Does anyone know of generalizations on what Mumford (Red Book) calls "uniformizing parameters"? For example, given a regular quasi-projective scheme over a *finite* field $\mathbb{F}$, is there an etale morphism into affine space over $\mathbb{F}$?. 



When does a projective morphism give an etale morphism (into affine space)? (Finite field) (normalization)
[This paper](http://arxiv.org/abs/math/0303382) by Kedlaya might be what you want, since it contains some rearrangement of the words you used, but I can't really tell from the question. If you want a proper F-scheme to have an etale map to affine space, it has to be a disjoint union of finite F-schemes, and the affine space has to be zero dimensional. 


