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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