Hopf algebra structure on topological Hochschild homology

Details Hopf algebra structure on topological Hochschild homology Hopf algebra structure on topological Hochschild homology

2005-02-09

arxiv.org/abs/math/0502195v2

Algebr. Geom. Topol. 5 (2005) 1223-1290

Mathematics

Algebraic Topology

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

Scientific paper

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.

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