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Tag 0BGA

Chapter 50: Resolution of Surfaces > Section 50.11: Base change to the completion

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}

    Comments (2)

    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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