Lemma 76.55.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. Then there is a maximal open subspace $X' \subset X$ such that $f|_{X'} : X' \to Y$ is at-worst-nodal of relative dimension $1$. Moreover, formation of $X'$ commutes with arbitrary base change.
Proof. Choose a commutative diagram
\[ \xymatrix{ U \ar[d] \ar[r]_ h & V \ar[d] \\ X \ar[r]^ f & Y } \]
where $U$, $V$ are schemes, the vertical arrows are étale, and $U \to X$ is surjective. By the lemma for the case of schemes (Algebraic Curves, Lemma 53.20.5) we find a maximal open subscheme $U' \subset U$ such that $h|_{U'} : U' \to V$ is at-worst-nodal of relative dimension $1$ and such that formation of $U'$ commutes with base change. Let $X' \subset X$ be the open subspace whose points correspond to the open subset $\mathop{\mathrm{Im}}(|U'| \to |X|)$. By Lemma 76.55.3 we see that $X' \to Y$ is at-worst-nodal of relative dimension $1$ and that $X'$ is the largest open subspace with this property (this also implies that $U'$ is the inverse image of $X'$ in $U$, but we do not need this). Since the same is true after base change the proof is complete. $\square$
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