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Hopf algebra structure on topological Hochschild homology

Vigleik Angeltveit and John Rognes

Algebraic & Geometric Topology 5 (2005) 1223–1290

arXiv: math.AT/0502195


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.

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 2000
Primary: 55P43, 55S10, 55S12, 57T05
Secondary: 13D03, 55T15
Forward citations
Received: 16 July 2004
Revised: 21 September 2005
Accepted: 29 September 2005
Published: 5 October 2005
Vigleik Angeltveit
Department of Mathematics
Massachusetts Institute of Technology
Cambridge MA 02139-4307
John Rognes
Department of Mathematics
University of Oslo
Blindern NO-0316