The Stacks project

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 (

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

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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