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35.30 Properties of morphisms of germs local on source-and-target

In this section we discuss the analogue of the material in Section 35.29 for morphisms of germs of schemes.

Definition 35.30.1. Let $\mathcal{Q}$ be a property of morphisms of germs of schemes. We say $\mathcal{Q}$ is étale local on the source-and-target if for any commutative diagram

\[ \xymatrix{ (U', u') \ar[d]_ a \ar[r]_{h'} & (V', v') \ar[d]^ b \\ (U, u) \ar[r]^ h & (V, v) } \]

of germs with étale vertical arrows we have $\mathcal{Q}(h) \Leftrightarrow \mathcal{Q}(h')$.

Lemma 35.30.2. Let $\mathcal{P}$ be a property of morphisms of schemes which is étale local on the source-and-target. Consider the property $\mathcal{Q}$ of morphisms of germs defined by the rule

\[ \mathcal{Q}((X, x) \to (S, s)) \Leftrightarrow \text{there exists a representative }U \to S \text{ which has }\mathcal{P} \]

Then $\mathcal{Q}$ is étale local on the source-and-target as in Definition 35.30.1.

Proof. If a morphism of germs $(X, x) \to (S, s)$ has $\mathcal{Q}$, then there are arbitrarily small neighbourhoods $U \subset X$ of $x$ and $V \subset S$ of $s$ such that a representative $U \to V$ of $(X, x) \to (S, s)$ has $\mathcal{P}$. This follows from Lemma 35.29.4. Let

\[ \xymatrix{ (U', u') \ar[r]_{h'} \ar[d]_ a & (V', v') \ar[d]^ b \\ (U, u) \ar[r]^ h & (V, v) } \]

be as in Definition 35.30.1. Choose $U_1 \subset U$ and a representative $h_1 : U_1 \to V$ of $h$. Choose $V'_1 \subset V'$ and an étale representative $b_1 : V'_1 \to V$ of $b$ (Definition 35.17.2). Choose $U'_1 \subset U'$ and representatives $a_1 : U'_1 \to U_1$ and $h'_1 : U'_1 \to V'_1$ of $a$ and $h'$ with $a_1$ étale. After shrinking $U'_1$ we may assume $h_1 \circ a_1 = b_1 \circ h'_1$. By the initial remark of the proof, we are trying to show $u' \in W(h'_1) \Leftrightarrow u \in W(h_1)$ where $W(-)$ is as in Lemma 35.23.3. Thus the lemma follows from Lemma 35.29.9. $\square$

Lemma 35.30.3. Let $\mathcal{P}$ be a property of morphisms of schemes which is étale local on source-and-target. Let $Q$ be the associated property of morphisms of germs, see Lemma 35.30.2. Let $f : X \to Y$ be a morphism of schemes. The following are equivalent:

  1. $f$ has property $\mathcal{P}$, and

  2. for every $x \in X$ the morphism of germs $(X, x) \to (Y, f(x))$ has property $\mathcal{Q}$.

Proof. The implication (1) $\Rightarrow $ (2) is direct from the definitions. The implication (2) $\Rightarrow $ (1) also follows from part (3) of Definition 35.29.3. $\square$

A morphism of germs $(X, x) \to (S, s)$ determines a well defined map of local rings. Hence the following lemma makes sense.

Lemma 35.30.4. The property of morphisms of germs

\[ \mathcal{P}((X, x) \to (S, s)) = \mathcal{O}_{S, s} \to \mathcal{O}_{X, x}\text{ is flat} \]

is étale local on the source-and-target.

Proof. Given a diagram as in Definition 35.30.1 we obtain the following diagram of local homomorphisms of local rings

\[ \xymatrix{ \mathcal{O}_{U', u'} & \mathcal{O}_{V', v'} \ar[l] \\ \mathcal{O}_{U, u} \ar[u] & \mathcal{O}_{V, v} \ar[l] \ar[u] } \]

Note that the vertical arrows are localizations of étale ring maps, in particular they are essentially of finite presentation, flat, and unramified (see Algebra, Section 10.143). In particular the vertical maps are faithfully flat, see Algebra, Lemma 10.39.17. Now, if the upper horizontal arrow is flat, then the lower horizontal arrow is flat by an application of Algebra, Lemma 10.39.10 with $R = \mathcal{O}_{V, v}$, $S = \mathcal{O}_{U, u}$ and $M = \mathcal{O}_{U', u'}$. If the lower horizontal arrow is flat, then the ring map

\[ \mathcal{O}_{V', v'} \otimes _{\mathcal{O}_{V, v}} \mathcal{O}_{U, u} \longleftarrow \mathcal{O}_{V', v'} \]

is flat by Algebra, Lemma 10.39.7. And the ring map

\[ \mathcal{O}_{U', u'} \longleftarrow \mathcal{O}_{V', v'} \otimes _{\mathcal{O}_{V, v}} \mathcal{O}_{U, u} \]

is a localization of a map between étale ring extensions of $\mathcal{O}_{U, u}$, hence flat by Algebra, Lemma 10.143.8. $\square$

