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Transactions of the American Mathematical Society

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


Authors: Joachim Kock and Wolfgang Pitsch
Journal: Trans. Amer. Math. Soc. 369 (2017), 223-261
MSC (2010): Primary 18E30; Secondary 06D22, 14A15
DOI: https://doi.org/10.1090/tran/6773
Published electronically: March 9, 2016
MathSciNet review: 3557773
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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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Additional Information

Joachim Kock
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
Email: kock@mat.uab.es

Wolfgang Pitsch
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
Email: pitsch@mat.uab.es

DOI: https://doi.org/10.1090/tran/6773
Keywords: Frames, Hochster duality, triangulated categories, localizing subcategories, reconstruction of schemes
Received by editor(s): February 14, 2014
Received by editor(s) in revised form: December 18, 2014
Published electronically: March 9, 2016
Additional Notes: Both authors were supported by FEDER/MEC grant MTM2010-20692 and SGR grant SGR119-2009.
Article copyright: © Copyright 2016 American Mathematical Society