On the Brun spectral sequence for topological Hochschild homology

TL;DR

This paper extends Brun's spectral sequence to compute the E-homology of topological Hochschild homology for certain algebraic structures, providing a new computational approach.

Contribution

It generalizes Brun's spectral sequence to a broader setting, enabling new computations of topological Hochschild homology with E-homology.

Findings

Derived a generalized spectral sequence for THH computations.

Provided an alternative method for computing mod p and v_1 THH of connective K-theory.

Simplified previous complex calculations using the generalized spectral sequence.

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 $A$-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.