nLab topological cyclic homology

Idea

Topological Hochschild homology (resp. topological cyclic homology) (see the survey 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.

One motivation for their study are computational considerations, as well in certain cases, these invariants are easier to compute than algebraic K-theory, while 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, called the Dennis trace map, whose fiber is relatively well-understood. Since Hochschild homology spectra are naturally cyclotomic spectra, this map factors through the topological cyclic homology spectrum via a map called the cyclotomic trace, which acts 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 do $TC$ and the cyclotomic trace map (Blumberg-Gepner-Tabuada 11)

Related concepts

• cyclic homology, dihedral homology

• Chern character

• norm cofibration

• prismatic cohomology

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

• Gunnar Carlsson, Christopher Douglas, Bjørn Ian Dundas, Higher topological cyclic homology and the Segal conjecture for tori,

  Adv. Math. 226 (2011), no. 2, 1823–1874, (arXiv:0803.2745, MR2737802, doi)

A $(\infty,1)$-category theoretic construction:

• Thomas Nikolaus, Peter Scholze, On topological cyclic homology, Acta Math. 221 2 (2018) 203-409 [arXiv:1707.01799, doi:10.4310/ACTA.2018.v221.n2.a1]

A relation to integral p-adic Hodge theory is discussed in

• Bhargav Bhatt, Matthew Morrow, Peter Scholze, Topological Hochschild Homology and Integral $p$-adic Hodge Theory [arXiv:1802.03261]

Review and exposition includes

• Peter May, Topological Hochschild and Cyclic Homology and Algebraic K-theory (pdf)

• Anthony Elmendorf, Igor Kriz, Michael Mandell, Peter May, chapter IX of Rings, modules and algebras in stable homotopy theory, AMS Mathematical Surveys and Monographs Volume 47 (1997) (pdf)

• Bjørn Dundas, Thomas Goodwillie, Randy McCarthy, The local structure of algebraic K-theory, Springer 2013 (pdf)

• Teena Gerhardt, Computations in algebraic K-theory, talk at CUNY Workshop on differential cohomologies 2014 (video recording)

• Ricardo Andrade, THH notes, MIT juvitop seminar pdf, babytop seminar pdf

• Anatoly Preygel, Hochschild homology notes, juvitop seminar, pdf

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

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

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

• Andrew Blumberg, David Gepner, Goncalo Tabuada, Uniqueness of the multiplicative cyclotomic trace, Advances in Mathematics 260 (2014) 191-232 (arXiv:1103.3923)

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 $\mathbf{Z}/p^k\mathbf{Z}$, 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

• Christian Ausoni, John Rognes, Algebraic K-theory of topological K-theory, Acta Mathematica March 2002, Volume 188, Issue 1, pp 1-39 (KTheory 0405)

• Vigleik Angeltveit, Michael Hill, Tyler Lawson, Topological Hochschild homology of $\ell$ and $ko$ (arXiv:0710.4368)

• Andrew Blumberg, Michael Mandell, Localization for $THH(ku)$ and the topological Hochschild and cyclic homology of Waldhausen categories (arXiv:1111.4003)

and of tmf in

• Robert Bruner, John Rognes, Topological Hochschild homology of topological modular forms, talk at Nordic Topology Meeting NTNU (2008) [pdf]

On the topological Hochschild homology of the Lubin-Tate ring spectrum via factorization homology:

• Geoffroy Horel, Higher Hochschild homology of the Lubin-Tate ring spectrum, Algebr. Geom. Topol. 15 (2015) 3215-3252 [arXiv:1311.2805, doi:10.2140/agt.2015.15.3215]