Lemma 53.20.10. Let $S = \mathop{\mathrm{lim}}\nolimits S_ i$ be a limit of a directed system of schemes with affine transition morphisms. Let $0 \in I$ and let $f_0 : X_0 \to Y_0$ be a morphism of schemes over $S_0$. Assume $S_0$, $X_0$, $Y_0$ are quasi-compact and quasi-separated. Let $f_ i : X_ i \to Y_ i$ be the base change of $f_0$ to $S_ i$ and let $f : X \to Y$ be the base change of $f_0$ to $S$. If
$f$ is at-worst-nodal of relative dimension $1$, and
$f_0$ is locally of finite presentation,
then there exists an $i \geq 0$ such that $f_ i$ is at-worst-nodal of relative dimension $1$.
Proof.
By Limits, Lemma 32.8.16 there exists an $i$ such that $f_ i$ is syntomic. Then $X_ i = \coprod _{d \geq 0} X_{i, d}$ is a disjoint union of open and closed subschemes such that $X_{i, d} \to Y_ i$ has relative dimension $d$, see Morphisms, Lemma 29.30.14. Because of the behaviour of dimensions of fibres under base change given in Morphisms, Lemma 29.28.3 we see that $X \to X_ i$ maps into $X_{i, 1}$. Then there exists an $i' \geq i$ such that $X_{i'} \to X_ i$ maps into $X_{i, 1}$, see Limits, Lemma 32.4.10. Thus $f_{i'} : X_{i'} \to Y_{i'}$ is syntomic of relative dimension $1$ (by Morphisms, Lemma 29.28.3 again). Consider the morphism $\text{Sing}(f_{i'}) \to Y_{i'}$. We know that the base change to $Y$ is an unramified morphism. Hence by Limits, Lemma 32.8.4 we see that after increasing $i'$ the morphism $\text{Sing}(f_{i'}) \to Y_{i'}$ becomes unramified. This finishes the proof.
$\square$
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