Koszul duality and semisimplicity of Frobenius
[Dualité de Koszul et semi-simplicité du Frobenius]
Annales de l'Institut Fourier, Tome 63 (2013) no. 4, pp. 1511-1612.

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.

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.

DOI : https://doi.org/10.5802/aif.2809
Classification : 16S37,  14F05,  14M15
Mots clés : Dualité de Koszul ; faisceaux pervers ; variété de drapeaux
@article{AIF_2013__63_4_1511_0,
     author = {Achar, Pramod N. and Riche, Simon},
     title = {Koszul duality and semisimplicity of~Frobenius},
     journal = {Annales de l'Institut Fourier},
     pages = {1511--1612},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {63},
     number = {4},
     year = {2013},
     doi = {10.5802/aif.2809},
     mrnumber = {3137361},
     zbl = {06359595},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2809/}
}
Achar, Pramod N.; Riche, Simon. Koszul duality and semisimplicity of Frobenius. Annales de l'Institut Fourier, Tome 63 (2013) no. 4, pp. 1511-1612. doi : 10.5802/aif.2809. http://archive.numdam.org/articles/10.5802/aif.2809/

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