Parametrizing recollement data

TL;DR

This paper provides a comprehensive parametrization of recollement data in triangulated categories, characterizes when such categories are decomposable, and explores the structure of smashing subcategories, connecting to the Generalized Smashing Conjecture.

Contribution

It introduces a general parametrization of recollement data, characterizes recollements of generated categories, and establishes a bijection between smashing subcategories and ideals of compact objects.

Findings

Parametrization of all recollement data for triangulated categories.

Characterization of when a triangulated category is a recollement of categories generated by a single compact object.

A bijection between smashing subcategories and certain ideals of compact objects, supporting a weak form of the Generalized Smashing Conjecture.

Abstract

We give a general parametrization of all the recollement data for a triangulated category with a set of generators. From this we deduce a characterization of when a perfectly generated (or aisled) triangulated category is a recollement of triangulated categories generated by a single compact object. Also, we use homological epimorphisms of dg categories to give a complete and explicit description of all the recollement data for (or smashing subcategories of) the derived category of a k-flat dg category. In the final part we give a bijection between smashing subcategories of compactly generated triangulated categories and certain ideals of the subcategory of compact objects, in the spirit of Henning Krause's work. This bijection implies the following weak version of the Generalized Smashing Conjecture: in a compactly generated triangulated category every smashing subcategory is generated by a set of Milnor colimits of compact objects.