The Stacks project

37.35 Étale neighbourhoods and Artin approximation

In this section we prove results of the form: if two pointed schemes have isomorphic complete local rings, then they have isomorphic étale neighbourhoods. We will rely on Popescu's theorem, see Smoothing Ring Maps, Theorem 16.12.1.

Lemma 37.35.1. 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 further we are given a local $\mathcal{O}_{S, s}$-algebra map

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

For every $N \geq 1$ there exists an elementary étale neighbourhood $(U, u) \to (X, x)$ and an $S$-morphism $f : U \to Y$ mapping $u$ to $y$ such that the diagram

\[ \xymatrix{ \mathcal{O}_{X, x}^\wedge \ar[r] & \mathcal{O}_{U, u}^\wedge \\ \mathcal{O}_{Y, y} \ar[r]^{f^\sharp _ u} \ar[u]^\varphi & \mathcal{O}_{U, u} \ar[u] } \]

commutes modulo $\mathfrak m_ u^ N$.

Proof. The question is local on $X$ hence we may assume $X$, $Y$, $S$ are affine. Say $S = \mathop{\mathrm{Spec}}(R)$, $X = \mathop{\mathrm{Spec}}(A)$, $Y = \mathop{\mathrm{Spec}}(B)$. Write $B = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$. Let $\mathfrak p \subset A$ be the prime ideal corresponding to $x$. The local ring $\mathcal{O}_{X, x} = A_\mathfrak p$ is a G-ring by More on Algebra, Proposition 15.50.10. The map $\varphi $ is a map

\[ B_\mathfrak q^\wedge \longrightarrow A_\mathfrak p^\wedge \]

where $\mathfrak q \subset B$ is the prime corresponding to $y$. Let $a_1, \ldots , a_ n \in A_\mathfrak p^\wedge $ be the images of $x_1, \ldots , x_ n$ via $R[x_1, \ldots , x_ n] \to B \to B_\mathfrak q^\wedge \to A_\mathfrak p^\wedge $. Then we can apply Smoothing Ring Maps, Lemma 16.13.4 to get an étale ring map $A \to A'$ and a prime ideal $\mathfrak p' \subset A'$ and $b_1, \ldots , b_ n \in A'$ such that $\kappa (\mathfrak p) = \kappa (\mathfrak p')$, $a_ i - b_ i \in (\mathfrak p')^ N(A'_{\mathfrak p'})^\wedge $, and $f_ j(b_1, \ldots , b_ n) = 0$ for $j = 1, \ldots , n$. This determines an $R$-algebra map $B \to A'$ by sending the class of $x_ i$ to $b_ i \in A'$. This finishes the proof by taking $U = \mathop{\mathrm{Spec}}(A') \to \mathop{\mathrm{Spec}}(B)$ as the morphism $f$ and $u = \mathfrak p'$. $\square$

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. $\square$

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. $\square$

We strongly encourage the reader to skip the rest of this section.

Lemma 37.35.4. Consider a diagram

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

where $S$ be a locally Noetherian scheme and the morphisms are locally of finite type. Assume $\mathcal{O}_{S, s}$ is a G-ring. Assume further we are given a local $\mathcal{O}_{S, s}$-algebra map

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

and a local $\mathcal{O}_{S, s}$-algebra map

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

where $Y_\sigma = Y \times _{T, \sigma } \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^\wedge )$ and $y_\sigma $ is the unique point of $Y_\sigma $ lying over $y$. For every $N \geq 1$ there exists a commutative diagram

