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I am reading Miln'es Etale Cohomology (1980 Princeton University Press). In page 46 to give some exampeles of $E$-morphisms he writes $E=(et)$ of all etale morphisms of finite-type. Next page he writes that $E$-morphisms are open and any open immersion is an $E$-morphism. enter image description here

Thus if I understand correctly, given any open immersion $U\to X$ it has to be an $E$-morphism. In this case $E$ is etale of finite-type. Therefore any open immersions needs to be of finite-type. But that is false.

Could you please tell me if my way of thinking is flawed?

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You are correct that there is an error here, and not every open immersion is etale of finite-type. But Milne is not saying that an open immersion has to be an $E$-morphism as a part of any definition; he is just observing that this property holds in these examples (except that he is wrong about one of them). For what it's worth, in most cases of interest the base scheme is Noetherian, and then it is true that all open immersions are finite type.

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  • $\begingroup$ He seems to use the fact that the etale site contains the open immersions in Proposition 1.5. I have seen lecture notes by other authors and they seem to drop the finite-type condition right away. Thus (et) for them is just etale of locally of finite type. I ask, therefore, is there any difference in behavior observed for the two (possibly) different sites for the cases of Noetherian Schemes(or finite-type over a field)? $\endgroup$ – Grobber Mar 7 '18 at 9:49
  • $\begingroup$ Hmmm, I think you're right that finiteness is usually not required. I think it doesn't make a difference though because every etale morphism can be covered by finite etale morphisms, and so the categories of sheaves you get are equivalent. I don't know this material very well though, so I might be wrong. $\endgroup$ – Eric Wofsey Mar 7 '18 at 17:18

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