Lemma 28.9.3. 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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