, K− N (Prj A) (resp., KN (Inj A)) is contained in KN (A) (resp., KN (A)). We have the following result which generalizes a classical result for the case N = 2 [7, 41]. Theorem 3.21. The following hold. (1) Assume that A is an Ab4-category with enough projectives. Then (KpN (A), KaN (A)) is a stable t-structure in KN (A) and we have a triangle equivalence KpN (A) ≃ DN (A). Moreover, any object in KpN (A) is isomorphic to an object in KpN (A) ∩ KN (Prj A), hence every object in KN (A) admits a projective N resolution. (2) Assume that A is an Ab4∗ -category with enough injectives. Then (KaN (A), KiN (A)) is a stable t-structure in KN (A) and we have a triangle equivalence KiN (A) ≃ DN (A). Moreover, any object in KiN (A) is isomorphic to an object in 18 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI KiN (A) ∩ KN (Inj A), hence every object in KN (A) admits an injective N resolution. To prove Theorem 3.21, we need the following easy observation. Lemma 3.22. Let A be an Ab3-category, and fi : Xi → Xi+1 (i = 0, 1, · · · ) a sequence of morphisms in CN (A). Assume that each j ∈ Z admits some n ∈ N j such that fij : Xij → Xi+1 is a split monomorphism for i ≥ n. Then we have an ` ` 1− fi ` exact sequence 0 → i≥0 Xi −−−−i−→ i≥0 Xi → lim Xi → 0 in (CN (A), SN (A)) −→ for the inductive limit lim Xi in CN (A). Therefore lim Xi is isomorphic to the −→ −→ homotopy colimit hlim Xi in KN (A). −→ ` j ` fi ` 1− j Proof. We have a split exact sequence 0 → i≥0 Xij −−−−i−→ Xj → i≥0 Xi → lim −→ i 0 in A for any j by our assumption. Thus the assertions follow.  of Theorem 3.21. We only prove (1) since (2) is the dual. By Lemma 1.6, it is enough to show KN (A) = KpN (A) ∗ KaN (A) to prove the first statement. For a complex X ∈ KN (A), we shall construct an N -quasi-isomorphism s : P → X with P ∈ KpN (A) ∩ KN (Prj A). Applying Lemma 3.22 to a sequence ιi : σ≤i X → σ≤i+1 X of morphisms, we have X = lim Xi ≃ hlim Xi in KN (A). By Theorem 3.16, −→ −→ there is an N -quasi-isomorphism si : Pi → σ≤i X with Pi ∈ K− N (Prj A). Since the mapping cone C(si+1 ) is N -acyclic, by Lemma 3.6 we have a commutative diagram in KN (A) si / σ≤i X Pi   ιi / σ≤i+1 X si+1 fi Pi+1 / C(si+1 ). Therefore we have a morphism between triangles in KN (A) ` 1− i Pi ` ` i fi si i `  i σ≤i X ` 1− ` i ιi / ` i Pi ` i si  / ` σ≤i X i u v /P s  /X /Σ ` i Pi ` Σ i si  / Σ` σ≤i X. i Since A is Ab4, i si is an N -quasi-isomorphism, hence so is s. The upper triangle shows P ∈ KpN (A) ∩ KN (Prj A). Now we prove the second statement. For any X ∈ KpN (A), the above construction s gives a triangle P − → X → Y → P [1] in KN (A) with P ∈ KpN (A) ∩ KN (Prj A) and Y ∈ KaN (A). Since KpN (A) is a triangulated subcategory of KN (A), we have Y ∈ KaN (A) ∩ KpN (A). Thus Y ≃ 0 and hence s is an isomorphism in KN (A).  Remark 3.23. Later we need a slightly more general version of Theorem 3.21 as follows. Let A be an Ab4-category with enough projectives and P an additive subcategory of Prj A closed under coproducts such that any object in Prj A is an epimorphic image from some object of P. Then the proof of Proposition 3.19 gives triangle equivalences − KN (P) ∩ KpN (A) ≃ DN (A) and K− N (P) ≃ DN (A). DERIVED CATEGORIES OF N -COMPLEXES 19 For example, the category Free R of free modules over a ring R satisfies this condition. d−2 d−1 Example 3.24. Take a projective 2-resolution · · · −−→ P −1 −−→ P 0 of X ∈ A. Then a projective N -resolution of X is given by the following. degree : PX : · · · −N −1 1 / P −3 −N d−3 / P −2 −N +1 d−2 / P −1 d−2 −N +2 1 / P −1 d−1 −1 1 / ··· 1 / P −1 d −1 0 1 2 / P0 /0 /0 / ··· . d0 Although the 2-complex · · · −−→ P −1 −−→ P 0 −→ X → 0 is 2-acyclic for some d0 : P 0 → X, the N -complex Y below is not N -acyclic for N > 2 since H1(1) (Y ) ≃ X. On the other hand, the following N -complex Z is N -acyclic. The truncation τ≤0 Z is not a projective N -resolution of X, but that of ΣΘ−1 (X) = µ0N −1 (X) since we have a triangle Θ−1 X → Z → τ≤0 Z → ΣΘ−1 X. degree : Y : ··· Z : ··· −N −1 1 1 / P −3 / P −2 −N d−3 1 / P −2 / P −2 −N +1 d−2 / P −1 d−2 / P −1 −N +2 1 d−1 / P −1 / P0 −1 1 1 / ··· / ··· 1 1 0 −1 / P −1 d / P 0 / P0 1 / P0 d0 d0 1 2 /X /X /0 /0 Let M be a full subcategory of A. We denote by CN,M (A) the full subcategory of CN (A) consisting of X such that Hi(r) (X) ∈ M for any 0 < r < N and i ∈ Z. Then KN,M (A) and DN,M (A) denote the corresponding full subcategories of KN (A) and DN (A) respectively. In the case that M is a Serre subcategory, that is, closed under subobjects, quotient objects and extensions, then KN,M (A) (resp., DN,M (A)) is a thick subcategory of KN (A) (resp., DN (A)). We use the notations ♯,♮ ♯,♮ ♯,♮ ♯,♮ C♯,♮ N,M (A) = CN (A) ∩ CN,M (A), KN,M (A) = KN (A) ∩ KN,M (A) and DN,M (A) = D♯,♮ N (A) ∩ D N,M (A) for ♯ =nothing, −, +, b and ♮ =nothing, −, +, b. By Proposition b 3.14, we have D♯,b N,M (A) ≃ DN,M (A) etc. Proposition 3.25. Let M be an additive full subcategory of A satisfying (V1 ) and (V2 ). (1) For any X ∈ C− N,M (A), there is an N -quasi-isomorphism L → X with − L ∈ CN (M). −,♮ −,a (2) K−,♮ N,M (A) ⊂ KN (M) ∗ KN (A) for ♮ =nothing, b. Proof. (1) There exists n0 such that X i = 0 for any i > n0 . Set Ln0 = X. We shall construct a sequence of N -quasi-isomorphisms vn : Ln−1 → Ln in CN (A) for n ≤ n0 such that Lin ∈ M c (i > n, 0 < r < N ) and vni = id (i > n) (i > n), Bi(r) (Ln ) ∈ M Then we get an N -quasi-isomorphism L = lim Ln → X with L ∈ C− N (M). Suppose ←− n < n0 and let Ln satisfy the conditions above. The exact sequence 0 → Hn(1) (Ln ) → c Applying Lemma 3.18(2) (Ln ) → Bn+1 (Ln ) → 0 implies Cn (Ln ) ∈ M. Cn (N −1) (1) (N −1) to the canonical epimorphism ρ : Lnn → Cn(N −1) (Ln ), we get a morphism v : M → Lnn with M ∈ M such that ρv is an epimorphism. Set Ln−1 = Vn (Ln , v) and / ··· / ··· 20 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI vn−1 = pn (v). {N −1} +1 Ln−N n−1  dL n−1 n−N +1 vn +1 Ln−N n / M = Ln n−1 v (E)  / Ln n {N −1} d Ln ρ / Cn (N −1) (Ln ) {N −1} {N −1} ) : Lnn−1 ⊕ and ρv is an epimorphism, (v dLn Since ρ is the cokernel of dLn n−N +1 n → Ln is an epimorphism, which shows (E) is an exact square. Thus Ln vn−1 = pn (v) is an N -quasi-isomorphism by Definition-Proposition 3.15. c for any i > n − 1 and 0 < r < N . If Now we show that Bi(r) (Ln−1 ) ∈ M i i i > n, then H(N −r) (Ln−1 ) = H(N −r) (Ln ) ∈ M holds. Moreover Zi(N −r) (Ln−1 ) = c since 0 → Zi Zi (Ln ) belongs to M (Ln ) → Li → Bi+N −r (Ln ) → 0 is (N −r) (N −r) n (N −r) c holds. To see Bn(r) (Ln−1 ) ∈ M, c it suffices to exact. Therefore Bi(r) (Ln−1 ) ∈ M c since Lnn−1 ∈ M. But this is clear since Bn+N −r (Ln−1 ) = show Cn(r) (Ln−1 ) ∈ M (N −r) c and Hn Bn+N −r (Ln ) ∈ M (Ln−1 ) ∈ M. (N −r) (N −r) (2) For given X ∈ K− N,M (A), there is an N -quasi-isomorphism s : L → X with L ∈ − KN (M) by (1). We get the first inclusion since C(s) ∈ KaN (A). If X ∈ K− N,M (A), −,a −,b the construction shows C(s) ∈ KN (A). If X ∈ KN,M (A), then obviously we have L ∈ K−,b  N (M). Theorem 3.26. If M is a Serre subcategory satisfying the condition (V1 ), then D♮N (M) ≃ D♮N,M (A) for ♮ = b, −. Proof. Since M is a Serre subcategory, it satisfies the condition (V2 ) and we have −,♮ −,♮ −,♮ KN (M) ⊂ KN,M (A). By Proposition 3.25(2), we have K−,♮ N,M (A) = KN (M) ∗ −,♮ −,a K−,a N (A). Applying Lemma 1.6 to U = KN (M) and V = KN (A), we have triangle equivalences D♮N (M) ≃ K−,♮ N (M) K−,a N (M) = U U ∩V ≃ U ∗V V = K−,♮ N,M (A) K−,a N (A) ≃ D♮N,M (A) as desired.  