Abstract. The paper gives a new proof that the model categories of stable modules for the rings Z/p 2 and Z/p[ɛ]/(ɛ 2) are not Quillen equivalent. The proof uses homotopy endomorphism ring spectra. Our considerations lead to an example of two differential graded algebras which are derived equivalent but whose associated model categories of modules are not Quillen equivalent. As a bonus, we also obtain derived equivalent dgas with non-isomorphic K-theories. Contents

and Hochschild homology

The notion of p-compact group [10] is a homotopy theoretic version of the geometric or analytic notion of compact Lie group, although the homotopy theory differs from the geometry is that there are parallel theories of p-compact groups, one for each prime number p. A key

Abstract We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established model categories of monoids, modules and algebras [SS00]. As an application we extend the Dold-Kan equivalence to show that the model categories of simplicial rings, modules and algebras are Quillen equivalent to the associated model categories of connected differential graded rings, modules and algebras. We also show that our classification results from [SS03] concerning stable model categories translate to any one of the known symmetric monoidal model categories of spectra. AMS Classification 55U35; 18D10, 55P43, 55P62

The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms. Such examples are called ‘algebraic ’ because they have underlying abelian (or at least additive) categories. Stable homotopy theory produces examples of triangulated categories by quite different means, and in this context the underlying categories are usually very ‘non-additive ’ before passing to homotopy classes of morphisms. We call such triangulated categories topological, compare Definition 3.1; this class includes the algebraic triangulated categories. The purpose of this paper is to explain some systematic differences between these two kinds of triangulated categories. There are certain properties – defined entirely in terms of the triangulated structure – which hold in all algebraic examples, but which can fail in general. These differences are all torsion phenomena, and rationally every topological triangulated category is algebraic (at least under mild size restrictions). Our main tool is a new numerical invariant, the n-order of an object in a triangulated category, for n a natural number (see Definition 1.1). The n-order is a nonnegative integer (or infinity), and an object Y has positive n-order if and only if n · Y = 0; the n-order can be thought of

Many triangulated categories arise from chain complexes in an additive or abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms. Such examples are called ‘algebraic ’ because they have underlying additive categories. Stable homotopy theory produces examples of triangulated categories by quite different means, and in this context the underlying categories are usually very ‘non-additive ’ before passing to homotopy classes of morphisms. We call such triangulated categories topological, and formalize this in Definition 1.4 via homotopy categories of stable cofibration categories. The purpose of this paper is to explain some systematic differences between algebraic and topological triangulated categories. There are certain properties – defined entirely in terms of the triangulated structure – which hold in all algebraic examples, but which can fail in general. The precise statements use the n-order of a triangulated category, for n a natural number (see Definition 2.1). The n-order is a non-negative integer (or infinity), and it measures, roughly speaking, ‘how strongly ’ the relation n · Y/n = 0 holds for the objects Y in a given triangulated category (where Y/n denotes a cone of multiplication by n on Y). Our main results are: • The n-order of every algebraic triangulated category is infinite (Theorem 3.3). • For every prime p, the p-order of every topological triangulated category is at least p − 1 (Theorem 8.2). • For every prime p, the p-order of the p-local stable homotopy category is exactly p − 1

We give a functorial construction of k–invariants for ring spectra and use these to classify extensions in the Postnikov tower of a ring spectrum. 55P43; 55S45 1

Abstract. We prove that for certain monoidal (Quillen) model categories, the category of comonoids therein also admits a model structure.

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