The Stacks project

Lemma 72.11.7. Let $k$ be a field. Let $f : X \to Y$ be a morphism of algebraic spaces over $k$. Let $x \in |X|$ be a point with image $y \in |Y|$.

  1. if $f$ is étale at $x$, then $X$ is geometrically reduced at $x$ $\Leftrightarrow $ $Y$ is geometrically reduced at $y$,

  2. if $f$ is surjective étale, then $X$ is geometrically reduced $\Leftrightarrow $ $Y$ is geometrically reduced.

Proof. Part (1) is clear because $\mathcal{O}_{X, \overline{x}} = \mathcal{O}_{Y, \overline{y}}$ if $f$ is étale at $x$. Part (2) follows immediately from part (1). $\square$

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