The Stacks project

Lemma 38.37.4. Let $X$ be a quasi-compact and quasi-separated scheme. Let $Z \subset X$ be a closed subscheme cut out by a finite type quasi-coherent sheaf of ideals. Then there exists an almost blow up square as in (

Proof. We may write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as a directed limit of an inverse system of Noetherian schemes with affine transition morphisms, see Limits, Proposition 32.5.4. We can find an index $i$ and a closed immersion $Z_ i \to X_ i$ whose base change to $X$ is the closed immersion $Z \to X$. See Limits, Lemmas 32.10.1 and 32.8.5. Let $b_ i : X'_ i \to X_ i$ be the blowing up with center $Z_ i$. This produces a blow up square

\[ \xymatrix{ E_ i \ar[r] \ar[d] & X'_ i \ar[d]^{b_ i} \\ Z_ i \ar[r] & X_ i } \]

where all the morphisms are finite type morphisms of Noetherian schemes and hence of finite presentation. Thus this is an almost blow up square. By Lemma 38.37.1 the base change of this diagram to $X$ produces the desired almost blow up square. $\square$

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