The $N$-Stable Category

Abstract

A well-known theorem of Buchweitz provides equivalences between three categories: the stable category of Gorenstein projective modules over a Gorenstein algebra, the homotopy category of acyclic complexes of projectives, and the singularity category. To adapt this result to N -complexes, one must find an appropriate candidate for the N -analogue of the stable category. We identify this "N -stable category" via the monomorphism category and prove Buchweitz's theorem for N -complexes over a Frobenius exact abelian category. We also compute the Serre functor on the N -stable category over a self-injective algebra and study the resultant fractional Calabi-Yau properties.Acknowledgements. V.M. is partially supported by EPSRC grant EP/S017216/1. Part of this research was carried out during a research visit of J.B. to the University of East Anglia, whose hospitality is gratefully acknowledged. Definitions and Notation2.1. Triangulated Categories. We shall assume the reader is familiar with the basic theory of triangulated categories. In lieu of a detailed explanation, we give a quick overview of the relevant topics and terminology; for more details, the reader may consult Neeman [22] or Gelfand-Manin [13].Let T be an additive category, and let Σ : T ∼ − → T be an additive automorphism of T . We shall call Σ the suspension functor on T . A triangle in T is any diagram of the formA triangulated category is the data of T , Σ, and a collection of triangles (called the distinguished triangles), satisfying certain axioms.If (T 1 , Σ 1 ) and (T 2 , Σ 2 ) are triangulated categories, a triangulated functor F : T 1 → T 2 is the data of an additive functor F and an isomorphism φ :Any morphism f : X → Y in a triangulated category T can be extended to a distinguished triangleWe refer to Z as the cone of f ; it is unique up to (non-canoncial) isomorphism. Similarly, we refer to X as the cocone of g.A full, replete, additive subcategory S ⊆ T is said to be a triangulated subcategory if S is closed under Σ ±1 and the cone of any morphism in S lies in S. A triangulated subcategory S is said to be thick if it is closed under direct summands. In this case, we can form a new triangulated category T /S, called the Verdier quotient, with the same objects and suspension functor as T . There is a natural triangulated functor T → T /S which is the identity on objects and whose kernel is precisely S. T /S can also be viewed as the localization of T with respect to the multiplicative set of morphisms with cone in S, hence morphisms in T /S can be expressed in terms of a calculus of left and right fractions. A triangle in T /S is distinguished if and only if it is isomorphic (in T /S) to a distinguished triangle in T .2.2. Serre Duality and Calabi-Yau Categories. Let F be a field and let (T , Σ) be an F -linear, Hom-finite triangulated category. A Serre functor on T is an equivalence of triangulated categories S : T ∼