Definable functors and Brown--Adams representability
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914255499952128 |
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| author | Bird, Isaac |
| author_facet | Bird, Isaac |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09443 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Definable functors and Brown--Adams representability Bird, Isaac Representation Theory Algebraic Topology Category Theory 18G80 (Primary), 16E35, 16E50, 18E45 (Secondary) 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. |
| title | Definable functors and Brown--Adams representability |
| topic | Representation Theory Algebraic Topology Category Theory 18G80 (Primary), 16E35, 16E50, 18E45 (Secondary) |
| url | https://arxiv.org/abs/2601.09443 |