The Stacks project

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. $\square$


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