The Stacks project

In the proof of 01HQ, the last word should be subspace instead of subscheme, since this lemma and the lemmas cited in the proof are about general locally ringed spaces.

Fixed here. Thanks!

There is a typo in the second sentence. "Functions" should be changed to "sections".

Slogan: closed immersions of locally ringed spaces are stable under base change.

Here's the proof of cartesianity of the square from the statement: we need to use

Exercise. Let $f:X\to Y$ be a morphism of ringed spaces and let $\varphi$ be a morphism of $\mathcal{O}_Y$-modules. Then $f^*\operatorname{Im}\varphi=\operatorname{Im}f^*\varphi$. (Hint: verify it on stalks and use that if $R$ is a commutative ring, then $L\otimes\operatorname{Im}\psi=\operatorname{Im}(L\otimes\psi)$, for $L$ an $R$-module and $\psi$ a morphism of $R$-modules.)

Suppose we have morphisms $g:W\to X$ and $h:W\to Z$ fitting into a commutative square which terminates in $Y$. If a factorization of this square through $Z'$ were to exist, then it would be unique for $i$ is monic (closed immersions of l.r.s. are monomorphisms). We see existence: We construct a map $W\to Z'$ factoring $g$ through $i'$ with 26.4.6: we must show that the map

$$g^*\operatorname{Im}(f^*\mathcal{I}\to f^*\mathcal{O}_Y)\to g^*\mathcal{O}_X$$ vanishes. But we have $$\begin{align*} g^*\operatorname{Im}(f^*\mathcal{I}\to f^*\mathcal{O}_Y) &=\operatorname{Im}(g^*f^*\mathcal{I}\to g^*f^*\mathcal{O}_Y), &\text{by the exercise,}\\ &=\operatorname{Im}(h^*i^*\mathcal{I}\to h^*i^*\mathcal{O}_Y)\\ &=0, \end{align*}$$ by 26.4.6. On the other hand, the induced map $W\to Z'$ can be seen to also factor $h$ through $f'$ using the monicity of $i$.

Going to leave as is.