On exponentiality of automorphisms of ${\mathbf{A}}^n$ of order $p$ in characteristic $p>0$

Abstract

Let $X$ be an integral affine scheme of characteristic $p>0$, and $\sigma$ a non-identity automorphism of $X$. If $\sigma$ is $\text{exponential}$, i.e., induced from a ${\mathbf{G}}_a$-action on $X$, then $\sigma$ is obviously of order $p$. It is easy to see that the converse is not true in general. In fact, there exists $X$ which admits an automorphism of order $p$, but admits no non-trivial ${\mathbf{G}}_a$-actions. However, the situation is not clear in the case where $X$ is the affine space ${\mathbf{A}}_R^n$, because ${\mathbf{A}}_R^n$ admits various ${\mathbf{G}}_a$-actions as well as automorphisms of order $p$. In this paper, we study exponentiality of automorphisms of ${\mathbf{A}}_R^n$ of order $p$, where the difficulty stems from the non-uniqueness of ${\mathbf{G}}_a$-actions inducing an exponential automorphism. Our main results are as follows. (1) We show that the triangular automorphisms of ${\mathbf{A}}_R^n$ of order $p$ are exponential in some low-dimensional cases. (2) We construct a non-exponential automorphism of ${\mathbf{A}}_R^n$ of order $p$ for each $n\ge 2$. Here, $R$ is any UFD which is not a field. (3) We investigate the ${\mathbf{G}}_a$-actions inducing an elementary automorphism of ${\mathbf{A}}_R^n$.