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 F-q and an open immersion, also defined over F-q. One important lemma is that a morphism f : Spec(B) -> 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: S L-n/k hooked right arrow A(k)(n2-1), S p(n/k) hooked right arrow A(k)(2n2+n), S O-n/k hooked right arrow A(k)(n2-n/2), PGL(n/k) hooked right arrow A(k)(n2-1), where k is an arbitrary field.