Triangular spectra and their applications to derived categories of noetherian schemes

2023/01/09 by Hiroki Matsui, Matsui, Hiroki · 1 citation
Mathematics · #14A15 #14F08 #14H52 #18G80 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2301.03168

openalex publication_date 2023/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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