The topological Hochschild homology of the Gaussian integers

The calculation in this paper gives the 2-torsion in the topological Hochschild homology of rings of integers in quadratic extensions of the rationals. In general, it is not hard to deduce the p-torsion in the topological Hochschild homology of rings of integers which do not ramify at p from the corresponding torsion for the integers. Thus, for the ring of Gaussian integers, and also for the integers with the square root of 2 or -2 adjoined, this paper completes the calculation of the topological Hochschild homology spectrum (which is a priori known to be a product of Eilenberg-MacLane spectra). Precisely because of this a priori knowledge, the homotopy groups can be found by doing a homology calculation. A spectral sequence arising from the filtration of topological Hochschild homology by simplicial 'skeleta' converges to the desired homology. If one reduces the rings in question module 2, the spectral sequence collapses at its second term. This comparison bounds the nontriviality of higher differentials in the spectral sequence for the original ring. It is explicitly demonstrated, using simplicial calculations, that the higher differentials are as nontrivial as they could be, given this bound.