Roughly speaking, the stable homotopy category of algebraic topology is obtained from the

Abstract not found

Abstract. We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. As an application we show that with an added assumption about underlying model structures Margolis ’ axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence. Also, the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, W-spaces, orthogonal spectra, simplicial functors) with symmetric spectra. The equivalences of modules, algebras and commutative algebras in these categories are also considered. 1.

Given a good homology theory E and a topological space X, E∗X is not just an E∗-module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a well-behaved Hopf algebroid (A, Γ). That is, we construct

Abstract. We revisit Freyd’s generating hypothesis in stable homotopy theory. We derive new equivalent forms of the generating hypothesis and some new consequences of it. A surprising one is that I, the Brown-Comenetz dual of the sphere and the source of many counterexamples in stable homotopy, is the cofiber of a self map of a wedge of spheres. We also show that a consequence of the generating hypothesis, that the homotopy of a finite spectrum that is not a wedge of spheres can never be finitely generated as a module over π∗S, is in fact true for many finite torsion spectra.

The stable homotopy category has been extensively studied by algebraic topologists for a long time. For many applications it is convenient or even necessary to work with point set level models of spectra as opposed to working up-to-homotopy, and the outcome of a calculation might depend

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

Abstract. We show how methods from K-theory of operator algebras can be applied in a completely algebraic setting to define a bivariant, M∞-stable, homotopy-invariant, excisive K-theory of algebras over a fixed unital ground ring H, (A, B) ↦ → kk∗(A, B), which is universal in the sense that it maps uniquely to any other such theory. It turns out kk is related to C. Weibel’s homotopy algebraic K-theory, KH. We prove that, if H is commutative and A is central as an H-bimodule, then kk∗(H, A) = KH∗(A).