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

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

  1. $v \mapsto s$, $\tau (v) = t$, $f(w) = (x, v)$, and $w \mapsto (y, v)$,

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

  3. the diagram

    \[ \xymatrix{ \mathcal{O}_{S, s}^\wedge \ar[r] & \mathcal{O}_{V, v}^\wedge \\ \mathcal{O}_{T, t} \ar[r]^{\tau ^\sharp _ v} \ar[u]_\sigma & \mathcal{O}_{V, v} \ar[u] } \]

    commutes module $\mathfrak m_ v^ N$,

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

  5. the diagram

    \[ \xymatrix{ \mathcal{O}_{X, x} \ar[r]_\varphi & \mathcal{O}_{Y_\sigma , y_\sigma }^\wedge \ar[r] & \mathcal{O}_{Y_\sigma , y_\sigma }/\mathfrak m_{y_\sigma }^ N \ar@{=}[r] & \mathcal{O}_{Y \times _{T, \tau } V, (y, v)}/\mathfrak m_{(y, v)}^ N \ar[d]_{\cong } \\ \mathcal{O}_{X, x} \ar[r] \ar@{=}[u] & \mathcal{O}_{X \times _ S V, (x, v)} \ar[r]^{f^\sharp _ w} & \mathcal{O}_{W, w} \ar[r] & \mathcal{O}_{W, w}/\mathfrak m_ w^ N } \]

    commutes. The equality comes from the fact that $Y_\sigma $ and $Y \times _{T, \tau } V$ are canonically isomorphic over $\mathcal{O}_{V, v}/\mathfrak m_ v^ N = \mathcal{O}_{S, s}/\mathfrak m_ s^ N$ by parts (2) and (3).

Proof. After replacing $X$, $S$, $T$, $Y$ by affine open subschemes we may assume the diagram in the statement of the lemma comes from applying $\mathop{\mathrm{Spec}}$ to a diagram

\[ \vcenter { \xymatrix{ A & B \\ R \ar[u] \ar[r] & C \ar[u] } } \quad \text{with primes}\quad \vcenter { \xymatrix{ \mathfrak p_ A & \mathfrak p_ B \\ \mathfrak p_ R \ar@{-}[u] \ar@{-}[r] & \mathfrak p_ C \ar@{-}[u] } } \]

of Noetherian rings and finite type ring maps. In this proof every ring $E$ will be a Noetherian $R$-algebra endowed with a prime ideal $\mathfrak p_ E$ lying over $\mathfrak p_ R$ and all ring maps will be $R$-algebra maps compatible with the given primes. Moreover, if we write $E^\wedge $ we mean the completion of the localization of $E$ at $\mathfrak p_ E$. We will also use without further mention that an étale ring map $E_1 \to E_2$ such that $\kappa (\mathfrak p_{E_1}) = \kappa (\mathfrak p_{E_2})$ induces an isomorphism $E_1^\wedge = E_2^\wedge $ by More on Algebra, Lemma 15.43.9.

With this notation $\sigma $ and $\varphi $ correspond to ring maps

\[ \sigma : C \to R^\wedge \quad \text{and}\quad \varphi : A \longrightarrow (B \otimes _{C, \sigma } R^\wedge )^\wedge \]

Here is a picture

\[ \xymatrix{ A \ar@/^1em/[rrr]^\varphi & B \ar[r] & B \otimes _{C, \sigma } R^\wedge \ar[r] & (B \otimes _{C, \sigma } R^\wedge )^\wedge \\ R \ar[r] \ar[u] & C \ar[r]^\sigma \ar[u] & R^\wedge \ar[u] \ar[ru] } \]

Observe that $R^\wedge $ is a G-ring by More on Algebra, Proposition 15.50.6. Thus $B \otimes _{C, \sigma } R^\wedge $ is a G-ring by More on Algebra, Proposition 15.50.10. By Lemma 37.35.1 (translated into algebra) there exists an étale ring map $B \otimes _{C, \sigma } R^\wedge \to B'$ inducing an isomorphism $\kappa (\mathfrak p_{B \otimes _{C, \sigma } R^\wedge }) \to \kappa (\mathfrak p_{B'})$ and an $R$-algebra map $A \to B'$ such that the composition

\[ A \to B' \to (B')^\wedge = (B \otimes _{C, \sigma } R^\wedge )^\wedge \]

