Extensions of the Frobenius to ring of differential operators on polynomial algebra in prime characteristic

TL;DR

This paper characterizes automorphisms of the ring of differential operators on polynomial algebras in prime characteristic, showing they are uniquely determined by their action on variables and describing the structure of Frobenius extensions.

Contribution

It explicitly describes the set of Frobenius extensions of differential operators, proving their relation to automorphisms and providing a detailed parametrization.

Findings

Automorphisms are uniquely determined by their action on variables.

The set of Frobenius extensions equals the automorphism group times a specific set.

Frobenius extensions depend on countably many parameters.

Abstract

Let $K$ be a field of characteristic $p>0$. It is proved that each automorphism $\s \in \Aut_K(\CDPn)$ of the ring $\CDPn$ of differential operators on a polynomial algebra $P_n= K[x_1, ..., x_n]$ is {\em uniquely} determined by the elements $\s (x_1), ... ,\s (x_n)$, and the set $\Frob (\CDPn)$ of all the extensions of the Frobenius from certain maximal commutative polynomial subalgebras of $\CDPn$, like $P_n$, is equal to $\Aut_K(\CDPn) \cdot \CF$ where $\CF$ is the set of all the extensions of the Frobenius from $P_n$ to $\CDPn$ that leave invariant the subalgebra of scalar differential operators. The set $\CF$ is found explicitly, it is large (a typical extension depends on {\em countably} many independent parameters).