Isomorphism problems and groups of automorphisms for Ore extensions $K[x][y; f\frac{d}{dx} ]$ (prime characteristic)

TL;DR

This paper explicitly describes the automorphism groups of Ore extensions of polynomial algebras over fields of prime characteristic, classifies their ideals, and analyzes their algebraic dimensions.

Contribution

It provides an explicit description of automorphism groups and eigengroups for Ore extensions in prime characteristic, and classifies their ideals and dimensions.

Findings

Automorphism group is a semidirect product of two explicit groups.

Every subgroup of Aut_K(K[x]) is an eigengroup of some polynomial.

Krull and global dimensions of the algebra are both 2.

Abstract

Let $\Lambda (f) = K[x][y; f\frac{d}{dx} ]$ be an Ore extension of a polynomial algebra $K[x]$ over an arbitrary field $K$ of characteristic $p>0$ where $f\in K[x]$. For each polynomial $f$, the automorphism group of the algebras $\Lambda (f)$ is explicitly described. The automorphism group ${\rm Aut}_K(\L (f))=S\rtimes G_f$ is a semidirect product of two explicit groups where $G_f$ is the {\em eigengroup} of the polynomial $f$ (the set of all automorphisms of $K[x]$ such that $f$ is their common eigenvector). For each polynomial $f$, the eigengroup $G_f$ is explicitly described. It is proven that every subgroup of ${\rm Aut}_K(K[x])$ is the eigengroup of a polynomial. It is proven that the Krull and global dimensions of the algebra $\Lambda (f)$ are 2. The prime, completely prime, primitive and maximal ideals of the algebra $\Lambda (f)$ are classified.