Lemma 35.29.6. Let $\mathcal{P}$ be a property of morphisms of schemes. Assume

1. $\mathcal{P}$ is étale local on the source,

2. $\mathcal{P}$ is étale local on the target, and

3. $\mathcal{P}$ is stable under postcomposing with open immersions: if $f : X \to Y$ has $\mathcal{P}$ and $Y \subset Z$ is an open subscheme then $X \to Z$ has $\mathcal{P}$.

Then $\mathcal{P}$ is étale local on the source-and-target.

Proof. Let $\mathcal{P}$ be a property of morphisms of schemes which satisfies conditions (1), (2) and (3) of the lemma. By Lemma 35.23.2 we see that $\mathcal{P}$ is stable under precomposing with étale morphisms. By Lemma 35.19.2 we see that $\mathcal{P}$ is stable under étale base change. Hence it suffices to prove part (3) of Definition 35.29.3 holds.

More precisely, suppose that $f : X \to Y$ is a morphism of schemes which satisfies Definition 35.29.3 part (3)(b). In other words, for every $x \in X$ there exists an étale morphism $a_ x : U_ x \to X$, a point $u_ x \in U_ x$ mapping to $x$, an étale morphism $b_ x : V_ x \to Y$, and a morphism $h_ x : U_ x \to V_ x$ such that $f \circ a_ x = b_ x \circ h_ x$ and $h_ x$ has $\mathcal{P}$. The proof of the lemma is complete once we show that $f$ has $\mathcal{P}$. Set $U = \coprod U_ x$, $a = \coprod a_ x$, $V = \coprod V_ x$, $b = \coprod b_ x$, and $h = \coprod h_ x$. We obtain a commutative diagram

$\xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ X \ar[r]^ f & Y }$

with $a$, $b$ étale, $a$ surjective. Note that $h$ has $\mathcal{P}$ as each $h_ x$ does and $\mathcal{P}$ is étale local on the target. Because $a$ is surjective and $\mathcal{P}$ is étale local on the source, it suffices to prove that $b \circ h$ has $\mathcal{P}$. This reduces the lemma to proving that $\mathcal{P}$ is stable under postcomposing with an étale morphism.

During the rest of the proof we let $f : X \to Y$ be a morphism with property $\mathcal{P}$ and $g : Y \to Z$ is an étale morphism. Consider the following statements:

1. With no additional assumptions $g \circ f$ has property $\mathcal{P}$.

2. Whenever $Z$ is affine $g \circ f$ has property $\mathcal{P}$.

3. Whenever $X$ and $Z$ are affine $g \circ f$ has property $\mathcal{P}$.

4. Whenever $X$, $Y$, and $Z$ are affine $g \circ f$ has property $\mathcal{P}$.

Once we have proved (-) the proof of the lemma will be complete.

Claim 1: (AAA) $\Rightarrow$ (AA). Namely, let $f : X \to Y$, $g : Y \to Z$ be as above with $X$, $Z$ affine. As $X$ is affine hence quasi-compact we can find finitely many affine open $Y_ i \subset Y$, $i = 1, \ldots , n$ such that $X = \bigcup _{i = 1, \ldots , n} f^{-1}(Y_ i)$. Set $X_ i = f^{-1}(Y_ i)$. By Lemma 35.19.2 each of the morphisms $X_ i \to Y_ i$ has $\mathcal{P}$. Hence $\coprod _{i = 1, \ldots , n} X_ i \to \coprod _{i = 1, \ldots , n} Y_ i$ has $\mathcal{P}$ as $\mathcal{P}$ is étale local on the target. By (AAA) applied to $\coprod _{i = 1, \ldots , n} X_ i \to \coprod _{i = 1, \ldots , n} Y_ i$ and the étale morphism $\coprod _{i = 1, \ldots , n} Y_ i \to Z$ we see that $\coprod _{i = 1, \ldots , n} X_ i \to Z$ has $\mathcal{P}$. Now $\{ \coprod _{i = 1, \ldots , n} X_ i \to X\}$ is an étale covering, hence as $\mathcal{P}$ is étale local on the source we conclude that $X \to Z$ has $\mathcal{P}$ as desired.

