  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)(R) 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.
