Let be the spectrum of a tensor-triangulated category . We show that there is a homeomorphism between the spectral space of radical thick tensor ideals in and the collection of open subsets of in inverse topology. In fact, we prove a more general result in terms of supports on and work by combining methods from commutative algebra, topology and tensor triangular geometry.
Soit le spectre d'une catégorie triangulée tensorielle et notons par la topologie inverse sur . Nous montrons qu'on dispose d'un homéomorphisme entre l'espace spectral des idéaux radicaux de et l'espace des sous-ensembles ouverts de . En fait, nous obtenons un résulat plus général en termes des données de support sur en utilisant des idées provenant d'algèbre commutative, de topologie et de géométrie triangulée tensorielle.
Accepted:
Published online:
Banerjee, Abhishek 1
@article{CRMATH_2018__356_4_365_0,
author = {Banerjee, Abhishek},
title = {A topological nullstellensatz for tensor-triangulated categories},
journal = {Comptes Rendus. Math\'ematique},
pages = {365--375},
year = {2018},
publisher = {Elsevier},
volume = {356},
number = {4},
doi = {10.1016/j.crma.2018.02.012},
language = {en},
url = {https://www.numdam.org/articles/10.1016/j.crma.2018.02.012/}
}
TY - JOUR AU - Banerjee, Abhishek TI - A topological nullstellensatz for tensor-triangulated categories JO - Comptes Rendus. Mathématique PY - 2018 SP - 365 EP - 375 VL - 356 IS - 4 PB - Elsevier UR - https://www.numdam.org/articles/10.1016/j.crma.2018.02.012/ DO - 10.1016/j.crma.2018.02.012 LA - en ID - CRMATH_2018__356_4_365_0 ER -
%0 Journal Article %A Banerjee, Abhishek %T A topological nullstellensatz for tensor-triangulated categories %J Comptes Rendus. Mathématique %D 2018 %P 365-375 %V 356 %N 4 %I Elsevier %U https://www.numdam.org/articles/10.1016/j.crma.2018.02.012/ %R 10.1016/j.crma.2018.02.012 %G en %F CRMATH_2018__356_4_365_0
Banerjee, Abhishek. A topological nullstellensatz for tensor-triangulated categories. Comptes Rendus. Mathématique, Volume 356 (2018) no. 4, pp. 365-375. doi: 10.1016/j.crma.2018.02.012
[1] The spectrum of prime ideals in tensor-triangulated categories, J. Reine Angew. Math., Volume 588 (2005), pp. 149-168
[2] Supports and filtrations in algebraic geometry and modular representation theory, Amer. J. Math., Volume 129 (2007), pp. 1227-1250
[3] Spectra, spectra, spectra – tensor triangular spectra versus Zariski spectra of endomorphism rings, Algebraic Geom. Topol., Volume 10 (2010), pp. 1521-1563
[4] Tensor triangular geometry (Bhatia, R., ed.), Proceedings of the International Congress of Mathematicians, vol. II, Hindustan Book Agency, New Delhi, 2010, pp. 85-112
[5] Gluing techniques in triangular geometry, Quart. J. Math., Volume 58 (2007), pp. 415-441
[6] Generalized tensor idempotents and the telescope conjecture, Proc. Lond. Math. Soc. (3), Volume 102 (2011), pp. 1161-1185
[7] Realizations of pairs and Oka families in tensor-triangulated categories, Eur. J. Math., Volume 2 (2016) no. 3, pp. 760-797
[8] Closure operators in Abelian categories and spectral spaces, Theory Appl. Categ., Volume 32 (2017) no. 20, pp. 719-735
[9] On some spectral spaces associated with tensor-triangulated categories, Arch. Math. (Basel), Volume 108 (2017) no. 6, pp. 581-591
[10] Thick subcategories of the stable module category, Fundam. Math., Volume 153 (1997), pp. 59-80
[11] Nilpotence and stable homotopy theory I, Ann. of Math. (2), Volume 128 (1988), pp. 207-241
[12] A guide to closure operations in commutative algebra, Progress in Commutative Algebra 2, Walter de Gruyter, Berlin, 2012, pp. 1-37
[13] A topological version of Hilbert's Nullstellensatz, J. Algebra, Volume 461 (2016), pp. 25-41
[14] Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., Volume 142 (1969), pp. 43-60
[15] Stone Spaces, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, Cambridge, 1982
[16] Les théorèmes de Chevalley–Tarski et remarques sur l'algèbre constructive, Cah. Topol. Géom. Différ. Catég., Volume XVI (1975) no. 3, pp. 256-258
[17] Chow groups of tensor-triangulated categories, J. Pure Appl. Algebra, Volume 220 (2016), pp. 1343-1381
[18] Intersection products for tensor triangular Chow groups, J. Algebra, Volume 449 (2016), pp. 497-538
[19] Hochster duality in derived categories and point-free reconstruction of schemes, Trans. Amer. Math. Soc., Volume 369 (2017) no. 1, pp. 223-261
[20] Deriving Auslander's formula, Doc. Math., Volume 20 (2015), pp. 669-688
[21] Prime ideals of mixed Artin–Tate motives, J. K-Theory, Volume 11 (2013), pp. 331-349
[22] Higher comparison maps for the spectrum of a tensor-triangulated category, Adv. Math., Volume 247 (2013), pp. 71-102
[23] Support theory via actions of tensor-triangulated categories, J. Reine Angew. Math., Volume 681 (2013), pp. 219-254
[24] Subcategories of singularity categories via tensor actions, Compos. Math., Volume 150 (2014), pp. 229-272
[25] The Stacks project, available online at stacks.math.columbia.edu.
[26] The classification of triangulated subcategories, Compos. Math., Volume 105 (1997), pp. 1-27
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