The Stacks project

Lemma 28.7.2. Let $X$ be a scheme. The following are equivalent:

  1. The scheme $X$ is normal.

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

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

  4. There exists an open covering $X = \bigcup X_ j$ such that each open subscheme $X_ j$ is normal.

Moreover, if $X$ is normal then every open subscheme is normal.

Proof. This is clear from the definitions. $\square$

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