Lemma 66.11.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of decent algebraic spaces over $S$. Let $x \in |X|$ be a point with image $y = f(x) \in |Y|$. The following are equivalent

1. $f$ induces an isomorphism $\kappa (y) \to \kappa (x)$, and

2. the induced morphism $\mathop{\mathrm{Spec}}(\kappa (x)) \to Y$ is a monomorphism.

Proof. Immediate from the discussion above. $\square$

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