is the same as $\varphi $ modulo $(\mathfrak p_{(B \otimes _{C, \sigma } R^\wedge )^\wedge })^ N$. Thus we may replace $\varphi $ by this composition because the only way $\varphi $ enters the conclusion is via the commutativity requirement in part (5) of the statement of the lemma. Picture:

\[ \xymatrix{ & & B' \ar[r] & (B')^\wedge \ar@{=}[d] \\ A \ar[rru] & B \ar[r] & B \otimes _{C, \sigma } R^\wedge \ar[r] \ar[u] & (B \otimes _{C, \sigma } R^\wedge )^\wedge \\ R \ar[r] \ar[u] & C \ar[r]^\sigma \ar[u] & R^\wedge \ar[u] \ar[ru] } \]

Next, we use that $R^\wedge $ is a filtered colimit of smooth $R$-algebras (Smoothing Ring Maps, Theorem 16.12.1) because $R_{\mathfrak p_ R}$ is a G-ring by assumption. Since $C$ is of finite presentation over $R$ we get a factorization

\[ C \to R' \to R^\wedge \]

for some $R \to R'$ smooth, see Algebra, Lemma 10.127.3. After increasing $R'$ we may assume there exists an étale $B \otimes _ C R'$-algebra $B''$ whose base change to $B \otimes _{C, \sigma } R^\wedge $ is $B'$, see Algebra, Lemma 10.143.3. Then $B'$ is the filtered colimit of these $B''$ and we conclude that after increasing $R'$ we may assume there is an $R$-algebra map $A \to B''$ such that $A \to B'' \to B'$ is the previously constructed map (same reference as above). Picture

\[ \xymatrix{ & & B'' \ar[r] & B' \ar[r] & (B')^\wedge \ar@{=}[d] \\ A \ar[rru] & B \ar[r] & B \otimes _ C R' \ar[r] \ar[u] & B \otimes _{C, \sigma } R^\wedge \ar[r] \ar[u] & (B \otimes _{C, \sigma } R^\wedge )^\wedge \\ R \ar[r] \ar[u] & C \ar[r] \ar[u] & R' \ar[r] \ar[u] & R^\wedge \ar[u] \ar[ru] } \]

and

\[ B' = B'' \otimes _{(B \otimes _ C R')} (B \otimes _{C, \sigma } R^\wedge ) \]

This means that we may replace $C$ by $R'$, $\sigma : C \to R^\wedge $ by $R' \to R^\wedge $, and $B$ by $B''$ so that we simplify to the diagram

\[ \xymatrix{ A \ar[r] & B \ar[r] & B \otimes _{C, \sigma } R^\wedge \\ R \ar[r] \ar[u] & C \ar[r]^\sigma \ar[u] & R^\wedge \ar[u] } \]

with $\varphi $ equal to the composition of the horizontal arrows followed by the canonical map from $B \otimes _{C, \sigma } R^\wedge $ to its completion. The final step in the proof is to apply Lemma 37.35.1 (or its proof) one more time to $\mathop{\mathrm{Spec}}(C)$ and $\mathop{\mathrm{Spec}}(R)$ over $\mathop{\mathrm{Spec}}(R)$ and the map $C \to R^\wedge $. The lemma produces a ring map $C \to D$ such that $R \to D$ is étale, such that $\kappa (\mathfrak p_ R) = \kappa (\mathfrak p_ D)$, and such that

\[ C \to D \to D^\wedge = R^\wedge \]

is equal to $\sigma : C \to R^\wedge $ modulo $(\mathfrak p_{R^\wedge })^ N$. Then we can take

\[ V = \mathop{\mathrm{Spec}}(D) \quad \text{and}\quad W = \mathop{\mathrm{Spec}}(B \otimes _ C D) \]

