Lemma 37.39.2. 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 37.39.1 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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