On Exponentiality of Automorphisms of $${\textbf {A}}^{n}$$ of Order p in Characteristic $$p>0$$

Abstract Let X be an integral affine scheme of characteristic $$p>0$$ p > 0 , and $$\sigma $$ σ a non-identity automorphism of X . If $$\sigma $$ σ is exponential , i.e., induced from a $$\textbf{G}_a$$ G a -action on X , then $$\sigma $$ σ 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 $$\textbf{G}_a$$ G a -actions. However, the situation is not clear in the case where X is the affine space $$\textbf{A}_R^n$$ A R n , because $$\textbf{A}_R^n$$ A R n admits various $$\textbf{G}_a$$ G a -actions as well as automorphisms of order p . In this paper, we study exponentiality of automorphisms of $$\textbf{A}_R^n$$ A R n of order p , where the difficulty stems from the non-uniqueness of $$\textbf{G}_a$$ G a -actions inducing an exponential automorphism. Our main results are as follows. (1) We show that the triangular automorphisms of $$\textbf{A}_R^n$$ A R n of order p are exponential in some low-dimensional cases. (2) We construct a non-exponential automorphism of $$\textbf{A}_R^n$$ A R n of order p for each $$n\ge 2$$ n ≥ 2 . Here, R is any UFD which is not a field. (3) We investigate the $$\textbf{G}_a$$ G a