Abstract. We study the homotopy category K N (B) of N -complexes of an additive category B and the derived category D N (A) of an abelian category A. First we show that both K N (B) and D N (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 D N (A) is triangle equivalent to the ordinary derived category D(Mor N−2 (A)) where Mor N−2 (A) is the category of sequential N − 2 morphisms of A. IntroductionThe notion of N -complexes, that is, graded objects with N -differentials d (d N = 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 C N (B) of N -complexes over an additive category B and the homotopy category K N (B). Precisely speaking, we introduce an exact structure on C N (B) to prove the following results.Theorem 0.1 (Theorems 2.1 and 2.6).(1) The category C N (B) has a structure of a Frobenius category.(2) The category K N (B) has a structure of a triangulated category.We give an explicit description of the suspension functor Σ and triangles in K N (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 K N (B).Theorem 0.2 (Theorem 2.7). There is a functorial isomorphismIn Section 3, we introduce the derived category D N (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 7]. Our main result is the following, where Prj A (resp., Inj A) is the subcategory of projective (resp., injective) objects in A andis the homotopy category of N -acyclic (resp., K-projective, K-injective) N -complexes (see Definitions 3.3, 3.20). We denote by KN (Prj A)) the subcategory of K N (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.N (A) and we have triangle equivalences) is a stable t-structure in K N (A) and we have a triangle equivalenceMoreover, we generalize a result of Krause [29] characterizing the compa...
We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. We show the existence of a recollement of the above quotient category and it has the homotopy category of acyclic complexes as a triangulated subcategory. In the case of the homotopy category of finitely generated projective modules over an Iwanaga-Gorenstein ring, we show that the above quotient category are triangle equivalent to the stable module category of Cohen-Macaulay T 2 (R)-modules.1991 Mathematics Subject Classification. 18E30, 18G35, 16G99.
Abstract. In a triangulated category T with a pair of triangulated subcategories X and Y, one may consider the subcategory of extensions X * Y.We give conditions for X * Y to be triangulated and use them to provide tools for constructing stable t-structures. In particular, we show how to construct so-called triangles of recollements, that is, triples of stable t-structures of the form (X, Y), (Y, Z), (Z, X). We easily recover some triangles of recollements known from the literature. IntroductionLet T be a triangulated category and let X, Y ⊆ T be subcategories. The subcategory of extensions is defined bythere is a distinguished triangle It has been well-known that X * Y is triangulated if Hom T (X, Y) = 0. We have examined this condition to see that a substantial generalization provides exact characterization of X * Y to be triangulated. Note that throughout, all subcategories are full and closed under isomorphisms in the ambient category, and that if Q : T → T ′ is a functor and X ⊆ T a subcategory, then QX denotes the isomorphism closure of { Qx | x ∈ X } in T ′ , viewed as a subcategory.Theorem A is a main ingredient in the proof of Theorem B below which is a tool for constructing stable t-structures. Recall that a stable t-structure in T is a pair (X, Y) of subcategories which are stable under Σ and satisfy Hom T (X, Y) = 0 and X * Y = T, see [9, def. 9.14]. Stable t-structures are important in several settings which involve triangulated categories, see for 2010 Mathematics Subject Classification. 18E30, 18E35, 18G35.
We answer a question posed by Auslander and Bridger. Every homomorphism of modules is projective-stably equivalent to an epimorphism but is not always to a monomorphism. We prove that a map is projective-stably equivalent to a monomorphism if and only if its kernel is torsionless, that is, a first syzygy. If it occurs, there can be various monomorphisms that are projective-stably equivalent to a given map. But in this case there uniquely exists a "perfect" monomorphism to which a given map is projectivestably equivalent.
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