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It seems to me that if maps in the etale topology are only required to be locally of finite presentation then contrary to the claim in the article an etale site is not equivalent to a small one. A cover of a scheme by the disjoint union of any number of copies of it is etale by this definition, and there are proper class many different isomorphism class of these uninteresting etale covers. Milne requires the maps in an etale site to be of finite type, so in his version it would be true to say etale sites are proper classes only because each isomorphism class of covers is that big. Am I missing something?Colin McLarty (talk) 06:24, 31 December 2010 (UTC)
- Yup, looks wrong to me too. SGA IV2, p. 343 says that, if we are working with schemes in a universe U, then the etale site is a U-site. That's not at all the same as being U-small or equivalent to a U-small category, though. So I've removed the claim from the article. Ozob (talk) 20:02, 31 December 2010 (UTC)
Todo edit
There are many things this page needs.
- examples of sheaves which are etale but not zariksi
- discuss constructible sheaves for the etale topology/ give examples