Lemma 79.11.1. Let S be a scheme. Let B be an algebraic space over S. Let f : X \to Y and g : X \to Z be morphisms of algebraic spaces over B. Assume
Y \to B is separated,
g is surjective, flat, and locally of finite presentation,
there is a scheme theoretically dense open V \subset Z such that f|_{g^{-1}(V)} : g^{-1}(V) \to Y factors through V.
Then f factors through g.
Proof. Set R = X \times _ Z X. By (2) we see that Z = X/R as sheaves. Also (2) implies that the inverse image of V in R is scheme theoretically dense in R (Morphisms of Spaces, Lemma 67.30.11). The we see that the two compositions R \to X \to Y are equal by Morphisms of Spaces, Lemma 67.17.8. The lemma follows. \square
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