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Definable functors and Brown--Adams representability

Isaac Bird

TL;DR

This work develops a unified, functorial approach to Brown–Adams representability in compactly generated triangulated categories via definable functors, showing that pure global dimension cannot drop under such transfers and that $pgdim(\mathsf{D}(R)) \ge pgdim(R)$ for any ring $R$. Central to the strategy are $PExt$-type isomorphisms induced by left-adjoint definable functors, which yield concrete criteria for when Brown representability holds or fails and facilitate tests in geometric contexts like $\mathsf{D}_{\mathrm{qc}}(X)$. A major advance is the detailed analysis of von Neumann regular rings, proving that they control Brown–Adams representability: the telescope conjecture holds, there are tight correspondences between localising subcategories and ring-theoretic ideals, and the Freyd localisation illuminates generating hypotheses. The results also extend to broader settings beyond derived categories, connecting purity, cosilting t-structures, and hearts to lower bounds on pure global dimension, with implications for schemes and other triangulated categories.

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.

Definable functors and Brown--Adams representability

TL;DR

This work develops a unified, functorial approach to Brown–Adams representability in compactly generated triangulated categories via definable functors, showing that pure global dimension cannot drop under such transfers and that for any ring . Central to the strategy are -type isomorphisms induced by left-adjoint definable functors, which yield concrete criteria for when Brown representability holds or fails and facilitate tests in geometric contexts like . A major advance is the detailed analysis of von Neumann regular rings, proving that they control Brown–Adams representability: the telescope conjecture holds, there are tight correspondences between localising subcategories and ring-theoretic ideals, and the Freyd localisation illuminates generating hypotheses. The results also extend to broader settings beyond derived categories, connecting purity, cosilting t-structures, and hearts to lower bounds on pure global dimension, with implications for schemes and other triangulated categories.

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.
Paper Structure (7 sections, 20 theorems, 26 equations)

This paper contains 7 sections, 20 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on modules
  3. Transfer of pure homological dimension
  4. Brown--Adams representability
  5. Pure global dimensions of derived categories
  6. von Neumann regular rings control Brown--Adams representability
  7. Beyond derived categories

Key Result

Theorem A

(prop:pgdimineq,prop:ringepi, prop:hringepi) Let $R$ be a ring, then there is an inequality $\mathsf{pgdim}(\mathsf{D}(R))\geq \mathsf{pgdim}(R)$. As such, $\mathsf{D}(R)$ is not a brown category whenever $\mathsf{pgdim}(R)\geq 2$. Moreover, if $f\colon R\to S$ a ring homomorphism, then

Theorems & Definitions (56)

  • Theorem A
  • Theorem B
  • Theorem C
  • Example 2.3
  • Example 2.7
  • Example 2.11
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Corollary 3.3
  • ...and 46 more