PRIME THICK SUBCATEGORIES AND SPECTRA OF DERIVED AND SINGULARITY CATEGORIES OF NOETHERIAN SCHEMES

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