The Stacks project

Lemma 54.11.1. Let $(A, \mathfrak m, \kappa )$ be a local ring with finitely generated maximal ideal $\mathfrak m$. Let $X$ be a scheme over $A$. Let $Y = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A^\wedge )$ where $A^\wedge $ is the $\mathfrak m$-adic completion of $A$. For a point $q \in Y$ with image $p \in X$ lying over the closed point of $\mathop{\mathrm{Spec}}(A)$ the local ring map $\mathcal{O}_{X, p} \to \mathcal{O}_{Y, q}$ induces an isomorphism on completions.

Proof. We may assume $X$ is affine. Then we may write $X = \mathop{\mathrm{Spec}}(B)$. Let $\mathfrak q \subset B' = B \otimes _ A A^\wedge $ be the prime corresponding to $q$ and let $\mathfrak p \subset B$ be the prime ideal corresponding to $p$. By Algebra, Lemma 10.96.3 we have

\[ B'/(\mathfrak m^\wedge )^ n B' = A^\wedge /(\mathfrak m^\wedge )^ n \otimes _ A B = A/\mathfrak m^ n \otimes _ A B = B/\mathfrak m^ n B \]

for all $n$. Since $\mathfrak m B \subset \mathfrak p$ and $\mathfrak m^\wedge B' \subset \mathfrak q$ we see that $B/\mathfrak p^ n$ and $B'/\mathfrak q^ n$ are both quotients of the ring displayed above by the $n$th power of the same prime ideal. The lemma follows. $\square$

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