论文标题
估值环和有价值场中的准季度模块
Quasi-quadratic modules in valuation ring and valued field
论文作者
论文摘要
这是以前版本的修订版，其新附录由特征性两种情况组成。我们在概括二次模块的概念的换向环中定义了准季度模块。主要定理是一个在$ 2 $ 2 $ -HENSELIAN REAGIEN的$（k，{\ bf val}）$的$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ f $ f protiant $ f $ \ neq 2 $的结构定理。我们进一步假设评估环$ b $包含在$ a $中。设置$ h = {\ bf val}（a^\ times）$和$ g _ {\ geq e} = \ {g \ in G \; | \;; | \; g \ geq e \} $。符号$ \ mathfrak x_r $表示通勤环$ r $中所有准季度模块的集合。我们的结构定理断言，存在$ \ mathfrak x_a $和一个子集$ \ Mathcal t_f^{我们明确构造映射$θ：\ Mathfrak x_a \ rightarrow \ Mathcal t_f^{h \ Cup g _ {\ geq e}} $及其倒数我们还提供$θ（\ Mathcal M \ CAP \ MATHCAL N）$和$θ（\ MATHCAL M+\ MATHCAL N）$的明确表达式，用于$ \ MATHCAL M，\ MATHCAL N \ in \ MATHFRAK x_A $。 此外，我们简要调查了该字段$ f $在附录中的特征两个。
This is a revised version of the previous version with a new appendix consisting of characteristic two case. We define quasi-quadratic modules in a commutative ring generalizing the notion of quadratic modules. The main theorem is a structure theorem of quasi-quadratic modules in a subring $A$ of a $2$-henselian valued field $(K,{\bf val})$ whose residue class field $F$ of characteristic $\neq 2$. We further assume that the valuation ring $B$ is contained in $A$. Set $H={\bf val}(A^\times)$ and $G_{\geq e}=\{g \in G\;|\; g \geq e\}$. The notation $\mathfrak X_R$ denotes the set of all the quasi-quadratic modules in a commutative ring $R$. Our structure theorem asserts that there exists a one-to-one correspondence between $\mathfrak X_A$ and a subset $\mathcal T_F^{ H \cup G_{\geq e}}$ of $\prod_{g \in H \cup G_{\geq e}}\mathfrak X_F$. We explicitly construct the map $Θ: \mathfrak X_A \rightarrow \mathcal T_F^{ H \cup G_{\geq e}}$ and its inverse. We also give explicit expressions of $Θ(\mathcal M \cap \mathcal N)$ and $Θ(\mathcal M+\mathcal N)$ for $\mathcal M, \mathcal N \in \mathfrak X_A$. In addition, we briefly investigate the case in which the field $F$ is of characteristic two in the appendix as well.
