#### Vol. 313, No. 2, 2021

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

### Hiroki Matsui

Vol. 313 (2021), No. 2, 433–457
DOI: 10.2140/pjm.2021.313.433
##### Abstract

For an essentially small triangulated category $\mathsc{𝒯}$, we introduce the notion of prime thick subcategories and define the spectrum of $\mathsc{𝒯}$, 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.

##### Keywords
complete intersection, derived category, hypersurface, noetherian scheme, prime thick subcategory, singularity category, spectrum, triangulated category
##### Mathematical Subject Classification
Primary: 14F08
Secondary: 13D09, 13H10, 18G80