Proposition 1.8.1

[solov-t]

Okay, this proposition is insane. The sentence on line 1112

...that ${}^a\vphi(D(f))=X_f$, which proves that ${}^a\vphi$ is a \emph{continuous map} $X\to S$.

is a little bit weird because the map goes from $X \to S$ but $X_f $ is a subset of $X$ and $D(f)$ is a subset of $S$. Now, since this is a proof of continuity I suppose it should be an inverse image.

Now according to 0.5.5.2 $X_f$ would make sense if $f$ was a section of $X$, but $f$ was declared as a section of $S$. Making some computations, I believe that the actual equation should be something like

${}^a\vphi^{-1}(D(f))=X_{\phi(f)}$.

Which checks out, and would prove continuity as desired.

Also on line 1119 the sentence

...$\Gamma(X_f,\sh{O}_X)=\Gamma(D(f),{}^a\vphi(\sh{O}_X))$

should actually be $\Gamma(X_f,\sh{O}X)=\Gamma(D(f),{}^a\vphi*(\sh{O}_X))$ I believe.

[ryankeleti]

Yes you are correct, I transcribed it wrong! It should be {}^a\vhpi^{-1} and {}^a\vphi_*. I'll correct it or if you want, you can submit a PR.

[ryankeleti]

Also https://stacks.math.columbia.edu/tag/01I1 gives a proof along your idea. I think EGA just abuses notation and writes X_f.

[ryankeleti]

fixed! 6d373aab98c7727b65679b46552f102ce422797e. If you would like, I can add a note about X_f.