# On the behavior of modules of $m$-integrable derivations in the sense of Hasse-Schmidt under base change

## TL;DR

This paper investigates how modules of $m$-integrable derivations, as defined by Hasse-Schmidt, behave under base change, especially in the context of separable extensions and polynomial rings over fields of positive characteristic.

## Contribution

It provides new insights into the behavior of $m$-integrable derivations under base change in specific algebraic settings, including separable extensions and polynomial rings.

## Key findings

- Modules of $m$-integrable derivations exhibit specific behaviors under base change.
- Results apply to separable extensions over fields of positive characteristic.
- Analysis includes polynomial rings in multiple variables.

## Abstract

We study the behavior of modules of $m$-integrable derivations of a commutative finitely generated algebra in the sense of Hasse-Schmidt under base change. We focus on the case of separable ring extensions over a field of positive characteristic and on the case where the extension is a polynomial ring in an arbitrary number of variables.