The Stacks project

Remark 32.9.2. We cannot do better than this if we do not assume more on $S$ and the morphism $f : X \to S$. For example, in general it will not be possible to find a closed immersion $X \to X'$ as in the lemma. The reason is that this would imply that $f$ is quasi-compact which may not be the case. An example is to take $S$ to be infinite dimensional affine space with $0$ doubled and $X$ to be one of the two infinite dimensional affine spaces.

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