Homotopy fixed points for profinite groups emulate homotopy fixed points for discrete groups

TL;DR

This paper demonstrates that homotopy fixed points for profinite groups can be understood as a profinite analogue of the classical fixed point spectra for discrete groups, by explicitly constructing a related spectrum.

Contribution

It establishes a precise analogy between homotopy fixed points for profinite groups and discrete groups, providing an explicit construction that emulates the classical case.

Findings

H_{G,X} is a profinite version of Map_*(EK_+, Z)

H_{G,X} replicates the stages of the classical fixed point construction

The pattern extends to the setting of discrete G-spectra and profinite groups

Abstract

If K is a discrete group and Z is a K-spectrum, then the homotopy fixed point spectrum Z^{hK} is Map_*(EK_+, Z)^K, the fixed points of a familiar expression. Similarly, if G is a profinite group and X is a discrete G-spectrum, then X^{hG} is often given by (H_{G,X})^G, where H_{G,X} is a certain explicit construction given by a homotopy limit in the category of discrete G-spectra. Thus, in each of two common equivariant settings, the homotopy fixed point spectrum is equal to the fixed points of an explicit object in the ambient equivariant category. We enrich this pattern by proving in a precise sense that the discrete G-spectrum H_{G,X} is just "a profinite version" of Map_*(EK_+, Z): at each stage of its construction, H_{G,X} replicates in the setting of discrete G-spectra the corresponding stage in the formation of Map_*(EK_+, Z) (up to a certain natural identification).