The Stacks project

Lemma 38.37.3. Consider an almost blow up square (38.37.0.1) with $X$ quasi-compact and quasi-separated. Then the blowup $X''$ of $X$ in $Z$ can be written as

\[ X'' = \mathop{\mathrm{lim}}\nolimits X'_ i \]

where the limit is over the directed system of closed subschemes $X'_ i \subset X'$ of finite presentation satisfying the equivalent conditions of Lemma 38.37.2.

Proof. Let $\mathcal{I} \subset \mathcal{O}_{X'}$ be the quasi-coherent sheaf of ideals corresponding to $X''$. By Properties, Lemma 28.22.3 we can write $\mathcal{I}$ as the filtered colimit $\mathcal{I} = \mathop{\mathrm{colim}}\nolimits \mathcal{I}_ i$ of its quasi-coherent submodules of finite type. Since these modules correspond $1$-to-$1$ to the closed subschemes $X'_ i$ the proof is complete. $\square$

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