Mathematische Annalen

, Volume 324, Issue 3, pp 557–580

Presheaves of triangulated categories and reconstruction of schemes

  • P. Balmer

DOI: 10.1007/s00208-002-0353-1

Cite this article as:
Balmer, P. Math. Ann. (2002) 324: 557. doi:10.1007/s00208-002-0353-1

Abstract.

To any triangulated category with tensor product \((K, \otimes)\), we associate a topological space \({\rm Spc}(K, \otimes)\), by means of thick subcategories of K, à la Hopkins-Neeman-Thomason. Moreover, to each open subset U of this space \(Spc(K, \otimes)\), we associate a triangulated category \({\cal K}(U)\), producing what could be thought of as a presheaf of triangulated categories. Applying this to the derived category \((K, \otimes):=({\rm D}^{\rm perf} (X), \otimes^L)\) of perfect complexes on a noetherian scheme X, the topological space \({\rm Spc}(K, \otimes)\) turns out to be the underlying topological space of X; moreover, for each open \(U \subset X\), the category \({\cal K}(U)\) is naturally equivalent to \({\rm D}^{\rm perf} (U)\). As an application, we give a method to reconstruct any reduced noetherian scheme X from its derived category of perfect complexes \({\rm D}^{\rm perf} (X)\), considering the latter as a tensor triangulated category with \(\otimes^L\).

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • P. Balmer
    • 1
  1. 1.SFB 478, Universität Münster, Hittorfstr. 27, 48149 Münster, Germany (e-mail: balmer@math.uni-muenster.de, URL: www.math.uni-muenster.de/u/balmer) DE

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