3.4. Homotopy categories of injective objects. In this subsection, we shall show that KN (Inj A) is compactly generated if A satisfies some conditions. An Ab5-category is an Ab3-category that has exact filtered colimits. A Grothendieck category is an Ab5-category with a generator. A Grothendieck category A is called locally noetherian if A has a generating set of noetherian objects. In this case, Inj A is closed under arbitrary coproducts [39, Theorem 8.7], and therefore the triangulated category KN (Inj A) has arbitrary coproducts. For an additive category B with arbitrary object C `is called ` coproducts, an ∼ Hom (C, X ) → Hom compact in B if the canonical morphism B i B (C, i Xi ) is i ` an isomorphism for any coproduct i Xi in B. We denote by B c the category of compact objects in B. A triangulated category D with arbitrary coproducts is called compactly generated by a set S of compact objects if any non-zero object of D has a non-zero morphism from a shift of some object of S. Let noeth A be the subcategory of A consisting of noetherian objects. For a locally noetherian Grothendieck category A, it is easy to see noeth A is a skeletally DERIVED CATEGORIES OF N -COMPLEXES 21 small Serre subcategory satisfying (V1 ) and (V2 ). By Theorem 3.26, we can identify DbN (noeth A) with DbN,noeth A (A). We aim to prove the N -complex version of a result of Krause [29]. Theorem 3.27. Let A be a locally noetherian Grothendieck category. Then KN (Inj A) is a compactly generated triangulated category such that the canonical functor KN (Inj A) → DN (A) induces an equivalence between KN (Inj A)c and DbN (noeth A). In the rest, A is a locally noetherian Grothendieck category. Recall that IX ∈ KiN (Inj A) stands for the injective N -resolution of an object X in KN (A). Lemma 3.28. (cf. [29, Lemma 2.1]) The object Iµsr (M) is compact in KN (Inj A) for any M ∈ noeth A, s ∈ Z and 0 < r < N . Proof. For any Y ∈ KN (Inj A), we have the following isomorphisms for sufficiently small t: HomKN (A) (Iµsr (M ) , Y ) ≃ HomKN (A) (Iµsr (M ) , τ≥t Y ) ≃ HomKN (A) (µsr (M ), τ≥t Y ) ≃ HomKN (A) (µsr (M ), Y ). The first and third isomorphisms come from Iµsr (M) , µsr (M ) ∈ K+ N (A) and the secs ond one from Lemma 3.6. Also we have HomKN (A) (µr (M ), Y ) ≃ Hs−r+1 (HomA (M, Y )) (r) by (3.7). This completes the proof since M ∈ noeth A is compact in A.  Let S stand for a set of representatives of isomorphism classes of objects {Iµsr (M) | M ∈ noeth A, s ∈ Z, 0 < r < N − 1} in KN (Inj A). Lemma 3.29. (cf. [29, Lemma 2.2]) KN (Inj A) is compactly generated by S. Proof. By Lemma 3.28, any object of S is compact in KN (Inj A). Let X ∈ KN (Inj A) be a non-zero object. Assume that Hi(r) (X) 6= 0 for some i ∈ Z and 0 < r < N . Since A is locally noetherian, there is a non-zero morphism M → Zi(r) (X) → Hi(r) (X) with M ∈ noeth A. Using the commutative diagram in Lemma 3.8(1), we have HomKN (A) (µri+r−1 (M ), X) 6= 0. Assume that X is N -acyclic. Since X 6= 0 in KN (Inj A), there are i ∈ Z and 0 < r < N with Zi(r) (X) 6∈ Inj A by Lemma 3.9(3). Baer criterion [28, Lemma A10] gives an object M of noeth A with Ext1A (M, Zi(r) (X)) 6= 0, which implies −1 HomKN (A) (µi+N  N −r (M ), X) 6= 0 by Lemma 3.8(3). Now we are ready to prove Theorem 3.27. of Theorem 3.27. Lemma 3.29 implies KN (Inj A) = Loc S (see [37, 1.6]). Hence by [37, Lemma 2.2], KN (Inj A)c coincides with thick S. On the other hand, the equivalence KiN (Inj A) ≃ DN (A) in Theorem 3.16(2) yields thickKiN (Inj A) S ≃ thickDN (A) (noeth A) ≃ DbN (noeth A).  3.5. Derived functor. In this subsection, we study the derived functor of a triangle functor KN (A) → KN ′ (A′ ) for abelian categories A, A′ . Definition 3.30. Let T be a triangulated category, U a full triangulated subcategory of T and Q : T → T /U the canonical functor. For a triangle functor F : T → T ′ , the right derived functor (resp., left derived functor ) of F with respect to U is a triangle functor RU F : T /U → T ′ (resp., LU F : T /U → T ′ ) 22 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI together with a functorial morphism of triangle functors ξ : F → (RU F )Q (resp., ξ : (LU F )Q → F ) with the following property: For a triangle functor G : T /U → T ′ and a functorial morphism of triangle functors ζ : F → GQ (resp., ζ : GQ → F ), there exists a unique functorial morphism η : RU F → G (resp., η : G → LU