论文标题
估价树的刚性
Rigidity of valuative trees under henselization
论文作者
论文摘要
令$（k，v）$为一个有价值的字段，让$（k^h，v^h）$是通过选择向代数关闭$ k $的$ v $的扩展而确定的henselization。考虑一个值组的嵌入$ v（k^*）\hookrightArrowλ$ $hookrightArrowλ$中的一个可划分的订购的阿贝利安组。令$ t（k，λ）$，$ t（k^h，λ）$为所有$ v $ $ v $，$ v^h $的树木，分别为$ v $，$ v^h $，分别分别为polyenmial Rings $ k [x] $，$ k^h [x] $。我们表明，自然限制映射$ t（k^h，λ）\ to t（k，λ）$是posets的同构。 结果，限制映射$ t_v \ to t_ {v^h} $也是posets的同构性，其中$ t_v $，$ t_v $，$ t_ {v^h} $是其节点的树木是对等价的估值类别的$ k [x] $，$ k^h [$ k^h [$ k $ k $ k $ k^$ k^h [ 分别。
Let $(K,v)$ be a valued field and let $(K^h,v^h)$ be the henselization determined by the choice of an extension of $v$ to an algebraic closure of $K$. Consider an embedding $v(K^*)\hookrightarrowΛ$ of the value group into a divisible ordered abelian group. Let $T(K,Λ)$, $T(K^h,Λ)$ be the trees formed by all $Λ$-valued extensions of $v$, $v^h$ to the polynomial rings $K[x]$, $K^h[x]$, respectively. We show that the natural restriction mapping $T(K^h,Λ)\to T(K,Λ)$ is an isomorphism of posets. As a consequence, the restriction mapping $T_v\to T_{v^h}$ is an isomorphism of posets too, where $T_v$, $T_{v^h}$ are the trees whose nodes are the equivalence classes of valuations on $K[x]$, $K^h[x]$ whose restriction to $K$, $K^h$ are equivalent to $v$, $v^h$, respectively.