Lemma 35.30.5. Consider a commutative diagram of morphisms of schemes

\[ \xymatrix{ U' \ar[r] \ar[d] & V' \ar[d] \\ U \ar[r] & V } \]

with étale vertical arrows and a point $v' \in V'$ mapping to $v \in V$. Then the morphism of fibres $U'_{v'} \to U_ v$ is étale.

Proof. Note that $U'_ v \to U_ v$ is étale as a base change of the étale morphism $U' \to U$. The scheme $U'_ v$ is a scheme over $V'_ v$. By Morphisms, Lemma 29.36.7 the scheme $V'_ v$ is a disjoint union of spectra of finite separable field extensions of $\kappa (v)$. One of these is $v' = \mathop{\mathrm{Spec}}(\kappa (v'))$. Hence $U'_{v'}$ is an open and closed subscheme of $U'_ v$ and it follows that $U'_{v'} \to U'_ v \to U_ v$ is étale (as a composition of an open immersion and an étale morphism, see Morphisms, Section 29.36). $\square$

Given a morphism of germs of schemes $(X, x) \to (S, s)$ we can define the fibre as the isomorphism class of germs $(U_ s, x)$ where $U \to S$ is any representative. We will often abuse notation and just write $(X_ s, x)$.

Lemma 35.30.6. Let $d \in \{ 0, 1, 2, \ldots , \infty \} $. The property of morphisms of germs

\[ \mathcal{P}_ d((X, x) \to (S, s)) = \text{the local ring } \mathcal{O}_{X_ s, x} \text{ of the fibre has dimension }d \]

is étale local on the source-and-target.

Proof. Given a diagram as in Definition 35.30.1 we obtain an étale morphism of fibres $U'_{v'} \to U_ v$ mapping $u'$ to $u$, see Lemma 35.30.5. Hence the result follows from Lemma 35.18.3. $\square$

Lemma 35.30.7. Let $r \in \{ 0, 1, 2, \ldots , \infty \} $. The property of morphisms of germs

\[ \mathcal{P}_ r((X, x) \to (S, s)) \Leftrightarrow \text{trdeg}_{\kappa (s)} \kappa (x) = r \]

is étale local on the source-and-target.

Proof. Given a diagram as in Definition 35.30.1 we obtain the following diagram of local homomorphisms of local rings

\[ \xymatrix{ \mathcal{O}_{U', u'} & \mathcal{O}_{V', v'} \ar[l] \\ \mathcal{O}_{U, u} \ar[u] & \mathcal{O}_{V, v} \ar[l] \ar[u] } \]

Note that the vertical arrows are localizations of étale ring maps, in particular they are unramified (see Algebra, Section 10.143). Hence $\kappa (u) \subset \kappa (u')$ and $\kappa (v) \subset \kappa (v')$ are finite separable field extensions. Thus we have $\text{trdeg}_{\kappa (v)} \kappa (u) = \text{trdeg}_{\kappa (v')} \kappa (u)$ which proves the lemma. $\square$

Let $(X, x)$ be a germ of a scheme. The dimension of $X$ at $x$ is the minimum of the dimensions of open neighbourhoods of $x$ in $X$, and any small enough open neighbourhood has this dimension. Hence this is an invariant of the isomorphism class of the germ. We denote this simply $\dim _ x(X)$.

Lemma 35.30.8. Let $d \in \{ 0, 1, 2, \ldots , \infty \} $. The property of morphisms of germs

\[ \mathcal{P}_ d((X, x) \to (S, s)) \Leftrightarrow \dim _ x (X_ s) = d \]

is étale local on the source-and-target.

Proof. Given a diagram as in Definition 35.30.1 we obtain an étale morphism of fibres $U'_{v'} \to U_ v$ mapping $u'$ to $u$, see Lemma 35.30.5. Hence now the equality $\dim _ u(U_ v) = \dim _{u'}(U'_{v'})$ follows from Lemma 35.18.2. $\square$


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