Definition 33.12.1. Let $k$ be a field. Let $X$ be a locally Noetherian scheme over $k$.

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

2. We say $X$ is geometrically regular over $k$ if $X$ is geometrically regular at all of its points.

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