The Stacks project

First proof using Artin approximation. We first use absolute Noetherian approximation to reduce to the case of schemes of finite type over $\mathbf{Z}$. The question is local on $X$ and $S$. Hence we may assume that $X$ and $S$ are affine. Then we can write $S = \mathop{\mathrm{Spec}}(R)$ and write $R$ as a filtered colimit $R = \mathop{\mathrm{colim}}\nolimits R_ i$ of finite type $\mathbf{Z}$-algebras. Using Limits, Lemma 32.10.1 we can find an $i$ and a morphism $f_ i : X_ i \to \mathop{\mathrm{Spec}}(R_ i)$ whose base change to $S$ is $f$. After increasing $i$ we may assume that $f_ i$ is at-worst-nodal of relative dimension $1$, see Lemma 53.20.10. The image $x_ i \in X_ i$ of $x$ will be a singular point of its fibre, for example because the formation of $\text{Sing}(f)$ commutes with base change (Divisors, Lemma 31.10.1). If we can prove the lemma for $f_ i : X_ i \to S_ i$ and $x_ i$, then the lemma follows for $f : X \to S$ by base change. Thus we reduce to the case studied in the next paragraph.

Assume $S$ is of finite type over $\mathbf{Z}$. Let $s \in S$ be the image of $x$. Recall that $\kappa (x)$ is a finite separable extension of $\kappa (s)$, for example because $\text{Sing}(f) \to S$ is unramified or because $x$ is a node of the fibre $X_ s$ and we can apply Lemma 53.19.7. Furthermore, let $\kappa '/\kappa (x)$ be the degree $2$ separable algebra associated to $\mathcal{O}_{X_ s, x}$ in Remark 53.19.8. By More on Morphisms, Lemma 37.35.2 we can choose an étale neighbourhood $(V, v) \to (S, s)$ such that the extension $\kappa (v)/\kappa (s)$ realizes either the extension $\kappa (x)/\kappa (s)$ in case $\kappa ' \cong \kappa (x) \times \kappa (x)$ or the extension $\kappa '/\kappa (s)$ if $\kappa '$ is a field. After replacing $X$ by $X \times _ S V$ and $S$ by $V$ we reduce to the situation described in the next paragraph.

Assume $S$ is of finite type over $\mathbf{Z}$ and $x \in X_ s$ is a split node, see Definition 53.19.10. By Lemma 53.20.11 we see that there exists an $\mathcal{O}_{S, s}$-algebra isomorphism

for some $h \in \mathfrak m_ s^\wedge \subset \mathcal{O}_{S, s}^\wedge $. In other words, if we consider the homomorphism

sending $a$ to $h$, then there exists an $\mathcal{O}_{S, s}$-algebra isomorphism

where

and $y_\sigma $ is the point of $Y_\sigma $ lying over the closed point of $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^\wedge )$ and having coordinates $u, v$ equal to zero. Since $\mathcal{O}_{S, s}$ is a G-ring by More on Algebra, Proposition 15.50.12 we may apply More on Morphisms, Lemma 37.39.3 to conclude. $\square$

Proof not using Artin approximation. We only sketch this proof; it was contributed by Mohan Swaminathan and it mimicks an argument of [Proposition X.2.1, ACG]. Contrary to the previous proof, this proof uses the actual equations.

In exactly the same manner as in the first proof we reduce to the case where $S$ is of finite type over $\mathbf{Z}$ and $x \in X_ s$ is a split node, see Definition 53.19.10. By Lemma 53.19.11 we see that there exists an $\kappa (s)$-algebra isomorphism

Note that the tangent space of $X_ s$ at $x$ has dimension $2$. Thus by Varieties, Lemma 33.18.5 after Zariski shrinking we may assume there exists a closed immersion $X \to Y$ of schemes over $S$ with $Y \to S$ smooth of relative dimension $2$. Since $X$ is syntomic of relative dimension $1$ (by definition), we see that $X$ is an effective Cartier divisor in $Y$ (follows from Divisors, Lemma 31.22.10). Thus after shrinking $Y$ and $X$ we may assume $Y = \mathop{\mathrm{Spec}}(B)$ is affine and that there exists a nonzerodivisor $g \in B$ such that $X = \mathop{\mathrm{Spec}}(B/gB)$.

