K-theory of semi-linear endomorphisms via the Riemann-Hilbert correspondence

Grayson, developing ideas of Quillen, has made computations of the K-theory of "semi-linear endomorphisms". In the present text we develop a technique to compute these groups in the case of Frobenius semi-linear actions. The main idea is to interpret the semi-linear modules as crystals and use a positive characteristic version of the Riemann-Hilbert correspondence. We also compute the K-theory of the category of etale constructible p-torsion sheaves.