Classifying spaces for \'etale algebras with generators

We construct varieties B(r;An) such that a map X -> B(r;An) corresponds to a degree-n 'etale algebra on X equipped with r generating global sections. We then show that when n = 2, i.e., in the quadratic 'etale case, that the singular cohomology of B(r; An)® can be used to reconstruct a famous example of S. Chase and to extend its application to showing that there is a smooth affine r-1-dimensional R-variety on which there are 'etale algebras An of arbitrary degrees n that cannot be generated by fewer than r elements. This shows that in the 'etale algebra case, a bound established by U. First and Z. Reichstein is sharp.