#### Algebraic and Geometric Topology 5 (2005),
paper no. 49, pages 1223-1290.

## Hopf algebra structure on topological Hochschild homology

### Vigleik Angeltveit, John Rognes

**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.
**Keywords**.
Topological Hochschild homology, commutative S-algebra, coproduct,
Hopf algebra, topological K-theory, image-of-J spectrum, Boekstedt
spectral sequence, Steenrod operations, Dyer-Lashof operations.

**AMS subject classification**.
Primary: 55P43, 55S10, 55S12, 57T05.
Secondary: 13D03, 55T15.

**E-print:** `arXiv:math.AT/0502195`

**DOI:** 10.2140/agt.2005.5.1223

Submitted: 16 July 2004.
(Revised: 21 September 2005.)
Accepted: 29 September 2005.
Published: 5 October 2005.

Notes on file formats
Vigleik Angeltveit, John Rognes

Department of Mathematics, Massachusetts Institute of Technology

Cambridge, MA 02139-4307, USA

and

Department of Mathematics, University of Oslo

Blindern NO-0316, Norway

Email: vigleik@math.mit.edu, rognes@math.uio.no

AGT home page

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