Presheaves of triangulated categories and reconstruction of schemes
By Paul Balmer. 2001
84 citations recorded by the source corpus.
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 $\otimes^L$.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/e6a5a6fe4ed50bf6540d99d4409837469dab435d
How researchers see this paper
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In Algebra and Number Theory
- A. Neeman
- Martin Brandenburg
- Michel van den Bergh
- L. Hille
- Paul Balmer
- M. Schlichting
- Osamu Iyama
- Tatiana Gateva-Ivanova
In Geometry and Topology
- A. Neeman
- L. Hille
- Martin Brandenburg
- Michel van den Bergh
- Paul Balmer
- J. Št’ovíček
- Henning Krause
- Dmitri Orlov
In Mathematical Physics
- A. Neeman
- L. Hille
- Martin Brandenburg
- Michel van den Bergh
- Paul Balmer
- J. Št’ovíček
- Henning Krause
- Dmitri Orlov
Citations in the corpus
- Handbook of Tilting Theory: Fourier-Mukai transforms (2004, background): “ existence of non-isomorphic Fourier-Mukai partners shows that this cannot be done in general, but it is possible if the canonical sheaf or the anticanonical sheaf is ample. Later Balmer and Rouquier [3, 67] have shown independently that one can reconstruct the variety from the category of coherent ”
- Tensor categorical foundations of algebraic geometry (2014, background): “In 2002 Balmer has reconstructed a noetherian scheme from its tensor triangulated category of perfect complexes ([Bal02]).”
- The spectrum of prime ideals in tensor triangulated categories (2004, methodology): “This was in turn inspired by Thomason’s theorem, see [1], Thm.”
- The homotopy category of flat modules, and Grothendieck duality (2008, background): “The category K(R-Flat) is a tensor triangulated category, in the terminology of Balmer [1,2].”
- Classifying thick subcategories of the stable category of Cohen-Macaulay modules (2009, unknown): “For example, see [7, 8, 9, 43, 15, 11, 38, 13, 20, 53].”