Lemma 10.43.3. Let $k$ be a field. If $R$ is geometrically reduced over $k$, and $S \subset R$ is a multiplicative subset, then the localization $S^{-1}R$ is geometrically reduced over $k$. If $R$ is geometrically reduced over $k$, then $R[x]$ is geometrically reduced over $k$.

**Proof.**
Omitted. Hints: A localization of a reduced ring is reduced, and localization commutes with tensor products.
$\square$

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## Comments (2)

Comment #3535 by Dario Weißmann on

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