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.

Proof. This follows immediately from Spaces over Fields, Lemma 72.11.6 and the definition of the equivalence relation defining $|X|$ given in Properties of Spaces, Section 66.4. $\square$

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