# Classifying spaces for \'etale algebras with generators

## TL;DR

This paper constructs algebraic varieties that classify étale algebras with generators, analyzes their topological properties over the real numbers, and demonstrates the sharpness of bounds on the number of generators needed.

## Contribution

It introduces new varieties parametrizing étale algebras with generators and explores their topological and algebraic properties, extending known bounds.

## Key findings

- The constructed varieties classify étale algebras with r generators.
- The real points' cohomology relates to classical examples like Chase's.
- The bounds on minimal generators are shown to be sharp.

## Abstract

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.