Lemma 26.17.6. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target.

1. If $f : X \to S$ is a closed immersion, then $X \times _ S Y \to Y$ is a closed immersion. Moreover, if $X \to S$ corresponds to the quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ S$, then $X \times _ S Y \to Y$ corresponds to the sheaf of ideals $\mathop{\mathrm{Im}}(g^*\mathcal{I} \to \mathcal{O}_ Y)$.

2. If $f : X \to S$ is an open immersion, then $X \times _ S Y \to Y$ is an open immersion.

3. If $f : X \to S$ is an immersion, then $X \times _ S Y \to Y$ is an immersion.

Proof. Assume that $X \to S$ is a closed immersion corresponding to the quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ S$. By Lemma 26.4.7 the closed subspace $Z \subset Y$ defined by the sheaf of ideals $\mathop{\mathrm{Im}}(g^*\mathcal{I} \to \mathcal{O}_ Y)$ is the fibre product in the category of locally ringed spaces. By Lemma 26.10.1 $Z$ is a scheme. Hence $Z = X \times _ S Y$ and the first statement follows. The second follows from Lemma 26.17.3 for example. The third is a combination of the first two. $\square$

Comment #2360 by Simon Pepin Lehalleur on

Suggested slogan: Immersions are stable under base change.

Comment #8465 by on

In the last sentence, I would give the hint "the third is a combination of the first two, plus the pasting law for pullbacks, https://stacks.math.columbia.edu/tag/001U#comment-3413 ."

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