On the abelianization of derived categories and a negative solution to Rosicky's problem

TL;DR

This paper demonstrates that for a broad class of rings, their derived categories do not satisfy certain representability properties, providing a negative answer to Rosicky's problem and revealing limitations in the abelianization process.

Contribution

It proves that for many rings, the lambda-pure global dimension exceeds one, showing that their derived categories lack Adams lambda-representability and highlighting limitations in lambda-abelianization.

Findings

Lambda-pure global dimension is greater than one for many rings.

Derived categories do not satisfy Adams lambda-representability.

Lambda-abelianization is not a full functor for these categories.

Abstract

We prove for a large family of rings R that their lambda-pure global dimension is greater than one for each infinite regular cardinal lambda. This answers in negative a problem posed by Rosicky. The derived categories of such rings then do not satisfy the Adams lambda-representability for morphisms for any lambda. Equivalently, they are examples of well generated triangulated categories whose lambda-abelianization in the sense of Neeman is not a full functor for any lambda. In particular we show that given a compactly generated triangulated category, one may not be able to find a Rosicky functor among the lambda-abelianization functors.