Frobenius height of prismatic cohomology with coefficients

TL;DR

This paper investigates how Frobenius operators affect prismatic F-crystals under smooth proper pushforwards, establishing bounds on Frobenius height increase using prismatic F-gauges, with a self-contained approach.

Contribution

It provides a new bound on Frobenius height increase for pushforwards of prismatic F-crystals, utilizing prismatic F-gauges without stacky methods.

Findings

Frobenius height increases by at most i under pushforward.

Introduction of a self-contained proof avoiding stacky formulations.

Application of prismatic F-gauges in analyzing Frobenius operators.

Abstract

We study the behavior of Frobenius operators on smooth proper pushforwards of prismatic F-crystals. In particular we show that the i-th pushforward has its Frobenius height increased by at most i. Our proof crucially uses the notion of prismatic F-gauges introduced by Drinfeld and Bhatt--Lurie and its relative version, and we give a self-contained treatment without using the stacky formulation.