Unnecessary assumption in Görtz-Wedhorn Proposition 4.20 (fiber products)
问题内容
(Note: I am aware about this post, my question is different)
Proposition 4.20 in Görtz-Wedhorn states the following:
All of the assumptions and assertions are local in $S,X,Y$ so we may assume they are affine (as we do in the proof of this proposition). However, except for injectivity of $g$, they are also local in $X'$, and injectivity of $g$ is easy to prove: if $g(z_1)=g(z_2)$ then $f(p'(z_1))=f(p'(z_2))$ and since $f$ is injective, $p'(z_1)=p'(z_2)$ so $z_1,z_2$ belong to the same open set of $X'\times_SY$ obtained from a fixed open cover of $X'$. Therefore we may assume $X'$ is affine. In this case the second part of (I) is automatic (choose $U':=X$).
Do I have a mistake? I believe that if all schemes are affine then we also have a much quicker proof.
回答 (1)
I found the mistake: proving that $g$ is a homeomorphism onto its image is equivalent to proving it's injective and open into its image. Openness in general is local at the source but "open into its image" is not, e.g. the obvious map from $\mathbb{R}\amalg\mathbb{R}$ to $\mathbb{R}\cup i\mathbb{R}$. We can even take zero out of one of the lines in the source to make this map injective.