Lemma 54.11.7 (0BGA)—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 June 30, 2017 at 18:18

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 on July 07, 2017 at 12:55

Thanks, fixed here.

Comments (2)

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