Presheaves of triangulated categories and reconstruction of schemes

Abstract

To any triangulated category with tensor product $(K,\otimes )$, we associate a topological space $Spc(K,\otimes )$, by means of thick subcategories of $K$, a la Hopkins-Neeman-Thomason. Moreover, to each open subset $U$ of $Spc(K,\otimes )$, 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 $(K,\otimes ):=(D^{perf}(X),{\otimes }^L)$ of perfect complexes on a noetherian scheme $X$, the topological space $Spc(K,\otimes )$ turns out to be the underlying topological space of $X$; moreover, for each open $U\subset X$, the category $K(U)$ is naturally equivalent to $D^{perf}(U)$. As an application, we give a method to reconstruct any reduced noetherian scheme $X$ from its derived category of perfect complexes $D^{perf}(X)$, considering the latter as a tensor triangulated category with ${\otimes }^L$.

18 pages; minor changes

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