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The Stacks project

Lemma 70.23.2. In Situation 70.23.1. Let X \to B be a quasi-separated and finite type morphism of algebraic spaces. Then there exists an i \in I and a diagram

70.23.2.1
\begin{equation} \label{spaces-limits-equation-good-diagram} \vcenter { \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ B \ar[r] & B_ i } } \end{equation}

such that W \to B_ i is of finite type and such that the induced morphism X \to B \times _{B_ i} W is a closed immersion.

Proof. By Lemma 70.11.6 we can find a closed immersion X \to X' over B where X' is an algebraic space of finite presentation over B. By Lemma 70.7.1 we can find an i and a morphism of finite presentation X'_ i \to B_ i whose pull back is X'. Set W = X'_ i. \square

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