The Stacks project

Lemma 37.35.3. Let $X \to S$, $Y \to T$, $x$, $s$, $y$, $t$, $\sigma $, $y_\sigma $, and $\varphi $ be given as follows: we have morphisms of schemes

\[ \vcenter { \xymatrix{ X \ar[d] & Y \ar[d] \\ S & T } } \quad \text{with points}\quad \vcenter { \xymatrix{ x \ar[d] & y \ar[d] \\ s & t } } \]

Here $S$ is locally Noetherian and $T$ is of finite type over $\mathbf{Z}$. The morphisms $X \to S$ and $Y \to T$ are locally of finite type. The local ring $\mathcal{O}_{S, s}$ is a G-ring. The map

\[ \sigma : \mathcal{O}_{T, t} \longrightarrow \mathcal{O}_{S, s}^\wedge \]

is a local homomorphism. Set $Y_\sigma = Y \times _{T, \sigma } \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^\wedge )$. Next, $y_\sigma $ is a point of $Y_\sigma $ mapping to $y$ and the closed point of $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^\wedge )$. Finally

\[ \varphi : \mathcal{O}_{X, x}^\wedge \longrightarrow \mathcal{O}_{Y_\sigma , y_\sigma }^\wedge \]

is an isomorphism of $\mathcal{O}_{S, s}^\wedge $-algebras. In this situation there exists a commutative diagram

\[ \xymatrix{ X \ar[d] & W \ar[l] \ar[rd] \ar[rr] & & Y \times _{T, \tau } V \ar[r] \ar[ld] & Y \ar[d] \\ S & & V \ar[ll] \ar[rr]^\tau & & T } \]

of schemes and points $w \in W$, $v \in V$ such that

  1. $(V, v) \to (S, s)$ is an elementary étale neighbourhood,

  2. $(W, w) \to (X, x)$ is an elementary étale neighbourhood, and

  3. $\tau (v) = t$.

Let $y_\tau \in Y \times _ T V$ correspond to $y_\sigma $ via the identification $(Y_\sigma )_ s = (Y \times _ T V)_ v$. Then

  1. $(W, w) \to (Y \times _{T, \tau } V, y_\tau )$ is an elementary étale neighbourhood.

Proof. Denote $X_\sigma = X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^\wedge )$ and $x_\sigma \in X_\sigma $ the unique point lying over $x$. Observe that $\mathcal{O}_{S, s}^\wedge $ is a G-ring by More on Algebra, Proposition 15.50.6. By Lemma 37.35.2 we can choose

\[ (X_\sigma , x_\sigma ) \leftarrow (U, u) \rightarrow (Y_\sigma , y_\sigma ) \]

where both arrows are elementary étale neighbourhoods.

After replacing $S$ by an open neighbourhood of $s$, we may assume $S = \mathop{\mathrm{Spec}}(R)$ is affine. Since $\mathcal{O}_{S, s}$ is a G-ring by Smoothing Ring Maps, Theorem 16.12.1 the ring $\mathcal{O}_{S, s}^\wedge $ is a filtered colimit of smooth $R$-algebras. Thus we can write

\[ \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^\wedge ) = \mathop{\mathrm{lim}}\nolimits S_ i \]

as a directed limit of affine schemes $S_ i$ smooth over $S$. Denote $s_ i \in S_ i$ the image of the closed point of $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^\wedge )$. Observe that $\kappa (s) = \kappa (s_ i)$. Set $X_ i = X \times _ S S_ i$ and denote $x_ i \in X_ i$ the unique point mapping to $x$. Note that $\kappa (x) = \kappa (x_ i)$. Since $T$ is of finite type over $\mathbf{Z}$ by Limits, Proposition 32.6.1 we can choose an $i$ and a morphism $\sigma _ i : (S_ i, s_ i) \to (T, t)$ of pointed schemes whose composition with $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) \to S_ i$ is equal to $\sigma $. Set $Y_ i = Y \times _ T S_ i$ and denote $y_ i$ the image of $y_\sigma $. Note that $\kappa (y_ i) = \kappa (y_\sigma )$. By Limits, Lemma 32.10.1 we can choose an $i$ and a diagram

\[ \xymatrix{ X_ i \ar[rd] & U_ i \ar[l] \ar[d] \ar[r] & Y_ i \ar[ld] \\ & S_ i } \]

whose base change to $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^\wedge )$ recovers $X_\sigma \leftarrow U \rightarrow Y_\sigma $. By Limits, Lemma 32.8.10 after increasing $i$ we may assume the morphisms $X_ i \leftarrow U_ i \rightarrow Y_ i$ are étale. Let $u_ i \in U_ i$ be the image of $u$. Then $u_ i \mapsto x_ i$ hence $\kappa (x) = \kappa (x_\sigma ) = \kappa (u) \supset \kappa (u_ i) \supset \kappa (x_ i) = \kappa (x)$ and we see that $\kappa (u_ i) = \kappa (x_ i)$. Hence $(X_ i, x_ i) \leftarrow (U_ i, u_ i)$ is an elementary étale neighbourhood. Since also $\kappa (y_ i) = \kappa (y_\sigma ) = \kappa (u)$ we see that also $(U_ i, u_ i) \to (Y_ i, y_ i)$ is an elementary étale neighbourhood.

At this point we have constructed a diagram

\[ \xymatrix{ X \ar[d] & X \times _ S S_ i \ar[l] \ar[rd] & U_ i \ar[l] \ar[r] \ar[d] & Y \times _ T S_ i \ar[r] \ar[ld] & Y \ar[d] \\ S & & S_ i \ar[ll] \ar[rr] & & T } \]

as in the statement of the lemma, except that $S_ i \to S$ is smooth. By Lemma 37.34.5 and after shrinking $S_ i$ we can assume there exists a closed subscheme $V \subset S_ i$ passing through $s_ i$ such that $V \to S$ is étale. Setting $W$ equal to the scheme theoretic inverse image of $V$ in $U_ i$ we conclude. $\square$


Comments (2)

Comment #6056 by Rachel Webb on

In the lemma statement, should be .

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

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