A homotopy orbit spectrum for profinite groups

Abstract

For a profinite group $G$, we define an $S[[G]]$-module to be a certain type of $G$-spectrum $X$ built from an inverse system $\{X_i\}_i$ of $G$-spectra, with each $X_i$ naturally a $G/N_i$-spectrum, where $N_i$ is an open normal subgroup and $G \cong \lim_i G/N_i$. We define the homotopy orbit spectrum $X_{hG}$ and its homotopy orbit spectral sequence. We give results about when its $E_2$-term satisfies $E_2^{p,q} \cong \lim_i H_p(G/N_i, \pi_q(X_i))$. Our main result is that this occurs if $\{\pi_\ast(X_i)\}_i$ degreewise consists of compact Hausdorff abelian groups and continuous homomorphisms, with each $G/N_i$ acting continuously on $\pi_q(X_i)$ for all $q$. If $\pi_q(X_i)$ is additionally always profinite, then the $E_2$-term is the continuous homology of $G$ with coefficients in the graded profinite $\widehat{\mathbb{Z}}[[G]]$-module $\pi_\ast(X)$. Other results include theorems about Eilenberg-Mac Lane spectra and about when homotopy orbits preserve weak equivalences.