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 10.107.4. 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.
$\square$

## Comments (2)

Comment #5820 by Laurent Moret-Bailly on

Comment #5844 by Johan on