Lemma 67.12.6. Let S be a scheme. Let Z \to X be an immersion of algebraic spaces over S. Assume Z \to X is quasi-compact. There exists a factorization Z \to \overline{Z} \to X where Z \to \overline{Z} is an open immersion and \overline{Z} \to X is a closed immersion.
Proof. Let U be a scheme and let U \to X be surjective étale. As usual denote R = U \times _ X U with projections s, t : R \to U. Set T = Z \times _ U X. Let \overline{T} \subset U be the scheme theoretic image of T \to U. Note that s^{-1}\overline{T} = t^{-1}\overline{T} as taking scheme theoretic images of quasi-compact morphisms commute with flat base change, see Morphisms, Lemma 29.25.16. Hence we obtain a closed subspace \overline{Z} \subset X whose pullback to U is \overline{T}, see Properties of Spaces, Lemma 66.12.2. By Morphisms, Lemma 29.7.7 the morphism T \to \overline{T} is an open immersion. It follows that Z \to \overline{Z} is an open immersion and we win. \square
Comments (0)