The Stacks project

Lemma 54.11.7. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local domain whose completion $A^\wedge $ is normal. Then given any sequence

\[ Y_ n \to Y_{n - 1} \to \ldots \to Y_1 \to \mathop{\mathrm{Spec}}(A^\wedge ) \]

of normalized blowups, there exists a sequence of (proper) normalized blowups

\[ X_ n \to X_{n - 1} \to \ldots \to X_1 \to \mathop{\mathrm{Spec}}(A) \]

whose base change to $A^\wedge $ recovers the given sequence.

Proof. Given the sequence $Y_ n \to \ldots \to Y_1 \to Y_0 = \mathop{\mathrm{Spec}}(A^\wedge )$ we inductively construct $X_ n \to \ldots \to X_1 \to X_0 = \mathop{\mathrm{Spec}}(A)$. 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 $\mathop{\mathrm{Spec}}(A^\wedge )$ is isomorphic to $Y'_ i$. By Lemma 54.11.6 the normalization $X_{i + 1} \to X'_ i$ is finite and its base change to $\mathop{\mathrm{Spec}}(A^\wedge )$ is isomorphic to $Y_{i + 1}$. $\square$

Comments (2)

Comment #2612 by Axel on

There's a typo in the first line of the proof. It should read instead of

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