The Stacks project

Lemma 54.11.3. Let $(A, \mathfrak m)$ be a Noetherian local ring with completion $A^\wedge $. Let $U \subset \mathop{\mathrm{Spec}}(A)$ and $U^\wedge \subset \mathop{\mathrm{Spec}}(A^\wedge )$ be the punctured spectra. If $Y \to \mathop{\mathrm{Spec}}(A^\wedge )$ is a $U^\wedge $-admissible blowup, then there exists a $U$-admissible blowup $X \to \mathop{\mathrm{Spec}}(A)$ such that $Y = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A^\wedge )$.

Proof. By definition there exists an ideal $J \subset A^\wedge $ such that $V(J) = \{ \mathfrak m A^\wedge \} $ and such that $Y$ is the blowup of $S^\wedge $ in the closed subscheme defined by $J$, see Divisors, Definition 31.34.1. Since $A^\wedge $ is Noetherian this implies $\mathfrak m^ n A^\wedge \subset J$ for some $n$. Since $A^\wedge /\mathfrak m^ n A^\wedge = A/\mathfrak m^ n$ we find an ideal $\mathfrak m^ n \subset I \subset A$ such that $J = I A^\wedge $. Let $X \to S$ be the blowup in $I$. Since $A \to A^\wedge $ is flat we conclude that the base change of $X$ is $Y$ by Divisors, Lemma 31.32.3. $\square$

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