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 $THH\left(A;B\right)$, where $E$ is a ring spectrum, $A$ is a commutative $S$–algebra and $B$ is a connective commutative $A$–algebra. The input of the spectral sequence are the topological Hochschild homology groups of $B$ with coefficients in the $E$–homology groups of $B{\wedge }_{A}B$. The mod $p$ and $v_1$ 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.