Hopf algebra structure on topological Hochschild homology

Abstract

The topological Hochschild homology THH (R) of a commu-tative S -algebra (E ∞ 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 Bökstedt 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 computa-tions of the algebraic K -theory of S -algebras, by means of the cyclotomic trace map to topological cyclic homology.