The Stacks project

Lemma 54.6.5. Let $S, s_ i, S_ i$ be as in ( and assume $f : X \to S$ corresponds to $g_ i : Y_ i \to S_ i$ under $F$. Assume every quasi-compact open of $S$ has finitely many irreducible components. Then there exists a factorization

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

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

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

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 $s_ i$.

Proof. The assumption on $S$ is used to assure us (successively) that the schemes we are normalizing have locally finitely many irreducible components so that the statement makes sense. Having said this the lemma follows by the exact same argument as used to prove Lemma 54.6.4. $\square$

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