F ) of triangle functors such that ζ = (ηQ)ξ (resp., ζ = ξ(ηQ)). F / ′ T ♥♥6 TI ♥ ♥ RU F ♥♥♥ ♥ Q ♥♥♥✴✴✴✴✴✴ ♥ ♥  ♥♥ ✴✴  T /U G We recover a classical Existence Theorem of derived functors as follows: Theorem 3.31 (Existence Theorem). Let T be a triangulated category, U its full triangulated subcategory, and Q : T → T /U the canonical functor. For a triangle functor F : T → T ′ , assume that there exists a full triangulated subcategory V of T such that T = U ∗ V and F (U ∩ V) = {0}. Then there exists the right derived functor (RU F, ξ) of F with respect to U such that ξX : F X → (RU F )QX is an isomorphism for X ∈ V. Proof. Let Q1 : T → − T /(U ∩ V) and Q2 : T /(U ∩ V) → T /U be the canonical functors. Then Q = Q2 Q1 holds. Since F (U ∩ V) = 0, the functor F : T → T ′ Q1 F′ factors as T −−→ T /(U ∩ V) −→ T ′ by universality. By Lemma 1.6, the functor Q2 : T /(U ∩ V) → T /U has a right adjoint R : T /U → T /(U ∩ V). We shall show that RU F = F ′ R satisfies the condition. We have only to give a functorial isomorphism Hom△ (F, GQ) ≃ Hom△ (F ′ R, G) for any triangle functor G : T /U → T ′ , where Hom△ is the class of morphisms between triangle functors. Indeed, we have Hom△ (F, GQ) ≃ Hom△ (F ′ , GQ2 ) by [18, Proposition 3.4], and Hom△ (F ′ , GQ2 ) ≃ Hom△ (F ′ R, G) by a triangle functor version of [31, Proposition X.7.3].  We apply these to the setting of N -complexes. Definition 3.32 (Derived Functor). Let A and A′ be abelian categories, and F : K♮N (A) → KN ′ (A′ ) a triangle functor where ♮ =nothing, −, +, b. We define the right (resp., left ) derived functor of F as R♮ F = RU (Q′ F ) : D♮N (A) → DN (A′ ) (resp., L♮ F = LU (Q′ F ) : D♮N (A) → DN (A′ )), where Q′ : KN (A′ ) → DN (A′ ) is the canonical functor, T = K♮N (A) and U = K♮,a N (A). According to Theorems 3.16, 3.21 and 3.31, we have the following N -complex version of classical results [18, 7, 41]. Corollary 3.33. Let A and A′ be abelian categories, and F : KN (A) → KN ′ (A′ ) a triangle functor. Then the following hold. ′ (1) If A has enough injectives, then R+ F : D+ N (A) → DN ′ (A ) exists. − − (2) If A has enough projectives, then L F : DN (A) → DN ′ (A′ ) exists. DERIVED CATEGORIES OF N -COMPLEXES 23 (3) If A is an Ab4∗ -category with enough injectives, then RF : DN (A) → DN ′ (A′ ) exists. (4) If A is an Ab4-category with enough projectives, then LF : DN (A) → DN ′ (A′ ) exists. We end this subsection with considering Ext and Tor groups. As we will see in Proposition 3.35, these homology groups are related to classical Tor and Ext. Definition 3.34. Let A be a ring, X a right A-module and Y a left A-module. We have triangle functors HomA (X, −) : KN (Mod A) → KN (Mod Z) and − ⊗A Y : KN (Mod A) → KN (Mod Z). By Corollary 3.33, we have derived functors R HomA (X, −) : DN (Mod A) → DN (Mod Z) and −⊗L A Y : DN (Mod A) → DN (Mod Z). For a right A-module Z, n ∈ Z and 0 < r < N , set r ExtnA (X, Z) = Hn(r) (R HomA (X, Z)) and r −n L TorA n (Z, Y ) = H(r) (Z⊗A Y ). Proposition 3.35. We have the following isomorphisms for i ≥ 0 and 0 < r < N . A iN 2i TorA iN (X, Y ) = Tor 2i (X,AY ) and r ExtA (X, Z) = ExtA (X, Z). Tor2i+1 (X, Y ) r = s. (2) r TorA iN +s (X, Y ) = 0 r 6= s  2i+1 ExtA (X, Z) r = N − s. iN +s (3) r ExtA (X, Z) = 0 r 6= N − s (1) r d−1 d−2 d0 Proof. We give a proof only for Tor. Let · · · −−→ P −1 −−→ P 0 −→ Y → 0 be a projective 2-resolution of Y ∈ Mod Aop . We have a projective N -resolution of Y by Example 3.24: 0 −1 −2 −N −1 −3 −N −2 −N +1 degree −N−3 d−1 1 1 d d 1 → ··· − →P −1 −−→P 0 . → P −3 −−→P −2 −−→ P −1 − ··· → P − Applying X ⊗A −, we can justify the assertions.  