Astérisque, n° 293. The Riemann-Hilbert correspondence for unit F-crystals
Let F<sub> q </sub> denote the finite field of order q (a power of a prime p ),
let X be a smooth scheme over a field k containing F<sub> q </sub>, and let A
be a finite F<sub> q </sub>-algebra. We study the relationship between constructible
A-sheaves on the étale site of X , and a certain class of
quasi-coherent (...)<sub> X </sub>(...)<sub>F</sub><sub> q </sub> A-modules equipped with a "unit" Frobenius
structure. We show that the two corresponding derived categories
are anti-equivalent as triangulated categories, and that this anti-equivalence
is compatible with direct and inverse images, tensor
products, and certain other operations.
We also obtain analogous results relating complexes of constructible
Z/ p<sup>n</sup> Z-sheaves on smooth W<sub>n</sub> ( k )-schemes, and complexes
of Berthelot's arithmetic D -modules, equipped with a unit
Frobenius.
