Delta-discrete $G$–spectra and iterated homotopy fixed points

Abstract

Let $G$ be a profinite group with finite virtual cohomological dimension and let $X$ be a discrete $G$–spectrum. If $H$ and $K$ are closed subgroups of $G$, with $H◃K$, then, in general, the $K∕H$–spectrum $X^{hH}$ is not known to be a continuous $K∕H$–spectrum, so that it is not known (in general) how to define the iterated homotopy fixed point spectrum ${\left(X^{hH}\right)}^{hK∕H}$. To address this situation, we define homotopy fixed points for delta-discrete $G$–spectra and show that the setting of delta-discrete $G$–spectra gives a good framework within which to work. In particular, we show that by using delta-discrete $K∕H$–spectra, there is always an iterated homotopy fixed point spectrum, denoted ${\left(X^{hH}\right)}^{h_{\delta }K∕H}$, and it is just $X^{hK}$.

Additionally, we show that for any delta-discrete $G$–spectrum $Y$, there is an equivalence $\left(\right.Y^{h_{\delta }H}{\left)\right.}^{h_{\delta }K∕H}\simeq Y^{h_{\delta }K}$. Furthermore, if $G$ is an arbitrary profinite group, there is a delta-discrete $G$–spectrum $X_{\delta }$ that is equivalent to $X$ and, though $X^{hH}$ is not even known in general to have a $K∕H$–action, there is always an equivalence ${\left({\left(X_{\delta }\right)}^{h_{\delta }H}\right)}^{h_{\delta }K∕H}\simeq {\left(X_{\delta }\right)}^{h_{\delta }K}.$ Therefore, delta-discrete $L$–spectra, by letting $L$ equal $H,K,$ and $K∕H$, give a way of resolving undesired deficiencies in our understanding of homotopy fixed points for discrete $G$–spectra.