Definable functors and Brown--Adams representability

TL;DR

This paper investigates when derived categories satisfy Brown--Adams representability, using definable functors to transfer properties and analyze the structure of derived categories of rings, including von Neumann regular rings.

Contribution

It introduces methods to transfer Brown--Adams representability via definable functors and provides new proofs related to the telescope conjecture and generating hypotheses.

Findings

Pure global dimension of derived categories bounds that of rings.

Conditions identified for Brown--Adams representability transfer.

Derived categories of von Neumann regular rings often satisfy Brown--Adams properties.

Abstract

The question of when the derived category of a ring satisfies Brown--Adams representability is revisited via studying the transfer of pure homological dimension along definable functors: it is shown that, for any ring, the pure global dimension of the derived category is at least the pure global dimension of the ring; expanding results of Beligiannis and Keller-Christensen-Neeman. This result is obtained by constructing `change of category' isomorphisms of PExt groups across definable functors. The same isomorphisms illustrate circumstances when one can transfer the property of Brown--Adams representability. We demonstrate how these methods can be used to test whether certain derived category of quasi-coherent sheaves are a Brown category. We also make an investigation into the structure of derived categories of von Neumann regular rings, which are shown in many cases to control Brown--Adams representability; this includes a new proof of the telescope conjecture, and a new and short proof that a coherent ring satisfies Freyd's (strong) generating hypothesis if and only if it is von Neumann regular.