The Stacks project

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

  1. The scheme $X$ is reduced, see Schemes, Definition 26.12.1.

  2. There exists an affine open covering $X = \bigcup U_ i$ such that each $\Gamma (U_ i, \mathcal{O}_ X)$ is reduced.

  3. For every affine open $U \subset X$ the ring $\mathcal{O}_ X(U)$ is reduced.

  4. For every open $U \subset X$ the ring $\mathcal{O}_ X(U)$ is reduced.

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  • 3 comment(s) on Section 28.3: Integral, irreducible, and reduced schemes

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