Hochster duality in derived categories and point-free reconstruction of schemes (Q2826759)

In the present paper the authors use localization techniques and point-free topologies in order to present an explicit realization of the Zariski frame of a commutative ring \(R\), and also to describe Hochster dual frame as lattices in the poset of localizing subcategories of the unbounded derived category \(D(R)\). The techniques used here are obtained after a careful analysis of some deep results about the classification of thick subcategories in the finite stable homotopy category proved by \textit{E. S. Devinatz} et al. [Ann. Math. (2) 128, No. 2, 207--241 (1988; Zbl 0673.55008)], and the correspondent algebraically results described by \textit{M. J. Hopkins} [Lond. Math. Soc. Lect. Note Ser. 117, 73--96 (1987; Zbl 0657.55008)] and \textit{A. Neeman} [Topology 31, No. 3, 519--532 (1992; Zbl 0793.18008)].NEWLINENEWLINEThis well written paper presents many interesting results and techniques. We mention here one of the main results, which is Theorem 4.1.10: if \(X\) is a coherent scheme, the Zarinski frame associated to the category of compact objects (i.e. perfect complexes) in the derived category of complexes of \(\mathcal{O}_X\)-modules with quasi-coherent homology is the Hochster dual of the Zariski frame of \(X\). Therefore every coherent scheme can be reconstructed from the tensor triangulated category of compact objects in the derived category of complexes of \(\mathcal{O}_X\)-modules with quasi-coherent homology.