Is there an open $S$-immersion from $S$ to $\mathbb A_S^n$

Question

Let $S$ be a scheme, then is there an open $S$-immersion from $S$ to $\mathbb A_S^n$ for some $n\in\mathbb{N}$?

Answer 1

I’m assuming that by “open $S$-immersion” you mean “a $S$- immersion which is an open map”. Let $n \geq 1$ and let $s \in S$ be a point. Consider $i: S \rightarrow \mathbb{A}^n_S$ an open $S$-immersion. Let $G \subset \mathbb{A}^n_S$ be its image, it is an open subset. Then $G \cap \mathbb{A}^n_{k(s)}$ is open in $\mathbb{A}^n_{k(s)}$ and is a point, a contradiction.

It is possible to find a scheme $S$ which is isomorphic to $\mathbb{A}^1_S$, though not over $S$. You can for instance take $S$ to be the spectrum of $\mathbb{Z}[x_1,\ldots,x_n,\ldots]$.