The Stacks project

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.