# nLab topological cyclic homology

### Context

#### Higher algebra

higher algebra

universal algebra

cohomology

# Contents

## Idea

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

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

$\array{ && \mathbf{TC}(R) \\ & \nearrow & \downarrow \\ \mathbf{K}(R) &\longrightarrow& \mathbf{THH}(R) }$

from the algebraic K-theory spectrum to the topological Hochschild homology spectrum and factoring through the topological cyclic homology spectrum called the cyclotomic trace which much like a Chern character map for algebraic K-theory.

## References

The originalreferences

• 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

Review and exposition includes

These theories are the target for the trace map from $K$-theory, so they can be viewed as an approximation to algebraic K-theory of a ring spectrum.

• 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, pdf

• 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

• Thomas Geisser, Motivic Cohomology, K-Theory and Topological Cyclic Homology, Handbook of K-theory II.1, pdf

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

• Gunnar Carlsson, Christopher L. Douglas, Bjørn Ian Dundas, Higher topological cyclic homology and the Segal conjecture for tori, Adv. Math. 226 (2011), no. 2, 1823–1874, MR2737802, doi

• Ib Madsen, Algebraic K-theory and traces, pdf

• Dundas, Goodwillie and McCarthy, The local structure of algebraic K-theory, book in progress, pdf

• 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

Revised on September 6, 2014 12:16:15 by Urs Schreiber (141.0.9.60)