nLab
topological cyclic homology

Context

Higher algebra

Algebraic theories

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Higher algebras

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Geometry on formal duals of algebras

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Cohomology

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Special notions

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  • differential cohomology

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Contents

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 comes from computational considerations, as in certain cases, these invariants are easier to compute than algebraic K-theory, while 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, 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)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 do 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

A general abstract construction is in

Review and exposition includes

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

An approach using only homotopy-invariant notions, which gives a construction of topological cyclic homology based on a new definition of the ∞-category of cyclotomic spectra is 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)

Last revised on September 2, 2017 at 21:45:34. See the history of this page for a list of all contributions to it.