  We explain some results concerning the topology of varieties and stacks
equipped with an action of the multiplicative group $\mathbb{G}_m$. We apply
these techniques to the moduli of Higgs bundles. Our main application is to
upgrade the cohomological Nonabelian Hodge Theorem in positive characteristic
[dCGZ23, HZ23b] to an isomorphism of cohomology rings compatible with cup
product.
