The shift-homological spectrum and parametrising kernels of rank functions

TL;DR

The paper develops a non-monoidal framework to classify thick subcategories in compactly generated triangulated categories by introducing the shift-homological spectrum $\mathsf{Spc}_{\Sigma}^{\mathsf{h}}(\mathsf{T}^{\mathrm{c}})$ and the shift-spectrum $\mathsf{Spc}_{\Sigma}(\mathsf{T}^{\mathrm{c}})$. It builds these spaces from Serre subcategories of the finitely presented functor category and connects them to the Ziegler spectrum and to rank-function theory, providing a coherent non-monoidal analogue of Balmer’s spectrum. The authors establish criteria (notably isolation and endofiniteness) under which radical thick subcategories coincide with intersections of primes and show several classes where every thick subcategory is radical (e.g., tame hereditary algebras, finite-type, and monogenic TT-categories); they also relate irreducible rank functions to shift-homological primes and give concrete computations in derived categories of hereditary algebras and singularity categories. The work reveals both parallels and distinctions with tensor-triangular geometry, offering new tools to analyze non-monoidal contexts and linking model-theoretic/topological methods (Ziegler, Kolmogorov quotients) to representation-theoretic structures with practical computations in classical settings.

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.

Figures (1)

• Figure 1: The start of the AR quiver of $\mathcal{U}_{n}$, which continues downwards indefinitely, and wraps horizontally with periodicity $n$. The red points are the indecomposable objects in $\mathcal{W}(R_{i_1}^{m_1},\dots,R_{i_l}^{m_l})$, the green points are the indecomposable objects with $\mathrm{Hom}_{\Lambda}(R,Z_1)\neq 0$ or $\text{Ext}_{\Lambda}^{1}(R,Z_1)\neq 0$, and the black points are the indecomposable objects in $\bigoplus_{u=1}^l \mathcal{W}(R_{i_u+1}^{1},R_{i_u+2}^{1} \dots,R_{i_{u+1}-2}^{1})$.

Theorems & Definitions (135)

• Theorem : \ref{['lem:downtohomological']}, \ref{['thm:maptobalmer']}
• Theorem : \ref{['homeomorphism']}
• Theorem : \ref{['thm:tamheredeverythingradical']}, \ref{['prop:LocNoeth']}, and \ref{['prop:rigideverythingradical']}
• Theorem : \ref{['lem:RankGivesPrime']}, \ref{['thm:whenisKahomeo']}
• Proposition 2.4: krspec
• Definition 3.1
• Lemma 3.3
• Lemma 3.5
• proof
• Proposition 3.8