Topological Hochschild homology of maximal orders in simple Q-algebras

TL;DR

This paper computes the topological Hochschild homology groups of maximal orders in simple Q-algebras by reducing the problem prime-by-prime and analyzing their structure through spectral sequences and torsion properties.

Contribution

It introduces a prime-by-prime approach to calculating THH of maximal orders in simple algebras over Q, including splitting results and spectral sequence applications.

Findings

THH groups of maximal orders are torsion in positive degrees.

The THH of A/(p) splits as a tensor product involving Hochschild and THH of F_p.

Homotopy groups of the p-adic completion of THH(A) are explicitly determined.

Abstract

We calculate the topological Hochschild homology groups of a maximal order in a central algebra over the rationals. Since the positive-dimensional THH groups consist only of torsion, we do this one prime ideal at a time for all the nonzero prime ideals in the center of the maximal order. This allows us to reduce the problem to studying the THH groups of maximal orders A in simple algebras over Q_p. We show that the topological Hochschild homology of A/(p) splits as the tensor product of its Hochschild homology and the topological Hochschild homology of F_p. We use this result in Brun's spectral sequence to calculate THH(A; A/(p)), and then we analyze the torsion to get the homotopy groups of the completion at p of THH(A).