On torsion torsionfree triples

TL;DR

This paper explores torsion torsionfree triples in abelian and triangulated categories, providing new characterizations, explicit descriptions, and extending classical theorems related to recollements and derived categories.

Contribution

It completes Jans' characterization of split TTF triples, proves a weak version of the Generalized Smashing Conjecture, and develops an unbounded approach to Koenig's theorem.

Findings

Complete characterization of split TTF triples in module categories

Explicit description of TTF triples in derived categories of dg categories

Extension of Koenig's theorem to unbounded derived categories

Abstract

We study torsion torsionfree(=TTF) triples in abelian and triangulated categories. (Notice that TTF triples in a triangulated category are essentially in bijection with recollement data for this triangulated category.) In particular, we complete Jans' characterization of split TTF triples on a category of modules, prove a weak version of the Generalized Smashing Conjecture, use homological epimorphisms of differential graded(=dg) categories to give an explicite description of all the TTF triples in the derived category of a k-flat dg category and develope an unbounded approach to Koenig's theorem on recollements of right bounded derived categories of ordinary algebras.