## 28.12 Serre's conditions

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 $k \geq 0$.

1. We say $X$ is regular in codimension $k$, or we say $X$ has property $(R_ k)$ if for every $x \in X$ we have

$\dim (\mathcal{O}_{X, x}) \leq k \Rightarrow \mathcal{O}_{X, x}\text{ is regular}$
2. 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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