The Stacks project

Lemma 70.11.6. Let $k$ be a field. Let $X$ be an algebraic space over $k$. Let $k'/k$ be a field extension. Let $x \in |X|$ be a point and let $x' \in |X_{k'}|$ be a point lying over $x$. The following are equivalent

  1. $X$ is geometrically reduced at $x$,

  2. $X_{k'}$ is geometrically reduced at $x'$.

In particular, $X$ is geometrically reduced over $k$ if and only if $X_{k'}$ is geometrically reduced over $k'$.

Proof. Choose an ├ętale morphism $U \to X$ where $U$ is a scheme and a point $u \in U$ mapping to $x \in |X|$. By Properties of Spaces, Lemma 64.4.3 we may choose a point $u' \in U_{k'} = U \times _ X X_{k'}$ mapping to both $u$ and $x'$. By Lemma 70.11.2 the lemma follows from the lemma for $U, u, u'$ which is Varieties, Lemma 33.6.6. $\square$

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