Lemma 35.29.11. Let $\mathcal{P}$ be a property of morphisms of schemes. Assume

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

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

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 $\mathcal{P}$.

Given a commutative diagram of schemes

$\vcenter { \xymatrix{ X' \ar[d]_{g'} \ar[r]_{f'} & Y' \ar[d]^ g \\ X \ar[r]^ f & Y } } \quad \text{with points}\quad \vcenter { \xymatrix{ x' \ar[d] \ar[r] & y' \ar[d] \\ x \ar[r] & y } }$

such that $g$ is smooth $y'$ and $X' \to X \times _ Y Y'$ is étale at $x'$, then $x \in W(f) \Leftrightarrow x' \in W(f')$ where $W(-)$ is as in Lemma 35.23.3.

Proof. Since $\mathcal{P}$ is étale local on the source we see that $x \in W(f)$ if and only if the image of $x$ in $X \times _ Y Y'$ is in $W(X \times _ Y Y' \to Y')$. Hence we may assume the diagram in the lemma is cartesian.

Assume $x \in W(f)$. Since $\mathcal{P}$ is smooth local on the target we see that $(g')^{-1}W(f) = W(f) \times _ Y Y' \to Y'$ has $\mathcal{P}$. Hence $(g')^{-1}W(f) \subset W(f')$. We conclude $x' \in W(f')$.

Assume $x' \in W(f')$. For any open neighbourhood $V' \subset Y'$ of $y'$ we may replace $Y'$ by $V'$ and $X'$ by $U' = (f')^{-1}V'$ because $V' \to Y'$ is smooth and hence the base change $W(f') \cap U' \to V'$ of $W(f') \to Y'$ has property $\mathcal{P}$. Thus we may assume there exists an étale morphism $Y' \to \mathbf{A}^ n_ Y$ over $Y$, see Morphisms, Lemma 29.36.20. Picture

$\xymatrix{ X' \ar[r] \ar[d] & Y' \ar[d] \\ \mathbf{A}^ n_ X \ar[r]_{f_ n} \ar[d] & \mathbf{A}^ n_ Y \ar[d] \\ X \ar[r]^ f & Y }$

By Lemma 35.29.6 (and because étale coverings are smooth coverings) we see that $\mathcal{P}$ is étale local on the source-and-target. By Lemma 35.29.9 we see that $W(f')$ is the inverse image of the open $W(f_ n) \subset \mathbf{A}^ n_ X$. In particular $W(f_ n)$ contains a point lying over $x$. After replacing $X$ by the image of $W(f_ n)$ (which is open) we may assume $W(f_ n) \to X$ is surjective. Claim: $W(f_ n) = \mathbf{A}^ n_ X$. The claim implies $f$ has $\mathcal{P}$ as $\mathcal{P}$ is local in the smooth topology and $\{ \mathbf{A}^ n_ Y \to Y\}$ is a smooth covering.

Essentially, the claim follows as $W(f_ n) \subset \mathbf{A}^ n_ X$ is a “translation invariant” open which meets every fibre of $\mathbf{A}^ n_ X \to X$. However, to produce an argument along these lines one has to do étale localization on $Y$ to produce enough translations and it becomes a bit annoying. Instead we use the automorphisms of Lemma 35.29.10 and étale morphisms of affine spaces. We may assume $n \geq 2$. Namely, if $n = 0$, then we are done. If $n = 1$, then we consider the diagram

$\xymatrix{ \mathbf{A}^2_ X \ar[r]_{f_2} \ar[d]_ p & \mathbf{A}^2_ Y \ar[d] \\ \mathbf{A}^1_ X \ar[r]^{f_1} & \mathbf{A}^1_ Y }$

We have $p^{-1}(W(f_1)) \subset W(f_2)$ (see first paragraph of the proof). Thus $W(f_2) \to X$ is still surjective and we may work with $f_2$. Assume $n \geq 2$.

For any $1 \leq i, j \leq n$ with $i \not= j$ and $d \geq 0$ denote $T_{i, j, d}$ the automorphism of $\mathbf{A}^ n$ defined in Lemma 35.29.10. Then we get a commutative diagram

$\xymatrix{ \mathbf{A}^ n_ X \ar[r]_{f_ n} \ar[d]_{T_{i, j, d}} & \mathbf{A}^ n_ Y \ar[d]^{T_{i, j, d}} \\ \mathbf{A}^ n_ X \ar[r]^{f_ n} & \mathbf{A}^ n_ Y }$

whose vertical arrows are isomorphisms. We conclude that $T_{i, j, d}(W(f_ n)) = W(f_ n)$. Applying Lemma 35.29.10 we conclude for any $x \in X$ the fibre $W(f_ n)_ x \subset \mathbf{A}^ n_ x$ is either $\mathbf{A}^ n_ x$ (this is what we want) or $\kappa (x)$ has characteristic $p > 0$ and $W(f_ n)_ x$ is the complement of a finite set $Z_ x \subset \mathbf{A}^ n_ x$ of closed points. The second possibility cannot occur. Namely, consider the morphism $T_ p : \mathbf{A}^ n \to \mathbf{A}^ n$ given by

$(x_1, \ldots , x_ n) \mapsto (x_1 - x_1^ p, \ldots , x_ n - x_ n^ p)$

As above we get a commutative diagram

$\xymatrix{ \mathbf{A}^ n_ X \ar[r]_{f_ n} \ar[d]_{T_ p} & \mathbf{A}^ n_ Y \ar[d]^{T_ p} \\ \mathbf{A}^ n_ X \ar[r]^{f_ n} & \mathbf{A}^ n_ Y }$

The morphism $T_ p : \mathbf{A}^ n_ X \to \mathbf{A}^ n_ X$ is étale at every point lying over $x$ and the morphism $T_ p : \mathbf{A}^ n_ Y \to \mathbf{A}^ n_ Y$ is étale at every point lying over the image of $x$ in $Y$. (Details omitted; hint: compute the derivatives.) We conclude that

$T_ p^{-1}(W) \cap \mathbf{A}^ n_ x = W \cap \mathbf{A}^ n_ x$

by Lemma 35.29.9 (we've already seen $\mathcal{P}$ is étale local on the source-and-target). Since $T_ p : \mathbf{A}^ n_ x \to \mathbf{A}^ n_ x$ is finite étale of degree $p^ n > 1$ we see that if $Z_ x$ is not empty then it contains $T_ p^{-1}(Z_ x)$ which is bigger. This contradiction finishes the proof. $\square$

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