The Riemann-Hilbert correspondence for unit F-crystals

Quatrième de couverture

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 A be a finite F_q-algebra. We study the relationship between constructible A-sheaves on the étale site of X, and a certain class of quasi-coherent (...)_X(...)_Fq 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 W_n(k)-schemes, and complexes of Berthelot's arithmetic D-modules, equipped with a unit Frobenius.

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