We show that the relative Auslander-Buchweitz context on a triangulated category T coincides with the notion of co-t-structure on certain triangulated subcategory of T (see Theorem 3.8). In the Krull-Schmidt case, we stablish a bijective correspondence between cot-structures and cosuspended, precovering subcategories (see Theorem 3.11). We also give a characterization of bounded co-t-structures...

The Hom closed colocalizing subcategories of the stable module category of a finite group are classified. Along the way, the colocalizing subcategories of the homotopy category of injectives over an exterior algebra, and the derived category of a formal commutative differential graded algebra, are classified. To this end, and with an eye towards future applications, a notion of local homology a...

We define the Grothendieck group of an n-angulated category and show that for odd n its properties are as in the special case of n = 3, i.e. the triangulated case. In particular, its subgroups classify the dense and complete n-angulated subcategories via a bijective correspondence. For a tensor n-angulated category, the Grothendieck group becomes a ring, whose ideals classify the dense and comp...

We give a criterion for cohomological symmetry in a triangulated category. As an application, we show that such cohomological symmetry holds for all pairs of modules over any exterior algebra.

We call a 2-Calabi-Yau triangulated category a cluster category if its cluster-tilting subcategories form a cluster structure as defined in [4]. In this paper, we show that the canonical orbit category of the bounded derived category of finite dimensional representations of a quiver without infinite paths of type A∞ or A∞ is a cluster category. Moreover, for a cluster category of type A∞, we gi...

Abstract Palu defined the index with respect to a cluster tilting object in suitable triangulated category, order better understand Caldero–Chapoton map that exhibits connection between algebras and representation theory. We push this further by proposing an contravariantly finite, rigid subcategory, we show behaves similarly classical index. Let ${\mathcal{C}}$ be skeletally small category spl...

We show that the Calabi-Yau dimension of the stable module category of a selfinjective algebra of finite representation type is determined by the action of the Nakayama and suspension functors on objects. Our arguments are based on recent results of C. Amiot, and hence apply more generally to triangulated categories having only finitely many indecomposable objects. Throughout, k is an algebraic...

We give a full exceptional collection in the triangulated category of singularities in the sense of Orlov [16] for a hypersurface singularity of Fermat type, and discuss its relation with homological mirror symmetry for simple elliptic hypersurface singularities.

Using representations of quivers of type A, we define an anticyclic cooperad in the category of triangulated categories, which is a categorification of the linear dual of the Diassociative operad.

We construct an isomorphism from Somekawa’s Kgroup associated to a finite collection of semi-abelian varieties (or more general sheaves) over a perfect field to a corresponding Hom group in Voevodsky’s triangulated category of effective motivic complexes.