S-modules in the category of schemes

Structured Ring Spectra, 2004

Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E * E is flat over E * . We wish to address the following question: given a commutative E * -algebra A in E * E-comodules, is there an E∞-ring spectrum X with E * X ∼ = A as comodule algebras? We will formulate this as a moduli problem, and give a way -suggested by work of Dwyer, Kan, and Stover -of dissecting the resulting moduli space as a tower with layers governed by appropriate André-Quillen cohomology groups. A special case is A = E * E itself. The final section applies this to discuss the Lubin-Tate or Morava spectra En. * The authors were partially supported by the National Science Foundation. map(X, Y ) is weakly equivalence to the simplicial mapping set out of cofibrant model for X into a fibrant model for Y . Alternatively, one can write down map(X, Y ) as the nerve of an appropriate diagram category, such as the Dwyer-Kan hammock localization .

Advances in Mathematics

Spectral Mackey functors are homotopy-coherent versions of ordinary Mackey functors as defined by Dress. We show that they can be described as excisive functors on a suitable ∞-category, and we use this to show that universal examples of these objects are given by algebraic K-theory. More importantly, we introduce the unfurling of certain families of Waldhausen ∞-categories bound together with suitable adjoint pairs of functors; this construction completely solves the homotopy coherence problem that arises when one wishes to study the algebraic K-theory of such objects as spectral Mackey functors. Finally, we employ this technology to lay the foundations of equivariant stable homotopy theory for profinite groups; the lack of such foundations has been a serious impediment to progress on the conjectures of Gunnar Carlsson. We also study fully functorial versions of A-theory, upside-down A-theory, and the algebraic K-theory of derived stacks. Contents 0. Summary 1. Preliminaries on ∞-categories 2. The twisted arrow ∞-category 3. The effective Burnside ∞-category 4. Disjunctive ∞-categories 5. Disjunctive triples 6. Mackey functors 7. Mackey stabilization 8. Representable spectral Mackey functors and assembly morphisms 9. Mackey stabilization via algebraic K-theory 10. Waldhausen bicartesian fibrations 11. Unfurling 12. Horn filling in effective Burnside ∞-categories 13. The Burnside Waldhausen bicartesian fibration Example A. Coherent n-topoi and the Segal-tom Dieck splitting Example B. Equivariant spectra for a profinite group Example C. A-theory and upside-down A-theory of ∞-topoi Example D. Algebraic K-theory of derived stacks References 2 CLARK BARWICK

Topology, 2004

It is now well known that the K-theory of a Waldhausen category depends on more than just its (triangulated) homotopy category (see [20]). The purpose of this note is to show that the K-theory spectrum of a (good) Waldhausen category is completely determined by its Dwyer-Kan simplicial localization, without any additional structure. As the simplicial localization is a refined version of the homotopy category which also determines the triangulated structure, our result is a possible answer to the general question: "To which extent K-theory is not an invariant of triangulated derived categories ?"

Under a certain normalization assumption we prove that the $\Pro^1$-spectrum $\mathrm{BGL}$ of Voevodsky which represents algebraic $K$-theory is unique over $\Spec(\mathbb{Z})$. Following an idea of Voevodsky, we equip the $\Pro^1$-spectrum $\mathrm{BGL}$ with the structure of a commutative $\Pro^1$-ring spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over $\Spec(\mathbb{Z})$. For an arbitrary Noetherian scheme $S$ of finite Krull dimension we pull this structure back to obtain a distinguished monoidal structure on $\mathrm{BGL}$. This monoidal structure is relevant for our proof of the motivic Conner-Floyd theorem. It has also been used by Gepner and Snaith to obtain a motivic version of Snaith's theorem.

Higher K-theories, 1973

2021

We provide a synthesis of different topos-theoretical approaches of the general construction of spectra, exploring jointly its purely categorical, model theoretic and geometric aspects. We detail also the exact relation with Diers notion of spectrum. Then we give a more detailed account of Cole's original construction from bicategorical universal properties. We also provide new formulas to compute the spectrum of arbitrary etale maps as pseudolimits, as well as generalizations of results on fibered topos we then apply to compute spectrum of a modelled topos.

2003

Advances in Mathematics, 2005

This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of ∞-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove in particular, that for an S-category T endowed with an S-topology, there exists a model category of stacks over T , generalizing the model category structure on simplicial presheaves over a Grothendieck site of A. Joyal and R. Jardine. We also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves, by proving that there exists a one-to-one correspodence between S-topologies on an S-category T , and certain left exact Bousfield localizations of the model category of pre-stacks on T. Based on the above results, we study the notion of model topos introduced by C. Rezk, and we relate it to our model categories of stacks over S-sites. In the second part of the paper, we present a parallel theory where S-categories, S-topologies and S-sites are replaced by model categories, model topologies and model sites. We prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over S-sites is provided by the simplicial localization construction of Dwyer and Kan. As an example of application, we propose a definition ofétale K-theory of ring spectra, extending theétale K-theory of commutative rings.

2017

In this survey paper on commutative ring spectra we present some basic features of commutative ring spectra and discuss model category structures. As a first interesting class of examples of such ring spectra we focus on (commutative) algebra spectra over commutative Eilenberg-MacLane ring spectra. We present two constructions that yield commutative ring spectra: Thom spectra associated to infinite loop maps and Segal's construction starting with bipermutative categories. We define topological Hochschild homology, some of its variants, and topological Andre-Quillen homology. Obstruction theory for commutative structures on ring spectra is described in two versions. The notion of etale extensions in the spectral world is tricky and we explain why. We define Picard groups and Brauer groups of commutative ring spectra and present examples.

Journal of Pure and Applied Algebra, 1991