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I believe I've encountered the statement below, but I've lost my reference and am unable to find another one. So, I'm posting this question to see if someone can give a reference, or at least confirm the statement is true (or make a correction).

Proposition: Let $\mathcal{C}$ be a class of schemes with the following properties:

  • $\mathcal{C}$ contains all affine schemes
  • If $X$ is a scheme, $\{ U, V \}$ is an open cover of $X$, and $\mathcal{C}$ contains the three schemes $U$, $V$, and $U \cap V$, then $\mathcal{C}$ contains $X$.

Then $\mathcal{C}$ contains every quasi-compact quasi-separated scheme. Conversely, the class of quasi-compact quasi-separated schemes has the properties above.

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This is proposition 3.3.1 and remark 3.3.2 of Generators and representability of functors in commutative and noncommutative geometry by A. Bondal, M. van den Bergh.

I originally found this reference via the MO question The biggest class of schemes which the reduction principle holds, and I have now found it again by browsing the the Related links to my question, so hurrah for the MO system.

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