Algebraic & Geometric Topology

Higher Hochschild cohomology of the Lubin–Tate ring spectrum

Geoffroy Horel

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We construct a spectral sequence computing factorization homology of an d–algebra in spectra using as an input an algebraic version of higher Hochschild homology due to Pirashvili. This induces a full computation of higher Hochschild cohomology when the algebra is étale. As an application, we compute higher Hochschild cohomology of the Lubin–Tate ring spectrum.

Article information

Algebr. Geom. Topol., Volume 15, Number 6 (2015), 3215-3252.

Received: 17 March 2014
Revised: 24 March 2015
Accepted: 6 April 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)
Secondary: 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.) 55P48: Loop space machines, operads [See also 18D50]

factorization homology Hochschild cohomology little disk operad Morava $E$ theory Lubin–Tate spectrum spectral sequence


Horel, Geoffroy. Higher Hochschild cohomology of the Lubin–Tate ring spectrum. Algebr. Geom. Topol. 15 (2015), no. 6, 3215--3252. doi:10.2140/agt.2015.15.3215.

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  • R Andrade, From manifolds to invariants of $E_n$–algebras, PhD thesis, Massachusetts Institute of Technology (2010) Available at \setbox0\makeatletter\@url {\unhbox0
  • V Angeltveit, Topological Hochschild homology and cohomology of $A\sb \infty$ ring spectra, Geom. Topol. 12 (2008) 987–1032
  • G Arone, V Turchin, Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots, preprint (2013)
  • D Ayala, J Francis, Factorization homology of topological manifolds, J. Topology (2015) online publication
  • C Berger, I Moerdijk, On the derived category of an algebra over an operad, Georgian Math. J. 16 (2009) 13–28
  • J Francis, The tangent complex and Hochschild cohomology of $\mathscr{E}\sb n$–rings, Compos. Math. 149 (2013) 430–480
  • G Ginot, Higher order Hochschild cohomology, C. R. Math. Acad. Sci. Paris 346 (2008) 5–10
  • G Ginot, T Tradler, M Zeinalian, Higher Hochschild homology, topological chiral homology and factorization algebras, Comm. Math. Phys. 326 (2014) 635–686
  • P,G Goerss, M,J Hopkins, Moduli spaces of commutative ring spectra, from: “Structured ring spectra”, (A Baker, B Richter, editors), London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 151–200
  • G Horel, Factorization homology and calculus à la Kontsevich Soibelman, preprint (2015)
  • G Horel, Operads, modules and topological field theories, preprint (2015)
  • M Hovey, Operations and co-operations in Morava $E$–theory, Homology Homotopy Appl. 6 (2004) 201–236
  • S,B Isaacson, Cubical homotopy theory and monoidal model categories, PhD thesis, Harvard (2009) Available at \setbox0\makeatletter\@url {\unhbox0
  • D Pavlov, J Scholbach, Symmetric operads in abstract symmetric spectra, preprint (2014)
  • T Pirashvili, Hodge decomposition for higher order Hochschild homology, Ann. Sci. École Norm. Sup. 33 (2000) 151–179
  • C Rezk, Notes on the Hopkins–Miller theorem, from: “Homotopy theory via algebraic geometry and group representations”, (M Mahowald, S Priddy, editors), Contemp. Math. 220, Amer. Math. Soc. (1998) 313–366
  • S Schwede, An untitled book project about symmetric spectra, preprint (2007) Available at \setbox0\makeatletter\@url {\unhbox0
  • C,A Weibel, S,C Geller, Étale descent for Hochschild and cyclic homology, Comment. Math. Helv. 66 (1991) 368–388
  • T Willwacher, M Kontsevich's graph complex and the Grothendieck–Teichmüller Lie algebra, Invent. Math. 200 (2015) 671–760