Homology, Homotopy and Applications

Volume 21 (2019)

Number 1

An algebraic model for rational naïve-commutative $G$-equivariant ring spectra for $\textrm{finite} \: G$

Pages: 73 – 93

DOI: http://dx.doi.org/10.4310/HHA.2019.v21.n1.a4

Authors

Barnes David (Mathematical Sciences Research Centre, School of Mathematics and Physics, Queen’s University Belfast, Belfast, United Kingdom)

J.P.C. Greenlees (Mathematics Institute, University of Warwick, Coventry, United Kingdom)

Magdalena Kȩdziorek (Max-Planck-Institut für Mathematik, Bonn, Germany)

Abstract

Equipping a non-equivariant topological $E_{\infty}$-operad with the trivial $G$-action gives an operad in $G$-spaces. The algebra structure encoded by this operad in $G$-spectra is characterised homotopically by having no non-trivial multiplicative norms. Algebras over this operad are called naïve-commutative ring $G$-spectra. In this paper we let $G$ be a finite group and we show that commutative algebras in the algebraic model for rational $G$-spectra model the rational naïve-commutative ring $G$-spectra. In other words, a rational naïve-commutative ring $G$-spectrum is given in the algebraic model by specifying a $\mathbb{Q} [W_G (H)]$-differential graded algebra for each conjugacy class of subgroups $H$ of $G$. Here $W_G (H) = N_G (H)/H$ is the Weyl group of $H$ in $G$.

Keywords

rational equivariant spectrum, commutative equivariant ring spectrum, left Bousfield localisation, model category, algebraic model

2010 Mathematics Subject Classification

55N91, 55P42, 55P60

Full Text (PDF format)

Received 12 September 2017

Received revised 12 March 2018

Published 22 August 2018