The Stacks project

Lemma 37.35.2. 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 37.35.1 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. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 37.35: Étale neighbourhoods and Artin approximation

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CAV. Beware of the difference between the letter 'O' and the digit '0'.