Triangular spectra and their applications to derived categories of noetherian schemes

TL;DR

This paper introduces the triangular spectrum of a triangulated category and demonstrates its use in reconstructing noetherian schemes from their derived categories, providing new insights and alternative proofs for existing theorems.

Contribution

The paper defines the triangular spectrum for triangulated categories and applies it to reconstruct noetherian schemes, offering an alternative proof of a key reconstruction theorem.

Findings

Reconstructed noetherian schemes from their perfect derived categories using the triangular spectrum.

Defined a structure sheaf on the triangular spectrum and compared it with the Balmer spectrum.

Provided an alternative proof of the Bondal-Orlov-Ballard reconstruction theorem in specific cases.

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.