Lemma 38.37.1. Consider an almost blow up square (38.37.0.1). Let $Y \to X$ be any morphism. Then the base change

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

is an almost blow up square too.

Proof. The morphism $Y \times _ X X' \to Y$ is proper and of finite presentation by Morphisms, Lemmas 29.41.5 and 29.21.4. The morphism $Y \times _ X Z \to Y$ is a closed immersion (Morphisms, Lemma 29.2.4) of finite presentation. The inverse image of $Y \times _ X Z$ in $Y \times _ X X'$ is equal to the inverse image of $E$ in $Y \times _ X X'$ and hence is locally principal (Divisors, Lemma 31.13.11). Let $X'' \subset X'$, resp. $Y'' \subset Y \times _ X X'$ be the closed subscheme corresponding to the quasi-coherent ideal of sections of $\mathcal{O}_{X'}$, resp. $\mathcal{O}_{Y \times _ Y X'}$ supported on $E$, resp. $Y \times _ X E$. Clearly, $Y'' \subset Y \times _ X X''$ is the closed subscheme corresponding to the quasi-coherent ideal of sections of $\mathcal{O}_{Y \times _ Y X''}$ supported on $Y \times _ X (E \cap X'')$. Thus $Y''$ is the strict transform of $Y$ relative to the blowing up $X'' \to X$, see Divisors, Definition 31.33.1. Thus by Divisors, Lemma 31.33.2 we see that $Y''$ is the blow up of $Y \times _ X Z$ on $Y$. $\square$

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