Lemma 27.3.2. Let $X$ be a scheme. The following are equivalent.

The scheme $X$ is reduced, see Schemes, Definition 25.12.1.

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.

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