as our solution to the problem posed by the lemma. Namely the diagram

\[ \xymatrix{ A \ar[r] & B \otimes _{C, \sigma } R^\wedge \ar[r] & B \otimes _{C, \sigma } R^\wedge /(\mathfrak p_{R^\wedge })^ N \ar@{=}[r] & B \otimes _ C D/(\mathfrak p_ D)^ N \\ A \ar@{=}[u] \ar[r] & A \otimes _ R D \ar[r] & B \otimes _ R D \ar[r] & B \otimes _ C D/(\mathfrak p_ D)^ N \ar@{=}[u] } \]

commutes because $C \to D \to D^\wedge = R^\wedge $ is equal to $\sigma $ modulo $(\mathfrak p_{R^\wedge })^ N$. This proves part (5) and the other properties are immediate from the construction. $\square$

Lemma 37.35.5. Let $T \to S$ be finite type morphisms of Noetherian schemes. Let $t \in T$ map to $s \in S$ and let $\sigma : \mathcal{O}_{T, t} \to \mathcal{O}_{S, s}^\wedge $ be a local $\mathcal{O}_{S, s}$-algebra map. For every $N \geq 1$ there exists a finite type morphism $(T', t') \to (T, t)$ such that $\sigma $ factors through $\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}$ and such that for every local $\mathcal{O}_{S, s}$-algebra map $\sigma ' : \mathcal{O}_{T, t} \to \mathcal{O}_{S, s}^\wedge $ which factors through $\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}$ the maps $\sigma $ and $\sigma '$ agree modulo $\mathfrak m_ s^ N$.

Proof. We may assume $S$ and $T$ are affine. Say $S = \mathop{\mathrm{Spec}}(R)$ and $T = \mathop{\mathrm{Spec}}(C)$. Let $c_1, \ldots , c_ n \in C$ be generators of $C$ as an $R$-algebra. Let $\mathfrak p \subset R$ be the prime ideal corresponding to $s$. Say $\mathfrak p = (f_1, \ldots , f_ m)$. After replacing $R$ by a principal localization (to clear denominators in $R_\mathfrak p$) we may assume there exist $r_1, \ldots , r_ n \in R$ and $a_{i, I} \in \mathcal{O}_{S, s}^\wedge $ where $I = (i_1, \ldots , i_ m)$ with $\sum i_ j = N$ such that

\[ \sigma (c_ i) = r_ i + \sum \nolimits _ I a_{i, I} f_1^{i_1} \ldots f_ m^{i_ m} \]

in $\mathcal{O}_{S, s}^\wedge $. Then we consider

\[ C' = C[t_{i, I}]/ \left(c_ i - r_ i - \sum \nolimits _ I t_{i, I} f_1^{i_1} \ldots f_ m^{i_ m}\right) \]

with $\mathfrak p' = \mathfrak pC' + (t_{i, I})$ and factorization of $\sigma : C \to \mathcal{O}_{S, s}^\wedge $ through $C'$ given by sending $t_{i, I}$ to $a_{i, I}$. Taking $T' = \mathop{\mathrm{Spec}}(C')$ works because any $\sigma '$ as in the statement of the lemma will send $c_ i$ to $r_ i$ modulo the maximal ideal to the power $N$. $\square$

Lemma 37.35.6. Let $Y \to T \to S$ be finite type morphisms of Noetherian schemes. Let $t \in T$ map to $s \in S$ and let $\sigma : \mathcal{O}_{T, t} \to \mathcal{O}_{S, s}^\wedge $ be a local $\mathcal{O}_{S, s}$-algebra map. There exists a finite type morphism $(T', t') \to (T, t)$ such that $\sigma $ factors through $\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}$ and such that for every local $\mathcal{O}_{S, s}$-algebra map $\sigma ' : \mathcal{O}_{T, t} \to \mathcal{O}_{S, s}^\wedge $ which factors through $\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}$ the closed immersions

\[ Y \times _{T, \sigma } \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^\wedge ) = Y_\sigma \longleftarrow Y_ t \longrightarrow Y_{\sigma '} = Y \times _{T, \sigma '} \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^\wedge ) \]

have isomorphic conormal algebras.

