Definition 33.6.1. Let $k$ be a field. Let $X$ be a scheme over $k$.

1. Let $x \in X$ be a point. We say $X$ is geometrically reduced at $x$ if for any field extension $k'/k$ and any point $x' \in X_{k'}$ lying over $x$ the local ring $\mathcal{O}_{X_{k'}, x'}$ is reduced.

2. We say $X$ is geometrically reduced over $k$ if $X$ is geometrically reduced at every point of $X$.

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