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arXiv 2202.00262math.ACmath.AG
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特征阶多项式自同构及其不变环

Polynomial automorphisms of characteristic order and their invariant rings

Shigeru Kuroda

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摘要：

令$k$是特征$p>0$的域。本文讨论了仿射空间${\bf A}_k^n$上的$p$阶多项式环$k[x_1,\ldots ,x_n]$的自同构，或等价于${\bf Z}/p{\bf Z}$ -作用.当$n=2$，这样的自同构被认为是固定变量的自同构的共轭。当使用$n\ge 3$时，是否同样成立，这是一个悬而未决的问题。本文（1）给出了$n=3$时这个问题的第一个反例。事实上，我们表明，对排名第三的${\bf A}_k^3$的每个${\bf G}_a$ -动作都会产生$n=3$的反例。通过构造${\bf A}_k^3$上秩为3的${\bf G}_a$ -作用族，给出了一族反例. (2)对于这类${\bf G}_a$ -作用所诱导的自同构，在一些温和的假设下，我们证明了不变环同构于$k[x_1,x_2,x_3]$当且仅当基座理想是主理想. (3)研究了$R[x_1,x_2]$的Nagata型自同构，其中$R$是特征$p>0$的UFD.这种类型的自同构是$p$阶的。给出了不变环同构于$R[x_1,x_2]$的一个充要条件。这个条件等价于基座理想为主的条件。

英文摘要：

Let $k$ be a field of characteristic $p>0$. We discuss the automorphisms of the polynomial ring $k[x_1,\ldots ,x_n]$ of order $p$, or equivalently the ${\bf Z}/p{\bf Z}$-actions on the affine space ${\bf A}_k^n$. When $n=2$, such an automorphism is know to be a conjugate of an automorphism fixing a variable. It is an open question whether the same holds when $n\ge 3$. In this paper, (1) we give the first counterexample to this question when $n=3$. In fact, we show that every ${\bf G}_a$-action on ${\bf A}_k^3$ of rank three yields counterexamples for $n=3$. We give a family of counterexamples by constructing a family of rank three ${\bf G}_a$-actions on ${\bf A}_k^3$. (2) For the automorphisms induced by this family of ${\bf G}_a$-actions, we show that the invariant ring is isomorphic to $k[x_1,x_2,x_3]$ if and only if the plinth ideal is principal, under some mild assumptions. (3) We study the Nagata type automorphisms of $R[x_1,x_2]$, where $R$ is a UFD of characteristic $p>0$. This type of automorphisms are of order $p$. We give a necessary and sufficient condition for the invariant ring to be isomorphic to $R[x_1,x_2]$. This condition is equivalent to the condition that the plinth ideal is principal.

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