67.22 Types of morphisms étale local on source-and-target
Given a property of morphisms of schemes which is étale local on the source-and-target, see Descent, Definition 35.32.3 we may use it to define a corresponding property of morphisms of algebraic spaces, namely by imposing either of the equivalent conditions of the lemma below.
Lemma 67.22.1. Let $\mathcal{P}$ be a property of morphisms of schemes which is étale local on the source-and-target. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Consider commutative diagrams
\[ \xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ X \ar[r]^ f & Y } \]
where $U$ and $V$ are schemes and the vertical arrows are étale. The following are equivalent
for any diagram as above the morphism $h$ has property $\mathcal{P}$, and
for some diagram as above with $a : U \to X$ surjective the morphism $h$ has property $\mathcal{P}$.
If $X$ and $Y$ are representable, then this is also equivalent to $f$ (as a morphism of schemes) having property $\mathcal{P}$. If $\mathcal{P}$ is also preserved under any base change, and fppf local on the base, then for representable morphisms $f$ this is also equivalent to $f$ having property $\mathcal{P}$ in the sense of Section 67.3.
Proof.
Let us prove the equivalence of (1) and (2). The implication (1) $\Rightarrow $ (2) is immediate (taking into account Spaces, Lemma 65.11.6). Assume
\[ \xymatrix{ U \ar[d] \ar[r]_ h & V \ar[d] \\ X \ar[r]^ f & Y } \quad \quad \xymatrix{ U' \ar[d] \ar[r]_{h'} & V' \ar[d] \\ X \ar[r]^ f & Y } \]
are two diagrams as in the lemma. Assume $U \to X$ is surjective and $h$ has property $\mathcal{P}$. To show that (2) implies (1) we have to prove that $h'$ has $\mathcal{P}$. To do this consider the diagram
\[ \xymatrix{ U \ar[d]_ h & U \times _ X U' \ar[l] \ar[d]^{(h, h')} \ar[r] & U' \ar[d]^{h'} \\ V & V \times _ Y V' \ar[l] \ar[r] & V' } \]
By Descent, Lemma 35.32.5 we see that $h$ has $\mathcal{P}$ implies $(h, h')$ has $\mathcal{P}$ and since $U \times _ X U' \to U'$ is surjective this implies (by the same lemma) that $h'$ has $\mathcal{P}$.
If $X$ and $Y$ are representable, then Descent, Lemma 35.32.5 applies which shows that (1) and (2) are equivalent to $f$ having $\mathcal{P}$.
Finally, suppose $f$ is representable, and $U, V, a, b, h$ are as in part (2) of the lemma, and that $\mathcal{P}$ is preserved under arbitrary base change. We have to show that for any scheme $Z$ and morphism $Z \to X$ the base change $Z \times _ Y X \to Z$ has property $\mathcal{P}$. Consider the diagram
\[ \xymatrix{ Z \times _ Y U \ar[d] \ar[r] & Z \times _ Y V \ar[d] \\ Z \times _ Y X \ar[r] & Z } \]
Note that the top horizontal arrow is a base change of $h$ and hence has property $\mathcal{P}$. The left vertical arrow is étale and surjective and the right vertical arrow is étale. Thus Descent, Lemma 35.32.5 once again kicks in and shows that $Z \times _ Y X \to Z$ has property $\mathcal{P}$.
$\square$
Definition 67.22.2. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of schemes which is étale local on the source-and-target. We say a morphism $f : X \to Y$ of algebraic spaces over $S$ has property $\mathcal{P}$ if the equivalent conditions of Lemma 67.22.1 hold.
Here are a couple of obvious remarks.
Given a property of morphisms of germs of schemes which is étale local on the source-and-target, see Descent, Definition 35.33.1 we may use it to define a corresponding property of morphisms of algebraic spaces at a point, namely by imposing either of the equivalent conditions of the lemma below.
Lemma 67.22.5. Let $\mathcal{Q}$ be a property of morphisms of germs which is étale local on the source-and-target. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$ be a point of $X$. Consider the diagrams
\[ \xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ X \ar[r]^ f & Y } \quad \quad \xymatrix{ u \ar[d] \ar[r] & v \ar[d] \\ x \ar[r] & y } \]
where $U$ and $V$ are schemes, $a, b$ are étale, and $u, v, x, y$ are points of the corresponding spaces. The following are equivalent
for any diagram as above we have $\mathcal{Q}((U, u) \to (V, v))$, and
for some diagram as above we have $\mathcal{Q}((U, u) \to (V, v))$.
If $X$ and $Y$ are representable, then this is also equivalent to $\mathcal{Q}((X, x) \to (Y, y))$.
Proof.
Omitted. Hint: Very similar to the proof of Lemma 67.22.1.
$\square$
Definition 67.22.6. Let $\mathcal{Q}$ be a property of morphisms of germs of schemes which is étale local on the source-and-target. Let $S$ be a scheme. Given a morphism $f : X \to Y$ of algebraic spaces over $S$ and a point $x \in |X|$ we say that $f$ has property $\mathcal{Q}$ at $x$ if the equivalent conditions of Lemma 67.22.5 hold.
The following lemma should not be used blindly to go from a property of morphisms to a property of morphisms at a point. For example if $\mathcal{P}$ is the property of being flat, then the property $Q$ in the following lemma means “$f$ is flat in an open neighbourhood of $x$” which is not the same as “$f$ is flat at $x$”.
Lemma 67.22.7. 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 associated to $\mathcal{P}$ in Descent, Lemma 35.33.2. Then
$\mathcal{Q}$ is étale local on the source-and-target.
given a morphism of algebraic spaces $f : X \to Y$ and $x \in |X|$ the following are equivalent
$f$ has $\mathcal{Q}$ at $x$, and
there is an open neighbourhood $X' \subset X$ of $x$ such that $X' \to Y$ has $\mathcal{P}$.
given a morphism of algebraic spaces $f : X \to Y$ the following are equivalent:
$f$ has $\mathcal{P}$,
for every $x \in |X|$ the morphism $f$ has $\mathcal{Q}$ at $x$.
Proof.
See Descent, Lemma 35.33.2 for (1). The implication (1)(a) $\Rightarrow $ (2)(b) follows on letting $X' = a(U) \subset X$ given a diagram as in Lemma 67.22.5. The implication (2)(b) $\Rightarrow $ (1)(a) is clear. The equivalence of (3)(a) and (3)(b) follows from the corresponding result for morphisms of schemes, see Descent, Lemma 35.33.3.
$\square$
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