The Stacks project

Proof. Using the embedding (99.15.1.1), the description of the image, and the corresponding fact for $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ (Lemma 99.13.7) this reduces to the following statement: given a morphism $X' \to S'$ of an algebraic space to $S'$ which is of finite presentation, flat, proper then $X' \to S'$ has relative dimension $\leq 1$ if and only if $S \times _{S'} X' \to S$ and $T' \times _{S'} X' \to T'$ have relative dimension $\leq 1$. One implication follows from the fact that having relative dimension $\leq 1$ is preserved under base change (Morphisms of Spaces, Lemma 67.34.3). The other follows from the fact that having relative dimension $\leq 1$ is checked on the fibres and that the fibres of $X' \to S'$ (over points of the scheme $S'$) are the same as the fibres of $S \times _{S'} X' \to S$ since $S \to S'$ is a thickening by More on Morphisms, Lemma 37.14.3. $\square$