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 ${\lbrace X_i \rbrace}_i$ 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 $E^{p,q}_2 \cong \lim_i H_p (G / N_i , \pi_q (X_i))$. Our main result is that this occurs if ${\lbrace \pi_\ast (X_i) \rbrace}_i$ 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.