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

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