The Stacks Project

Tag 0BGA

Lemma 50.11.7. Let $(A, \mathfrak m, \kappa)$ be a Noetherian local domain whose completion $A^\wedge$ is normal. Then given any sequence $$Y_n \to Y_{n - 1} \to \ldots \to Y_1 \to \mathop{\rm Spec}(A^\wedge)$$ of normalized blowups, there exists a sequence of (proper) normalized blowups $$X_n \to X_{n - 1} \to \ldots \to X_1 \to \mathop{\rm Spec}(A)$$ whose base change to $A^\wedge$ recovers the given sequence.

Proof. Given the sequence $Y_n \to \ldots \to Y_1 \to Y_0 = \mathop{\rm Spec}(A^\wedge)$ we inductively construct $X_n \to \ldots \to X_1 \to X_0 = \mathop{\rm Spec}(A)$. The base case is $i = 0$. Given $X_i$ whose base change is $Y_i$, let $Y'_i \to Y_i$ be the blowing up in the closed point $y_i \in Y_i$ such that $Y_{i + 1}$ is the normalization of $Y_i$. Since the closed fibres of $Y_i$ and $X_i$ are isomorphic, the point $y_i$ corresponds to a closed point $x_i$ on the special fibre of $X_i$. Let $X'_i \to X_i$ be the blowup of $X_i$ in $x_i$. Then the base change of $X'_i$ to $\mathop{\rm Spec}(A^\wedge)$ is isomorphic to $Y'_i$. By Lemma 50.11.6 the normalization $X_{i + 1} \to X'_i$ is finite and its base change to $\mathop{\rm Spec}(A^\wedge)$ is isomorphic to $Y_{i + 1}$. $\square$

The code snippet corresponding to this tag is a part of the file resolve.tex and is located in lines 2987–2999 (see updates for more information).

\begin{lemma}
\label{lemma-normalized-blowup-completion}
Let $(A, \mathfrak m, \kappa)$ be a Noetherian local domain whose completion
$A^\wedge$ is normal. Then given any sequence
$$Y_n \to Y_{n - 1} \to \ldots \to Y_1 \to \Spec(A^\wedge)$$
of normalized blowups, there exists a sequence of (proper) normalized blowups
$$X_n \to X_{n - 1} \to \ldots \to X_1 \to \Spec(A)$$
whose base change to $A^\wedge$ recovers the given sequence.
\end{lemma}

\begin{proof}
Given the sequence $Y_n \to \ldots \to Y_1 \to Y_0 = \Spec(A^\wedge)$ we
inductively construct $X_n \to \ldots \to X_1 \to X_0 = \Spec(A)$.
The base case is $i = 0$. Given $X_i$ whose base change is $Y_i$,
let $Y'_i \to Y_i$ be the blowing up in the closed point $y_i \in Y_i$
such that $Y_{i + 1}$ is the normalization of $Y_i$.
Since the closed fibres of $Y_i$ and $X_i$ are isomorphic, the point
$y_i$ corresponds to a closed point $x_i$ on the special fibre of $X_i$.
Let $X'_i \to X_i$ be the blowup of $X_i$ in $x_i$. Then the base change
of $X'_i$ to $\Spec(A^\wedge)$ is isomorphic to $Y'_i$.
By Lemma \ref{lemma-normalization-completion}
the normalization $X_{i + 1} \to X'_i$ is finite and its base change
to $\Spec(A^\wedge)$ is isomorphic to $Y_{i + 1}$.
\end{proof}

Comment #2612 by Axel on June 30, 2017 a 6:18 pm UTC

There's a typo in the first line of the proof. It should read $Spec (A^\wedge)$ instead of $Spec (A)^\wedge)$

Comment #2634 by Johan (site) on July 7, 2017 a 12:55 pm UTC

Thanks, fixed here.

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