The Riemann-Hilbert correspondence for unit F-crystals

Let F-q 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-q, and let Lambda be a finite F-q-algebra. We study the relationship between constructible Lambda-sheaves on the etale site of X, and a certain class of quasi-coherent O(X)circle times(Fq) Lambda-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(n) Z-sheaves on smooth W-n(k)-schemes, and complexes of Berthelot's arithmetic D-modules, equipped with a unit Frobenius.