Surjective etale morphism from an open subscheme.

Question

I'm looking for an example of a surjective etale morphism from an open subscheme of $\mathbb{A}^d$ (which is not the whole $\mathbb{A}^d$) to $\mathbb{A}^d$. The base field is a finite field or its algebraic closure. I am just wondering whether something like this is possible.

Answer 1

There is an infinitude of examples. Namely, let us assume that $f:\mathbb{A}^d_k\to \mathbb{A}^d_k$ is a non-trivial finite etale cover (e.g. an Artin-Schreier cover) where $k$ is some finite extension of $\mathbb{F}_p$. Choose any closed point $p$ of $\mathbb{A}^d_k$. After possibly taking a finite extension of $k$, we can assume that $f^{-1}(p)$ consists of $n:=\deg(f)$ $k$-points. Let us write $f^{-1}(p)=\{p_1,\ldots,p_n\}$. Then, evidently

$$f:U\to \mathbb{A}^d_k,\qquad U:=\mathbb{A}^d_k-\{p_1\}$$

is an example of what you're after.

Answer 2

If you're particularly interested in positive characteristic, you may also be interested in Proposition 5.2 of Piotr Achinger's "Wild ramification and $K(\pi,1)$ spaces" (link): Let $k$ be a field of characteristic $p > 0$, let $U$ be an affine scheme of finite type over $k$, let $g : U \to \mathbb{A}_{k}^{n}$ be an etale map. Then there is a finite etale map $f : U \to \mathbb{A}_{k}^{n}$.