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Abstract. We define a relative version of the Loday construction for a sequence of commutative S-algebras A → B → C and a pointed simplicial subset Y ⊂ X. We use this to construct several spectral sequences for the calculation of higher topological Hochschild homology and apply those for calculations in some examples that could not be treated before.

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.

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Ramification for commutative ring spectra can be detected by relative topological Hochschild homology and by topological André-Quillen homology. In the classical algebraic context it is important to distinguish between tame and wild ramification. Noether's theorem characterizes tame ramification in terms of a normal basis and tame ramification can also be detected via the surjectivity of the trace map. We transfer the latter fact to ring spectra and use the Tate cohomology spectrum to detect wild ramification in the context of commutative ring spectra. We study ramification in examples in the context of topological K-theory and topological modular forms.