arXiv 1406.09492014-06math.NTmath.GR

代数学再访

Algebraic tori revisited

Ming-Chang Kang

备注：: To appear in Asian J. Math. ; the title is changed

摘要：

设$K/k$是有限Galois扩张，$π= \fn{Gal}(K/k)$ .定义在$k$上的代数环面$T$称为$π$ -环面，如果对于某个整数$n$有$T\times_{\fn{Spec}(k)} \fn{Spec}(K)\simeq \bm{G}_{m,K}^n$.在稳定同构下定义在$k$上的所有代数$π$ -环面的集合形成一个半群，记为$T(π)$。我们将给出Endo和Miyata \cite{EM5}的以下定理的完整证明。定理设$π$是一个有限群。然后是$T(π)\simeq C(Ω_{\bm{Z}π})$，其中$Ω_{\bm{Z}π}$是包含$\bm{Z}π$的$\bm{Q}π$中的最大$\bm{Z}$ -阶，$C(Ω_{\bm{Z}π})$是$Ω_{\bm{Z}π}$的局部自由类群，前提是$π$同构于以下四种类型的群：$C_n$（$n$是任何正整数），$D_m$（$m$是任何奇数$\ge 3$），$C_{q^f}\times D_m$（$m$是任何奇整数$\ge 3$，$q$是不整除$m$、$f\ge 1$和$(\bm{Z}/q^f\bm{Z})^{\times}=\langle \bar{p}\rangle$的奇素数，对于$m$的任何素数$p$），$Q_{4m}$（$m$是任何奇数$\ge 3$，$p\equiv 3 \pmod{4}$是$m$的任何素因子$p$）。

英文摘要：

Let $K/k$ be a finite Galois extension and $π= \fn{Gal}(K/k)$. An algebraic torus $T$ defined over $k$ is called a $π$-torus if $T\times_{\fn{Spec}(k)} \fn{Spec}(K)\simeq \bm{G}_{m,K}^n$ for some integer $n$. The set of all algebraic $π$-tori defined over $k$ under the stably isomorphism form a semigroup, denoted by $T(π)$. We will give a complete proof of the following theorem due to Endo and Miyata \cite{EM5}. Theorem. Let $π$ be a finite group. Then $T(π)\simeq C(Ω_{\bm{Z}π})$ where $Ω_{\bm{Z}π}$ is a maximal $\bm{Z}$-order in $\bm{Q}π$ containing $\bm{Z}π$ and $C(Ω_{\bm{Z}π})$ is the locally free class group of $Ω_{\bm{Z}π}$, provided that $π$ is isomorphic to the following four types of groups : $C_n$ ($n$ is any positive integer), $D_m$ ($m$ is any odd integer $\ge 3$), $C_{q^f}\times D_m$ ($m$ is any odd integer $\ge 3$, $q$ is an odd prime number not dividing $m$, $f\ge 1$, and $(\bm{Z}/q^f\bm{Z})^{\times}=\langle \bar{p}\rangle$ for any prime divisor $p$ of $m$), $Q_{4m}$ ($m$ is any odd integer $\ge 3$, $p\equiv 3 \pmod{4}$ for any prime divisor $p$ of $m$).

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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