Claim 2: (AAA) $\Rightarrow$ (A). Namely, let $f : X \to Y$, $g : Y \to Z$ be as above with $Z$ affine. Choose an affine open covering $X = \bigcup X_ i$. As $\mathcal{P}$ is étale local on the source we see that each $f|_{X_ i} : X_ i \to Y$ has $\mathcal{P}$. By (AA), which follows from (AAA) according to Claim 1, we see that $X_ i \to Z$ has $\mathcal{P}$ for each $i$. Since $\{ X_ i \to X\}$ is an étale covering and $\mathcal{P}$ is étale local on the source we conclude that $X \to Z$ has $\mathcal{P}$.

Claim 3: (AAA) $\Rightarrow$ (-). Namely, let $f : X \to Y$, $g : Y \to Z$ be as above. Choose an affine open covering $Z = \bigcup Z_ i$. Set $Y_ i = g^{-1}(Z_ i)$ and $X_ i = f^{-1}(Y_ i)$. By Lemma 35.19.2 each of the morphisms $X_ i \to Y_ i$ has $\mathcal{P}$. By (A), which follows from (AAA) according to Claim 2, we see that $X_ i \to Z_ i$ has $\mathcal{P}$ for each $i$. Since $\mathcal{P}$ is local on the target and $X_ i = (g \circ f)^{-1}(Z_ i)$ we conclude that $X \to Z$ has $\mathcal{P}$.

Thus to prove the lemma it suffices to prove (AAA). Let $f : X \to Y$ and $g : Y \to Z$ be as above $X, Y, Z$ affine. Note that an étale morphism of affines has universally bounded fibres, see Morphisms, Lemma 29.36.6 and Lemma 29.56.9. Hence we can do induction on the integer $n$ bounding the degree of the fibres of $Y \to Z$. See Morphisms, Lemma 29.56.8 for a description of this integer in the case of an étale morphism. If $n = 1$, then $Y \to Z$ is an open immersion, see Lemma 35.22.2, and the result follows from assumption (3) of the lemma. Assume $n > 1$.

Consider the following commutative diagram

$\xymatrix{ X \times _ Z Y \ar[d] \ar[r]_{f_ Y} & Y \times _ Z Y \ar[d] \ar[r]_-{\text{pr}} & Y \ar[d] \\ X \ar[r]^ f & Y \ar[r]^ g & Z }$

Note that we have a decomposition into open and closed subschemes $Y \times _ Z Y = \Delta _{Y/Z}(Y) \amalg Y'$, see Morphisms, Lemma 29.35.13. As a base change the degrees of the fibres of the second projection $\text{pr} : Y \times _ Z Y \to Y$ are bounded by $n$, see Morphisms, Lemma 29.56.5. On the other hand, $\text{pr}|_{\Delta (Y)} : \Delta (Y) \to Y$ is an isomorphism and every fibre has exactly one point. Thus, on applying Morphisms, Lemma 29.56.8 we conclude the degrees of the fibres of the restriction $\text{pr}|_{Y'} : Y' \to Y$ are bounded by $n - 1$. Set $X' = f_ Y^{-1}(Y')$. Picture

$\xymatrix{ X \amalg X' \ar@{=}[d] \ar[r]_-{f \amalg f'} & \Delta (Y) \amalg Y' \ar@{=}[d] \ar[r] & Y \ar@{=}[d] \\ X \times _ Z Y \ar[r]^{f_ Y} & Y \times _ Z Y \ar[r]^-{\text{pr}} & Y }$

As $\mathcal{P}$ is étale local on the target and hence stable under étale base change (see Lemma 35.19.2) we see that $f_ Y$ has $\mathcal{P}$. Hence, as $\mathcal{P}$ is étale local on the source, $f' = f_ Y|_{X'}$ has $\mathcal{P}$. By induction hypothesis we see that $X' \to Y$ has $\mathcal{P}$. As $\mathcal{P}$ is local on the source, and $\{ X \to X \times _ Z Y, X' \to X \times _ Y Z\}$ is an étale covering, we conclude that $\text{pr} \circ f_ Y$ has $\mathcal{P}$. Note that $g \circ f$ can be viewed as a morphism $g \circ f : X \to g(Y)$. As $\text{pr} \circ f_ Y$ is the pullback of $g \circ f : X \to g(Y)$ via the étale covering $\{ Y \to g(Y)\}$, and as $\mathcal{P}$ is étale local on the target, we conclude that $g \circ f : X \to g(Y)$ has property $\mathcal{P}$. Finally, applying assumption (3) of the lemma once more we conclude that $g \circ f : X \to Z$ has property $\mathcal{P}$. $\square$

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