Lemma 38.31.1. Let $S$ be a quasi-compact and quasi-separated scheme. Let $X$ be a scheme over $S$. Let $U \subset S$ be a quasi-compact open. Assume

1. $X \to S$ is of finite type and quasi-separated, and

2. $X_ U \to U$ is flat and locally of finite presentation.

Then there exists a $U$-admissible blowup $S' \to S$ such that the strict transform of $X$ is flat and of finite presentation over $S'$.

Proof. Since $X \to S$ is quasi-compact and quasi-separated by assumption, the strict transform of $X$ with respect to a blowing up $S' \to S$ is also quasi-compact and quasi-separated. Hence to prove the lemma it suffices to find a $U$-admissible blowup such that the strict transform is flat and locally of finite presentation. Let $X = W_1 \cup \ldots \cup W_ n$ be a finite affine open covering. If we can find a $U$-admissible blowup $S_ i \to S$ such that the strict transform of $W_ i$ is flat and locally of finite presentation, then there exists a $U$-admissible blowing up $S' \to S$ dominating all $S_ i \to S$ which does the job (see Divisors, Lemma 31.34.4; see also Remark 38.30.1). Hence we may assume $X$ is affine.

Assume $X$ is affine. By Morphisms, Lemma 29.39.2 we can choose an immersion $j : X \to \mathbf{A}^ n_ S$ over $S$. Let $V \subset \mathbf{A}^ n_ S$ be a quasi-compact open subscheme such that $j$ induces a closed immersion $i : X \to V$ over $S$. Apply Theorem 38.30.7 to $V \to S$ and the quasi-coherent module $i_*\mathcal{O}_ X$ to obtain a $U$-admissible blowup $S' \to S$ such that the strict transform of $i_*\mathcal{O}_ X$ is flat over $S'$ and of finite presentation over $\mathcal{O}_{V \times _ S S'}$. Let $X'$ be the strict transform of $X$ with respect to $S' \to S$. Let $i' : X' \to V \times _ S S'$ be the induced morphism. Since taking strict transform commutes with pushforward along affine morphisms (Divisors, Lemma 31.33.4), we see that $i'_*\mathcal{O}_{X'}$ is flat over $S$ and of finite presentation as a $\mathcal{O}_{V \times _ S S'}$-module. This implies the lemma. $\square$

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