The Stacks project

Remark 98.27.18. The proof of Theorem 98.27.17 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 98.27.7 (this is the usual bootstrap procedure for the diagonal). Thus the proof of Lemma 98.27.14 is the only place in our proof of Theorem 98.27.17 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 76.43.4. 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.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GIC. Beware of the difference between the letter 'O' and the digit '0'.