The shift-homological spectrum and parametrising kernels of rank functions

TL;DR

This paper introduces the shift-homological and shift-spectrum topological spaces for compactly generated triangulated categories, parametrising radical thick subcategories and relating them to rank functions and the Ziegler spectrum.

Contribution

It develops new topological invariants for triangulated categories, generalising spectral concepts from tensor-triangular geometry using module category quotients.

Findings

The shift-homological spectrum is related to irreducible rank functions.

In certain categories, all thick subcategories are radical.

The introduced spectra can be described via the Ziegler spectrum.

Abstract

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 thick subcategories of the compact objects, which we call radical. These spaces can be viewed as non-monoidal analogues of the Balmer and homological spectra arising in tensor-triangular geometry: we prove that for monogenic tensor-triangulated categories the Balmer spectrum is a subspace of the shift-spectrum. To construct these analogues we utilise quotients of the module category, rather than the lattice theoretic methods which have been adopted in other approaches. We characterise radical thick subcategories and show in certain cases, such as the perfect derived categories of tame hereditary algebras or monogenic tensor-triangulated categories, that every thick subcategory is radical. We establish a close relationship between the shift-homological spectrum and the set of irreducible integral rank functions, and provide necessary and sufficient conditions for every radical thick subcategory to be given by an intersection of kernels of rank functions. In order to facilitate these results, we prove that both spaces we introduce may equivalently be described in terms of the Ziegler spectrum.