Rational torus-equivariant stable homotopy. II: Algebra of the standard model (Q1934968)

As its title suggests, this is a continuation of earlier work by the author, cf. [J. Pure Appl. Algebra 212, No. 1, 72--98 (2008; Zbl 1128.55011)]. Let \(G=S^{1}\times \dots \times S^{1}\) be a torus of rank~\(r\). In earlier work a certain category \({\mathcal A}(G)\) was introduced and the author together with Shipley showed that it provided an algebraic model for rational \(G\)-equivariant cohomology theories. In fact they showed that the category of differential graded objects of \({\mathcal A}(G)\) is Quillen equivalent to rational \(G\)-spectra. The present paper studies a number of algebraic properties of \({\mathcal A}(G)\). In particular, it is shown that \({\mathcal A}(G)\) has injective dimension equal to \(r\), flatness properties are proved and right adjoints are constructed for the inclusion of \({\mathcal A}(G)\) into certain larger categories, giving concrete constructions of limits in \({\mathcal A}(G)\). The author promises a future systematic study of \({\mathcal A}(G)\) and its similarities to categories of sheaves on projective varieties.