On the Algebraic Classification of Module Spectra

Using methods developed by Franke, we obtain algebraic classification results for modules over certain symmetric ring spectra ($S$-algebras). In particular, for any symmetric ring spectrum $R$ whose graded homotopy ring $\pi_*R$ has graded global homological dimension 2 and is concentrated in degrees divisible by some natural number $N \geq 4$, we prove that the homotopy category of $R$-modules is equivalent to the derived category of the homotopy ring $\pi_*R$. This improves the Bousfield-Wolbert algebraic classification of isomorphism classes of objects of the homotopy category of $R$-modules. The main examples of ring spectra to which our result applies are the $p$-local real connective $K$-theory spectrum $ko_{(p)}$, the Johnson-Wilson spectrum E(2), and the truncated Brown-Peterson spectrum $BP$, for an odd prime $p$.Comment: 39 page

Rigidity in Equivariant Stable Homotopy Theory

For any finite group G, we show that the 2-local G-equivariant stable homotopy category, indexed on a complete G-universe, has a unique equivariant model in the sense of Quillen model categories. This means that the suspension functor, homotopy cofiber sequences and the stable Burnside category determine all "higher order structure" of the 2-local G-equivariant stable homotopy category, such as the equivariant homotopy types of function G-spaces. The theorem can be seen as an equivariant version of Schwede's rigidity theorem at the prime 2

Topological Hochschild homology and the cyclic bar construction in symmetric spectra

The cyclic bar construction in symmetric spectra and B\"okstedt's original construction are two possible ways to define the topological Hochschild homology of a symmetric ring spectrum. In this short note we explain how to correct an error in Shipley's original comparison of these two approaches.Comment: v2: 7 pages; exposition improved, accepted for publication in Proceedings of the AM

Rigidity and exotic models for v1v_1-local GG-equivariant stable homotopy theory

We prove that the $v_1$-local $G$-equivariant stable homotopy category for $G$ a finite group has a unique $G$-equivariant model at $p=2$. This means that at the prime $2$ the homotopy theory of $G$-spectra up to fixed point equivalences on $K$-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for $K$-local spectra of the second author with the equivariant rigidity result for $G$-spectra of the first author. Further, when the prime $p$ is at least $5$ and does not divide the order of $G$, we provide an algebraic exotic model as well as a $G$-equivariant exotic model for the $v_1$-local $G$-equivariant stable homotopy category, showing that for primes $p \ge 5$ equivariant rigidity fails in general.Comment: 34 page

On the de Rham-Witt complex over perfectoid rings

Acknowledgments The 1st author is very grateful to Lars Hesselholt, who introduced and explained many aspects of this project to him. (The project began around 2014 when the 1st author was a postdoc of Lars Hesselholt at the University of Copenhagen.) The 1st author would also like to especially thank Bhargav Bhatt for assistance at many different points, especially during a visit to the University of Michigan. Furthermore, both authors thank Johannes Anschütz, Bryden Cais, Dustin Clausen, Elden Elmanto, Kiran Kedlaya, Arthur-César Le Bras, Thomas Nikolaus, Peter Scholze, and David Zureick-Brown for useful conversations regarding this paper. The authors also thank the anonymous referee of an earlier version of this paper; the referee provided careful feedback and many suggestions for improvement, especially in Section 7. Both authors thank the Department of Mathematical Sciences of the University of Copenhagen for its hospitality and pleasant working environment.Peer reviewedPostprin