The homological spectrum via definable subcategories
Category Theory
2025-01-13 v3 Algebraic Topology
Representation Theory
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.
Keywords
Cite
@article{arxiv.2304.05179,
title = {The homological spectrum via definable subcategories},
author = {Isaac Bird and Jordan Williamson},
journal= {arXiv preprint arXiv:2304.05179},
year = {2025}
}
Comments
v3: 20pp. To appear in Bulletin of the LMS (this is the Author Accepted Manuscript version)