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Let $U$ be a scheme. Let $\{\phi_i:U_i \rightarrow U\}_i$ be a covering (in any topology).

Standard references say this means (in particular) that

$$\bigcup_i \phi_i(U_i) = U$$

In most cases $\phi$ has some property so that $\phi_i(U_i)$ is an open subscheme, so that $\bigcup_i$ could mean colimit of locally ringed spaces.

Question:

  1. Is the above an equality of topological spaces or one of locally ringed spaces?

  2. For example, is the single map $Spec(\mathbb{Q}[i]) \rightarrow Spec(\mathbb{Q})$ a covering (in any topology) or not?

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  • $\begingroup$ 2. $\operatorname{Spec} \mathbb{Q}[i] \to \operatorname{Spec} \mathbb{Q}$ is a finite étale covering. In particular, it is a covering morphism in the étale topology. $\endgroup$ – Zhen Lin Sep 13 '15 at 20:43
  • $\begingroup$ 1. As sets. A morphism (or family of morphisms) of schemes is said to be surjective if it is surjective on the underlying sets. $\endgroup$ – anon Sep 13 '15 at 22:45
  • $\begingroup$ You should include the reference where this is written. Any scheme can be written as a colimit in the category of schemes or locally ringed spaces. $\endgroup$ – user40276 Sep 13 '15 at 23:50

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