Write $S = \mathop{\mathrm{Spec}}(R)$ and denote $\mathfrak p \subset R$ be the prime corresponding to $s$. Let $\mathfrak q \subset B$ be the prime ideal corresponding to $x$ (viewed as a point of $Y$). We have maps

where the second arrow becomes an isomorphism after completion. After replacing $B$ by a suitable princpal localization, we may assume that there exist $U, V \in B$ which map to $u$ and $v$ up to $(u, v)^2$. Then the completion of $B_\mathfrak q/\mathfrak p B_\mathfrak q$ is $\kappa (s)[[U, V]]$. Furthermore, after shrinking $Y$ and after multiplying $g$ by a unit we may assume that $g$ maps to $UV$ plus higher order terms in $\kappa (s)[[U, V]]$. After shrinking $Y$ we may further assume

1. $\mathfrak q = \mathfrak pB + (U, V)$

2. $\Omega _{B/R}$ is free on $\text{d}U$ and $\text{d}V$

Write $\text{d}(g) = g_2 \text{d}U + g_1 \text{d}V$ with $g_1, g_2 \in B$. Note that the ideal $J = (g_1, g_2) \subset B$ is independent of the choice of the “coordinates” $U$ and $V$ (for fixed $g$). Looking modulo $\mathfrak p B_\mathfrak q + (U, V)^3B_\mathfrak q$ we see that $g_1$ and $g_2$ are congruent to $U$ and $V$ modulo $\mathfrak p B_\mathfrak q + (U, V)^2B_\mathfrak q$. Thus we may redo the argument above with $U$ replaced by $g_1$ and $V$ replaced by $g_2$. Then we find that $J = (U, V)$ which implies that $g_1, g_2 \in (U, V) = J$ (of course these are different from the $g_1$ and $g_2$ with our previous choice of coordinates).

Note that $\mathop{\mathrm{Spec}}(B/J) \to S = \mathop{\mathrm{Spec}}(R)$ is étale at $x$. Thus after replacing $S$ by an étale neighbourhood and shrinking $Y$ again, we may assume that $R \to B \to B/J$ is an isomorphism. Denote $r \in R$ the element whose image in $B/J$ is the image of $-g$. Then we can write

for some $a, b \in R$. Taking derivatives using that $\text{d}g \in J\Omega _{B/R}$ we find $a = b = 0$. Thus we can write

for some $\alpha , \beta , \gamma \in B$. Then $\beta $ maps to $1$ in $\kappa (x)$ and $\alpha $ and $\gamma $ map to $0$ in $\kappa (x)$. Thus we may assume $\beta $ and $\alpha + \beta + \gamma $ are invertible in $B$. Consider the cover $Y'$ of $Y$ given by

with point $x'$ the unique point over $x$ with $t = 1$. Note that $Y' \to Y$ is étale at $x'$. Consider the elements

of $\Gamma (Y', \mathcal{O}_{Y'})$. Then we see that

Since $T_1$ and $T_2$ are also coordinates in the sense above, we see that the morphism $Y' \to \mathbf{A}^2_ S$ given by $T_1$ and $T_2$ is étale in a neighbourhood of $x'$ and that the pullback $X' \subset Y'$ of $X$ is the inverse image of the closed subscheme of $\mathbf{A}^2_ S = \mathop{\mathrm{Spec}}(R[x_1, x_2])$ given by the equation $x_1x_2 - r = 0$ (again after possibly shrinking $Y'$). We leave it to the reader to put everything together to obtain the desired diagram. $\square$