The homological spectrum via definable subcategories

TL;DR

This paper introduces a new approach to understanding the homological spectrum of tensor-triangulated categories by using definable subcategories, establishing a homeomorphism with a quotient of the Ziegler spectrum and exploring related structures.

Contribution

It provides an alternative framework for the homological spectrum via definable subcategories and characterizes injective objects in homological residue fields.

Findings

Homological spectrum is homeomorphic to a quotient of the Ziegler spectrum.

Injective objects in homological residue fields are characterized via definable subcategories.

A purity perspective links the homological and Balmer spectra.

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.