Topological Hochschild homology of maximal orders in simple $\mathbb{Q}$–algebras

Abstract

We calculate the topological Hochschild homology groups of a maximal order in a simple 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 topological Hochschild homology groups of maximal orders $A$ in simple ${\mathbb{Q}}_p$–algebras. We show that the topological Hochschild homology of $A∕\left(p\right)$ splits as the tensor product of its Hochschild homology with ${THH}_{\ast }\left({\mathbb{F}}_p\right)$. We use this result in Brun’s spectral sequence to calculate ${THH}_{\ast }\left(A,A∕\left(p\right)\right)$, and then we analyze the torsion to get ${\pi }_{\ast }\left(THH{\left(A\right)}_p^{\wedge }\right)$.