The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 25.17.6. 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 25.4.7 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 25.10.1 $Z$ is a scheme. Hence $Z = X \times _ S Y$ and the first statement follows. The second follows from Lemma 25.17.3 for example. The third is a combination of the first two. $\square$


Comments (1)

Comment #2360 by Simon Pepin Lehalleur on

Suggested slogan: Immersions are stable under base change.

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  • 2 comment(s) on Section 25.17: Fibre products of schemes

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