Hopf algebra structure on topological Hochschild homology

Angeltveit, Vigleik¹ ; Rognes, John²

¹Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139-4307, USA
²Department of Mathematics, University of Oslo, Blindern NO-0316, Norway

Algebraic and Geometric Topology, Tome 5 (2005) no. 3, pp. 1223-1290

Résumé

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.

Détail

DOI : 10.2140/agt.2005.5.1223

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

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