Triangular spectra and their applications to derived categories of noetherian schemes

Abstract

In recent work, for a triangulated category $\cT$, the author introduced a topological space $\tSpec(\cT)$ which we call the triangular spectrum of $\cT$ as a tensor-free analog of the Balmer spectrum for a tensor triangulated category. In this paper, we use the triangular spectrum to reconstruct a noetherian scheme $X$ from its perfect derived category $\dpf(X)$. As an application, we give an alternative proof of the Bondal-Orlov-Ballard reconstruction theorem in the special case (when both varieties have ample or anti-ample canonical bundles). Moreover, we define the structure sheaf on $\tSpec(\cT)$ and compare the triangular spectrum and the Balmer spectrum as ringed spaces.