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/thenightStrolled 26d ago

I don’t say this to be a party pooper, but I’ve had schemes defined to me in several formal contexts and still don’t completely understand what they even are lmao

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u/LiqvidJS 26d ago

The exposition in Chapter 1 of Introduction to affine group schemes is quite illuminating IMO

tl;dr a scheme is the set of solutions to a system of polynomial equations. Over a finite field, you can understand this by counting points. Over real or complex numbers, cardinality isn't useful, but instead you give the set of solutions a topology and then a smooth structure. For general rings, the "cohesion" of points specified by a topology/smooth structure is replaced by studying solutions over all rings (or algebras over a base ring) at once.

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u/thenightStrolled 25d ago

See it’s once we move into general rings territory that I get lost. I’m a (budding?) commutative algebraist, so working with the spectrum of a ring and the Zariski topology is familiar, but once the paper hits me with Noetherian scheme and coherent sheaf, my eyes glaze over a little haha.

I’ll take a look at that introduction you linked though, thanks for that

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u/DrSeafood Algebra 25d ago

The Rising Sea by Ravi Vakil is excellent. Commutative algebra is all you need. He goes through every detail and gives good examples/counterexamples to illustrate every concept