Rational torus-equivariant stable homotopy III: comparison of models

TL;DR

This paper develops and compares various algebraic models for rational torus-equivariant stable homotopy theory, establishing their equivalences and applications to spectra construction and algebraic models.

Contribution

It provides detailed comparison functors and proves the equivalence of different models based on subgroups and flags, advancing the algebraic understanding of rational torus-equivariant spectra.

Findings

Models based on all subgroups, connected subgroups, or dimensions are equivalent.

Comparison functors establish the equivalence of models.

Applications include algebraic models for spectra and geometric data construction.

Abstract

We give details of models for rational torus equivariant homotopy theory based on (a) all subgroups, connected subgroups or dimensions of subgroups and (b) on pairs or general flags. We provide comparison functors and show the models are equivalent. This is used in (A) An algebraic model for rational torus equivariant spectra (with B.Shipley; the new version of 1101.2511), (B) An algebraic model for the toral part of rational equivariant G-spectra for an arbitrary compact Lie group (1501.03425) and (C) construction of torus equivariant spectra from algebraic geometric data. It also highlights the role of the localization theorem in these models.