Intrinsic homological algebra for triangulated categories

TL;DR

This paper introduces a new intrinsic framework for analyzing homological properties of triangulated categories, generalizing classical results and applying to various categories including derived and singularity categories.

Contribution

It develops a novel approach based on far-away orthogonality and intrinsic subcategories to study homological properties in triangulated categories, extending previous theories.

Findings

Generalizes properties like regularity and Gorensteinness to broader categories

Provides new tools for analyzing recollements of triangulated categories

Applies the theory to derived categories, homotopy categories, and singularity categories

Abstract

We propose a new framework for the study of homological properties for (compactly generated) triangulated categories such as regularity, finiteness of global or finitistic dimension, gorensteinness or injective generation and the relation between them. Our approach focuses on distinguished, intrinsically defined, subcategories and our main tool is the new notion of far-away orthogonality. We observe that these homological properties generalise previously studied properties on derived categories of modules over rings, and we use the generality of our theory to also examine those same attributes for the homotopy category of injectives and the big singularity category (in the sense of Krause) of an Artin algebra, as well as the derived category of a non-positive differential graded algebra. Finally, using our theory we recover and generalise various results in the theory of recollements of triangulated categories.