Lemma 10.43.3 (04KN)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.43: Geometrically reduced algebras
Lemma 10.43.3 (cite)

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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. $\square$

Comments (2)

Comment #3535 by Dario Weißmann on August 21, 2018 at 17:29

The first part of the lemma is already mentioned in lemma 030T, (4).

Comment #3667 by Johan on October 22, 2018 at 10:06

Yes, OK, but let's leave it for now (neither lemma has a written out proof).

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