The Stacks project

Lemma 37.38.2. Let $S$ be a locally Noetherian scheme. Let $X$, $Y$ be schemes locally of finite type over $S$. Let $x \in X$ and $y \in Y$ be points lying over the same point $s \in S$. Assume $\mathcal{O}_{S, s}$ is a G-ring. Assume we have an $\mathcal{O}_{S, s}$-algebra isomorphism

\[ \varphi : \mathcal{O}_{Y, y}^\wedge \longrightarrow \mathcal{O}_{X, x}^\wedge \]

between the complete local rings. Then for every $N \geq 1$ there exists morphisms

\[ (X, x) \leftarrow (U, u) \rightarrow (Y, y) \]

of pointed schemes over $S$ such that both arrows define elementary étale neighbourhoods and such that the diagram

\[ \xymatrix{ & \mathcal{O}_{U, u}^\wedge \\ \mathcal{O}_{Y, y}^\wedge \ar[rr]^\varphi \ar[ru] & & \mathcal{O}_{X, x}^\wedge \ar[lu] } \]

commutes modulo $\mathfrak m_ u^ N$.

Proof. We may assume $N \geq 2$. Apply Lemma 37.38.1 to get $(U, u) \to (X, x)$ and $f : (U, u) \to (Y, y)$. We claim that $f$ is étale at $u$ which will finish the proof. In fact, we will show that the induced map $\mathcal{O}_{Y, y}^\wedge \to \mathcal{O}_{U, u}^\wedge $ is an isomorphism. Having proved this, Lemma 37.12.1 will show that $f$ is smooth at $u$ and of course $f$ is unramified at $u$ as well, so Morphisms, Lemma 29.36.5 tells us $f$ is étale at $u$. For a local ring $(R, \mathfrak m)$ we set $\text{Gr}_\mathfrak m(R) = \bigoplus _{n \geq 0} \mathfrak m^ n/\mathfrak m^{n + 1}$. To prove the claim we look at the induced diagram of graded rings

\[ \xymatrix{ & \text{Gr}_{\mathfrak m_ u}(\mathcal{O}_{U, u}) \\ \text{Gr}_{\mathfrak m_ y}(\mathcal{O}_{Y, y}) \ar[rr]^\varphi \ar[ru] & & \text{Gr}_{\mathfrak m_ x}(\mathcal{O}_{X, x}) \ar[lu] } \]

Since $N \geq 2$ this diagram is actually commutative as the displayed graded algebras are generated in degree $1$! By assumption the lower arrow is an isomorphism. By More on Algebra, Lemma 15.43.9 (for example) the map $\mathcal{O}_{X, x}^\wedge \to \mathcal{O}_{U, u}^\wedge $ is an isomorphism and hence the north-west arrow in the diagram is an isomorphism. We conclude that $f$ induces an isomorphism $\text{Gr}_{\mathfrak m_ x}(\mathcal{O}_{X, x}) \to \text{Gr}_{\mathfrak m_ y}(\mathcal{O}_{U, u})$. Using induction and the short exact sequences

\[ 0 \to \text{Gr}^ n_{\mathfrak m}(R) \to R/\mathfrak m^{n + 1} \to R/\mathfrak m^ n \to 0 \]

for both local rings we conclude (from the snake lemma) that $f$ induces isomorphisms $\mathcal{O}_{Y, y}/\mathfrak m_ y^ n \to \mathcal{O}_{U, u}/\mathfrak m_ u^ n$ for all $n$ which is what we wanted to show. $\square$

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