Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZ-algebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZ-algebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that bascially any rational stable model category is Quillen equivalent to modules over a differential graded Q-algebra (with many objects). 1.

Abstract. We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an Eilenberg-Mac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasiisomorphic dgas are topologically equivalent, but we produce explicit counterexamples of the converse. We also develop an associated notion of topological Morita equivalence using a homotopical version of tilting. Contents

graded objects in A models the whole rational S 1-equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between dgA and the model category of rational S 1-equivariant spectra, before the quasi-isomorphisms or stable equivalences have been inverted. This implies that all of the higher order structures such as mapping spaces, function spectra and homotopy (co)limits are reflected in the algebraic model. The construction of this equivalence involves calculations with Massey products. In an appendix we show that Toda brackets, and hence also Massey products, are determined by the derived category.

Abstract. We define the notion of an additive model category and prove that

Greenlees defined an abelian category A whose derived category is equivalent to the rational S 1 -equivariant stable homotopy category whose objects represent rational S 1 - equivariant cohomology theories. We show that in fact the model category of di#erential graded objects in A models the whole rational S 1 -equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between dgA and the model category of rational S 1 -equivariant spectra, before the quasi-isomorphisms or stable equivalences have been inverted. This implies that all of the higher order structures such as mapping spaces, function spectra and homotopy (co)limits are reflected in the algebraic model. The construction of this equivalence involves calculations with Massey products. In an appendix we show that Toda brackets, and hence also Massey products, are determined by the derived category. 1.

We construct an abelian category A(G) of sheaves over a category of closed subgroups of the r-torus G. The category A(G) is of injective dimension r, and can be used as a model for rational G-spectra. Indeed, we show that there is a homology theory A : G-spectra ! 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 nite stage. This is the rst paper in a series of three. It culminates in [8] where the author and B.E.Shipley combine the Adams spectral sequence constructed here with the enriched Morita equivalence of Schwede and Shipley [9] to deduce that the category of dierential graded objects of A(G) is Quillen equivalent to the category of rational G-spectra. Contents Part 1.

We discuss an analogue of Morita theory for ring spectra, a thickening of the category of rings inspired by stable homotopy theory. This follows work by Rickard and Keller on Morita theory for derived categories. We also discuss two results for derived equivalences of DGAs which show they differ from derived equivalences of rings.

Abstract. We give a functorial construction of a rational S 1-equivariant cohomology theory from an elliptic curve A equipped with suitable coordinate data. The elliptic curve may be recovered from the cohomology theory; indeed, the value of the cohomology theory on the compactification of an S 1-representation is given by the sheaf cohomology of a suitable line bundle on A. This suggests the construction: by considering functions on the elliptic curve with specified poles one may write down the representing S 1-spectrum in the author’s algebraic model of rational S 1-spectra [9]. The construction extends to give an equivalence of categories between the homotopy

We give a new proof that for a finite group G, the category of rational G-equivariant spectra is Quillen equivalent to the product of the model categories of chain complexes of modules over the rational group ring of the Weyl group of H in G, as H runs over the conjugacy classes of subgroups of G. Furthermore the Quillen equivalences of our proof are all symmetric monoidal. Thus we can understand categories of algebras or modules over a ring spectrum in terms of the algebraic model. 1