1.1. Simplicial sets 5 1.2. Symmetric spectra 6

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.

A finite group scheme G over a field k is equivalent to its coordinate algebra, a finite dimensional commutative Hopf algebra k[G] over k. In many contexts, it is natural to consider the rational (or Hochschild) cohomology of G with coefficients in a k[G]-comodule M. This is naturally isomorphic to the cohomology of the dual cocommutative Hopf algebra k[G] # with coefficients in the k[G] #-module M. In this latter formulation, we encounter familiar examples of the cohomology of group algebras kπ of a finite groups π and of restricted enveloping algebras V (g) of finite dimensional restricted Lie algebras g. In recent years, the representation theory of the algebras kπ and V (g) has been studied by considering the spectrum of the cohomology algebra with coefficients in the ground field k and the support in this spectrum of the cohomology with coefficients in various modules. This approach relies on the fact that H ∗ (π, k) and H ∗ (V (g), k) are finitely generated k-algebras as proved in [G], [E], [V], [FP2]. Rational representations of algebraic groups in positive characteristic correspond to representations of a hierarchy of finite group schemes. In order to begin the process of introducing geometric methods to the study of these other group schemes, finite generation must be proved. Such a proof has proved surprisingly elusive (though partial results can be found in [FP2]). The main theorem of this paper is the following: Theorem 1.1. Let G be a finite group scheme and M a finite dimensional rational G-module. Then H ∗ (G, k) is a finitely generated k-algebra and H ∗ (G, M) is a finite H ∗ (G, k)-module. Work in progress by C. Bendel (and the authors) reveals that Theorem 1.1 and its proof will provide interesting theorems of a geometric nature concerning the representation theory of finite group schemes. In a sense that is made explicit in section 1, our proof of finite generation is quite constructive. We embed G in some general linear group GLn and establish the existence of universal extension classes for GLn of specified degrees. In a direct manner, these classes provide the generators of H ∗ (G, k). In order to construct these universal extension classes, we follow closely the approach of V. Franjou, J. Lannes, and L. Schwartz [FLS]. This entails the study of

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The purpose of this paper and its sequel is to determine the homotopy groups of the spectrum THH(l). Here p is an odd prime, l is the Adams summand of p-local connective K-theory (see for example [25]) and THH is the topological Hochschild homology

A functor is defined which detects stable equivalences of symmetric spectra. As an application, the definition of topological Hochschild homology on symmetric ring spectra using the Hochschild complex is shown to agree with Bokstedt's original ad hoc definition. In particular, this shows that Bokstedt's definition is correct even for non-connective, non-convergent symmetric ring spectra.

The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure. We show that the associated stable homotopy theory is completely determined by a ring spectrum functorially associated with the algebraic theory. For several familiar algebraic theories we can identify the parameterizing ring spectrum; for other theories we obtain new examples of ring spectra. For the theory of commutative algebras we obtain a ring spectrum which is related to AndreH}Quillen homology via certain spectral sequences. We show that the (co-)homology of an algebraic theory is isomorphic to the topological Hochschild (co-)homology of the parameterizing ring spectrum. # 2000 Elsevier Science Ltd. All rights reserved. MSC: 55U35; 18C10 Keywords: Algebraic theories; Ring spectra; AndreH}Quillen homology; #-spaces The original motivation for this paper came from the attempt to generalize a rational result about the homotopy theory of commutative rings. For...

The notion of motivic functors refers to a motivic homotopy theoretic analog of continuous functors. In this paper we lay the foundations for a homotopical study of these functors. Of particular interest is a model structure suitable for studying motivic functors which preserve motivic weak equivalences and a model structure suitable for motivic stable homotopy theory. The latter model is Quillen equivalent to the category of motivic symmetric spectra. There is a symmetric monoidal smash product of motivic functors, and all model structures constructed are compatible with the smash product in the sense that we can do homotopical algebra on the various categories of modules and algebras. In particular, motivic cohomology

Abstract. We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a “global ” construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofiber sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of Thomason-Trobaugh in K-theory. We also deduce versions of Thomason’s blow-up formula and the projective bundle formula for THH and TC. 1.