Ideal cotorsion theories in triangulated categories

TL;DR

This paper develops a theory of ideal cotorsion pairs in triangulated categories, extending existing approximation theories and providing simplified proofs of key lemmas, with applications to projective classes and cohomological functors.

Contribution

It introduces ideal cotorsion pairs in triangulated categories, extending ideal approximation theory and simplifying proofs of fundamental lemmas, with applications to projective classes.

Findings

Extended ideal approximation theory to triangulated categories.

Provided simplified proofs for Salce's, Wakamatsu's, and Christensen's lemmas.

Connected projective classes with cohomological functors in compactly generated categories.

Abstract

We study ideal cotorsion pairs associated to weak proper classes of triangles in extension closed subcategories of triangulated categories. This approach allows us to extend the recent ideal approximations theory developed by Fu, Herzog et al. for exact categories in the above mentioned context, and to provide simplified proofs for the ideal versions of some standard results as Salce's Lemma, Wakamatsu's Lemma and Christensen's Ghost Lemma. In the last part of the paper we apply the theory in order to study connections between projective classes (in particular localization or smashing subcategories) in compactly generated categories and cohomological functors into Grothendieck categories.