Equivariant techniques in stable homotopy theory

    The chromatic support of a ($p$-local) $\mathbb{E}_\infty$-ring $R$ are the $n$ for which $R\otimes K(n)$ is nonzero. A theorem of Jeremy Hahn [arXiv:1612.04386] classifies the chromatic supports of an $\mathbb{E}_\infty$-ring.

There is a notion of chromatic support in equivariant homotopy theory coming from the fact that we know the Balmer spectrum (as a set) of equivariant homotopy theory.

    A wonderful feature of the prime $2$ is that there is a great model for the action of $C_2$ on Lubin–Tate theories, coming from quotients of $\mathrm{MU}_{\mathbb{R}}$.

    Any $\mathbb{E}_{\infty}$-ring $R$ has a Nikolaus–Scholze Frobenius map $R \to R^{tC_p}$, which is a map of $\mathbb{E}_{\infty}$-rings. In the case $R=\tau_{\geq 0}E_1$, the map $R \to \tau_{\geq 0}R^{tC_p}$ on $\pi_0$ is the map encoding the total power operation of $E_1$. In this way, the map can be considered an $\mathbb{E}_{\infty}$-refinement of the total power operation.

    Given a complex oriented ring $R$, $\pi_{2*}R$ can be interpreted as the line bundle on the associated formal group law.

    The $\mathbb{E}_{\infty}$-ring $\mathbb{F}_p^{tC_p}$ has two special properties that are related to each other. It admits no nontrivial $\mathbb{E}_{1}$ maps to an even ring, and its rational $K$-theory vanishes.

    Blueshift suggests that taking $tC_p$ lowers height. However given an $\mathbb{E}_n$-ring $R$, if $n<\infty$ of height $h$, $R^{tC_p}$ may not be of height $h-1$.

    Given a $G$ acting on an $X$ in a $1$-semiadditive stable symmetric monoidal category $C$, the $1$-semiadditive structure gives rise to a power operation $\theta_{G,X}$. In the case $C$ is height $1$ (such as $K(1)$-local spectra), this operation satisfies a bilinear relation with respect to products corresponding to the fact that the $\theta$ operation gives rise to a ring homomorphism at height $1$.