Proof. A useful observation is that $\kappa (s) = \kappa (t)$ by the existence of $\sigma $. Observe that the statement makes sense as the fibres of $Y_\sigma $ and $Y_{\sigma '}$ over $s \in \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^\wedge )$ are both canonically isomorphic to $Y_ t$. We will think of the property “$\sigma '$ factors through $\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}$” as a constraint on $\sigma '$. If we have several such constraints, say given by $(T'_ i, t'_ i) \to (T, t)$, $i = 1, \ldots , n$ then we can combined them by considering $(T'_1 \times _ T \ldots \times _ T T'_ n, (t'_1, \ldots , t'_ n)) \to (T, t)$. We will use this without further mention in the following.

By Lemma 37.35.5 we can assume that any $\sigma '$ as in the statement of the lemma is the same as $\sigma $ modulo $\mathfrak m_ s^2$. Note that the conormal algebra of $Y_ t$ in $Y_\sigma $ is just the quasi-coherent graded $\mathcal{O}_{Y_ t}$-algebra

\[ \bigoplus \nolimits _{n \geq 0} \mathfrak m_ s^ n\mathcal{O}_{Y_\sigma }/ \mathfrak m_ s^{n + 1}\mathcal{O}_{Y_\sigma } \]

and similarly for $Y_{\sigma '}$. Since $\sigma $ and $\sigma '$ agree modulo $\mathfrak m_ s^2$ we see that these two algebras are the same in degrees $0$ and $1$. On the other hand, these conormal algebras are generated in degree $1$ over degree $0$. Hence if there is an isomorphism extending the isomorphism just constructed in degrees $0$ and $1$, then it is unique.

We may assume $S$ and $T$ are affine. Let $Y = Y_1 \cup \ldots \cup Y_ n$ be an affine open covering. If we can construct $(T_ i', t'_ i) \to (T, t)$ as in the lemma such that the desired isomorphism (see previous paragraph) exists for $Y_ i \to T \to S$ and $\sigma $, then these glue by uniqueness to prove the result for $Y \to T$. Thus we may assume $Y$ is affine.

Write $S = \mathop{\mathrm{Spec}}(R)$, $T = \mathop{\mathrm{Spec}}(C)$, and $Y = \mathop{\mathrm{Spec}}(B)$. Choose a presentation $B = C[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$. Denote $R^\wedge = \mathcal{O}_{S, s}^\wedge $. Let $a_{kj} \in R^\wedge [x_1, \ldots , x_ n]$ be polynomials such that

\[ \sum \nolimits _{j = 1, \ldots , m} a_{kj}\sigma (f_ j) = 0,\quad \text{for }k = 1, \ldots , K \]

is a set of generators for the module of relations among the $\sigma (f_ j) \in R^\wedge [x_1, \ldots , x_ n]$. Thus we have an exact sequence

37.35.6.1
\begin{equation} \label{more-morphisms-equation-resolution} R^\wedge [x_1, \ldots , x_ n]^{\oplus K} \to R^\wedge [x_1, \ldots , x_ n]^{\oplus m} \to R^\wedge [x_1, \ldots , x_ n] \to B \otimes _{C, \sigma } R^\wedge \to 0 \end{equation}

