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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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.
References
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Additional Information
  • Joachim Kock
  • Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
  • MR Author ID: 617085
  • Email: kock@mat.uab.es
  • Wolfgang Pitsch
  • Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
  • Email: pitsch@mat.uab.es
  • 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.
  • © Copyright 2016 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 223-261
  • MSC (2010): Primary 18E30; Secondary 06D22, 14A15
  • DOI: https://doi.org/10.1090/tran/6773
  • MathSciNet review: 3557773