A topological nullstellensatz for tensor-triangulated categories

Abhishek Banerjee

doi:10.1016/j.crma.2018.02.012

Homological algebra/Algebraic geometry

A topological nullstellensatz for tensor-triangulated categories
[Un théorème des zéros topologique pour les catégories triangulées tensorielles]

Présenté par : Claire Voisin

Abhishek Banerjee ¹

¹ Department of Mathematics, Indian Institute of Science, Bangalore-560012, India

Comptes Rendus. Mathématique, Volume 356 (2018) no. 4, pp. 365-375.

Résumé
Abstract

Soit $Spec (T)$ le spectre d'une catégorie triangulée tensorielle $(T, \otimes, 1)$ et notons par $Spec {(T)}^{inv}$ la topologie inverse sur $Spec (T)$ . Nous montrons qu'on dispose d'un homéomorphisme entre l'espace spectral des idéaux radicaux de $(T, \otimes, 1)$ et l'espace des sous-ensembles ouverts de $Spec {(T)}^{inv}$ . En fait, nous obtenons un résulat plus général en termes des données de support sur $(T, \otimes, 1)$ en utilisant des idées provenant d'algèbre commutative, de topologie et de géométrie triangulée tensorielle.

Let $Spec (T)$ be the spectrum of a tensor-triangulated category $(T, \otimes, 1)$ . We show that there is a homeomorphism between the spectral space of radical thick tensor ideals in $(T, \otimes, 1)$ and the collection of open subsets of $Spec (T)$ in inverse topology. In fact, we prove a more general result in terms of supports on $(T, \otimes, 1)$ and work by combining methods from commutative algebra, topology and tensor triangular geometry.

Reçu le : 2017-05-12
Accepté le : 2018-02-27
Publié le : 2018-03-09

DOI : 10.1016/j.crma.2018.02.012

Affiliations des auteurs :

Abhishek Banerjee ¹

¹ Department of Mathematics, Indian Institute of Science, Bangalore-560012, India

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Abhishek Banerjee. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.02.012/

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