• Corpus ID: 50623237

Topological cyclic homology via the norm

@article{Angeltveit2014TopologicalCH,
  title={Topological cyclic homology via the norm},
  author={Vigleik Angeltveit and Andrew J. Blumberg and Teena Gerhardt and Matthew Hill and Tyler Lawson and Michael A. Mandell},
  journal={arXiv: K-Theory and Homology},
  year={2014}
}
We describe a construction of the cyclotomic structure on topological Hochschild homology ($THH$) of a ring spectrum using the Hill-Hopkins-Ravenel multiplicative norm. Our analysis takes place entirely in the category of equivariant orthogonal spectra, avoiding use of the Bokstedt coherence machinery. We are able to define versions of topological cyclic homology ($TC$) and TR-theory relative to a cyclotomic commutative ring spectrum $A$. We describe spectral sequences computing this relative… 

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References

SHOWING 1-10 OF 47 REFERENCES

Localization theorems in topological Hochschild homology and topological cyclic homology

We construct localization cofibration sequences for the topological Hochschild homology (THH ) and topological cyclic homology (TC ) of small spectral categories. Using a global construction of the

On the topological Hochschild homology of $DX$

We begin a systematic study of the topological Hochschild homology of the commutative ring spectrum $DX$, the dual of a finite CW-complex $X$. We prove that the "Atiyah duality" between $THH(DX)$ and

The cyclotomic trace and algebraic K-theory of spaces

The cyclotomic trace is a map from algebraic K-theory of a group ring to a certain topological refinement of cyclic homology. The target is naturally mapped to topological Hochschild homology, and

Topological Hochschild Homology

In the appendix to [20] Waldhausen discussed a trace map tr:K(R)→HH(R), from the algebraic K‐theory of a ring to its Hochschild homology, which can be used to obtain information about K(R) from

Rings, Modules, and Algebras in Stable Homotopy Theory

Introduction Prologue: the category of ${\mathbb L}$-spectra Structured ring and module spectra The homotopy theory of $R$-modules The algebraic theory of $R$-modules $R$-ring spectra and the

Cyclic homology and equivariant homology

The purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of Connes [9-11], see also Loday and Quillen [20], and "IF equivariant homology and

Symmetric Spectra and 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

Localization for THH(ku) and the topological Hochschild and cyclic homology of Waldhausen categories

We prove a conjecture of Hesselholt and Ausoni-Rognes, establishing localization cofiber sequences of spectra for THH(ku) and TC(ku). These sequences support Hesselholt's view of the map l to ku as a

Relative algebraic K-theory and topological cyclic homology

In recent years, the study of the algebraic K-theory space K(R) of a ring R has been approached by the introduction of spaces with a more homological flavor. One collection of such spaces is