Lemma 76.3.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume
$f$ is locally of finite type,
for every étale morphism $V \to Y$ the map $|X \times _ Y V| \to |V|$ is injective.
Then $f$ is universally injective.
Proof. The question is étale local on $Y$ by Morphisms of Spaces, Lemma 67.19.6. Hence we may assume that $Y$ is a scheme. Then $Y$ is in particular decent and by Decent Spaces, Lemma 68.18.9 we see that $f$ is locally quasi-finite. Let $y \in Y$ be a point and let $X_ y$ be the scheme theoretic fibre. Assume $X_ y$ is not empty. By Spaces over Fields, Lemma 72.10.8 we see that $X_ y$ is a scheme which is locally quasi-finite over $\kappa (y)$. Since $|X_ y| \subset |X|$ is the fibre of $|X| \to |Y|$ over $y$ we see that $X_ y$ has a unique point $x$. The same is true for $X_ y \times _{\mathop{\mathrm{Spec}}(\kappa (y))} \mathop{\mathrm{Spec}}(k)$ for any finite separable extension $k/\kappa (y)$ because we can realize $k$ as the residue field at a point lying over $y$ in an étale scheme over $Y$, see More on Morphisms, Lemma 37.35.2. Thus $X_ y$ is geometrically connected, see Varieties, Lemma 33.7.11. This implies that the finite extension $\kappa (x)/\kappa (y)$ is purely inseparable.
We conclude (in the case that $Y$ is a scheme) that for every $y \in Y$ either the fibre $X_ y$ is empty, or $(X_ y)_{red} = \mathop{\mathrm{Spec}}(\kappa (x))$ with $\kappa (y) \subset \kappa (x)$ purely inseparable. Hence $f$ is radicial (some details omitted), whence universally injective by Lemma 76.3.2. $\square$
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