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I always had the intuition that on a complex projective variety, the étale topology can be thought as the complex topology of the analytification, so every closed point has a "small" neighbourhood.

However, I do not see how something like $B_\epsilon(0)$ for the projective line can be represented by an étale map $U \to \mathbb{P}^1$ as I do not think an epsilon ball can be given a scheme-structure. Could somebody explain to me how this can be constructed as an étale neighbourhood or if this is not possible, what kind of neighbourhoods, which do not come from open immersions, are added?

Best,

Matthias

Matthias
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You're maybe not quite on the right track here. It's not the case that one literally adds new open sets (so you can't represent a small ball like you're asking about), but one considers generalizations of "open coverings".

For a Zariski open covering of a scheme $X$, we mean a map $\prod U_i\to X$ where each $U_i\to X$ is an open immersion. For an etale covering, we instead ask for each $U_i\to X$ to be etale - one should think of this as "analytically locally an isomorphism". For an example of an etale covering which is not a Zariski covering, consider the squaring map on $\Bbb A^1_k\setminus 0$ for $k$ a field not of characteristic two: this is etale, but it's not an open immersion because it's not an isomorphism.

One slogan which is believable and can lend some credence to your impression that every point has a "small" neighborhood is that every smooth variety is etale-locally like affine space. This is explained by Zhen Lin here. So in this sense, one can think of each point in a smooth variety as having an etale neighborhood like $\Bbb A^n$.

KReiser
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  • So is it true that an for an étale cover, which is not a Zariski cover, there is always a point that has more than one preimage? – Matthias Feb 08 '22 at 21:58
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    @Matthias Appropriately construed, yes - it's not always appropriate to count the literal number of points, because something like the spectrum of the natural inclusion $\Bbb Q\subset\Bbb Q(i)$ is an etale map between two one-point spaces which is not an isomorphism. Instead one should count geometric points or the degree of the fiber, and an etale morphism with all fibers of degree one is actually an open immersion. – KReiser Feb 08 '22 at 22:31