Let $c$ be an integer which works in the Artin-Rees lemma for both the first and the second map in this sequence and the ideal $\mathfrak m_{R^\wedge }R^\wedge [x_1, \ldots , x_ n]$ as defined in More on Algebra, Section 15.4. Write

\[ a_{kj} = \sum \nolimits _{I \in \Omega } a_{kj, I} x^ I \quad \text{and}\quad f_ j = \sum \nolimits _{I \in \Omega } f_{j, I} x^ I \]

in multiindex notation where $a_{kj, I} \in R^\wedge $, $f_{j, I} \in C$, and $\Omega $ a finite set of multiindices. Then we see that

\[ \sum \nolimits _{j = 1, \ldots , m,\ I, I' \in \Omega ,\ I + I' = I''} a_{kj, I} \sigma (f_{j, I'}) = 0,\quad I''\text{ a multiindex} \]

in $R^\wedge $. Thus we take

\[ C' = C[t_{jk, I}]/ \left( \sum \nolimits _{j = 1, \ldots , m,\ I, I' \in \Omega ,\ I + I' = I''} t_{kj, I} f_{j, I'},\ I''\text{ a multiindex}\right) \]

Then $\sigma $ factors through a map $\tilde\sigma : C' \to R^\wedge $ sending $t_{kj, I}$ to $a_{jk, I}$. Thus $T' = \mathop{\mathrm{Spec}}(C')$ comes with a point $t' \in T'$ such that $\sigma $ factors through $\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}$. Let $t_{kj} = \sum t_{kj, I} x^ I$ in $C'[x_1, \ldots , x_ n]$. Then we see that we have a complex

37.35.6.2
\begin{equation} \label{more-morphisms-equation-resolution-new} C'[x_1, \ldots , x_ n]^{\oplus K} \to C'[x_1, \ldots , x_ n]^{\oplus m} \to C'[x_1, \ldots , x_ n] \to B \otimes _ C C' \to 0 \end{equation}

which is exact at $C'[x_1, \ldots , x_ n]$ and whose base change by $\tilde\sigma $ gives (37.35.6.1).

By Lemma 37.35.5 we can find a further morphism $(T'', t'') \to (T', t')$ such that $\tilde\sigma $ factors through $\mathcal{O}_{T', t'} \to \mathcal{O}_{T'', t''}$ and such that if $\sigma ' : C \to R^\wedge $ factors through $\mathcal{O}_{T'', t''}$, then the induced map $\tilde\sigma ' : C' \to R^\wedge $ agrees modulo $\mathfrak m_ s^{c + 1}$ with $\tilde\sigma $. Thus if $\sigma '$ is such a map, then we obtain a complex

\[ R^\wedge [x_1, \ldots , x_ n]^{\oplus K} \to R^\wedge [x_1, \ldots , x_ n]^{\oplus m} \to R^\wedge [x_1, \ldots , x_ n] \to B \otimes _{C, \sigma '} R^\wedge \to 0 \]

over $R^\wedge [x_1, \ldots , x_ n]$ by applying $\tilde\sigma '$ to the polynomials $t_{kj}$ and $f_ j$. In other words, this is the base change of the complex (37.35.6.2) by $\tilde\sigma '$. The matrices defining this complex are congruent modulo $\mathfrak m_ s^{c + 1}$ to the matrices defining the complex (37.35.6.1) because $\tilde\sigma $ and $\tilde\sigma '$ are congruent modulo $\mathfrak m_ s^{c + 1}$. Since (37.35.6.1) is exact, we can apply More on Algebra, Lemma 15.4.2 to conclude that

\[ \text{Gr}_{\mathfrak m_ s}(B \otimes _{C, \sigma '} R^\wedge ) \cong \text{Gr}_{\mathfrak m_ s}(B \otimes _{C, \sigma } R^\wedge ) \]

as desired. $\square$

Lemma 37.35.7. With notation an assumptions as in Lemma 37.35.4 assume that $\varphi $ induces an isomorphism on completions. Then we can choose our diagram such that $f$ is étale.

Proof. We may assume $N \geq 2$ and we may replace $(T, t)$ with $(T', t')$ as in Lemma 37.35.6. Since $(V, v) \to (S, s)$ is an elementary étale neighbourhood, so is $(X \times _ S V, (x, v)) \to (X, x)$. Thus $\mathcal{O}_{X, x} \to \mathcal{O}_{X \times _ S V, (x, v)}$ induces an isomorphism on completions by More on Algebra, Lemma 15.43.9. We claim $\mathcal{O}_{X, x} \to \mathcal{O}_{W, w}$ induces an isomorphism on completions. Having proved this, Lemma 37.12.1 will show that $f$ is smooth at $w$ and of course $f$ is unramified at $u$ as well, so Morphisms, Lemma 29.36.5 tells us $f$ is étale at $w$.

