Separable extensions  in tensor-triangular geometry  and generalized Quillen stratification

Balmer, Paul

doi:10.24033/asens.2298

Separable extensions in tensor-triangular geometry and generalized Quillen stratification
[Extensions séparables en géométrie triangulaire tensorielle et stratification de Quillen généralisée]

Balmer, Paul

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 4, pp. 907-925.

Résumé
Abstract

Nous montrons un lien entre le théorème du going-up en algèbre commutative et le théorème de stratification de Quillen en théorie des représentations modulaires. Dans ce but, nous étudions l'application continue induite sur les spectres par une extension séparable de catégories triangulées tensorielles. Nous en déterminons l'image et bornons le cardinal de ses fibres par le degré de l'extension. Nous prouvons alors une forme faible de descente, « à nilpotence près », qui nous permet de généraliser la stratification de Quillen à d'autres catégories dérivées équivariantes.

We exhibit a link between the Going-Up Theorem in commutative algebra and Quillen Stratification in modular representation theory. To this effect, we study the continuous map induced on spectra by a separable extension of tensor-triangulated categories. We determine the image of this map and bound the cardinality of its fibers by the degree of the extension. We then prove a weak form of descent, “up-to-nilpotence,” which allows us to generalize Quillen Stratification to equivariant derived categories.

MR Zbl

DOI : 10.24033/asens.2298

Classification : 18E30, 20J05, 13B24, 55U35.
Keywords: Separable, tt-category, descent, nilpotence, Quillen stratification.
Mot clés : Séparable, tt-catégorie, descente, nilpotence, stratification de Quillen.

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     title = {Separable extensions  in tensor-triangular geometry  and generalized {Quillen} stratification},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {907--925},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 49},
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     year = {2016},
     doi = {10.24033/asens.2298},
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Balmer, Paul. Separable extensions  in tensor-triangular geometry  and generalized Quillen stratification. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 4, pp. 907-925. doi : 10.24033/asens.2298. http://www.numdam.org/articles/10.24033/asens.2298/

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