Presheaves of triangulated categories and reconstruction of schemes

Paul Balmer

Published 2001-11-05, updated 2002-05-28Version 2

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 $\otimes^L$.