Lemma 25.17.6. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target.

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 $\mathop{\mathrm{Im}}(g^*\mathcal{I} \to \mathcal{O}_ Y)$.

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

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

## Comments (1)

Comment #2360 by Simon Pepin Lehalleur on

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