29.14 Types of morphisms defined by properties of ring maps

In this section we study what properties of ring maps allow one to define local properties of morphisms of schemes.

Definition 29.14.1. Let $P$ be a property of ring maps.

1. We say that $P$ is local if the following hold:

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

2. For any rings $R$, $A$, any $f \in R$, $a\in A$, and any ring map $R_ f \to A$ we have $P(R_ f \to A) \Rightarrow P(R \to A_ a)$.

3. For any ring map $R \to A$, and $a_ i \in A$ such that $(a_1, \ldots , a_ n) = A$ then $\forall i, P(R \to A_{a_ i}) \Rightarrow P(R \to A)$.

2. We say that $P$ is stable under base change if for any ring maps $R \to A$, $R \to R'$ we have $P(R \to A) \Rightarrow P(R' \to R' \otimes _ R A)$.

3. We say that $P$ is stable under composition if for any ring maps $A \to B$, $B \to C$ we have $P(A \to B) \wedge P(B \to C) \Rightarrow P(A \to C)$.

Definition 29.14.2. Let $P$ be a property of ring maps. Let $f : X \to S$ be a morphism of schemes. We say $f$ is locally of type $P$ if for any $x \in X$ there exists an affine open neighbourhood $U$ of $x$ in $X$ which maps into an affine open $V \subset S$ such that the induced ring map $\mathcal{O}_ S(V) \to \mathcal{O}_ X(U)$ has property $P$.

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

Lemma 29.14.3. Let $f : X \to S$ be a morphism of schemes. Let $P$ be a property of ring maps. Let $U$ be an affine open of $X$, and $V$ an affine open of $S$ such that $f(U) \subset V$. If $f$ is locally of type $P$ and $P$ is local, then $P(\mathcal{O}_ S(V) \to \mathcal{O}_ X(U))$ holds.

Proof. As $f$ is locally of type $P$ for every $u \in U$ there exists an affine open $U_ u \subset X$ mapping into an affine open $V_ u \subset S$ such that $P(\mathcal{O}_ S(V_ u) \to \mathcal{O}_ X(U_ u))$ holds. Choose an open neighbourhood $U'_ u \subset U \cap U_ u$ of $u$ which is standard affine open in both $U$ and $U_ u$, see Schemes, Lemma 26.11.5. By Definition 29.14.1 (1)(b) we see that $P(\mathcal{O}_ S(V_ u) \to \mathcal{O}_ X(U'_ u))$ holds. Hence we may assume that $U_ u \subset U$ is a standard affine open. Choose an open neighbourhood $V'_ u \subset V \cap V_ u$ of $f(u)$ which is standard affine open in both $V$ and $V_ u$, see Schemes, Lemma 26.11.5. Then $U'_ u = f^{-1}(V'_ u) \cap U_ u$ is a standard affine open of $U_ u$ (hence of $U$) and we have $P(\mathcal{O}_ S(V'_ u) \to \mathcal{O}_ X(U'_ u))$ by Definition 29.14.1 (1)(a). Hence we may assume both $U_ u \subset U$ and $V_ u \subset V$ are standard affine open. Applying Definition 29.14.1 (1)(b) one more time we conclude that $P(\mathcal{O}_ S(V) \to \mathcal{O}_ X(U_ u))$ holds. Because $U$ is quasi-compact we may choose a finite number of points $u_1, \ldots , u_ n \in U$ such that

$U = U_{u_1} \cup \ldots \cup U_{u_ n}.$

By Definition 29.14.1 (1)(c) we conclude that $P(\mathcal{O}_ S(V) \to \mathcal{O}_ X(U))$ holds. $\square$

Lemma 29.14.4. Let $P$ be a local property of ring maps. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

1. The morphism $f$ is locally of type $P$.

2. For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ we have $P(\mathcal{O}_ S(V) \to \mathcal{O}_ X(U))$.

3. There exists an open covering $S = \bigcup _{j \in J} V_ j$ and open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that each of the morphisms $U_ i \to V_ j$, $j\in J, i\in I_ j$ is locally of type $P$.

4. There exists an affine open covering $S = \bigcup _{j \in J} V_ j$ and affine open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that $P(\mathcal{O}_ S(V_ j) \to \mathcal{O}_ X(U_ i))$ holds, for all $j\in J, i\in I_ j$.

Moreover, if $f$ is locally of type $P$ then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_ U : U \to V$ is locally of type $P$.

Proof. This follows from Lemma 29.14.3 above. $\square$

Lemma 29.14.5. Let $P$ be a property of ring maps. Assume $P$ is local and stable under composition. The composition of morphisms locally of type $P$ is locally of type $P$.

