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.

A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent `the same homotopy theory'. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a `ring spectrum with several objects', i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard's work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R. 1.

We characterize ring spectra morphism from the algebraic cobordism spectrum MGL to an oriented spectrum E (in the sense of Morel [Mo]) via formal group laws on the ”topological” subring E ∗ = ⊕iE 2i,i of E ∗ ∗. This result is then used to construct BP-spectra in essentially the same way Quillen ([Q1]) did for the complex-oriented topological case. 1

Abstract. We construct an algebraic commutative ring T-spectrum BO which is stably fibrant and (8,4)-periodic and such that on SmOp/S the cohomology theory (X,U) ↦→ (X,U) are BO p,q (X+/U+) and Schlichting’s hermitian K-theory functor (X,U) ↦ → KO [q] 2q−p canonically isomorphic. We use the motivic weak equivalence Z×HGr ∼ − → KSp relating the infinite quaternionic Grassmannian to symplectic K-theory to equip BO with the structure of a commutative monoid in the motivic stable homotopy category. When the base scheme is SpecZ [ 1 2], this monoid structure and the induced ring structure on the cohomology theory BO ∗, ∗ are the unique structures compatible with the products

This paper is concerned with importing the stable homotopy theory of symmetric spectra [4] and more generally presheaves of symmetric spectra [8] into the MorelVoevodsky stable category [9], [11], [12]. Loosely speaking, the latter is the result of formally inverting the functor X

Abstract. We construct algebraic cobordism spectra MSL and MSp. They are commutative monoids in the category of symmetric T ∧2-spectra. The spectrum MSp comes with a natural symplectic orientation given either by a tautological Thom class th MSp ∈ MSp 4,2 (MSp2), a tautological Pontryagin class p MSp 1 ∈ MSp 4,2 (HP ∞ ) or any of six other equivalent structures. For a commutative monoid E in the category SH(S) we prove that assignment ϕ ↦ → ϕ(th MSp) identifies the set of homomorphisms of monoids ϕ: MSp → E in the motivic stable homotopy category SH(S) with the set of tautological Thom elements of symplectic orientations of E. A weaker universality result is obtained for MSL and special linear orientations. 1.