The Stacks project

Lemma 87.5.3. Let $X, x_ i, U_ i \to X, u_ i$ be as in ( and assume $f : Y \to X$ corresponds to $g_ i : Y_ i \to U_ i$ under $F$. Assume $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma 65.49.1. Then there exists a factorization

\[ Y = Z_ m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = X \]

of $f$ where $Z_{j + 1} \to Z_ j$ is the normalized blowing up of $Z_ j$ at a closed point $z_ j$ lying over $\{ x_1, \ldots , x_ n\} $ if and only if for each $i$ there exists a factorization

\[ Y_ i = Z_{i, m_ i} \to Z_{i, m_ i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = U_ i \]

of $g_ i$ where $Z_{i, j + 1} \to Z_{i, j}$ is the normalized blowing up of $Z_{i, j}$ at a closed point $z_{i, j}$ lying over $u_ i$.

Proof. This follows by the exact same argument as used to prove Lemma 87.4.2. $\square$

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