Timeline for The SGA1 version of Riemann's Existence Theorem is about analytic spaces. How does one relate it to topological covering spaces?

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Nov 28, 2015 at 21:27	comment	added	nfdc23		In the link you give, Corollary 2.5.2 is the local structure theorem for analytic maps near a point isolated in its fiber, the "Grauert-Remmert theorem" on finite morphisms alluded to in the Introduction, and it has been known for a very long time (well-documented in the book of G&R on coherent analytic sheaves, for example; look up their notion of "locally quasi-finite" morphism). The CM hypothesis there ensures flatness by Theorem 23.1 in Matsumura's book "Commutative Ring Theory". If you also require $x$ to be reduced in its fiber then it becomes what I called "inverse function theorem".
Nov 28, 2015 at 21:17	comment	added	nfdc23		What is your definition of "etale" in the complex-analytic setting? The usual definition is "local analytic isomorphism", and there is an "inverse function theorem" allowing arbitrary singularities: a map of complex-analytic spaces $X \rightarrow Y$ is etale near $x$ if it is flat at $x$ and $x$ is an isolated reduced point in $X_{f(x)}$ (for smooth $X$ and $Y$, in case of an isomorphism on tangent spaces we get an isomorphism of completed local rings). The main content in the proof is to show that if $x$ is isolated in its fiber then $f$ is finite between opens around $x$ and $f(x)$.
Nov 28, 2015 at 20:57	history	answered	Ben Webster♦	CC BY-SA 3.0