Our Definition 3.34 is slightly different from Ext and Tor groups introduced by Kassel and Wambst [23]. As we discussed in Example 3.24, their definitions are interpreted as r ExtnA (X, Z)KW = Hn(r) (HomA (PΣΘ−1 X , Z)) and r −n KW TorA = H(r) (PΣΘ−1 X ⊗A Y ). n (X, Y ) 4. Triangle equivalence between derived categories In this section, we show that the derived category DN (A) of N -complexes is triangle equivalent to the ordinary derived category D(MorN −2 (A)) where MorN −2 (A) is the category of sequences of N − 2 morphisms in A. Definition 4.1. Let B be an additive category. The category MorN −2 (B) (resp., se Morsm N −2 (B), MorN −2 (B)) is defined as follows. (1) An object is a sequence of N − 2 morphisms (resp., split monomorphisms, α1 α2 αN −2 split epimorphisms) X : X 1 −−X → X 2 −−X → · · · −−X−−→ X N −1 in B. 24 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI (2) A morphism from X to Y is an (N − 1)-tuple f = (f 1 , · · · , f N −1 ) of morphisms f i : X i → Y i which makes the following diagram commutative. X1  f1 Y1 α1X α1Y / X2  f2 / Y2 α2X α2Y / ··· / ··· −3 αN X −2 αN X αY αY / X N −2 / X N −1  f N −2  f N −1 / Y N −2 / Y N −1 N −3 N −2 We can identify MorN −2 (B) with a full subcategory of CN (B) (and KN (B)) consisting of N -complexes concentrated in degrees 1, . . . , N − 1. Indeed, we have isomorphisms HomMorsm (X, Y ) = HomCN (B) (X, Y ) = HomKN (B) (X, Y ) N −2 (B) for any X, Y ∈ Morsm N −2 (B). As usual, a set S of objects in an abelian category A is a set of generators if any object X ∈ A admits an epimorphism from a coproduct of objects in S to X. Theorem 4.2. Let A be an Ab3-category with a small full subcategory C of compact projective generators. Then we have a triangle equivalence DN (A) ≃ D(MorN −2 (A)) which restricts to the identity functor on Morsm N −2 (C). We start with the following basic observations. Lemma 4.3. Let B be an additive category. (1) Assume that B is idempotent complete, that is, for any X ∈ B and any idempotent e ∈ EndB (X), there are an object Y ∈ B, and morphisms p : X → Y and q : Y → X such that e = qp and pq = 1Y . Then for every object P of Morsm N −2 (B), there are objects C1 , · · · , CN −1 of B such that P ≃ `N −1 N −1 µ (C ). i i i=1 j (2) For any P, Q ∈ Morsm N −2 (B), we have HomKN (B) (P, Σ Q) = 0 (j 6= 0). b sm (3) KN (B) = tri MorN −2 (B). c (4) Assume that B has arbitrary coproducts. Then every object in Morsm N −2 (B ) is compact in CN (B) (resp., KN (B)). Proof. (1) This is clear. (2) Let Be be the idempotent completion of B (e.g. [2, Definition 1.2]). Since e we can assume that B is idemKN (B) is a full triangulated subcategory of KN (B), −1 potent complete. By (1), we have only to consider the case P = µN (C) and r N −1 ′ ′ ′ Q = µr′ (C ) for C, C ∈ B and 0 < r, r < N . For the case j = 1, we have −1 N −r ′ ′ ΣµN (C ′ ) = µN r′ −r ′ (C ) by Lemma 2.9(1), and it is easy to check that any morN −r ′ N −1 ′ phism from µr (C) to µN −r ′ (C ) is null-homotopic. Now we consider the case j 6= −1 −1 (C) and Σj µN 0, 1. Since Σ2 = ΘN , there is no degree in which both µN (C ′ ) r r′ j N −1 N −1 ′ have non-zero terms. Thus we have HomCN (B) (µr (C), Σ µr′ (C )) = 0. −1 (3) For any C ∈ B and 0 < r < N , we have a triangle µr1 (C) → µN N −r (C) → b sm −1 r r µN N −r−1 (C) → Σµ1 (C) in KN (B). Thus µ1 (C) ∈ tri MorN −2 (B) holds. By Lemma 2.9(2), the assertion follows. −1 (4) Taking idempotent completion of B, it suffices to show that µN (C) is compact r c in CN (B) (resp. KN (B)) for C ∈ B . This follows from (3.7).  DERIVED CATEGORIES OF N -COMPLEXES 25 Definition 4.4. Let T be a triangulated category with arbitrary coproducts. A small full subcategory S of T c is called a tilting subcategory if the following conditions are satisfied. (1) HomT (S, Σi S) = 0 for any i 6= 0. (2) If X ∈ T satisfies HomT (S, Σi X) = 0 for any i ∈ Z, then X = 0. The following general result by Keller is basic, where we always regard S as a full subcategory of Mod S and D(Mod S) by Yoneda embedding. Proposition 4.5. Let T be an algebraic triangulated category with arbitrary coproducts and S a tilting subcategory. Then we have a triangle equivalence F : T ≃ D(Mod S), which restricts to the identity functor on S. Proof. Although this is well-known, we include