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Although I agree with Peter's comments, I believe I can add a few helpful comments of my own. First for $G$ finite, every rational Mackey functor is both injective and projective, so chain complexes a …

Some clarifications: 1) You need that $X$ is grouplike (so the induced multiplication makes $\pi_0 X$ a group). This condition is always satisfied for a loop space, but not satisfied by the discrete …

I'm going to assume that your Hopf algebras are connected in which case this follows from Theorem 4.10 of Milnor-Moore (On the structure of Hopf-algebras). That result shows that $A\cong B\otimes A//B …

This question already has been answered in the comments. (Tilson) We regard a commutative ring as an $E_\infty$ spectrum via the EM functor $H$. This is definitely what Jacob is doing. One could als …

Your identification with the geometric fixed points and the calculation for $C_2$ is correct. As Akhil remarked, one obtains the same answer for $C_{2^n}$ and this is calculated in Hill-Hopkins-Ravene …

The spectral sequence I constructed with Niles Johnson was precisely designed to handle questions of this sort (here is a version that is closer to the publication version: T-algebra SS). A special ca …

Assume that $R$ is a connective $E_\infty$ ring spectrum. Typically $GL_1(R)$ denotes the set of components in $\Omega^\infty R$ which span $GL_1(\pi_0 R)=\pi_0 R^\times$. I would call the unit compon …

First let me describe the norm in degree 0. Let $k$ be a commutative ring, given $\alpha\in H^0(BH,k)$, which corresponds to a an $H$ equivariant homotopy class of a map $EH_+\rightarrow k$ where $k$ …

I'll answer a related question: in the $K(n)$-local stable category, $BG$ is dualizable for all finite groups $G$, moreover, each is self-dual. You can find this in Hovey and Strickland's 'Morava $K$- …

I would like to add a few points: You can define $RO(G)$-graded homotopy groups of $G$-spectra, see for example Stefan Schwede's course notes on equivariant homotopy theory. These groups are intere …