Lemma 10.45.6. Let $k$ be a perfect field. Any reduced $k$ algebra is geometrically reduced over $k$. Let $R$, $S$ be $k$-algebras. Assume both $R$ and $S$ are reduced. Then the $k$-algebra $R \otimes _ k S$ is reduced.

**Proof.**
The first statement follows from Lemma 10.44.3. For the second statement use the first statement and Lemma 10.43.5.
$\square$

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