Here are two technical notions that are often useful. See also Cohomology of Schemes, Section 30.11.
Definition 28.12.1. Let X be a locally Noetherian scheme. Let k \geq 0.
We say X is regular in codimension k, or we say X has property (R_ k) if for every x \in X we have
We say X has property (S_ k) if for every x \in X we have \text{depth}(\mathcal{O}_{X, x}) \geq \min (k, \dim (\mathcal{O}_{X, x})).
The phrase “regular in codimension k” makes sense since we have seen in Section 28.11 that if Y \subset X is irreducible closed with generic point x, then \dim (\mathcal{O}_{X, x}) = \text{codim}(Y, X). For example condition (R_0) means that for every generic point \eta \in X of an irreducible component of X the local ring \mathcal{O}_{X, \eta } is a field. But for general Noetherian schemes it can happen that the regular locus of X is badly behaved, so care has to be taken.
Lemma 28.12.2. Let X be a locally Noetherian scheme. Then X is regular if and only if X has (R_ k) for all k \geq 0.
Proof. Follows from Lemma 28.9.2 and the definitions. \square
Lemma 28.12.3. Let X be a locally Noetherian scheme. Then X is Cohen-Macaulay if and only if X has (S_ k) for all k \geq 0.
Proof. By Lemma 28.8.2 we reduce to looking at local rings. Hence the lemma is true because a Noetherian local ring is Cohen-Macaulay if and only if it has depth equal to its dimension. \square
Lemma 28.12.4. Let X be a locally Noetherian scheme. Then X is reduced if and only if X has properties (S_1) and (R_0).
Proof. This is Algebra, Lemma 10.157.3. \square
Lemma 28.12.5. Let X be a locally Noetherian scheme. Then X is normal if and only if X has properties (S_2) and (R_1).
Proof. This is Algebra, Lemma 10.157.4. \square
Lemma 28.12.6. Let X be a locally Noetherian scheme which is normal and has dimension \leq 1. Then X is regular.
Proof. This follows from Lemma 28.12.5 and the definitions. \square
Lemma 28.12.7. Let X be a locally Noetherian scheme which is normal and has dimension \leq 2. Then X is Cohen-Macaulay.
Proof. This follows from Lemma 28.12.5 and the definitions. \square
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