Hochster duality in derived categories and point-free reconstruction of schemes

TL;DR

This paper uses localization and point-free topology to explicitly realize the Zariski frame and its Hochster dual within the derived category of a ring, providing new proofs and a reconstruction method for schemes.

Contribution

It introduces a novel realization of the Zariski frame and Hochster dual in derived categories, simplifying existing theories and enabling scheme reconstruction from tensor triangulated structures.

Findings

Explicit realization of Zariski frame and Hochster dual as lattices in derived categories

New proofs of Hopkins-Neeman and Thomason theorems

Reconstruction of coherent schemes from derived categories

Abstract

For a commutative ring $R$, we exploit localization techniques and point-free topology to give an explicit realization of both the Zariski frame of $R$ (the frame of radical ideals in $R$) and its Hochster dual frame, as lattices in the poset of localizing subcategories of the unbounded derived category $D(R)$. This yields new conceptual proofs of the classical theorems of Hopkins-Neeman and Thomason. Next we revisit and simplify Balmer's theory of spectra and supports for tensor triangulated categories from the viewpoint of frames and Hochster duality. Finally we exploit our results to show how a coherent scheme $(X,\mathcal{O}_X)$ can be reconstructed from the tensor triangulated structure of its derived category of perfect complexes.