a proof for convenience of the reader, because of the lack of proper reference in this setting (cf. [26, Theorem 8.3.3] for the one-object version). Replacing objects in T with their complete resolutions in the Frobenius category (cf. [25, Theorem 4.3], [30, Theorem 7.5]), we obtain a DG category R and a triangle functor G : T → D(R) satisfying the following conditions. • H0 (R) = S and Hi (R) = 0 for any i 6= 0. b • G commutes with arbitrary coproducts and induces an equivalence S → R, b is the full subcategory of D(R) consisting of representable DG where R functors. b Since Loc S = T and Then G induces a triangle equivalence Loc S → Loc R. b Loc R = D(R) hold by Brown representability, G : T → D(R) is a triangle equivalence. On the other hand, DG functors σ≤0 (R) → R and σ≤0 (R) → H0 (R) = S are quasi-equivalences [27] where σ≤0 (R) is the DG category with the same objects as R and the morphism spaces given as Homσ≤0 (R) (X, Y ) = σ≤0 HomR (X, Y ). Hence we have triangle equivalences D(R) ≃ D(σ≤0 (R)) ≃ D(Mod S) by [25, 9.1] (cf. [27, Lemma 3.10]). Thus the assertion follows.  We need the following general observation. Proposition 4.6. Let A be an Ab3-category with a small full subcategory C of compact projective generators. Then we have an equivalence A ≃ Mod C given by X 7→ HomA (−, X)|C . In particular, A is a Grothendieck category which satisfies the condition Ab4∗ . Proof. See [35, Chapter IV, Theorem 5.3] and [39, 3.4].  Now we give the following crucial results. Proposition 4.7. Let A be an Ab3-category with a small full subcategory C of compact projective generators. (1) DN (A) has a tilting subcategory Morsm N −2 (C). (2) We have a triangle equivalence DN (A) ≃ D(Mod(Morsm N −2 (C))), which restricts to the identity functor on Morsm (C). N −2 p c c Proof. (1) Set S = Morsm N −2 (C). Lemma 4.3(4) gives S ⊂ KN (Prj A) ≃ DN (A) . Also, S satisfies (1) of Definition 4.4 by Lemma 4.3(2). To show (2) of Definition 4.4, let X be a non-zero object in DN (A). It suffices to find some C ∈ C and r, s ∈ Z with 26 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI 0 < r < N such that HomD(A) (µsr (C), X) 6= 0. Indeed, there exist i ∈ Z and 0 < r < N such that Hi(r) (X) 6= 0. Since C generates A, we have HomA (C, Hi(r) (X)) 6= 0 for some C ∈ C. So HomD(A) (µri+r−1 (C), X) = HomKN (A) (µri+r−1 (C), X) 6= 0 by Lemma 3.8(2). (2) This is immediate from (1) and Proposition 4.5.  We also need the following observation for abelian categories. Lemma 4.8. Let A be an abelian category. (1) Any object in Morsm N −2 (Prj A) is projective in MorN −2 (A). (2) If P is a subcategory of A of projective generators, then Morsm N −2 (P) is a subcategory of MorN −2 (A) of projective generators. Assume that A is an Ab3-category with a small full subcategory C of compact projective generators. (3) MorN −2 (A) is an Ab3-category with a small full subcategory Morsm N −2 (C) of compact projective generators. (4) We have an equivalence MorN −2 (A) ≃ Mod(Morsm N −2 (C)) given by X 7→ HomMorN −2 (A) (−, X)|Morsm . (C) N −2 −1 Proof. (1) By Lemma 4.3(1), it suffices to prove that µN (C) is projective in i MorN −2 (A) for C ∈ Prj A and 1 ≤ i ≤ N − 1. Indeed, let an epimorphism Y → X in MorN −2 (A) be given. Then it induces an epimorphism HomA (C, Y N −i ) → −i −i N −i HomA (C, X N −i ). Since X N −i = HN = HN (i) (X) and Y (i) (Y ), we get an epi−1 −1 morphism HomKN (A) (µN (C), Y ) → HomKN (A) (µN (C), X) from Lemma 3.8(2). i i α1 αN −2 (2) Let X = (X 1 −→ · · · −−−−→ X N −1 ) be any object in MorN −2 (A). For each 1 ≤ i ≤ N − 1, we take an epimorphism Pi → X i with Pi ∈ P. Then we have an ` −1 N −1 epimorphism N i=1 µN −i (Pi ) → X. (3) The assertion follows from (1), (2) and Lemma 4.3(4). (4) This is immediate from (3) and Proposition 4.6.  Now we are ready to prove Theorem 4.2. of Theorem 4.2. By Proposition 4.7 and Lemma 4.8, we have triangle equivalences DN (A) ≃ D(Mod(Morsm N −2 (C))) ≃ D(MorN −2 (A)), which restrict to the identity functor on Morsm (C).  