Lemma 53.20.5. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Then there is a maximal open subscheme $U \subset X$ such that $f|_ U : U \to S$ is at-worst-nodal of relative dimension $1$. Moreover, formation of $U$ commutes with arbitrary base change.
Proof. By Morphisms, Lemma 29.30.12 we find that there is such an open where $f$ is syntomic. Hence we may assume that $f$ is a syntomic morphism. In particular $f$ is a Cohen-Macaulay morphism (Duality for Schemes, Lemmas 48.25.5 and 48.25.4). Thus $X$ is a disjoint union of open and closed subschemes on which $f$ has given relative dimension, see Morphisms, Lemma 29.29.4. This decomposition is preserved by arbitrary base change, see Morphisms, Lemma 29.29.2. Discarding all but one piece we may assume $f$ is syntomic of relative dimension $1$. Let $\text{Sing}(f) \subset X$ be the closed subscheem defined by the first fitting ideal of $\Omega _{X/S}$. There is a maximal open subscheme $W \subset \text{Sing}(f)$ such that $W \to S$ is unramified and its formation commutes with base change (Morphisms, Lemma 29.35.15). Since also formation of $\text{Sing}(f)$ commutes with base change (Divisors, Lemma 31.10.1), we see that
\[ U = (X \setminus \text{Sing}(f)) \cup W \]
is the maximal open subscheme of $X$ such that $f|_ U : U \to S$ is at-worst-nodal of relative dimension $1$ and that formation of $U$ commutes with base change. $\square$
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