On the nonexistence of some open immersions

In this paper, we will prove a sufficient condition for that there does not exist an open immersion between two affine schemes of finite type over a field $k$ with the same dimension. The proof is based on the following fact: the complement of an open affine subset in a noetherian integral separated scheme has pure codimension 1. We will first prove it when $k$ is a finite field, the key to the proof of this part is Lang-Weil estimation. Then we'll prove a general result over an arbitrary field by reducing to the case when $k$ is finite. And the proof of the general result is much more complicated. In order to use the result over a finite field, at some point we must produce a scheme that is defined over $\mathbf{F}_{q}$ and an open immersion, also defined over $\mathbf{F}_{q}$. One important lemma is that a morphism $f:\text{Spec}(B) \longrightarrow \text{Spec}(A)$ between two integral domains with the same quotient field $K$ is an open immersion if and only if $B$ is a birational extension of $A$ in $K$ and $B$ is flat over $A$. According to the general result, the following open immersions do not exist: $SL_{n/k} \hookrightarrow \mathbf{A}_{k}^{n^{2}-1}$, $Sp_{n/k} \hookrightarrow \mathbf{A}_{k}^{2n^{2}+n}$, $SO_{n/k} \hookrightarrow \mathbf{A}_{k}^{\frac{n^{2}-n}{2}}$, $PGL_{n/k} \hookrightarrow \mathbf{A}_{k}^{n^{2}-1}$, where $k$ is an arbitrary field.