论文标题
étale开放拓扑何时是现场拓扑？
When is the étale open topology a field topology?
论文作者
论文摘要
我们调查以下问题：给定一个字段$ k $，什么时候étale开放拓扑$ \ MATHCAL {E} _K $由现场拓扑引起？从积极的一面来看，当$ k $是本地域$ r \ neq k $的分数时，使用薄弱的奇异性，我们表明$ \ mathcal {e} _k $与$ r $ - adic-r $ - adic topology一致。删除准脱位假设时会出现各种病理。对于本地界面拓扑，我们介绍了Prestel和Ziegler之后的广义T-Henselianity（GT-Henselianity）的概念。我们建立以下内容：对于本地界面拓扑$τ$，étale开放拓扑是由$τ$引起的，并且仅当$τ$是GT-HENSELIAN，并且某些非空的oftery of the the the topology是$τ$的，并且某些非空的of of the offecty offer-the the togology且一些非空的of the the the the tobology upenty obledy of the togology。负面的一面，我们可以获得伪用封闭的封闭字段$ k $，$ \ mathcal {e} _k $从未通过现场拓扑引起的。
We investigate the following question: Given a field $K$, when is the étale open topology $\mathcal{E}_K$ induced by a field topology? On the positive side, when $K$ is the fraction field of a local domain $R\neq K$, using a weak form of resolution of singularities due to Gabber, we show that $\mathcal{E}_K$ agrees with the $R$-adic topology when $R$ is quasi-excellent and henselian. Various pathologies appear when dropping the quasi-excellence assumption. For locally bounded field topologies, we introduce the notion of generalized t-henselianity (gt-henselianity) following Prestel and Ziegler. We establish the following: For a locally bounded field topology $τ$, the étale open topology is induced by $τ$ if and only if $τ$ is gt-henselian and some non-empty étale image is $τ$-bounded open. On the negative side, we obtain that for a pseudo-algebraically closed field $K$, $\mathcal{E}_K$ is never induced by a field topology.
