nLab
topological cyclic homology

Context

Higher algebra

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

Topological Hochschild (resp. topological cyclic, survey is in (May) homology 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 RR, then there is a natural morphism of spectra

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

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 TCTC and the cyclotomic trace map (Blumberg-Gepner-Tabuada 11)

References

General

The original references are

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

See also

  • 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 Z/p kZ\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 KK-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 bub 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 June 9, 2016 10:59:18 by Urs Schreiber (131.220.184.222)