28.4 Types of schemes defined by properties of rings

In this section we study what properties of rings allow one to define local properties of schemes.

Definition 28.4.1. Let $P$ be a property of rings. We say that $P$ is local if the following hold:

1. For any ring $R$, and any $f \in R$ we have $P(R) \Rightarrow P(R_ f)$.

2. For any ring $R$, and $f_ i \in R$ such that $(f_1, \ldots , f_ n) = R$ then $\forall i, P(R_{f_ i}) \Rightarrow P(R)$.

Definition 28.4.2. Let $P$ be a property of rings. Let $X$ be a scheme. We say $X$ is locally $P$ if for any $x \in X$ there exists an affine open neighbourhood $U$ of $x$ in $X$ such that $\mathcal{O}_ X(U)$ has property $P$.

This is only a good notion if the property is local. Even if $P$ is a local property we will not automatically use this definition to say that a scheme is “locally $P$” unless we also explicitly state the definition elsewhere.

Lemma 28.4.3. Let $X$ be a scheme. Let $P$ be a local property of rings. The following are equivalent:

1. The scheme $X$ is locally $P$.

2. For every affine open $U \subset X$ the property $P(\mathcal{O}_ X(U))$ holds.

3. There exists an affine open covering $X = \bigcup U_ i$ such that each $\mathcal{O}_ X(U_ i)$ satisfies $P$.

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

Moreover, if $X$ is locally $P$ then every open subscheme is locally $P$.

Proof. Of course (1) $\Leftrightarrow$ (3) and (2) $\Rightarrow$ (1). If (3) $\Rightarrow$ (2), then the final statement of the lemma holds and it follows easily that (4) is also equivalent to (1). Thus we show (3) $\Rightarrow$ (2).

Let $X = \bigcup U_ i$ be an affine open covering, say $U_ i = \mathop{\mathrm{Spec}}(R_ i)$. Assume $P(R_ i)$. Let $\mathop{\mathrm{Spec}}(R) = U \subset X$ be an arbitrary affine open. By Schemes, Lemma 26.11.6 there exists a standard covering of $U = \mathop{\mathrm{Spec}}(R)$ by standard opens $D(f_ j)$ such that each ring $R_{f_ j}$ is a principal localization of one of the rings $R_ i$. By Definition 28.4.1 (1) we get $P(R_{f_ j})$. Whereupon $P(R)$ by Definition 28.4.1 (2). $\square$

Here is a sample application.

Lemma 28.4.4. Let $X$ be a scheme. Then $X$ is reduced if and only if $X$ is “locally reduced” in the sense of Definition 28.4.2.

Proof. This is clear from Lemma 28.3.2. $\square$

Lemma 28.4.5. The following properties of a ring $R$ are local.

1. (Cohen-Macaulay.) The ring $R$ is Noetherian and CM, see Algebra, Definition 10.104.6.

2. (Regular.) The ring $R$ is Noetherian and regular, see Algebra, Definition 10.110.7.

3. (Absolutely Noetherian.) The ring $R$ is of finite type over $Z$.

4. Add more here as needed.1

Proof. Omitted. $\square$

 But we only list those properties here which we have not already dealt with separately somewhere else.

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