Lemma 74.12.2. The property \mathcal{P}(f) =“f is locally separated” is fppf local on the base.
Proof. A base change of a locally separated morphism is locally separated, see Morphisms of Spaces, Lemma 67.4.4. Hence the direct implication in Definition 74.10.1.
Let \{ Y_ i \to Y\} _{i \in I} be an fppf covering of algebraic spaces over S. Let f : X \to Y be a morphism of algebraic spaces over S. Assume each base change X_ i := Y_ i \times _ Y X \to Y_ i is locally separated. This means that each of the morphisms
is an immersion. The base change of a fppf covering is an fppf covering, see Topologies on Spaces, Lemma 73.7.3 hence \{ Y_ i \times _ Y (X \times _ Y X) \to X \times _ Y X\} is an fppf covering of algebraic spaces. Moreover, each \Delta _ i is the base change of the morphism \Delta : X \to X \times _ Y X. Hence it follows from Lemma 74.12.1 that \Delta is a immersion, i.e., f is locally separated. \square
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