On the Brun spectral sequence for topological Hochschild homology

Abstract

We generalize a spectral sequence of Brun for the computation of topological Hochschild homology. The generalized version computes the E-homology of THH (A; B), where E is a ring spectrum, A is a commutative S-algebra and B is a connective commutative Aalgebra. The input of the spectral sequence are the topological Hochschild homology groups of B with coefficients in the E-homology groups of B ∧A B. The mod p and v1 topological Hochschild homology of connective complex K-theory has been computed by Ausoni and later again by Rognes, Sagave and Schlichtkrull. We present an alternative, short computation using the generalized Brun spectral sequence. (−) denotes ordinary Hochschild homology over the ground ring F p . The spectral sequence is particularly useful if the pages E r * , * are flat over the mod p homology of B. By [1] the spectral sequence is an (HF p ) * HF p -comodule (HF p ) * B-bialgebra spectral sequence in this case. This structure can be very helpful to compute the differentials. However, if the flatness condition is not satisfied, computations with the Bökstedt spectral sequence can be harder. For example, if B = ku the flatness condition is not satisfied, and if B = K(F q ) it is not always satisfied. The Bökstedt spectral can be generalized from HF p to a more general ring spectrum E, if we have a Künneth isomorphism for E. But, for example for the mod p Moore spectrum V (0) and for V (1) one does not have a Künneth isomorphism, so that the Bökstedt spectral sequence does not allow a direct computation of V (0)-or V (1)-homotopy.

Topological Hochschild homology of truncated Brown-Peterson spectra I

“…Here we write taf D /x for the cofiber of a representative of an element x ∈ π 2k taf D regarded as a taf Dmodule map Σ 2k taf D → taf D . In the sequels to this paper, we plan to compute THH * (taf D ; M) for M = taf D /3 and M = taf D /v 1 by comparing faces of the cube of Bockstein spectral sequences to the topological Hochschild-May spectral sequence [1] and the Brun spectral sequence [21], which compute the diagonals of the faces of the cube directly. Finally, we plan to compute THH * (taf D ) by again comparing the topological Hochschild-May spectral sequence to the relevant Bockstein spectral sequences.…”