Lemma 33.6.7. Let $k$ be a field. Let $X$, $Y$ be schemes over $k$.

If $X$ is geometrically reduced at $x$, and $Y$ reduced, then $X \times _ k Y$ is reduced at every point lying over $x$.

If $X$ geometrically reduced over $k$ and $Y$ reduced, then $X \times _ k Y$ is reduced.

If $X$ and $Y$ are geometrically reduced over $k$, then $X \times _ k Y$ is geometrically reduced.

If $k$ is perfect and $X$ and $Y$ are reduced, then $X \times _ k Y$ is reduced.

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