Rational torus-equivariant homotopy I: calculating groups of stable maps

TL;DR

This paper develops an algebraic model for rational G-spectra over an r-torus, enabling explicit calculations of stable maps and providing a convergent Adams spectral sequence.

Contribution

It constructs an abelian category of sheaves over subgroups of the r-torus and demonstrates its effectiveness as a model for rational G-spectra, including a collapsing spectral sequence.

Findings

Constructed the abelian category A(G) of sheaves with finite injective dimension.

Established a homology theory πA_* for rational G-spectra.

Proved the Adams spectral sequence converges and collapses at a finite stage.

Abstract

We construct an abelian category A(G) of sheaves over a category of closed subgroups of the r-torus G and show it is of finite injective dimension. It can be used as a model for rational $G$-spectra in the sense that there is a homology theory \piA_*: G-spectra/Q --> A(G) on rational G-spectra with values in A(G), and the associated Adams spectral sequence converges for all rational $G$-spectra and collapses at a finite stage.