Lemma 72.11.4 (0E02)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 72: Algebraic Spaces over Fields
Section 72.11: Geometrically reduced algebraic spaces
Lemma 72.11.4 (cite)

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Lemma 72.11.4. Let $X$ be an algebraic space over a perfect field $k$ (for example $k$ has characteristic zero).

For $x \in |X|$ , if $\mathcal{O}_{X, \overline{x}}$ is reduced, then $X$ is geometrically reduced at $x$ .
If $X$ is reduced, then $X$ is geometrically reduced over $k$ .

Proof. The first statement follows from Algebra, Lemma 10.43.6 and the definition of a perfect field (Algebra, Definition 10.45.1). The second statement follows from the first. $\square$

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