Tannakian properties of unit Frobenius-modules

TL;DR

This paper explores the tannakian properties of unit Frobenius-modules, establishing their analogy to locally constant sheaves and demonstrating their tannakian category structure over various coefficient algebras.

Contribution

It introduces a tannakian framework for unit Frobenius-modules, extending previous work to a broad class of coefficient algebras and base schemes.

Findings

Unit Frobenius-modules form a tannakian category over field coefficients.

The results apply to a wide class of coefficient algebras including Drinfeld rings.

The framework generalizes to arbitrary locally noetherian base schemes.

Abstract

We show that unit $\mathcal{O}_{F,X}^\Lambda$-modules of Emerton and Kisin provide an analogue of locally constant sheaves in the context of B\"ockle-Pink $\Lambda$-crystals. For example they form a tannakian category if the coefficient algebra $\Lambda$ is a field. Our results hold for a big class of coefficien algebras which includes Drinfeld rings, and for arbitary locally noetherian base schemes.