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I'm reading Milne's Lecture notes on étale cohomology to get an understanding of the étale fundamental group (my undergrad thesis is looking at étale fundamental groups of elliptic curves).

On page 19, Milne defines an étale map for varieties over arbitrary fields:

Let $\phi: W\rightarrow V$ be a regular map of varieties over a field $k$. We say that $\phi$ is étale at $w\in W$ if for some algebraic closure $k^\text{al}$ of $k$, $\phi_{k^\text{al}} : W_{k^\text{al}} \rightarrow V_{k^\text{al}}$ is étale at the points of $W_{k^\text{al}}$ mapping to $w$.

Here the notion of a morphism being étale over an algebraically closed field has already been defined - for a smooth point of the domain (which is all I need with elliptic curves) a morphism is étale at a point if the induced linear map on tangent spaces is an isomorphism.

I can't find any reference to what these subscripted $\phi_{k^\text{al}}$,$W_{k^\text{al}}$ and $V_{k^\text{al}}$ are. I'd assume they are the varieties defined over $k^\text{al}$ by the same equations as the original varieties - is this right?

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Yes, that's right. More abstractly, they are the "base change" or "pullback" of all these objects along the map from the spectrum of the algebraic closure to the spectrum of the field.

General advice with Milne is that if something is unclear, he almost certainly defined it in an earlier text. So keep his algebraic geometry, and probably also commutative algebra, notes on hand.

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    $\begingroup$ Thanks for the advice about reading Milne - I'll have a look at his other notes next time! $\endgroup$ – Alex Saad Dec 31 '15 at 17:20

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