Differentiably simple rings and ring extensions defined by $p$-basis

TL;DR

This paper reviews differentiably simple rings, provides a new proof of Harper's theorem, and introduces the Yuan scheme to parametrize Galois subextensions, establishing its smoothness and fiber dimensions.

Contribution

It offers a new proof of Harper's theorem and introduces the Yuan scheme for Galois subextensions, advancing understanding of differentiably simple rings and their extensions.

Findings

Proof of Harper's theorem on differentiably simple rings.

Introduction of the Yuan scheme for Galois subextensions.

Proof that the Yuan scheme is smooth with computed fiber dimensions.

Abstract

We review the concept of differentiably simple ring and we give a new proof of Harper's Theorem on the characterization of Noetherian differentiably simple rings in positive characteristic. We then study flat families of differentiably simple rings, or equivalently, finite flat extensions of rings which locally admit $p$-basis. These extensions are called "Galois extensions of exponent one". For such an extension $A\subset C$, we introduce an $A$-scheme, called the "Yuan scheme", which parametrizes subextensions $A\subset B\subset C$ such that $B\subset C$ is Galois of a fixed rank. So, roughly, the Yuan scheme can be thought of as a kind of Grassmannian of Galois subextensions. We finally prove that the Yuan scheme is smooth and compute the dimension of the fibers.