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Hochster duality in derived categories and point-free reconstruction of
By Joachim Kock and Wolfgang Pitsch
Trans. Amer. Math. Soc. 369 (2017), 223-261.
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
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,OX) can be reconstructed from the tensor triangulated structure of its
derived category of perfect complexes.
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