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KOSZUL DUALITY AND SEMISIMPLICITY OF FROBENIUS PRAMOD N. ACHAR AND SIMON RICHE
 

Summary: KOSZUL DUALITY AND SEMISIMPLICITY OF FROBENIUS
PRAMOD N. ACHAR AND SIMON RICHE
Abstract. A fundamental result of Beilinson­Ginzburg­Soergel 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 cor-
responding functors on the modified category or its triangulated analogue. In
particular, we show that these functors preserve semisimplicity of the Frobe-
nius action.
1. Introduction
Let X be a variety over a finite field Fq. In Deligne's work on the Weil conjec-
tures [D1, D2], a central role is played by the category of "mixed constructible com-
plexes of Q -sheaves" on X, denoted DWeil
(X) in the present paper. (Henceforth,
we will avoid calling this category "mixed," as that conflicts with the terminology
of [BGS].) In order to belong to DWeil
(X), a complex F must have the property
that the eigenvalues of the Frobenius action on stalks of F at Fqn -points of X are of

  

Source: Achar, Pramod - Department of Mathematics, Louisiana State University

 

Collections: Mathematics