Hopf algebra structure on topological Hochschild homology

Abstract

The topological Hochschild homology THH(R) of a commutative S-algebra (E_infty ring spectrum) R naturally has the structure of a commutative R-algebra in the strict sense, and of a Hopf algebra over R in the homotopy category. We show, under a flatness assumption, that this makes the Boekstedt spectral sequence converging to the mod p homology of THH(R) into a Hopf algebra spectral sequence. We then apply this additional structure to the study of some interesting examples, including the commutative S-algebras ku, ko, tmf, ju and j, and to calculate the homotopy groups of THH(ku) and THH(ko) after smashing with suitable finite complexes. This is part of a program to make systematic computations of the algebraic K-theory of S-algebras, by means of the cyclotomic trace map to topological cyclic homology.

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-49.abs.html

References in corpus (2)

• The smooth Whitehead spectrum of a point at odd regular primes
• Differentials in the homological homotopy fixed point spectral sequence

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• Differentials in the homological homotopy fixed point spectral sequence
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• On the Brun spectral sequence for topological Hochschild homology
• The Adams spectral sequence for the image-of-$J$ spectrum
• The topological Hochschild homology of algebraic $K$-theory of finite fields
• Towards topological Hochschild homology of Johnson-Wilson spectra
• On topological Hochschild homology of the $K(1)$-local sphere
• The circle action on topological Hochschild homology of complex cobordism and the Brown-Peterson spectrum
• Algebraic K-theory of elliptic cohomology
• Topological Hochschild homology of truncated Brown-Peterson spectra I
• Higher chromatic Thom spectra via unstable homotopy theory
• The $H{\underline{\mathbb{F}}}_2$-homology of $C_2$-equivariant Eilenberg-MacLane spaces
• Whitehead Filtrations for Computations in Topological Hochschild Homology