Rational torus-equivariant stable homotopy V: the torsion Adams spectral sequence

TL;DR

This paper develops a calculational framework for rational stable equivariant homotopy theory with a focus on torus groups, introducing a torsion model and a spectral sequence for computations.

Contribution

It introduces an abelian torsion model and a homology theory for rational G-spectra, enabling explicit calculations via a finite Adams spectral sequence.

Findings

Defined the abelian torsion model A_t(G) with finite injective dimension.

Constructed a homology theory t_* based on Borel construction homology.

Established a finite Adams spectral sequence converging to G-equivariant homotopy classes.

Abstract

We provide a calculational method for rational stable equivariant homotopy theory for a torus G based on the homology of the Borel construction on fixed points. More precisely we define an abelian torsion model, A_t(G) of finite injective dimension, a homology theory \piAt_* taking values in A_t(G) based on the homology of the Borel construction, and a finite Adams spectral sequence Ext_{A_t(G)}^{*,*}(\piAt_*(X), \piAt_*(Y)) ==> [X,Y]^G_* for rational G-spectra X and Y.