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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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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. –  mephisto Apr 16 '11 at 22:06
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.) –  Donu Arapura Apr 16 '11 at 22:19
It seems our comments crossed. –  Donu Arapura Apr 16 '11 at 22:19
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? –  Mariano Suárez-Alvarez Apr 17 '11 at 3:37
Thanks, everyone. –  user14449 Apr 17 '11 at 19:49

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