Functors on triangulated tensor categories

TL;DR

This paper introduces a functorial spectrum for triangulated tensor categories, providing new reconstruction results for schemes and offering alternative proofs for known theorems in algebraic geometry.

Contribution

It defines a new functorial spectrum for triangulated tensor categories and proves a reconstruction theorem for schemes, also giving an alternative proof of Bondal-Orlov's theorem.

Findings

Reconstruction of topologically noetherian schemes using the functorial spectrum.

Alternative proof of Bondal-Orlov's theorem for smooth projective varieties.

Establishment of the functorial spectrum as a tool in tensor triangulated geometry.

Abstract

We define and study the functorial spectrum for every triangulated tensor category. A reconstruction result for topologically noetherian schemes similar to (and based on) a theorem by Balmer is proved. An alternative proof of the reconstruction theorem by Bondal-Orlov for smooth projective varieties with ample (anti-)canonical bundles is given.