Lemma 38.37.5 (0EVB)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.37: Almost blow up squares and the h topology
Lemma 38.37.5 (cite)

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Lemma 38.37.5. Let $X$ be a quasi-compact and quasi-separated scheme and let $Z \subset X$ be a closed subscheme cut out by a finite type quasi-coherent sheaf of ideals. Suppose given almost blow up squares (38.37.0.1)

$\xymatrix{ E_ k \ar[r] \ar[d] & X_ k' \ar[d] \\ Z \ar[r] & X }$

for $k = 1, 2$ , then there exists an almost blow up square

$\xymatrix{ E \ar[r] \ar[d] & X' \ar[d] \\ Z \ar[r] & X }$

and closed immersions $i_ k : X' \to X'_ k$ over $X$ with $E = i_ k^{-1}(E_ k)$ .

Proof. Denote $X'' \to X$ the blowing up of $Z$ in $X$ . We view $X''$ as a closed subscheme of both $X'_1$ and $X'_2$ . Write $X'' = \mathop{\mathrm{lim}}\nolimits X'_{1, i}$ as in Lemma 38.37.3. By Limits, Proposition 32.6.1 there exists an $i$ and a morphism $h : X'_{1, i} \to X'_2$ agreeing with the inclusions $X'' \subset X'_{1, i}$ and $X'' \subset X'_2$ . By Limits, Lemma 32.4.20 the restriction of $h$ to $X'_{1, i'}$ is a closed immersion for some $i' \geq i$ . This finishes the proof. $\square$

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