The Stacks project

27.4 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 27.4.1. 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 27.4.2. 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 27.4.3. 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 25.11.6 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 27.4.1 (1) we get $P(R_{f_ j})$. Whereupon $P(R)$ by Definition 27.4.1 (2). $\square$

Here is a sample application.

Lemma 27.4.4. Let $X$ be a scheme. Then $X$ is reduced if and only if $X$ is “locally reduced” in the sense of Definition 27.4.2.

Proof. This is clear from Lemma 27.3.2. $\square$

Lemma 27.4.5. The following properties of a ring $R$ are local.

  1. (Cohen-Macaulay.) The ring $R$ is Noetherian and CM, see Algebra, Definition 10.103.6.

  2. (Regular.) The ring $R$ is Noetherian and regular, see Algebra, Definition 10.109.7.

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

  4. Add more here as needed.1

Proof. Omitted. $\square$

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

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