AI 中文总结
针对特征p>0的F有限域上函数域的赋值环，证明其Frobenius分裂等价于对应赋值为Abhyankar赋值，还构造反例说明基域非F有限时该等价性不成立，方法不依赖局部一致化结果。
AI 中文摘要
设K是特征p>0的F有限域k上的函数域，ν是K/k的一个赋值。我们证明：ν是K/k的Abhyankar赋值当且仅当对应的赋值环是Frobenius分裂的。该证明通过分析赋值环的理想何时是一致F兼容的，并建立F有限域的Frobenius分裂赋值环的一般分歧理论刻画来完成。此外，当基域k不是F有限时，我们通过构造一个函数域的优秀Frobenius分裂离散赋值环（DVR）的例子，说明Frobenius分裂与Abhyankar性质之间的等价性可能不成立，该例子对应的赋值不是除子型的。我们的方法不依赖于局部一致化结果。
英文摘要
Let $K$ be a function field over an $F$-finite field $k$ of characteristic $p > 0$ and let $ν$ be a valuation of $K/k$. We show $ν$ is an Abhyankar valuation of $K/k$ if and only if the corresponding valuation ring is Frobenius split. The proof proceeds by analyzing when ideals of a valuation ring are uniformly $F$-compatible and establishing a general ramification-theoretic characterization of Frobenius split valuation rings of $F$-finite fields. In addition, when the ground field $k$ is not $F$-finite, we show that the equivalence between Frobenius splitting and the Abhyankar property can fail by constructing an example of an excellent Frobenius split DVR of a function field whose corresponding valuation is not divisorial. Our methods do not rely on local uniformization results.
Comments17 pages, comments welcome!
