The Stacks project

Lemma 76.29.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $y \in |Y|$. The following are equivalent

1. for some morphism $\mathop{\mathrm{Spec}}(k) \to Y$ in the equivalence class of $y$ the algebraic space $X_ k$ is geometrically reduced over $k$,

2. for every morphism $\mathop{\mathrm{Spec}}(k) \to Y$ in the equivalence class of $y$ the algebraic space $X_ k$ is geometrically reduced over $k$,

3. for every morphism $\mathop{\mathrm{Spec}}(k) \to Y$ in the equivalence class of $y$ the algebraic space $X_ k$ is reduced.