论文标题
étale同喻理论中的基本纤维序列
The fundamental fiber sequence in étale homotopy theory
论文作者
论文摘要
令$ k $为具有可分离闭合$ \ bar {k} \ supset k $的字段，让$ x $为qcqs $ k $ -scheme。我们使用Barwick-Glasman-Haine开发的Profinite Galois类别的理论，提供了一个快速的概念证明，即序列\ BEGIN \ BEGIN {等式*}π_ {<\ infty}^{\ Mathrm {\ Mathrm {\ Acute {accute {E}可\wideHatπ{} _ {\ infty}^{\ mathrm {\ acute {e} \ mathrm {bgal}（\ bar {k}/k）\ End {equation*}的突出和profiniteÉtale同拷贝类型是光纤序列。这给出了以下两种现象的共同概念原因：首先，$ x $的较高典型同型组和几何纤维$ x _ {\ bar {k}} $是同构的，第二，如果$ x _ {\ bar {k bar {k}} $，则是相互连接的，然后是相互关联的，然后将$ 1 \ to \hatπ{} _ {1}^{\ mathrm {\ acute {e} t}}}}}}（x _ {\ bar {k}}）\ to \hatπ {} _ {1}^{\ mathrm {\ acute {e} t}}}}（x）\ to \ mathrm {gal}（\ bar {k}/k）\ to 1 $是准确的。这也证明了SGA3的“ groupe fondamentalélargi”的类似结果。
Let $k$ be a field with separable closure $\bar{k}\supset k$, and let $X$ be a qcqs $k$-scheme. We use the theory of profinite Galois categories developed by Barwick-Glasman-Haine to provide a quick conceptual proof that the sequences \begin{equation*} Π_{<\infty}^{\mathrm{\acute{e}t}}(X_{\bar{k}}) \to Π_{<\infty}^{\mathrm{\acute{e}t}}(X) \to \mathrm{BGal}(\bar{k}/k) \qquad \text{and} \qquad \widehatΠ{}_{\infty}^{\mathrm{\acute{e}t}}(X_{\bar{k}}) \to \widehatΠ{}_{\infty}^{\mathrm{\acute{e}t}}(X) \to \mathrm{BGal}(\bar{k}/k) \end{equation*} of protruncated and profinite étale homotopy types are fiber sequences. This gives a common conceptual reason for the following two phenomena: first, the higher étale homotopy groups of $X$ and the geometric fiber $X_{\bar{k}}$ are isomorphic, and second, if $X_{\bar{k}}$ is connected, then the sequence of profinite étale fundamental groups $1\to\hatπ{}_{1}^{\mathrm{\acute{e}t}}(X_{\bar{k}})\to\hatπ{}_{1}^{\mathrm{\acute{e}t}}(X)\to\mathrm{Gal}(\bar{k}/k)\to 1$ is exact. It also proves the analogous results for the `groupe fondamental élargi' of SGA3.
