二维Hensel域分数域上二次型的局部-整体原理
Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains
摘要：
设$R$是2维正规优良Hensel局部域，其中2是可逆的，$L$和$k$分别是它的分数域和剩余域.令$Ω_R$是$L$的秩1离散赋值的集合，其对应于$\Spec R$的正则真模型的余维1点。证明了$L$上的二次型$q$在以下两种情况下满足关于$Ω_R$的局部-整体原理：（1）$q$的秩为3或4;（2）$q$的秩为$\ge 5$和$R=A[y]$，其中$A$是一个完全离散赋值环，对剩余域$k$有一个不太严格的限制条件，当$k$是$C_1$时，这个限制条件满足.
英文摘要：
Let $R$ be a 2-dimensional normal excellent henselian local domain in which 2 is invertible and let $L$ and $k$ be respectively its fraction field and residue field. Let $Ω_R$ be the set of rank 1 discrete valuations of $L$ corresponding to codimension 1 points of regular proper models of $\Spec R$. We prove that a quadratic form $q$ over $L$ satisfies the local-global principle with respect to $Ω_R$ in the following two cases: (1) $q$ has rank 3 or 4; (2) $q$ has rank $\ge 5$ and $R=A[y]$, where $A$ is a complete discrete valuation ring with a not too restrictive condition on the residue field $k$, which is satisfied when $k$ is $C_1$.
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