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

TL;DR

This paper develops a method to compute the K-theory of semi-linear endomorphisms, especially Frobenius actions, by interpreting modules as crystals and applying a positive characteristic Riemann-Hilbert correspondence.

Contribution

It introduces a new technique to compute K-theory for Frobenius semi-linear actions using crystals and a positive characteristic Riemann-Hilbert correspondence.

Findings

Computed K-theory of semi-linear modules with Frobenius actions

Determined K-theory of etale constructible p-torsion sheaves

Established a link between crystals and semi-linear endomorphisms

Abstract

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.