I recently read that Grothendieck originally introduced the etale site of a scheme as an analog of the formation of Riemann surfaces over the complex numbers (the salient point being that the latter is the proper setting in which to consider multi-valued functions). However, morphisms of Riemann surfaces generally have ramification points, and so in particular they aren't etale. On the other hand, if we know that a morphism of Riemann surfaces is a ramified cover, then its restriction to the unramified locus uniquely determines the full ramified covering. So, my questions are:
- Should this additional fact be taken as part of the analogy? That is, does "importing the theory of Riemann surfaces" really mean "importing the theory of unramified maps of (possibly punctured) Riemann surfaces"?
- Is the analogous fact true for schemes (i.e., something along the lines of "an etale cover of the complement of a closed set [perhaps satisfying certain properties$^*$] extends uniquely to a ramified covering")?
Of course, I suppose a positive answer to #2 would make #1 irrelevant...
$^*$ for Riemann surfaces, I think this should just be that the covering has finitely many sheets