Let $G$ be a non-finite profinite group and let $G-Sets_{df}$ be the canonical site of finite discrete $G$-sets. Then the category $R^+_G$, defined by Devinatz and Hopkins, is the category obtained by considering $G-Sets_{df}$ together with the profinite $G$-space $G$ itself, with morphisms being continuous $G$-equivariant maps. We show that $R^+_G$ is a site when equipped with the pretopology of epimorphic covers. We point out that presheaves of spectra on $R^+_G$ are an efficient way of organizing the data that is obtained by taking the homotopy fixed points of a continuous $G$-spectrum with respect to the open subgroups of $G$. Additionally, utilizing the result that $R^+_G$ is a site, we describe various model category structures on the category of presheaves of spectra on $R^+_G$ and make some observations about them.
Keywords: site, profinite group, finite discrete $G$-sets, presheaves of spectra, Lubin-Tate spectrum, continuous $G$-spectrum
@article{TAC_2009_22_a15,
author = {Daniel G. Davis},
title = {Epimorphic covers make $R^+_G$ a site, for profinite $G$},
journal = {Theory and applications of categories},
pages = {388--400},
year = {2009},
volume = {22},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2009_22_a15/}
}
Daniel G. Davis. Epimorphic covers make $R^+_G$ a site, for profinite $G$. Theory and applications of categories, Tome 22 (2009), pp. 388-400. http://geodesic.mathdoc.fr/item/TAC_2009_22_a15/