The Stacks project

Lemma 29.55.11. Let $X$ be a scheme such that every quasi-compact open has finitely many irreducible components. The following are equivalent:

  1. The scheme $X$ is weakly normal.

  2. For every affine open $U \subset X$ the ring $\mathcal{O}_ X(U)$ satisfies conditions (1) and (2) of Lemma 29.55.10.

  3. There exists an affine open covering $X = \bigcup U_ i$ such that each ring $\mathcal{O}_ X(U_ i)$ satisfies conditions (1) and (2) of Lemma 29.55.10.

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

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

Proof. The condition to $X$ be weakly normal is that the morphism $X^{wn} = X^{X^\nu /wn} \to X$ is an isomorphism. Since the construction of $X^\nu \to X$ commutes with base change to open subschemes and since the construction of $X^{X^\nu /wn}$ commutes with base change to open subschemes of $X$ (Lemma 29.55.5) the lemma is clear. $\square$

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