Geometry & Topology

Topological Hochschild homology and cohomology of $A_\infty$ ring spectra

Vigleik Angeltveit

Full-text: Open access


Let A be an A ring spectrum. We use the description from our preprint [math.AT/0612165] of the cyclic bar and cobar construction to give a direct definition of topological Hochschild homology and cohomology of A using the Stasheff associahedra and another family of polyhedra called cyclohedra. This construction builds the maps making up the A structure into THH(A), and allows us to study how THH(A) varies over the moduli space of A structures on A.

As an example, we study how topological Hochschild cohomology of Morava K–theory varies over the moduli space of A structures and show that in the generic case, when a certain matrix describing the noncommutativity of the multiplication is invertible, topological Hochschild cohomology of 2–periodic Morava K–theory is the corresponding Morava E–theory. If the A structure is “more commutative”, topological Hochschild cohomology of Morava K–theory is some extension of Morava E–theory.

Article information

Geom. Topol., Volume 12, Number 2 (2008), 987-1032.

Received: 5 April 2007
Accepted: 8 February 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)
Secondary: 18D50: Operads [See also 55P48] 55S35: Obstruction theory

structured ring spectra Morava K-theory associahedra cyclohedra topological Hochschild homology


Angeltveit, Vigleik. Topological Hochschild homology and cohomology of $A_\infty$ ring spectra. Geom. Topol. 12 (2008), no. 2, 987--1032. doi:10.2140/gt.2008.12.987.

Export citation


  • V Angeltveit, The cyclic bar construction on ${A}_\infty$ H–spaces
  • V Angeltveit, Enriched Reedy categories, to appear in Proc. Amer. Math. Soc.
  • V Angeltveit, J Rognes, Hopf algebra structure on topological Hochschild homology, Algebr. Geom. Topol. 5 (2005) 1223–1290
  • C Ausoni, J Rognes, Algebraic K–theory of the fraction field of topological K–theory, in preparation
  • N A Baas, I Madsen, On the realization of certain modules over the Steenrod algebra, Math. Scand. 31 (1972) 220–224
  • A Baker, A Jeanneret, Brave new Bockstein operations, preprint available at\char'176ajb/dvi-ps.html
  • A Baker, A Lazarev, On the Adams spectral sequence for $R$–modules, Algebr. Geom. Topol. 1 (2001) 173–199
  • A Baker, A Lazarev, Topological Hochschild cohomology and generalized Morita equivalence, Algebr. Geom. Topol. 4 (2004) 623–645
  • M Basterra, M A Mandell, Multiplicative structures on topological Hochschild homology, to appear
  • J M Boardman, Conditionally convergent spectral sequences, from: “Homotopy invariant algebraic structures (Baltimore, MD, 1998)”, Contemp. Math. 239, Amer. Math. Soc. (1999) 49–84
  • J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Math. 347, Springer, Berlin (1973)
  • M Bökstedt, Topological Hochschild homology, unpublished
  • M Bökstedt, The topological Hochschild homology of $\mathbb{Z}$ and $\mathbb{Z}/p$, unpublished
  • R R Bruner, J P May, J E McClure, M Steinberger, $H\sb \infty $ ring spectra and their applications, Lecture Notes in Math. 1176, Springer, Berlin (1986)
  • D Dugger, B Shipley, Postnikov extensions of ring spectra, Algebr. Geom. Topol. 6 (2006) 1785–1829
  • W G Dwyer, J P C Greenlees, Complete modules and torsion modules, Amer. J. Math. 124 (2002) 199–220
  • A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs 47, Amer. Math. Soc. (1997) With an appendix by M. Cole
  • Z Fiedorowicz, R Vogt, Topological Hochschild homology of $E_n$–ring spectra
  • P G Goerss, Associative $MU$–algebras, preprint available at\char'176pgoerss/
  • P G Goerss, M J Hopkins, Moduli spaces of commutative ring spectra, from: “Structured ring spectra”, London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 151–200
  • M J Hopkins, H R Miller, Lubin–Tate deformations in algebraic topology, preprint
  • S O Kochman, Symmetric Massey products and a Hirsch formula in homology, Trans. Amer. Math. Soc. 163 (1972) 245–260
  • A Lazarev, Hoschschild cohomology and moduli spaces of strongly homotopy associative algebras, Homology Homotopy Appl. 5 (2003) 73–100
  • A Lazarev, Towers of $M$U–algebras and the generalized Hopkins–Miller theorem, Proc. London Math. Soc. $(3)$ 87 (2003) 498–522
  • T Leinster, Higher operads, higher categories, London Math. Soc. Lecture Note Series 298, Cambridge University Press (2004)
  • J E McClure, J H Smith, A solution of Deligne's Hochschild cohomology conjecture, from: “Recent progress in homotopy theory (Baltimore, MD, 2000)”, Contemp. Math. 293, Amer. Math. Soc. (2002) 153–193
  • C Nassau, On the structure of $P(n)_{*}P((n))$ for $p=2$, Trans. Amer. Math. Soc. 354 (2002) 1749–1757
  • D C Ravenel, Complex cobordism and stable homotopy groups of spheres. 2nd ed., AMS Chelsea Pub. (2004)
  • C Rezk, Notes on the Hopkins–Miller theorem, from: “Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997)”, Contemp. Math. 220, Amer. Math. Soc. (1998) 313–366
  • C Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006) 969–1014
  • A Robinson, Obstruction theory and the strict associativity of Morava $K$–theories, from: “Advances in homotopy theory (Cortona, 1988)”, London Math. Soc. Lecture Note Ser. 139, Cambridge Univ. Press (1989) 143–152
  • A Robinson, S Whitehouse, Operads and $\Gamma$–homology of commutative rings, Math. Proc. Cambridge Philos. Soc. 132 (2002) 197–234
  • J Rognes, Galois extensions of structured ring spectra, Mem. Amer. Math. Soc. 192 (2008) 1–97
  • N P Strickland, Products on ${\rm MU}$–modules, Trans. Amer. Math. Soc. 351 (1999) 2569–2606
  • M van den Bergh, A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc. 126 (1998) 1345–1348
  • M van den Bergh, Erratum to: “A relation between Hochschild homology and cohomology for Gorenstein rings” [Proc. Amer. Math. Soc. 126 (1998) 1345–1348], Proc. Amer. Math. Soc. 130 (2002) 2809–2810 (electronic)
  • C A Weibel, S C Geller, Étale descent for Hochschild and cyclic homology, Comment. Math. Helv. 66 (1991) 368–388