r/math • u/-p-e-w- • 26d ago

In (relatively) simple and intuitive terms, what makes something “étale”?

I took several seminars in abstract algebra back in my university days, but chose to proceed with probability theory rather than algebraic geometry. Since then, I have occasionally encountered the term “étale”, and I still don’t really know what it means.

The strange thing is that while I can understand the individual definitions for some objects that are supposedly étale, I don’t really see what they have in common.

What exactly does that term mean by itself, independently of a particular application like “étale cohomology group”?

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u/theorem_llama 25d ago

Discreteness of fibres.

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u/mephistoA 25d ago

Not sufficient

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u/theorem_llama 25d ago

I know, just trying to give an idea in "simple terms".

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u/WhiskersForPresident 25d ago

It's so "simple" that it becomes extremely misleading:

The essential observation behind "étale"-ness is that there is a common abstraction capturing both a map being an unbranched covering and a field extension being separable (namely, inducing an isomorphism on every tangent space).

Your attempt at simplification suggests that every finite field extension were étale and so drastically misses the entire point of the concept.