The homological spectrum via definable subcategories

Isaac Bird; Jordan Williamson

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arXiv · 2304.05179

The homological spectrum via definable subcategories

Isaac Bird·Jordan Williamson

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Abstract

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 homological spectrum is homeomorphic to a quotient of the Ziegler spectrum. Along the way, we characterise injective objects in homological residue fields in terms of the definable subcategory corresponding to a given homological prime. We use these results to give a purity perspective on the relationship between the homological and Balmer spectrum.

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Isaac Bird, Jordan Williamson. 2023-04-11. The homological spectrum via definable subcategories. https://arxiv.org/abs/2304.05179

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