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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Proof. To prove (1) combine Lemma 33.6.2 with Algebra, Lemma 10.43.5. To prove (2) combine Lemma 33.6.4 with Algebra, Lemma 10.43.5. To prove (3) note that $(X \times _ k Y)_{\overline{k}} = X_{\overline{k}} \times _{\overline{k}} Y_{\overline{k}}$ and use (2) as well as Lemma 33.6.4. To prove (4) use (3) combined with Lemma 33.6.3. $\square$
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