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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