Koszul duality and semisimplicity of Frobenius

Achar, Pramod N.; Riche, Simon

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 $\ell$-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 $\ell$-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.

Classification : 16S37, 14F05, 14M15
Keywords: Koszul duality, perverse sheaves, flag variety
Mot clés : Dualité de Koszul, faisceaux pervers, variété de drapeaux

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