Classifying spaces for étale algebras with generators

Abstract

Abstract We construct a scheme $B(r; {\mathbb {A}}^n)$ such that a map $X \to B(r; {\mathbb {A}}^n)$ corresponds to a degree-n étale algebra on X equipped with r generating global sections. We then show that when $n=2$ , i.e., in the quadratic étale case, the singular cohomology of $B(r; {\mathbb {A}}^n)({\mathbb {R}})$ 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 ${\mathbb {R}}$ -variety on which there are étale algebras ${\mathcal {A}}_n$ of arbitrary degrees n that cannot be generated by fewer than r elements. This shows that in the étale algebra case, a bound established by U. First and Z. Reichstein in [6] is sharp.