$K(π, 1)$ -邻域与比较定理
$K(π, 1)$-neighborhoods and comparison theorems
摘要：
在Faltings最初的p-adic比较定理方法中的一个技术成分涉及到在具有完美剩余域的混合特征dvr上构造光滑概形X的$K(π, 1)$ -邻域：X的每个点都有一个开邻域，其一般纤维是$K(π, 1)$方案（类似于具有可收缩的泛覆盖的概念）。我们展示了如何将这一结果推广到几何光滑的情况，这可能有助于简化p-adic Hodge理论中的一些证明。主要成分的证明是一个变种的技巧永田用在他的证明诺特规范化引理。
英文摘要：
A technical ingredient in Faltings' original approach to p-adic comparison theorems involves the construction of $K(π, 1)$-neighborhoods for a smooth scheme X over a mixed characteristic dvr with a perfect residue field: every point of X has an open neighborhood whose general fiber is a $K(π, 1)$ scheme (a notion analogous to having a contractible universal cover). We show how to extend this result to the logarithmically smooth case, which might help to simplify some proofs in p-adic Hodge theory. The main ingredient of the proof is a variant of a trick of Nagata used in his proof of the Noether Normalization Lemma.
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