Homological support of big objects in tensor-triangulated categories

Objects in Triangulated Categories

We introduce the Calabi-Yau (CY) objects in a Hom-finite Krull-Schmidt triangulated k-category, and notice that the structure of the minimal, consequently all the CY objects, can be described. The relation between indecomposable CY objects and Auslander-Reiten triangles is provided. Finally we classify all the CY modules of selfinjective Nakayama algebras, determining this way the self-injectiv...

Injective objects in triangulated categories

Strongly copure projective objects in triangulated categories

In this paper, we introduce and investigate the notions of ξ-strongly copure projective objects in a triangulated category. This extends Asadollahi’s notion of ξ-Gorenstein projective objects. Then we study the ξ-strongly copure projective dimension and investigate the existence of ξ-strongly copure projective precover.

Dimensions of Triangulated Categories via Koszul Objects

Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin algebras.

A Note on Support in Triangulated Categories

In this note, I define a notion of a compactly supported object in a triangulated category. I prove a number of propositions relating this to traditional notions of support and give an application to the theory of derived Morita equivalence. I also discuss a connection to supersymmetric gauge theories arising from D-branes at a singularity.