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

Kock, Joachim; Pitsch, Wolfgang

doi:10.1090/tran/6773

Hochster duality in derived categories and point-free reconstruction of schemes
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by Joachim Kock and Wolfgang Pitsch PDF

Trans. Amer. Math. Soc. 369 (2017), 223-261 Request permission

Abstract:

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 $D(R)$. 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 $(X,\mathcal {O}_X)$ can be reconstructed from the tensor triangulated structure of its derived category of perfect complexes.

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