Homology, Homotopy and Applications

Volume 10 (2008)

Number 3

Proceedings of a Conference in Honor of Douglas C. Ravenel and W. Stephen Wilson

Explicit fibrant replacement for discrete $G$-spectra

Pages: 137 – 150

DOI: http://dx.doi.org/10.4310/HHA.2008.v10.n3.a7

Author

Daniel G. Davis (Department of Mathematics, University of Louisiana, Lafayette, La., U.S.A.)

Abstract

If $C$ is the model category of simplicial presheaves on a site with enough points, with fibrations equal to the global fibrations, then it is well-known that the fibrant objects are, in general, mysterious. Thus, it is not surprising that, when $G$ is a profinite group, the fibrant objects in the model category of discrete $G$-spectra are also difficult to get a handle on. However, with simplicial presheaves, it is possible to construct an explicit fibrant model for an object in $C$, under certain finiteness conditions. Similarly, in this paper, we show that if $G$ has finite virtual cohomological dimension and $X$ is a discrete $G$-spectrum, then there is an explicit fibrant model for $X$. Also, we give several applications of this concrete model related to closed subgroups of $G$.

Keywords

homotopy fixed points; discrete $G$-spectrum; homotopy limit

2010 Mathematics Subject Classification

55P42, 55P91

Full Text (PDF format)