Locally well generated homotopy categories of complexes

TL;DR

This paper proves that the homotopy category of complexes over any finitely accessible additive category is locally well generated, and characterizes when it is globally well generated based on pure semisimplicity.

Contribution

It establishes the local well generation of homotopy categories over finitely accessible additive categories and links global well generation to pure semisimplicity of the underlying category.

Findings

Homotopy category K(B) is locally well generated.

K(B) is well generated iff B is pure semisimple.

Generalizes pure semisimplicity concept from rings to additive categories.

Abstract

We show that the homotopy category of complexes K(B) over any finitely accessible additive category B is locally well generated. That is, any localizing subcategory L in K(B) which is generated by a set is well generated in the sense of Neeman. We also show that K(B) itself being well generated is equivalent to B being pure semisimple, a concept which naturally generalizes right pure semisimplicity of a ring R for B = Mod-R.