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# 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: https://dx.doi.org/10.4310/HHA.2019.v21.n1.a4

#### Authors

#### 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

Received 12 September 2017

Received revised 12 March 2018

Published 22 August 2018