The Stacks project

Lemma 29.47.4. Let $X$ be a scheme. The following are equivalent:

  1. The scheme $X$ is seminormal.

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

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

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

Moreover, if $X$ is seminormal then every open subscheme is seminormal. The same statements are true with “seminormal” replaced by “absolutely weakly normal”.

Proof. Combine Properties, Lemma 28.4.3 and Lemma 29.47.2. $\square$


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