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

The scheme $X$ is seminormal.

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

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

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”.

## Comments (0)