Explicit fibrant replacement for discrete G-spectra

TL;DR

This paper constructs an explicit fibrant model for discrete G-spectra when G has finite virtual cohomological dimension, simplifying the understanding of fibrant objects in this context.

Contribution

It provides a concrete fibrant replacement method for discrete G-spectra under specific finiteness conditions, advancing the comprehension of these objects.

Findings

Explicit fibrant model for discrete G-spectra with finite virtual cohomological dimension

Applications to closed subgroups of G

Simplification of fibrant object analysis in this setting

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.