The Stacks project

Remark 97.27.18. The proof of Theorem 97.27.17 uses that $X'$ and $W$ are separated over $S$ in two places. First, the proof uses this in showing $\Delta : F \to F \times F$ is representable by algebraic spaces. This use of the assumption can be entirely avoided by proving that $\Delta $ is representable by applying the theorem in the separated case to the triples $E'$, $(E' \to V)^{-1}Z$, and $E'_{/Z} \to E_ W$ found in Remark 97.27.7 (this is the usual bootstrap procedure for the diagonal). Thus the proof of Lemma 97.27.14 is the only place in our proof of Theorem 97.27.17 where we really need to use that $X' \to S$ is separated. The reader checks that we use the assumption only to obtain the morphism $x' : V' \to X'$. The existence of $x'$ can be shown, using results in the literature, if $X' \to S$ is quasi-separated, see More on Morphisms of Spaces, Remark 75.43.4. We conclude the theorem holds as stated with “separated” replaced by “quasi-separated”. If we ever need this we will precisely state and carefully prove this here.

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