A homotopy orbit spectrum for profinite groups

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.

Keywords

homotopy orbit spectrum, profinite group, continuous group homology

2010 Mathematics Subject Classification

55P42, 55P91, 55T25

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The initial version of this paper [5] was written while the first author was partially supported by a VIGRE NSF grant of the Purdue University Mathematics Department.

Received 16 July 2021

Received revised 5 July 2023

Accepted 17 July 2023

Published 29 May 2024