Local rings of affine space in etale topology.

Question

This is probably a very straightforward question. What does the etale local rings of $\mathbb{A}_{\mathbb{F}_q}^n$ look like? (In other words the strict henselization of local rings at closed points)

Answer 1

The strict henselization of a DVR is (up to isomorphism) the smallest DVR extending the valuation where Hensel lemma holds and with separably closed residue field.

• The strict henselization of $\Bbb{F}_q[T]_{(T)}$ is $\overline{k(T)}\cap k[[T]]$ where $k=\overline{\Bbb{F}_q}$:

  If $a(T^p)\in k[[T^p]]$ is algebraic over $k(T)$ then $\sum_{n=0}^N c_n(T)a(T^p)^n=0$. Write $c_n(T)=\sum_{m=0}^{p-1} T^m c_{n,m}(T^p)$ with $c_{n,m}(T^p)\in k(T^p)\subset k((T^p))$ then $\sum_{m=0}^{p-1} T^m \sum_{n=0}^N c_{n,m}(T^p)a(T^p)^n=0$ implies that each $\sum_{n=0}^N c_{n,m}(T^p)a(T^p)^n=0$. Whence the minimal polynomial of $a(T^p)$ is in $k(T^p)[Y]$ from which $\overline{k(T)}\cap k[[T]]$ is separable over $k(T)$.

  I doubt there is any simple characterization of the elements of $\overline{k(T)}\cap k[[T]]$.

• For $f(T)\in \Bbb{F}_q[T]$ irreducible, $f(T)=(T-\alpha)g(T)\in k[T]$, the strict henselization of $\Bbb{F}_q[T]_{(f(T))}$ is the same as the strict henselization of $\Bbb{F}_q[T]_{(f(T))}[\alpha,g(T)^{-1}]\cong \Bbb{F}_q(\alpha)[T]_{(T)}$ which is again $\overline{k(T)}\cap k[[T]]$.

• For the local ring $\Bbb{F}_q[T]_{\{0\}}=\Bbb{F}_q(T)$ at the generic point, its strict henselization is $\Bbb{F}_q(T)^{sep}$.