7 papers
Definable functoriality of tensor-triangular spectra
Isaac Bird, Jordan Williamson
We prove that the homological and Balmer spectra in tensor-triangular geometry are functorial in certain definable functors, thereby providing an alternative perspective on functor…
Definable functors and Brown--Adams representability
Isaac Bird
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 fun…
The shift-homological spectrum and parametrising kernels of rank functions
Isaac Bird, Jordan Williamson, Alexandra Zvonareva
For any compactly generated triangulated category we introduce two topological spaces, the shift-spectrum and the shift-homological spectrum. We use them to parametrise a family of…
Purity, ascent, and periodicity for Gorenstein flat cotorsion modules
Isaac Bird
We investigate purity within the Frobenius category of Gorenstein flat cotorsion modules, which can be seen as an infinitely generated analogue of the Frobenius category of Gorenst…
Definable functors between triangulated categories
Isaac Bird, Jordan Williamson
We systematically develop the theory of definable functors between compactly generated triangulated categories. Such functors preserve pure triangles, pure injective objects, and d…
The homological spectrum via definable subcategories
Isaac Bird, Jordan Williamson
We develop an alternative approach to the homological spectrum of a tensor-triangulated category through the lens of definable subcategories. This culminates in a proof that the ho…