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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)