Rational torus-equivariant stable homotopy theory II: the algebra of the standard model

TL;DR

This paper advances rational torus-equivariant stable homotopy theory by analyzing the algebraic structure of the model category, establishing its injective dimension, and developing tools for algebraic constructions.

Contribution

It proves that the abelian category A(G) has injective dimension equal to the torus rank and constructs torsion functors for algebraic adjunctions.

Findings

A(G) has injective dimension exactly r.

Constructed torsion functors for right adjoint constructions.

Proved a flatness result and described algebraic change of groups adjunctions.

Abstract

In previous work it is shown that there is an abelian category A(G) constructed to model rational G-equivariant cohomology theories, where G is a torus of rank r together with a homology functor \piA_* : Gspectra ---> A(G), and an Adams spectral sequence Ext_{A (G)} (\piA_*(X), \piA_*(Y)) ===> [X,Y]^G_* In joint work with Shipley (arxiv:1101.2511), it is shown that the Adams spectral sequence can be lifted to a Quillen equivalence Rational-Gspectra = DG-A (G). The purpose of the present paper is to prove that A(G) has injective dimension precisely r, and to construct certain torsion functors allowing us to make certain right adjoint constructions (such as products) in A(G). Along the way, we have an opportunity to prove a flatness result, and describe algebraic counterparts of some basic change of groups adjunctions.