Algebraic & Geometric Topology 5 (2005) 1223–1290

DOI: 10.2140/agt.2005.5.1223

Abstract

The topological Hochschild homology THH(R) of a commutative 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 computations of the algebraic K–theory of S–algebras, by means of the cyclotomic trace map to topological cyclic homology.

Keywords

topological Hochschild homology, commutative S–algebra, coproduct, Hopf algebra, topological K–theory, image-of-J spectrum, Bök­stedt spectral sequence, Steenrod operations, Dyer–Lashof operations

Mathematical Subject Classification

Primary: 55P43, 55S10, 55S12, 57T05

Secondary: 13D03, 55T15

References

Publication

Received: 16 July 2004
Revised: 21 September 2005
Accepted: 29 September 2005
Published: 5 October 2005

Authors