Proposition 1.8.1
solov-t opened this issue · 3 comments
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
Now according to 0.5.5.2
${}^a\vphi^{-1}(D(f))=X_{\phi(f)}$
.
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.
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.
Also https://stacks.math.columbia.edu/tag/01I1 gives a proof along your idea. I think EGA just abuses notation and writes X_f
.
fixed! 6d373aa. If you would like, I can add a note about X_f
.