Hopf algebra structure on topological Hochschild homology

Angeltveit, Vigleik; Rognes, John

doi:10.2140/agt.2005.5.1223

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Hopf algebra structure on topological Hochschild homology

Vigleik Angeltveit and John Rognes

Algebraic & Geometric Topology 5 (2005) 1223–1290

DOI: 10.2140/agt.2005.5.1223

arXiv: math.AT/0502195

Abstract

The topological Hochschild homology $T H H (R)$ 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 Bökstedt spectral sequence converging to the mod $p$ homology of $T H H (R)$ into a Hopf algebra spectral sequence. We then apply this additional structure to the study of some interesting examples, including the commutative $S$ –algebras $k u$ , $k o$ , $t m f$ , $j u$ and $j$ , and to calculate the homotopy groups of $T H H (k u)$ and $T H H (k o)$ 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 2000

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

Secondary: 13D03, 55T15

References

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Publication

Received: 16 July 2004

Revised: 21 September 2005

Accepted: 29 September 2005

Published: 5 October 2005

Authors

Vigleik Angeltveit
Department of Mathematics Massachusetts Institute of Technology Cambridge MA 02139-4307 USA

John Rognes
Department of Mathematics University of Oslo Blindern NO-0316 Norway

Volume 5, issue 3 (2005)