The spectrum of the Chern subring

TL;DR

This paper characterizes the prime ideal spectrum of certain subrings of the mod-p cohomology of compact Lie groups, focusing on the Chern subring, and establishes conditions for when this subring captures the cohomology ring's structure.

Contribution

It provides a description of the spectrum of the Chern subring and identifies conditions under which the inclusion into the cohomology ring is an F-isomorphism.

Findings

The spectrum of the Chern subring can be explicitly described similarly to Quillen's spectrum.

The inclusion of the Chern subring into the cohomology ring is an F-isomorphism under specific conjugation conditions.

The results apply to both compact Lie groups and finite groups with respect to Chern classes.

Abstract

For certain subrings of the mod-p cohomology ring of a compact Lie group, we give a description of the prime ideal spectrum, analogous to Quillen's description of the spectrum of the whole ring. Examples of such subrings include the Chern subring (the subring generated by Chern classes of all unitary representations), and for finite groups the subring generated by Chern classes of representations realizable over any specified field. As a corollary, we deduce that the inclusion of the Chern subring in the cohomology ring is an F-isomorphism for a compact Lie group G if and only if the following condition holds: For any homomorphism f between elementary abelian p-subgroups of G such that f(v) is always conjugate to v, there is an element g of G such that f is equal to conjugation by g.