Prime thick subcategories and spectra of derived and singularity categories of noetherian schemes

TL;DR

This paper introduces the concept of prime thick subcategories and their spectra in triangulated categories, particularly in derived and singularity categories of noetherian schemes, linking geometric subspaces to spectral topology.

Contribution

It defines prime thick subcategories and spectra for triangulated categories, extending Balmer’s framework to algebraic geometry contexts like derived and singularity categories.

Findings

Prime thick subcategories are characterized in derived and singularity categories.

Certain subspaces of schemes are embedded into the spectra as topological spaces.

The spectrum construction generalizes Balmer’s tensor triangulated spectrum.

Abstract

For an essentially small triangulated category $\mathcal{T}$, we introduce the notion of prime thick subcategories and define the spectrum of $\mathcal{T}$, which shares the basic properties with the spectrum of a tensor triangulated category introduced by Balmer. We mainly focus on triangulated categories that appear in algebraic geometry such as the derived and the singularity categories of a noetherian scheme $X$. We prove that certain classes of thick subcategories are prime thick subcategories of these triangulated categories. Furthermore, we use this result to show that certain subspaces of $X$ are embedded into their spectra as topological spaces.