Lemma 28.3.2 (01OL)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 28: Properties of Schemes
Section 28.3: Integral, irreducible, and reduced schemes
Lemma 28.3.2 (cite)

Lemma 28.3.2. Let $X$ be a scheme. The following are equivalent.

The scheme $X$ is reduced, see Schemes, Definition 26.12.1.
There exists an affine open covering $X = \bigcup U_ i$ such that each $\Gamma (U_ i, \mathcal{O}_ X)$ is reduced.
For every affine open $U \subset X$ the ring $\mathcal{O}_ X(U)$ is reduced.
For every open $U \subset X$ the ring $\mathcal{O}_ X(U)$ is reduced.

Proof. See Schemes, Lemmas 26.12.2 and 26.12.3. $\square$

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