First we use the commutativity in part (5) of Lemma 37.35.4 to see that for $i \leq N$ there is a commutative diagram

\[ \xymatrix{ \text{Gr}^ i_{\mathfrak m_ x}(\mathcal{O}_{X, x}) \ar[r]_-\varphi & \text{Gr}^ i_{\mathfrak m_{y_\sigma }}(\mathcal{O}_{Y_\sigma , y_\sigma }^\wedge ) \ar@{=}[r] & \text{Gr}^ i_{\mathfrak m_{(y, v)}}(\mathcal{O}_{Y \times _{T, \tau } V, (y, v)}) \ar[d]_{\cong } \\ \text{Gr}^ i_{\mathfrak m_ x}(\mathcal{O}_{X, x}) \ar[r]^-{\cong } \ar@{=}[u] & \text{Gr}^ i_{\mathfrak m_{(x, v)}}(\mathcal{O}_{X \times _ S V, (x, v)}) \ar[r]^{f^\sharp _ w} & \text{Gr}^ i_{\mathfrak m_ w}(\mathcal{O}_{W, w}) } \]

This implies that $f^\sharp _ w$ defines an isomorphism $\kappa (x) \to \kappa (w)$ on residue fields and an isomorphism $\mathfrak m_ x/\mathfrak m_ x^2 \to \mathfrak m_ w/\mathfrak m_ w^2$ on cotangent spaces. Hence $f^\sharp _ w$ defines a surjection $\mathcal{O}_{X, x}^\wedge \to \mathcal{O}_{W, w}^\wedge $ on complete local rings.

By Lemma 37.35.6 there is an isomorphism of $\text{Gr}_{\mathfrak m_ s}(\mathcal{O}_{(Y \times _{T, \tau } V, (y, v)})$ with $\text{Gr}_{\mathfrak m_ s}(\mathcal{O}_{Y_\sigma , y_\sigma })$. This follows by taking stalks of the isomorphism of conormal sheaves at the point $y$. Since our local rings are Noetherian taking associated graded with respect to $\mathfrak m_ s$ commutes with completion because completion with respect to an ideal is an exact functor on finite modules over Noetherian rings. This produces the right vertical isomorphism in the diagram of graded rings

\[ \xymatrix{ \text{Gr}_{\mathfrak m_ s}(\mathcal{O}_{W, w}^\wedge ) & \text{Gr}_{\mathfrak m_ s} (\mathcal{O}_{(Y \times _{T, \tau } V, (y, v)}^\wedge ) \ar[l] \\ \text{Gr}_{\mathfrak m_ s}(\mathcal{O}_{X, x}^\wedge ) \ar[r]^\varphi \ar[u] & \text{Gr}_{\mathfrak m_ s}(\mathcal{O}_{Y_\sigma , y_\sigma }^\wedge ) \ar[u]_{\cong } } \]

We do not claim the diagram commutes. By the result of the previous paragraph the left arrow is surjective. The other three arrows are isomorphisms. It follows that the left arrow is a surjective map between isomorphic Noetherian rings. Hence it is an isomorphism by Algebra, Lemma 10.31.10 (you can argue this directly using Hilbert functions as well). In particular $\mathcal{O}_{X, x}^\wedge \to \mathcal{O}_{W, w}^\wedge $ must be injective as well as surjective which finishes the proof. $\square$


Comments (2)

Comment #2819 by Matthieu Romagny on

Typo in first sentence : if two schemes pointed schemes...


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 0CAT. Beware of the difference between the letter 'O' and the digit '0'.