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

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Hochster duality in derived categories and point-free reconstruction of schemes. / Kock, Joachim; Pitsch, Wolfgang.

In: Transactions of the American Mathematical Society, Vol. 369, No. 1, 2017, p. 223-261.

Research output: Contribution to journal › Journal article › Research › peer-review

Harvard

Kock, J & Pitsch, W 2017, 'Hochster duality in derived categories and point-free reconstruction of schemes', Transactions of the American Mathematical Society, vol. 369, no. 1, pp. 223-261. https://doi.org/10.1090/tran/6773

APA

Kock, J., & Pitsch, W. (2017). Hochster duality in derived categories and point-free reconstruction of schemes. Transactions of the American Mathematical Society, 369(1), 223-261. https://doi.org/10.1090/tran/6773

Vancouver

Kock J, Pitsch W. Hochster duality in derived categories and point-free reconstruction of schemes. Transactions of the American Mathematical Society. 2017;369(1):223-261. https://doi.org/10.1090/tran/6773

Author

Kock, Joachim ; Pitsch, Wolfgang. / Hochster duality in derived categories and point-free reconstruction of schemes. In: Transactions of the American Mathematical Society. 2017 ; Vol. 369, No. 1. pp. 223-261.

Bibtex

@article{7f036d8756934f79adefde84a400f880,
title = "Hochster duality in derived categories and point-free reconstruction of schemes",
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{\textquoteright}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.",
keywords = "Frames, Hochster duality, Localizing subcategories, Reconstruction of schemes, Triangulated categories",
author = "Joachim Kock and Wolfgang Pitsch",
note = "Publisher Copyright: {\textcopyright} 2016 American Mathematical Society.",
year = "2017",
doi = "10.1090/tran/6773",
language = "English",
volume = "369",
pages = "223--261",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "1",

}

RIS

TY - JOUR

T1 - Hochster duality in derived categories and point-free reconstruction of schemes

AU - Kock, Joachim

AU - Pitsch, Wolfgang

N1 - Publisher Copyright: © 2016 American Mathematical Society.

PY - 2017

Y1 - 2017

N2 - 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, OX) can be reconstructed from the tensor triangulated structure of its derived category of perfect complexes.

AB - 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, OX) can be reconstructed from the tensor triangulated structure of its derived category of perfect complexes.

KW - Frames

KW - Hochster duality

KW - Localizing subcategories

KW - Reconstruction of schemes

KW - Triangulated categories

UR - http://www.scopus.com/inward/record.url?scp=84992065241&partnerID=8YFLogxK

U2 - 10.1090/tran/6773

DO - 10.1090/tran/6773

M3 - Journal article

AN - SCOPUS:84992065241

VL - 369

SP - 223

EP - 261

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -

ID: 331494739