Presheaves of Triangulated Categories and Reconstruction of Schemes

To any triangulated category with tensor product (K,⊗), we associate a topological space Spc(K,⊗), by means of thick subcategories of K, à la Hopkins-Neeman-Thomason. Moreover, to each open subset U of this space Spc(K,⊗), we associate a triangulated category K(U), producing what could be thought of as a presheaf of triangulated categories. Applying this to the derived category (K,⊗) := (D(X),⊗L) of perfect complexes on a noetherian scheme X, the topological space Spc(K,⊗) turns out to be the underlying topological space of X; moreover, for each open U ⊂ X, the category K(U) is naturally equivalent to D(U). As an application, we give a method to reconstruct any reduced noetherian scheme X from its derived category of perfect complexes D(X), considering the latter as a tensor triangulated category with ⊗.