N −2 Next, to restrict the above equivalence to the subcategories of bounded complexes, we give the following preliminary result. Lemma 4.9. Let A be an abelian category and C a full subcategory of projective generators. Then the following conditions are equivalent for X ∈ DN (A). + (1) X belongs to DbN (A) (resp., D− N (A), DN (A)). s (2) For every 0 < r < N , HomDN (A) (µr (C), X) = 0 holds for all but finitely many (resp., sufficiently large, sufficiently small) s ∈ Z. i (3) HomDN (A) (Morsm N −2 (C), Σ X) = 0 holds for all but finitely many (resp., sufficiently large, sufficiently small) i ∈ Z. Proof. (1) and (2) are equivalent by Lemma 3.8(2). + Since Σ2 = ΘN holds and DbN (A) (resp., D− N (A), DN (A)) is closed under Σ, the condition (2) is equivalent to the following condition. DERIVED CATEGORIES OF N -COMPLEXES 27 • For any 0 < r < N and 0 ≤ s < N , HomDN (A) (µsr (C), Σi X) = 0 holds for all but finitely many (resp., sufficiently large, sufficiently small) i ∈ Z. This is equivalent to the condition (3) since tri{µsr (P ) | P ∈ C, 0 < r < N, 0 ≤ s < N } = KbN (C) = tri Morsm  N −2 (C) holds by Lemmas 2.9(2) and 4.3(3). Now we are able to prove the following result. Theorem 4.10. Let A be an Ab3-category with a small full subcategory of compact projective generators. Then the triangle equivalence in Theorem 4.2 restricts to those for ♮ = +, −, b D♮N (A) ≃ D♮ (MorN −2 (A)). Proof. This is immediate from Theorem 4.2 and Lemma 4.9.  In the case A = Mod R for a ring R, MorN −2 (A) is nothing but the category of modules over the upper triangular matrix ring TN −1 (R) of size N − 1 over R. Then we have the following precise description of homologies. Proposition 4.11. Let R be a ring. Then we have a triangle equivalence G : DN (Mod R) ≃ D(Mod TN −1 (R)) which gives the following for X ∈ DN (Mod R) and i ∈ Z:   +1 iN +2 iN +N −1 H2i (GX) = HiN (X) → H (X) → · · · → H (X) , (N −1) (N −2) (1)   (i+1)N (i+1)N (i+1)N (X) → · · · → H(N −1) (X) , (X) → H(2) H2i+1 (GX) = H(1) where each morphism is a canonical one between homologies. Proof. By Theorem 4.2, we have a triangle equivalence G : DN (Mod R) ≃ D(Mod TN −1 (R)) which is the identity on Morsm N −2 (prj R). We shall show the equalities only for i = 0, 1 since for others it follow from ΘN = Σ2 . For 0 < r < N , we have −1 HomMod TN −1 (R) (µN (R), H0 (GX)) ≃ r −1 ≃ HomD(Mod TN −1 (R)) (µN (R), GX) ≃ r −1 HomK(Mod TN −1 (R)) (µN (R), GX) r −r −1 HomDN (Mod R) (µN (R), X) ≃ HN r (r) (X). −1 The first isomorphism is from Lemma 4.8(1), the second from µN (R) ∈ KpN (Prj R), r and the the third by G. The last is from Lemma 3.8(2). Thus the morphism −r−1 N −r HN (r+1) (X) → H(r) (X) is the canonical one since it is induced from the canonical −1 −1 morphism µN (R) → µN r r+1 (R). Similarly we have −1 −1 HomMod TN −1 (R) (µN (R), H1 (GX)) ≃ HomD(Mod TN −1 (R)) (Σ−1 µN (R), GX) r r −r−1 −1 ≃ HomDN (Mod R) (Σ−1 µN (R), X) ≃ H0(N −r) (X) (R), X) ≃ HomDN (Mod R) (µN r N −r as desired.  As an application, we have the following results for homotopy categories. Corollary 4.12. Let B be an additive category with arbitrary coproducts. If B c is skeletally small and satisfies B = Add(B c ), then we have triangle equivalences − sm b b sm K− N (B) ≃ K (MorN −2 (B)) and KN (B) ≃ K (MorN −2 (B)). 28 OSAMU IYAMA, KIRIKO KATO AND JUN-ICHI MIYACHI Proof. Let A = Mod B c . Then A (resp., MorN −2 (A)) is an Ab3-category with a subcategory B (resp., Morsm N −2 (B)) of projective generators by Lemma 4.8(2). Thus we have triangle equivalences − − − sm K− N (B) ≃ DN (A) ≃ D (MorN −2 (A)) ≃ K (MorN −2 (B)). where the first and the third equivalence by Remark 3.23 and the second by Theorem 4.10. Since these equivalences restrict to the identity functor on Morsm N −2 (B), we have a triangle equivalence b sm KbN (B) = triK− (B) Morsm Morsm N −2 (B) ≃ triK− (Morsm N −2 (B) = K (MorN −2 (B)) N −2 (B)) N by Lemma 4.3(3).  