The Stacks project

Lemma 33.6.9. Let $k$ be a field. Let $X$ be a scheme over $k$. Let $x \in X$. Assume $X$ locally Noetherian and geometrically reduced at $x$. Then there exists an open neighbourhood $U \subset X$ of $x$ which is geometrically reduced over $k$.

Proof. Assume $X$ locally Noetherian and geometrically reduced at $x$. By Properties, Lemma 28.29.8 we can find an affine open neighbourhood $U \subset X$ of $x$ such that $R = \mathcal{O}_ X(U) \to \mathcal{O}_{X, x}$ is injective. By Lemma 33.6.2 the assumption means that $\mathcal{O}_{X, x}$ is geometrically reduced over $k$. By Algebra, Lemma 10.43.2 this implies that $R$ is geometrically reduced over $k$, which in turn implies that $U$ is geometrically reduced. $\square$

Comments (1)

Comment #766 by Keenan Kidwell on

The equality should be .

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