Lemma 29.26.1. Let $X$ be a scheme. The rule which associates to a closed subscheme of $X$ its underlying closed subset defines a bijection

$\left\{ \begin{matrix} \text{closed subschemes }Z \subset X \\ \text{such that }Z \to X\text{ is flat} \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{closed subsets }Z \subset X \\ \text{closed under generalizations} \end{matrix} \right\}$

If $Z \subset X$ is such a closed subscheme, every morphism of schemes $g : Y \to X$ with $g(Y) \subset Z$ set theoretically factors (scheme theoretically) through $Z$.

Proof. The affine case of the bijection is Algebra, Lemma 10.108.4. For general schemes $X$ the bijection follows by covering $X$ by affines and glueing. Details omitted. For the final assertion, observe that the projection $Z \times _{X, g} Y \to Y$ is a flat (Lemma 29.25.8) closed immersion which is bijective on underlying topological spaces and hence must be an isomorphism by the bijection esthablished in the first part of the proof. $\square$

Comment #5820 by Laurent Moret-Bailly on

Suggested addition: if $Z$ is such a subscheme, every morphism $Y\to X$ mapping $\vert Y\vert$ into $\vert Z\vert$ factors through $Z$. (Proof: for $y\in Y$, any section of the ideal sheaf of $Z$ vanishes in $\mathcal{O}_{X,f(y)}$). This simplifies the proof of 0B7R, for instance.

Comment #5844 by on

Thanks for this suggestion. I have made the corresponding changes here.

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