Lemma 28.8.3. Let X be a scheme. The following are equivalent:
The scheme X is Cohen-Macaulay.
For every affine open U \subset X the ring \mathcal{O}_ X(U) is Noetherian and Cohen-Macaulay.
There exists an affine open covering X = \bigcup U_ i such that each \mathcal{O}_ X(U_ i) is Noetherian and Cohen-Macaulay.
There exists an open covering X = \bigcup X_ j such that each open subscheme X_ j is Cohen-Macaulay.
Moreover, if X is Cohen-Macaulay then every open subscheme is Cohen-Macaulay.
Comments (0)