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Classifying spaces for étale algebras with generators

Part of: (Co)homology theory Fiber spaces and bundles Chain conditions, finiteness conditions

Published online by Cambridge University Press:  30 March 2020

Abhishek Kumar Shukla  and
Ben Williams
Abhishek Kumar Shukla
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BCV6T 1Z2, Canada e-mail: abhisheks@math.subc.ca
Ben Williams*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BCV6T 1Z2, Canada e-mail: abhisheks@math.subc.ca
*
e-mail: tbjw@math.ubc.ca
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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.

Keywords

Étale algebrasalgebra generatorsclassifying spaces

MSC classification

Primary: 13E15: Rings and modules of finite generation or presentation; number of generators
Secondary: 14F25: Classical real and complex (co)homology 14F42: Motivic cohomology; motivic homotopy theory 55R40: Homology of classifying spaces, characteristic classes
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 3 , June 2021 , pp. 854 - 874
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The first author was partially supported by a graduate fellowship from the Science and Engineering Research Board, India. The second author was partially supported by an NSERC discovery grant.

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