Astérisque N° 293/2004 - Grand Format

Let Fq denote the finite field of order q (a power of a prime p), let X be a smooth scheme over a field k containing Fq and let A be a finite Fq-algebra. We study the relationship between constructible A-sheaves on the etale site of X, and a certain class of quasi-coherent OxOFq 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/pnZ-sheaves on smooth Wn(k)-schemes, and complexes of Berthelot's arithmetic D-modules, equipped with a unit Frobenius.