Derived categories of $N$-complexes

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.

Polygon of recollements and $N$-complexes

“…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.…”

“…By Lemma 6.9, we have K(Mor sm N −1 (P))/ K φ (Mor sm N −1 (P)) ≃ D(Mor N −1 (A)). (2) According to [IKM2] Theorem 2.23, for any complex X ∈ K N (A), there is a quasi-isomorphism P → X with P ∈ K N (P). By Lemma 6.9, we have K N (P)/ K φ N (P) ≃ D N (A).…”

“…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 D N (A) of N -complexes of an abelian category A, and showD N (A) is a triangle equivalent to the derived category D(Mor N −1 (A)) of ordinary complexes over an abelian category Mor N −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.…”

“…− (Mor sm N −1 (P)) ≃ K − N (P), K b (Mor sm N −1 (P)) ≃ K b N (P), D − (Mor N −1 (A)) ≃ D − N (A).Moreover ifA is an Ab4 category, then Then we have triangle equivalences:D(Mor N −1 (A)) ≃ D N (A).Proof. According to[IKM2] Lemma 4.8, we have a triangle equivalenceK − (Mor sm N −1 (P)) ≃ D − (Mor N −1 (A)). By[IKM2] Theorem 2.18, we have a triangle equivalence K − N (P) ≃ D − N (A).…”

“…According to[IKM2] Lemma 4.8, we have a triangle equivalenceK − (Mor sm N −1 (P)) ≃ D − (Mor N −1 (A)). By[IKM2] Theorem 2.18, we have a triangle equivalence K − N (P) ≃ D − N (A). By Theorem 6.8, we have the statement.…”