Example 4.13. Let R be a graded ring, and GrMod R the category of graded right R-modules. Then GrMod R satisfies the condition of Theorem 4.2. Hence we have a triangle equivalence for ♮ =nothing, −, b: D♮N (GrMod R) ≃ D♮ (MorN −2 (GrMod R)). Finally we study the bounded derived category of N -complexes in the case of coherent rings. We prepare the following easy observation. Lemma 4.14. Let G : DN (A) → D(MorN −2 (A)) be the triangle equivalence given in Theorem 4.2. For any P ∈ C and i, r ∈ Z with 0 ≤ r < N , we have  −1 if r = 0, · · · → 0 → µN iN +r N −1 (P ) → 0 → · · · G(µ1 (P )) = N −1 −1 · · · → 0 → µN −r−1 (P ) → µN (P ) → 0 → · · · if 0 < r < N. N −r which is a complex concentrated in degree 2i − 1 if r = 0, in 2i − 1 and 2i otherwise. Proof. Since Σ2 = ΦN , we have only to show them for the case i = 0 by an induction −1 N −1 on r. If r = 0, then we have G(P ) = ΣµN N −1 (P ) since P = ΣµN −1 (P ). Assume N −1 N −1 0 < r < N . Then an exact sequence 0 → µN −r−1 (P ) → µN −r (P ) → µr1 (P ) → 0 N −1 N −1 −1 r in CN (A) induces a triangle µN N −r−1 (P ) → µN −r (P ) → µ1 (P ) → ΣµN −r−1 (P ) N −1 in DN (A) by Proposition 3.11(1). Applying G, we have a triangle µN −r−1 (P ) → N −1 −1 r  µN N −r (P ) → Gµ1 (P ) → ΣµN −r−1 (P ) in DN (A). Proposition 4.15. Let R be a ring. (1) We have triangle equivalences for ♮ = −, b, (−, b): K♮N (prj R) ≃ K♮ (prj TN −1 (R)). (2) If R is right coherent, then we have triangle equivalences for ♮ = −, b: D♮N (mod R) ≃ D♮ (mod TN −1 (R)). − Proof. (1) According to Theorem 3.16, we regard K− N (Prj R) (resp., K (Prj TN −1 (R)) as a full subcategory of DN (Mod R) (resp., D(Mod TN −1 (R))). We shall show that the triangle equivalence G : DN (Mod R) ≃ D(Mod TN −1 (R)) in Theorem 4.2 restricts to the desired equivalence. Indeed, G induces a triangle equivalence KbN (prj R) = triDN (Mod R) Morsm N −2 (prj R) ≃ triD(Mod TN −1 (R)) prj TN −1 (R) = Kb (prj TN −1 (R)). To get the triangle equivalence for ♮ = −, we shall show GP ∈ K− (prj TN −1 (R)) − for each P ∈ K− N (prj R). We may assume P ∈ CN (prj R) and τ≥1 P = 0. Set DERIVED CATEGORIES OF N -COMPLEXES 29 Pn = τ≥−n P for each n > 0. Then we have a term-wise split exact sequence 0 → Pn−1 → Pn → Θn P −n → 0 in CbN (prj R), and a triangle in DN (Mod R) ϕn Pn−1 → Pn → Θn P −n → ΣPn−1 . Applying G, we have a triangle in D(Mod TN −1 (R)) Gϕn GPn−1 → GPn → GΘn P −n → ΣGPn−1 . There exists a term-wise split exact sequence 0 → Qn−1 → Qn → GΘn P −n → 0 in Cb (prj TN −1 (R)) such that GP0 → GP1 → GP2 → · · · is isomorphic to Q0 → Q1 → Q2 → · · · . Then Lemma 4.14 gives a triangle GPn−1 → GPn → GΘn P −n → ΣGPn−1 such that GΘn P −n has only non-zero terms at degrees 2⌊n/N ⌋ and 2⌊n/N ⌋ − 1, where ⌊n/N ⌋ is the largest integer m satisfying m ≤ n/N . Therefore τ>2⌊n/N ⌋ Qn−1 = τ>2⌊n/N ⌋ Qn hence lim Qn ∈ K− (prj TN −1 (R)). Since P ≃ hlim Pn −→ −→ in DN (Mod R) by Lemma 3.22, GP ≃ hlim GPn ≃ lim Qn in D(Mod TN −1 (R)). −→ −→ Thus GP ∈ K− (prj TN −1 (R)) holds. By a similar argument, a quasi-inverse functor G−1 : D(Mod TN −1 (R)) ≃ DN (Mod R) induces a functor K− (prj TN −1 (R)) ≃ K− N (prj R). Hence G restricts to a triangle − equivalence K− (prj R) ≃ K (prj T (R)). By Lemma 4.9, this restricts to a triN −1 N −,b angle equivalence K−,b (prj R) ≃ K (prj T N −1 (R)). N (2) When R is right coherent, TN −1 (R) is also right coherent. In fact, let A be TN −1 (R) and ei (1 ≤ i ≤ N − 1) the idempotent of A whose (i, i)-entry is 1 and others are zero. Let 0 → Z → Y → X be an exact sequence of A-modules such that X and Y are finitely presented. Since ei Aei = R, we have an exact sequence 0 → Zei → Y ei → Xei of R-modules. The R-modules Xei and Y ei are finitely presented and R is coherent, hence so is the R-module Zei for any 1 ≤ i ≤ N − 1. Therefore the A-module Z is finitely generated. 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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]