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When discussing presheaves on a topological space, we define an étale map as one which is locally a homeomorphism (see étalé space). However in other places (see Milne’s notes on etale cohomology or etale morphism of schemes), an etale map between schemes is defined as one which is an isomorphism on tangent spaces (or tangent cones, or is smooth and unramified, apparently equivalent conditions). In the smooth category, the two definitions are equivalent by the inverse function theorem. However in the category of schemes or varieties, etale in the second sense (local homeo) is strictly weaker ($x\mapsto x^n$ is an iso of tangent spaces but never a local homeomorphism).
What is the general definition? One expects the weaker one (tangent iso) to apply more generally, and the stronger one (local homeo) to imply the weaker one, however it’s not clear that this even makes sense generally, for example in topological spaces where we do not have access to a tangent space.
Also, is this word pronounced with a silent ’e’ at the end [e:tal] as I’ve heard it in conversation, or does the accent aigu on the final ’e’ found sometimes, but not always, indicate another vowel [etal:e]?
There are two words here: ‘étalé’ and ‘étale’. The topological space associated with a (pre)sheaf is called ‘espace étalé’, and it is probably on this basis that a local homeomorphism is (perhaps wrongly) called ‘étale’. On the other hand, étale morphisms of schemes really are analogous to local isomorphisms (which can be made tautologically precise in the étale topology).
Technically ’étalé’ is a past participle, sort of more-or-less’smoothed’ or ’spread-out’, whilst ’étale’ is an adjective. The final ‘e’ is silent, but the ’é’ comes across as an ’eh’ sound, so perhaps something like ’aye-tal-eh’, if you see what I mean.
@zhen lin: so all the definitions are equivalent, if you use the etale topology instead of the Zariski topology on your schemes? Ok I see, thank you.
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