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arXiv · 1902.07745

Classifying spaces for \'etale algebras with generators

Abhishek Kumar Shukla·Ben Williams
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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.

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Abhishek Kumar Shukla, Ben Williams. 2019-02-20. Classifying spaces for \'etale algebras with generators. https://doi.org/10.4153/s0008414x20000206

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