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.
I: Transactions of the American Mathematical Society, Bind 369, Nr. 1, 2017, s. 223-261.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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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