Classifying spaces for étale algebras with generators

Abhishek Kumar Shukla, Ben Williams
2020 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
We construct varieties B(r; A n ) such that a map X → B(r; A n ) corresponds to a degree-nétale algebra on X equipped with r generating global sections. We then show that when n = 2, i.e., in the quadraticétale case, that the singular cohomology of B(r; A n )(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 areétale algebras An of arbitrary degrees n that cannot be
more » ... d by fewer than r elements. This shows that in theétale algebra case, a bound established by U. First and Z. Reichstein in [2] is sharp.
doi:10.4153/s0008414x20000206 fatcat:ut5lsvlp7rfvfenmtucjxpfwfm