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In Milne's EC, he defines an etale neighbourhood of a point $x \in X$ by a pair $(Y,y)$ with an etale morphism $f: Y \to X$ where $f(y) = x$, such that $k(x) = k(y)$.

What I don't understand is this last condition. Is there an intuitive reason why this condition is necessary? Explanations are welcome, although examples would be more appreciated.

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    $\begingroup$ The condition is needed so that when you take the direct limit of the rings $\mathcal{O}(Y)$ over the etale nhds $Y$ of $x$ you get the henselization of the local ring $\mathcal{O}_x$. A more natural notion is the etale nhd of a geometric point, where the direct limit gives the strict henselization. $\endgroup$
    – mephisto
    Commented Apr 16, 2011 at 22:06
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    $\begingroup$ If you look further down the page (p. 38), you'll see that both versions, with and without the condition $k(x)=k(y)$, are considered. If you include it and take the direct limit of regular functions over all etale neighbourhoods you get the Henselization $\mathcal{O}_x^h$. If you do this but drop the condition $k(x)=k(y)$, you get the strict Henselization. (By the way, you want to consider a different alias.) $\endgroup$
    – Donu Arapura
    Commented Apr 16, 2011 at 22:19
  • $\begingroup$ It seems our comments crossed. $\endgroup$
    – Donu Arapura
    Commented Apr 16, 2011 at 22:19
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    $\begingroup$ While you may well be stupid, your user name makes it a bit hard to talk to you seriously... Could you consider something more neutral? $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Apr 17, 2011 at 3:37

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