Proof. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms locally of type $P$. Let $x \in X$. Choose an affine open neighbourhood $W \subset Z$ of $g(f(x))$. Choose an affine open neighbourhood $V \subset g^{-1}(W)$ of $f(x)$. Choose an affine open neighbourhood $U \subset f^{-1}(V)$ of $x$. By Lemma 29.14.4 the ring maps $\mathcal{O}_ Z(W) \to \mathcal{O}_ Y(V)$ and $\mathcal{O}_ Y(V) \to \mathcal{O}_ X(U)$ satisfy $P$. Hence $\mathcal{O}_ Z(W) \to \mathcal{O}_ X(U)$ satisfies $P$ as $P$ is assumed stable under composition. $\square$

Lemma 29.14.6. Let $P$ be a property of ring maps. Assume $P$ is local and stable under base change. The base change of a morphism locally of type $P$ is locally of type $P$.

Proof. Let $f : X \to S$ be a morphism locally of type $P$. Let $S' \to S$ be any morphism. Denote $f' : X_{S'} = S' \times _ S X \to S'$ the base change of $f$. For every $s' \in S'$ there exists an open affine neighbourhood $s' \in V' \subset S'$ which maps into some open affine $V \subset S$. By Lemma 29.14.4 the open $f^{-1}(V)$ is a union of affines $U_ i$ such that the ring maps $\mathcal{O}_ S(V) \to \mathcal{O}_ X(U_ i)$ all satisfy $P$. By the material in Schemes, Section 26.17 we see that $f^{-1}(U)_{V'} = V' \times _ V f^{-1}(V)$ is the union of the affine opens $V' \times _ V U_ i$. Since $\mathcal{O}_{X_{S'}}(V' \times _ V U_ i) = \mathcal{O}_{S'}(V') \otimes _{\mathcal{O}_ S(V)} \mathcal{O}_ X(U_ i)$ we see that the ring maps $\mathcal{O}_{S'}(V') \to \mathcal{O}_{X_{S'}}(V' \times _ V U_ i)$ satisfy $P$ as $P$ is assumed stable under base change. $\square$

Lemma 29.14.7. The following properties of a ring map $R \to A$ are local.

1. (Isomorphism on local rings.) For every prime $\mathfrak q$ of $A$ lying over $\mathfrak p \subset R$ the ring map $R \to A$ induces an isomorphism $R_{\mathfrak p} \to A_{\mathfrak q}$.

2. (Open immersion.) For every prime $\mathfrak q$ of $A$ there exists an $f \in R$, $\varphi (f) \not\in \mathfrak q$ such that the ring map $\varphi : R \to A$ induces an isomorphism $R_ f \to A_ f$.

3. (Reduced fibres.) For every prime $\mathfrak p$ of $R$ the fibre ring $A \otimes _ R \kappa (\mathfrak p)$ is reduced.

4. (Fibres of dimension at most $n$.) For every prime $\mathfrak p$ of $R$ the fibre ring $A \otimes _ R \kappa (\mathfrak p)$ has Krull dimension at most $n$.

5. (Locally Noetherian on the target.) The ring map $R \to A$ has the property that $A$ is Noetherian.

6. Add more here as needed1.

Proof. Omitted. $\square$

Lemma 29.14.8. The following properties of ring maps are stable under base change.

1. (Isomorphism on local rings.) For every prime $\mathfrak q$ of $A$ lying over $\mathfrak p \subset R$ the ring map $R \to A$ induces an isomorphism $R_{\mathfrak p} \to A_{\mathfrak q}$.

2. (Open immersion.) For every prime $\mathfrak q$ of $A$ there exists an $f \in R$, $\varphi (f) \not\in \mathfrak q$ such that the ring map $\varphi : R \to A$ induces an isomorphism $R_ f \to A_ f$.

3. Add more here as needed2.

Proof. Omitted. $\square$

Lemma 29.14.9. The following properties of ring maps are stable under composition.

1. (Isomorphism on local rings.) For every prime $\mathfrak q$ of $A$ lying over $\mathfrak p \subset R$ the ring map $R \to A$ induces an isomorphism $R_{\mathfrak p} \to A_{\mathfrak q}$.

2. (Open immersion.) For every prime $\mathfrak q$ of $A$ there exists an $f \in R$, $\varphi (f) \not\in \mathfrak q$ such that the ring map $\varphi : R \to A$ induces an isomorphism $R_ f \to A_ f$.

3. (Locally Noetherian on the target.) The ring map $R \to A$ has the property that $A$ is Noetherian.

4. Add more here as needed3.

Proof. Omitted. $\square$

[1] But only those properties that are not already dealt with separately elsewhere.
[2] But only those properties that are not already dealt with separately elsewhere.
[3] But only those properties that are not already dealt with separately elsewhere.

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