# nLab topological cyclic homology

### Context

#### Higher algebra

higher algebra

universal algebra

cohomology

# Contents

## Idea

Topological Hochschild homology (resp. topological cyclic homology) survey is in (May) is a refinement of Hochschild homology/cyclic homology from commutative rings/algebras to the higher algebra of ring spectra/E-∞ rings/E-∞ algebras.

In particular, given a ring $R$, then there is a natural morphism of spectra

$\array{ && \mathbf{TC}(R) \\ & {}^{\mathllap{cyclotomic \atop trace}}\nearrow & \downarrow \\ \mathbf{K}(R) &\underset{}{\longrightarrow}& \mathbf{THH}(R) }$

from the algebraic K-theory spectrum to the topological Hochschild homology spectrum, the Dennis trace map. Since Hochschild homology spectra are naturally cyclotomic spectra, that factors through the topological cyclic homology spectrum via a map called the cyclotomic trace which much like a Chern character map for algebraic K-theory.

The spectra $THH(R)$ and $TC(R)$ are typically easier to analyze than $K(R)$. Moreover, the difference between them and $K(R)$ is “locally constant” (Dundas-Goodwillie-McCarthy13) and often otherwise bounded in complexity. Accordingly, $THH$ and $TC$ are in practice computationally useful approximations to $K$.

There are various generalizations:

1. Just as for basic Hochschild homology, there is higher topological Hochschild homology (Carlsson-Douglas-Dundas 08) given not just by derived loop spaces but by derived mapping spaces out of higher dimensional tori.

2. Just as algebraic K-theory generalizes from E-∞ rings to stable ∞-categories, so does $TC$ and the cyclotomic trace map (Blumberg-Gepner-Tabuada 11)

## References

### General

The original references are

• Marcel Bökstedt, Topological Hochschild homology, Bielefeld, 1985, 1988

• Marcel Bökstedt, W.C. Hsiang, Ib Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), 463-539, MR94g:55011, doi

• Marcel Bökstedt, Ib Madsen, Topological cyclic homology of the integers, $K$-theory (Strasbourg, 1992). Astérisque 226 (1994), 7–8, 57–143.

and a further generalization is defined in

• Bjørn Ian Dundas, Randy McCarthy, Topological Hochschild homology of ring functors and exact categories, J. Pure Appl. Algebra 109 (1996), no. 3, 231–294, MR97i:19001, doi

Higher topological Hochschild Homology is discussed in

Review and exposition includes

Abstract characterization of the Dennis trace and cyclotomic trace is discussed in

• T. Pirashvili, F. Waldhausen, Mac Lane homology and topological Hochschild homology, J. Pure Appl. Algebra 82 (1992), 81-98, MR96d:19005, doi

• T. Pirashvili, On the topological Hochschild homology of $\mathbf{Z}/p^k\mathbf{Z}$, Comm. Algebra 23 (1995), no. 4, 1545–1549, MR97h:19007, doi

• Z. Fiedorowicz, T. Pirashvili, R. Schwänzl, R. Vogt, F. Waldhausen, Mac Lane homology and topological Hochschild homology, Math. Ann. 303 (1995), no. 1, 149–164, MR97h:19007, doi

• Bjørn Ian Dundas, Relative K-theory and topological cyclic homology, Acta Math. 179 (1997), 223-242, (publisher)

• Thomas Geisser, Lars Hesselhoft, Topological cyclic homology of schemes, in: Algebraic $K$-theory (Seattle, WA, 1997), 41–87, Proc. Sympos. Pure Math. 67, Amer. Math. Soc. 1999, MR2001g:19003; K-theory archive

• R. McCarthy, Relative algebraic K-theory and topological cyclic homology, Acta Math. 179 (1997), 197-222.

• J. McClure, R. Staffeldt, On the topological Hochschild homology of $b u$, I, pdf

• Daniel Joseph Vera, Topological Hochschild homology of twisted group algebra, MIT Ph. D. thesis 2006, pdf

• V. Angeltveit, A. Blumberg, T. Gerhardt, M. Hill, T. Lawson, M. Mandell, Topological cyclic homology via the norm (arXiv:1401.5001)

### Examples

THH and TC specifically of ku and ko is discussed in

and of tmf in

• Bob Bruner?, John Rognes, Topological Hochschild homology of topological modular forms 2008 (pdf)

Revised on August 30, 2016 11:35:25 by Urs Schreiber (89.204.155.205)