Topological Hochschild homology and cohomology of A∞ ring spectra

Angeltveit, Vigleik¹

¹ University of Chicago, Department of Mathematics, 5734 S University Ave, Chicago IL 60637, USA

Geometry & topology, Tome 12 (2008) no. 2, pp. 987-1032.

Résumé

Let $A$ be an $A_{\infty }$ 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 $A_{\infty }$ structure into $THH\left(A\right)$, and allows us to study how $THH\left(A\right)$ varies over the moduli space of $A_{\infty }$ structures on $A$.

As an example, we study how topological Hochschild cohomology of Morava $K$–theory varies over the moduli space of $A_{\infty }$ 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 $A_{\infty }$ structure is “more commutative”, topological Hochschild cohomology of Morava $K$–theory is some extension of Morava $E$–theory.

Détail

DOI : 10.2140/gt.2008.12.987

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

Bibliographie

[1] V Angeltveit, The cyclic bar construction on ${A}_\infty$ H–spaces

[2] V Angeltveit, Enriched Reedy categories, to appear in Proc. Amer. Math. Soc.

[3] V Angeltveit, J Rognes, Hopf algebra structure on topological Hochschild homology, Algebr. Geom. Topol. 5 (2005) 1223

[4] C Ausoni, J Rognes, Algebraic K–theory of the fraction field of topological K–theory, in preparation

[5] N A Baas, I Madsen, On the realization of certain modules over the Steenrod algebra, Math. Scand. 31 (1972) 220

[6] A Baker, A Jeanneret, Brave new Bockstein operations, preprint available at http://www.maths.gla.ac.uk/ ajb/dvi-ps.html

[7] A Baker, A Lazarev, On the Adams spectral sequence for $R$–modules, Algebr. Geom. Topol. 1 (2001) 173

[8] A Baker, A Lazarev, Topological Hochschild cohomology and generalized Morita equivalence, Algebr. Geom. Topol. 4 (2004) 623

[9] M Basterra, M A Mandell, Multiplicative structures on topological Hochschild homology, to appear

[10] J M Boardman, Conditionally convergent spectral sequences, from: "Homotopy invariant algebraic structures (Baltimore, MD, 1998)", Contemp. Math. 239, Amer. Math. Soc. (1999) 49

[11] J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Math. 347, Springer (1973)

[12] M Bökstedt, Topological Hochschild homology, unpublished

[13] M Bökstedt, The topological Hochschild homology of $\mathbb{Z}$ and $\mathbb{Z}/p$, unpublished

[14] R R Bruner, J P May, J E Mcclure, M Steinberger, $H_\infty $ ring spectra and their applications, Lecture Notes in Math. 1176, Springer (1986)

[15] D Dugger, B Shipley, Postnikov extensions of ring spectra, Algebr. Geom. Topol. 6 (2006) 1785

[16] W G Dwyer, J P C Greenlees, Complete modules and torsion modules, Amer. J. Math. 124 (2002) 199

[17] 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)

[18] Z Fiedorowicz, R Vogt, Topological Hochschild homology of $E_n$–ring spectra

[19] P G Goerss, Associative $MU$–algebras, preprint available at http://www.math.northwestern.edu/ pgoerss/

[20] 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

[21] M J Hopkins, H R Miller, Lubin–Tate deformations in algebraic topology, preprint

[22] S O Kochman, Symmetric Massey products and a Hirsch formula in homology, Trans. Amer. Math. Soc. 163 (1972) 245

[23] A Lazarev, Hoschschild cohomology and moduli spaces of strongly homotopy associative algebras, Homology Homotopy Appl. 5 (2003) 73

[24] A Lazarev, Towers of $M$U–algebras and the generalized Hopkins–Miller theorem, Proc. London Math. Soc. $(3)$ 87 (2003) 498

[25] T Leinster, Higher operads, higher categories, London Math. Soc. Lecture Note Series 298, Cambridge University Press (2004)

[26] 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

[27] C Nassau, On the structure of $P(n)_{*}P((n))$ for $p=2$, Trans. Amer. Math. Soc. 354 (2002) 1749

[28] D C Ravenel, Complex cobordism and stable homotopy groups of spheres. 2nd ed., AMS Chelsea Pub. (2004)

[29] 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

[30] C Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006) 969

[31] 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

[32] A Robinson, S Whitehouse, Operads and $\Gamma$–homology of commutative rings, Math. Proc. Cambridge Philos. Soc. 132 (2002) 197

[33] J Rognes, Galois extensions of structured ring spectra, Mem. Amer. Math. Soc. 192 (2008) 1

[34] N P Strickland, Products on $\mathrm{MU}$–modules, Trans. Amer. Math. Soc. 351 (1999) 2569

[35] M Van Den Bergh, A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc. 126 (1998) 1345

[36] 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)

[37] C A Weibel, S C Geller, Étale descent for Hochschild and cyclic homology, Comment. Math. Helv. 66 (1991) 368