n-Abelian quotient categories

Abstract

Let C be an (n + 2)-angulated category with shift functor Σ and X be a cluster-tilting subcategory of C . Then we show that the quotient category C /X is an n-abelian category.If C has a Serre functor, then C /X is equivalent to an n-cluster tilting subcategory of an abelian category mod(Σ −1 X ). Moreover, we also prove that mod(Σ −1 X ) is Gorenstein of Gorenstein dimension at most n. As an application, we generalize recent results of Jacobsen-Jørgensen and Koenig-Zhu.Definition 1.1. Let C be an (n + 2)-angulated category with an autoequivalence Σ and X an additive subcategory of C . X is called cluster-tilting if (1) Hom C (X , ΣX ) = 0.(2) For any object C ∈ C , there exists an (n + 2)-angleAn object X is called cluster-tilting in the sense of Opperman-Thomas [OT, Definition 5.3] if add(X) is clsuter-tilting.Recently, Jacobsen-Jørgensen showed that the following result, which was obtained in a special case in the first part of [OT, Theorem 5.6].Theorem 1.2. [JJ, Theorem 0.6] Let C be a k-linear Hom-finite (n + 2)-angulated category with split idempotents, where k is an algebraically closed field and n is a positive integer.Assume that C has a Serre functor S, that is, an autoequivlence for which there are natural equivalenceswhere D(−) = Hom k (−, k) is the k-linear duality functor and Y, Z ∈ C . Assume that X is a cluster tilting object in C with Γ = End C (X). Then

From $n$-exangulated categories to $n$-abelian categories

“…Since an (n+2)-angulated category can be viewed as an n-exangulated category with enough projectives and enough injectives, when C is an (n + 2)-angulated category, our main result is just the Theorem 3.4 in [ZZ3]. Moreover, this result is completely new when it is applied to n-exact categories.…”

“…Corollary 3.10. [ZZ3,Theorem 3.4] Let C be an (n + 2)-angulated category with split idempotents and X a cluster tilting subcategory of C . Then C /X is an n-abelian category.…”

“…For n = 1 they specialise to abelian and exact categories. Zhou and Zhu [ZZ3] proved that any quotient of an (n + 2)-angulated category modulo a cluster tilting subcategory is an n-abelian category. When n = 1, it is just the Koenig-Zhu's result.…”

“…Remark 3.5. When C is an (n + 2)-angulated category, this definition is just the Definition 3.3 in [ZZ3].…”

Modules of infinite projective dimension

“…Theorem 2.3. [JJ1, Theorem 0.5] and[ZZ, Theorem 3.8] Consider the essential image D of the functor C(X, −) : C → modΓ. Then D is an n-cluster tilting subcategory of modΓ where modΓ is the category of finite dimensional right Γ-modules.…”