Lemma 27.9.3. Let $X$ be a scheme. The following are equivalent:

The scheme $X$ is regular.

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

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

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

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

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