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In Topology, by Definition, when we have a compact space $X$ and an open cover $(U_i)_{i\in I}$, there exists a finite subcover $U_1,\ldots, U_n$ which covers $X$.

Is there an analogous definition for general Grothendieck topologies? For instance, does there exist the definition of properness in the étale topology, by which I mean that if $(U_i \rightarrow X)_i$ is an étale cover of $X$, there exists a finite subset of étale open $(U_1\rightarrow X), \ldots, (U_n \rightarrow X)$?

Furthermore, if $k$ is a field and $X$ a proper $k$-scheme, is it proper in the étale topology?

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For the general statement, it sounds like what you are looking for is a coherent topos. I'm not an expert on this, so there may be better terminology suggestions or references from people who are.

For the specific question of when an etale covering of a scheme has a finite subcover, it suffices for $X$ to be quasicompact. SGA4, section VIII (available here at MSRI's website) or Stacks 021E explains that the etale topos is equivalent to the topos of sheaves on the site obtained from the full subcategory of affine finitely presented etale schemes over $X$. As every affine morphism is quasicompact, we see that each such cover has a finite subcover.

For the more specific case of $X$ proper over a field, since proper morphisms are universally closed and universally closed morphisms are quasicompact, we see that the previous paragraph applies and any etale covering should have a finite subcover.

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  • $\begingroup$ In your second paragraph, I think you need that $X$ is also quasi-separated to have the equivalence you claim (SGA 4, VII, 3.1 & 3.2). $\endgroup$
    – user187567
    Aug 19, 2019 at 8:00
  • $\begingroup$ This restriction can be removed - the equivalence between the relevant topoi is completely general for all schemes, as proven in Stacks 021E. If I have the time later, I'll try and find an explanation from SGA, but I make no guarantees (I am fairly slow at reading French). $\endgroup$
    – KReiser
    Aug 19, 2019 at 21:49

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