Abstract
We show that the category of free rational G-spectra for a connected compact Lie group G is Quillen equivalent to the category of torsion differential graded modules over the polynomial ring H*(BG). The ingredients are the enriched Morita equivalences of Schwede and Shipley (Topology 42(1):103–153, 2003), the functors of Shipley (Am J Math 129:351–379, 2007) making rational spectra algebraic, Koszul duality and thick subcategory arguments based on the simplicity of the derived category of a polynomial ring.
Highlights
ArXiv:0906.5575v2 [math.AT] 10 Jun 2010 as one in the naive sense
It follows that i∗ corresponds to r∗, but the right adjoint i! corresponds to a new functor; at the derived level r! is equivalent to a left adjoint r!′, and r!′ has a right adjoint r!
Our task is to obtain a Quillen equivalence between the category of rational free G-spectra and the category of differential graded (DG) torsion H∗(BG)-modules. In joining these two categories we have three main boundaries to cross: first, we have to pass from the realm of apparently unstructured homotopy theory to a category of modules, secondly we have to pass from a category of topological objects to a category of algebraic objects (DG vector spaces), and we have to pass from modules over an arbitrary ring to modules over a commutative ring. These three steps could in principle be taken in any order, but we have found it convenient to begin by moving to modules over a ring spectrum, to modules over a commutative ring spectrum and passing from modules over a commutative ring spectrum to modules over a commutative DGA
Summary
ArXiv:0906.5575v2 [math.AT] 10 Jun 2010 as one in the naive sense (i.e., a contravariant functor on the category of free G-spaces satisfying the Eilenberg-Steenrod axioms and the wedge axiom). The category of free rational G-spectra is equivalent to the category of Q-cellular modules over the commutative ring spectrum Rtop (Proposition 5.2).
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