It is known that the little étale topos of a scheme $X$ is coherent, i.e. the Grothendieck topology of the étale site is generated by a basis of finite coverings. For example, Butz and Moerdijk say that in the introduction to Representing topoi by topological groupoids.
I am trying to prove why: if the scheme is quasicompact, every element of the Grothendieck pretopology on the étale site, namely a covering of $X$ made by étale schemes, is an open cover, hence admits a finite subcovering.
But in general? Why is the theorem true if $X$ is not quasicompact?
Best Answer
Solved! SGA 4, VIII, explains that the étale topos is equivalent to the topos of sheaves on a smaller site, namely the one obtained as the full subcategory of affine finitely presented étale schemes over $X$. Now affine implies quasicompact, so every covering has a finite subcovering.