The Stacks project

Lemma 33.6.3. Let $X$ be a scheme over a perfect field $k$ (e.g. $k$ has characteristic zero). Let $x \in X$. If $\mathcal{O}_{X, x}$ 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 Lemma 33.6.2 and 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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