These questions appeared while studying Vakil's AG notes July 3123 version. This section really confuses me. Let me give the definition in this context first:
Definition (Smooth of relative dimension n). A morphism $\pi:X\to Y$ is smooth of relative dimension $n$ if there exists open covers $\{U_i\}$ of $X$ and $\{V_i\}$ of $Y$ such that $\pi(U_i)\subseteq V_i$ and the diagram commutes
\begin{array}{}
U_i & \xrightarrow{\cong} & W& \xrightarrow{open} \operatorname{Spec}B[x_1,\ldots,x_{n+r}]/(f_1,\ldots,f_r)\\
\downarrow & &\downarrow_{\rho|_W}&\swarrow_{\rho}\\
V_i&\xrightarrow{\cong}&\operatorname{Spec}B
\end{array}
where $\rho$ comes from the obvious ring map and $W$ is an open subscheme, such that the determinant $det(\frac{\partial{f_j}}{\partial{x_i}})_{i,j\leq r}$ of the Jacobian of the $f_i$'s with respect to the $x_i$'s is an invertible function on $W$.
Definition (etale morphism). An Etale morphism is a morphism smooth of relative dimension 0.
My question.
- How do we show that, if $\pi:X\to Y$ is smooth of relative dimension $n$, then for every
$p\in X$, there is an open neighborhood $U$ such that $\pi|_U=\beta\circ\alpha$ where $\alpha: U\to \mathbb{A}_Y^n$ is Etale and $\beta:\mathbb{A}^n_Y\to Y$ is the projection. - Suppose $k$ is a field with $char(k)\neq 2$. Consider the morphism $\pi:X=\operatorname{Spec}k[u,1/u]\to Y=\operatorname{Spec}k[t]$ corresponding to the ring homomorphism $k[t]\to k[u,1/u],t\mapsto u^2$. How do we show that it is Etale, and there is no non-empty open subset $U\subseteq X$ on which $\pi$ is an isomorphism? (exercise 13.6.E)
My Attempts.
- I only proved that $\mathbb{A}^n_Y\to Y$ is smooth of relative dimension $n$, simply by regarding it as the pullback of $\mathbb{A}^n_{\mathbb{Z}}\to\operatorname{Spec}\mathbb{Z}$, which is smooth of relative dimension $n$. Then I got confused.
- I sketched this as a map from the punctured line to the affine line. And the map only sends points to the "positive" part of the affine line. Namely, $[(u-a)]\in\operatorname{Spec}k[u,1/u]$ is taken to $[(t-a^2)]\in\operatorname{Spec}k[t]$. The generic point is taken to the generic point. I have no idea how to prove it is Etale using the definition he gave. I tried to prove the second part: given $\emptyset\neq U\subseteq X$ an open subset, we must have the generic point $[(0)]\in U$. If $\pi|_U$ is an isomorphism, we would have an isomorphism on stalk
$$\mathcal{O}_{Y,{\pi([(0)])}}\to \mathcal{O}_{X,[(0)]}$$
which is a field homomorphism $\varphi:k(t)\to k(u)$. I think this is not an isomorphism, as $u$ does not have a preimage. However, I am not so sure how to formalize this idea (maybe if there exists some $f\in k(t)$ such that $\varphi(f)=u$, then $\varphi(f^2)=u^2$ and the injectivity forces $f^2=t$, which is not possible).
Any help is appreciated!
Best Answer
For the first question, it is almost just an immediate consequence of the definition. Given a smooth morphism $\pi \colon X\longrightarrow Y$ of relative dimension zero, by definition, locally, it is $W \hookrightarrow \operatorname{Spec}(B[x_1,...,x_{n+r}]/(f_1,...,f_n)) \longrightarrow \operatorname{Spec}(B)$. By the invertibility of the jacobian, you can show that there is an étale morphism $$\operatorname{Spec}(B[x_1,...,x_{n+r}]/(f_1,...,f_n)) \longrightarrow \operatorname{Spec}(B[x_{n+1},...,x_{n+r}])$$ (again, this should follow from the definition). Now note that a composition of an étale morphism with an open morphism (which is also étale) is étale, you get the desired factorization.
For the second question, what you are trying to show is that the morphism $$\mathbb{G}_{m,k} \overset{x \mapsto x^2}{\longrightarrow} \mathbb{G}_{m,k} \hookrightarrow \mathbb{A}^1_k$$ is étale, where the second morphism is the canonical inclusion, hence étale. Therefore it remains to prove that the former one is étale provided that the field has characteristic different from $2$. It corresponds to the morphism $$k[u,1/u] \longrightarrow k[u,1/u], u \longmapsto u^2.$$ Via this morphism we can view $k[u,1/u]$ is an algebra over itself, which can be seen to be isomorphic to $k[u,1/u][t]/(t^2-u)$. The jacobian here is simply $2t$ which is invertible (the inverse is simply $t/(2u)$ because $t^2=u$ in this ring).