Lemma 54.6.1. The functor $F$ (54.6.0.1) is an equivalence.
54.6 Modifying over local rings
Let $S$ be a scheme. Let $s_1, \ldots , s_ n \in S$ be pairwise distinct closed points. Assume that the open embedding
\[ U = S \setminus \{ s_1, \ldots , s_ n\} \longrightarrow S \]
is quasi-compact. Denote $FP_{S, \{ s_1, \ldots , s_ n\} }$ the category of morphisms $f : X \to S$ of finite presentation which induce an isomorphism $f^{-1}(U) \to U$. Morphisms are morphisms of schemes over $S$. For each $i$ set $S_ i = \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s_ i})$ and let $V_ i = S_ i \setminus \{ s_ i\} $. Denote $FP_{S_ i, s_ i}$ the category of morphisms $g_ i : Y_ i \to S_ i$ of finite presentation which induce an isomorphism $g_ i^{-1}(V_ i) \to V_ i$. Morphisms are morphisms over $S_ i$. Base change defines an functor
54.6.0.1
\begin{equation} \label{resolve-equation-equivalence} F : FP_{S, \{ s_1, \ldots , s_ n\} } \longrightarrow FP_{S_1, s_1} \times \ldots \times FP_{S_ n, s_ n} \end{equation}
To reduce at least some of the problems in this chapter to the case of local rings we have the following lemma.
Proof. For $n = 1$ this is Limits, Lemma 32.21.1. For $n > 1$ the lemma can be proved in exactly the same way or it can be deduced from it. For example, suppose that $g_ i : Y_ i \to S_ i$ are objects of $FP_{S_ i, s_ i}$. Then by the case $n = 1$ we can find $f'_ i : X'_ i \to S$ of finite presentation which are isomorphisms over $S \setminus \{ s_ i\} $ and whose base change to $S_ i$ is $g_ i$. Then we can set
\[ f : X = X'_1 \times _ S \ldots \times _ S X'_ n \to S \]
This is an object of $FP_{S, \{ s_1, \ldots , s_ n\} }$ whose base change by $S_ i \to S$ recovers $g_ i$. Thus the functor is essentially surjective. We omit the proof of fully faithfulness. $\square$
Lemma 54.6.2. Let $S, s_ i, S_ i$ be as in (54.6.0.1). If $f : X \to S$ corresponds to $g_ i : Y_ i \to S_ i$ under $F$, then $f$ is separated, proper, finite, if and only if $g_ i$ is so for $i = 1, \ldots , n$.
Proof. Follows from Limits, Lemma 32.21.2. $\square$
Lemma 54.6.3. Let $S, s_ i, S_ i$ be as in (54.6.0.1). If $f : X \to S$ corresponds to $g_ i : Y_ i \to S_ i$ under $F$, then $X_{s_ i} \cong (Y_ i)_{s_ i}$ as schemes over $\kappa (s_ i)$.
Proof. This is clear. $\square$
Lemma 54.6.4. Let $S, s_ i, S_ i$ be as in (54.6.0.1) and assume $f : X \to S$ corresponds to $g_ i : Y_ i \to S_ i$ under $F$. Then there exists a factorization of $f$ where $Z_{j + 1} \to Z_ j$ is the blowing up of $Z_ j$ at a closed point $z_ j$ lying over $\{ s_1, \ldots , s_ n\} $ if and only if for each $i$ there exists a factorization of $g_ i$ where $Z_{i, j + 1} \to Z_{i, j}$ is the blowing up of $Z_{i, j}$ at a closed point $z_{i, j}$ lying over $s_ i$.
\[ X = Z_ m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = S \]
\[ Y_ i = Z_{i, m_ i} \to Z_{i, m_ i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = S_ i \]
Proof. Let's start with a sequence of blowups $Z_ m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = S$. The first morphism $Z_1 \to S$ is given by blowing up one of the $s_ i$, say $s_1$. Applying $F$ to $Z_1 \to S$ we find a blowup $Z_{1, 1} \to S_1$ at $s_1$ is the blowing up at $s_1$ and otherwise $Z_{i, 0} = S_ i$ for $i > 1$. In the next step, we either blow up one of the $s_ i$, $i \geq 2$ on $Z_1$ or we pick a closed point $z_1$ of the fibre of $Z_1 \to S$ over $s_1$. In the first case it is clear what to do and in the second case we use that $(Z_1)_{s_1} \cong (Z_{1, 1})_{s_1}$ (Lemma 54.6.3) to get a closed point $z_{1, 1} \in Z_{1, 1}$ corresponding to $z_1$. Then we set $Z_{1, 2} \to Z_{1, 1}$ equal to the blowing up in $z_{1, 1}$. Continuing in this manner we construct the factorizations of each $g_ i$.
Conversely, given sequences of blowups $Z_{i, m_ i} \to Z_{i, m_ i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = S_ i$ we construct the sequence of blowing ups of $S$ in exactly the same manner. $\square$
Here is the analogue of Lemma 54.6.4 for normalized blowups.
Lemma 54.6.5. Let $S, s_ i, S_ i$ be as in (54.6.0.1) 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 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 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 $s_ i$.
\[ X = Z_ m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = S \]
\[ Y_ i = Z_{i, m_ i} \to Z_{i, m_ i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = S_ i \]
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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