关于某些值域的结构
On the structure of certain valued fields
摘要：
本文研究了一类非分歧混合特征值域的结构。对任意两个具有完全剩余域的混合特征的完备离散值域$K_1$和$K_2$，证明了若每个$n\ge 1$的$n$次剩余环同构，则$K_1$和$K_2$是等距同构的.更一般地，对于$n_1\ge 1$，存在仅取决于$K_1$和$K_2$的分歧指数的$n_2$，使得从$K_1$的第$n_1$个残基环到$K_2$的第$n_2$个残基环的任何同态可以提升为赋值环之间的同态。此外，我们还得到了从某些长为$n$的主Artin局部环范畴到某些具有完全剩余域的混合特征的完备离散赋值环范畴的函子，它自然地推广了非分歧完备离散赋值环的函子性质.我们的提升结果改进了Basarab关于完全分歧的Hensel值域的相对完备性定理，解决了Basarab在完全剩余域情况下提出的一个问题。
英文摘要：
In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the $n$-th residue rings are isomorphic for each $n\ge 1$, then $K_1$ and $K_2$ are isometric and isomorphic. More generally, for $n_1\ge 1$, there is $n_2$ depending only on the ramification indices of $K_1$ and $K_2$ such that any homomorphism from the $n_1$-th residue ring of $K_1$ to the $n_2$-th residue ring of $K_2$ can be lifted to a homomorphism between the valuation rings. Moreover, we get a functor from the category of certain principal Artinian local rings of length $n$ to the category of certain complete discrete valuation rings of mixed characteristic with perfect residue fields, which naturally generalizes the functorial property of unramified complete discrete valuation rings. Our lifting result improves Basarab's relative completeness theorem for finitely ramified henselian valued fields, which solves a question posed by Basarab, in the case of perfect residue fields.
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