# Lift of Frobenius and Descent to Constants

## TL;DR

This paper extends Buium's differential algebra results to the arithmetic setting by demonstrating how schemes with a lift of Frobenius descend to constants using Witt vectors and arithmetic Taylor expansions.

## Contribution

It establishes an arithmetic analogue of Buium's descent theorem, replacing derivations with lifts of Frobenius in equal characteristic.

## Key findings

- Schemes with a lift of Frobenius descend to constants in the arithmetic setting.
- Uses Witt vectors to develop an arithmetic Taylor expansion.
- Provides an analogous result to differential algebra in the context of arithmetic geometry.

## Abstract

In differential algebra, a proper scheme $X$ defined over an algebraically closed field $K$ with a derivation $\partial$ on it descends to the field of constants $K^\partial$ if $X$ itself lifts the derivation $\partial$. This is a result by A. Buium. Now in the arithmetic case, the notion of a derivation is replaced by the notion of a $\pi$-derivation $\delta$ or equivalently in the flat case, a lift of Frobenius $\phi$. We will show an analogous result in the arithmetic case of equal characteristic. We show our results using the arithmetic analogue of Taylor expansion using Witt vectors.