Lemma 29.3.2. Let h : Z \to X be an immersion. If h is quasi-compact, then we can factor h = i \circ j with j : Z \to \overline{Z} an open immersion and i : \overline{Z} \to X a closed immersion.
Proof. Note that h is quasi-compact and quasi-separated (see Schemes, Lemma 26.23.8). Hence h_*\mathcal{O}_ Z is a quasi-coherent sheaf of \mathcal{O}_ X-modules by Schemes, Lemma 26.24.1. This implies that \mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ X \to h_*\mathcal{O}_ Z) is a quasi-coherent sheaf of ideals, see Schemes, Section 26.24. Let \overline{Z} \subset X be the closed subscheme corresponding to \mathcal{I}, see Lemma 29.2.3. By Schemes, Lemma 26.4.6 the morphism h factors as h = i \circ j where i : \overline{Z} \to X is the inclusion morphism. To see that j is an open immersion, choose an open subscheme U \subset X such that h induces a closed immersion of Z into U. Then it is clear that \mathcal{I}|_ U is the sheaf of ideals corresponding to the closed immersion Z \to U. Hence we see that Z = \overline{Z} \cap U. \square
Comments (0)
There are also: