On exponentiality of automorphisms of ${\bf A}^n$ of order $p$ in characteristic $p>0$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917741259128832 |
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| author | Kuroda, Shigeru |
| author_facet | Kuroda, Shigeru |
| contents | Let $X$ be an integral affine scheme of characteristic $p>0$, and $σ$ a non-identity automorphism of $X$. If $σ$ is $\textit{exponential}$, i.e., induced from a ${\bf G}_a$-action on $X$, then $σ$ 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 ${\bf G}_a$-actions. However, the situation is not clear in the case where $X$ is the affine space ${\bf A}_R^n$, because ${\bf A}_R^n$ admits various ${\bf G}_a$-actions as well as automorphisms of order $p$.
In this paper, we study exponentiality of automorphisms of ${\bf A}_R^n$ of order $p$, where the difficulty stems from the non-uniqueness of ${\bf G}_a$-actions inducing an exponential automorphism. Our main results are as follows.
(1) We show that the triangular automorphisms of ${\bf A}_R^n$ of order $p$ are exponential in some low-dimensional cases.
(2) We construct a non-exponential automorphism of ${\bf A}_R^n$ of order $p$ for each $n\ge 2$. Here, $R$ is any UFD which is not a field.
(3) We investigate the ${\bf G}_a$-actions inducing an elementary automorphism of ${\bf A}_R^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02204 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On exponentiality of automorphisms of ${\bf A}^n$ of order $p$ in characteristic $p>0$ Kuroda, Shigeru Algebraic Geometry Commutative Algebra Primary 14R10, Secondary 14R20, 13A50 Let $X$ be an integral affine scheme of characteristic $p>0$, and $σ$ a non-identity automorphism of $X$. If $σ$ is $\textit{exponential}$, i.e., induced from a ${\bf G}_a$-action on $X$, then $σ$ 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 ${\bf G}_a$-actions. However, the situation is not clear in the case where $X$ is the affine space ${\bf A}_R^n$, because ${\bf A}_R^n$ admits various ${\bf G}_a$-actions as well as automorphisms of order $p$. In this paper, we study exponentiality of automorphisms of ${\bf A}_R^n$ of order $p$, where the difficulty stems from the non-uniqueness of ${\bf G}_a$-actions inducing an exponential automorphism. Our main results are as follows. (1) We show that the triangular automorphisms of ${\bf A}_R^n$ of order $p$ are exponential in some low-dimensional cases. (2) We construct a non-exponential automorphism of ${\bf A}_R^n$ of order $p$ for each $n\ge 2$. Here, $R$ is any UFD which is not a field. (3) We investigate the ${\bf G}_a$-actions inducing an elementary automorphism of ${\bf A}_R^n$. |
| title | On exponentiality of automorphisms of ${\bf A}^n$ of order $p$ in characteristic $p>0$ |
| topic | Algebraic Geometry Commutative Algebra Primary 14R10, Secondary 14R20, 13A50 |
| url | https://arxiv.org/abs/2408.02204 |