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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