Towards topological Hochschild homology of Johnson–Wilson spectra

Ausoni, Christian¹ ; Richter, Birgit²

¹LAGA (UMR7539), Institut Galilée, Université Paris 13, Villetaneuse, France
²Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany

Algebraic and Geometric Topology, Tome 20 (2020) no. 1, pp. 375-393

Résumé

We present computations in Hochschild homology that lead to results on the K(i)–local behaviour of THH(E(n)) for all n ≥ 2 and 0 ≤ i ≤ n, where E(n) is the Johnson–Wilson spectrum at an odd prime. This permits a computation of K(i)∗THH(E(n)) under the assumption that E(n) is an E3–ring spectrum. We offer a complete description of THH(E(2)) as an E(2)–module in the form of a splitting into chromatic localizations of E(2), under the assumption that E(2) carries an E∞–structure. If E(2) is admits an E3–structure, we obtain a similar splitting of the cofiber of the unit map E(2) → THH(E(2)).

Détail

DOI : 10.2140/agt.2020.20.375

Classification : 55N35, 55P43
Keywords: topological Hochschild homology, Johnson–Wilson spectra, $E_\infty$–structures on ring spectra, chromatic squares

Bibliographie

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

[2] O Antolín-Camarena, T Barthel, Chromatic fracture cubes, preprint (2014)

[3] C Ausoni, Topological Hochschild homology of connective complex K–theory, Amer. J. Math. 127 (2005) 1261 | DOI

[4] A Baker, B Richter, On the Γ–cohomology of rings of numerical polynomials and E∞ structures on K–theory, Comment. Math. Helv. 80 (2005) 691 | DOI

[5] C Barwick, From operator categories to higher operads, Geom. Topol. 22 (2018) 1893 | DOI

[6] M Basterra, M A Mandell, Homology of En ring spectra and iterated THH, Algebr. Geom. Topol. 11 (2011) 939 | DOI

[7] M Basterra, M A Mandell, The multiplication on BP, J. Topol. 6 (2013) 285 | DOI

[8] T Bauer, Bousfield localization and the Hasse square, from: "Topological modular forms" (editors C L Douglas, J Francis, A G Henriques, M A Hill), Mathematical Surveys and Monographs 201, Amer. Math. Soc. (2014) 112

[9] M Bökstedt, The topological Hochschild homology of Z and of Z∕pZ, unpublished manuscript (1987)

[10] M Brun, Z Fiedorowicz, R M Vogt, On the multiplicative structure of topological Hochschild homology, Algebr. Geom. Topol. 7 (2007) 1633 | DOI

[11] R Bruner, J Rognes, Topological Hochschild homology of topological modular forms, talk slides (2008)

[12] A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, 47, Amer. Math. Soc. (1997)

[13] M Hill, T Lawson, Automorphic forms and cohomology theories on Shimura curves of small discriminant, Adv. Math. 225 (2010) 1013 | DOI

[14] P S Landweber, BP∗(BP) and typical formal groups, Osaka Math. J. 12 (1975) 357

[15] T Lawson, Secondary power operations and the Brown–Peterson spectrum at the prime 2, Ann. of Math. 188 (2018) 513 | DOI

[16] T Lawson, N Naumann, Commutativity conditions for truncated Brown–Peterson spectra of height 2, J. Topol. 5 (2012) 137 | DOI

[17] A Mathew, N Naumann, J Noel, On a nilpotence conjecture of J P May, J. Topol. 8 (2015) 917 | DOI

[18] J E Mcclure, R E Staffeldt, On the topological Hochschild homology of bu, I, Amer. J. Math. 115 (1993) 1 | DOI

[19] D C Ravenel, Complex cobordism and stable homotopy groups of spheres, 121, Academic (1986)

[20] B Richter, A lower bound for coherences on the Brown–Peterson spectrum, Algebr. Geom. Topol. 6 (2006) 287 | DOI

[21] B Richter, B Shipley, An algebraic model for commutative HZ–algebras, Algebr. Geom. Topol. 17 (2017) 2013 | DOI

[22] A Robinson, Gamma homology, Lie representations and E∞ multiplications, Invent. Math. 152 (2003) 331 | DOI

[23] A Robinson, S Whitehouse, Operads and Γ–homology of commutative rings, Math. Proc. Cambridge Philos. Soc. 132 (2002) 197 | DOI

[24] M O Ronco, Smooth algebras473

[25] A Senger, The Brown–Peterson spectrum is not E2(p2+2) at odd primes, preprint (2017)

[26] B Shipley, HZ–algebra spectra are differential graded algebras, Amer. J. Math. 129 (2007) 351 | DOI

[27] B Stonek, Higher topological Hochschild homology of periodic complex K–theory, preprint (2018)

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

[29] N Yagita, On the Steenrod algebra of Morava K–theory, J. London Math. Soc. 22 (1980) 423 | DOI