Koszul duality and semisimplicity of Frobenius
Annales de l'Institut Fourier, Volume 63 (2013) no. 4, pp. 1511-1612.

A fundamental result of Beĭlinson–Ginzburg–Soergel states that on flag varieties and related spaces, a certain modified version of the category of -adic perverse sheaves exhibits a phenomenon known as Koszul duality. The modification essentially consists of discarding objects whose stalks carry a nonsemisimple action of Frobenius. In this paper, we prove that a number of common sheaf functors (various pull-backs and push-forwards) induce corresponding functors on the modified category or its triangulated analogue. In particular, we show that these functors preserve semisimplicity of the Frobenius action.

D’après un résultat fondamental de Beĭlinson–Ginzburg–Soergel, sur les variétés de drapeaux et certains autres espaces, une version modifiée de la catégorie des faisceaux pervers -adiques possède des propriétés liées à la dualité de Koszul. Cette catégorie modifiée est obtenue en éliminant les objets où l’action du Frobenius sur les fibres n’est pas semi-simple. Dans cet article, nous démontrons que de nombreuses opérations faisceautiques s’étendent à cette catégorie modifiée et sa version triangulée. En particulier, ces foncteurs préservent la semi-simplicité de l’action du Frobenius.

DOI: 10.5802/aif.2809
Classification: 16S37, 14F05, 14M15
Keywords: Koszul duality, perverse sheaves, flag variety
Mot clés : Dualité de Koszul, faisceaux pervers, variété de drapeaux
Achar, Pramod N. 1; Riche, Simon 2

1 Department of Mathematics Louisiana State University Baton Rouge, LA 70803 USA
2 Clermont Université, Université Blaise Pascal, Laboratoire de Mathématiques, BP 10448, F-63000 Clermont-Ferrand. CNRS, UMR 6620, Laboratoire de Mathématiques, F-63177 Aubière.
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Achar, Pramod N.; Riche, Simon. Koszul duality and semisimplicity of Frobenius. Annales de l'Institut Fourier, Volume 63 (2013) no. 4, pp. 1511-1612. doi : 10.5802/aif.2809. http://www.numdam.org/articles/10.5802/aif.2809/

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