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

Henry Yi-Wei Chan and Ayelet Lindenstrauss

Algebraic & Geometric Topology 19 (2019) 31–75

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 p–algebras. We show that the topological Hochschild homology of A(p) splits as the tensor product of its Hochschild homology with THH(Fp). We use this result in Brun’s spectral sequence to calculate THH(A,A(p)), and then we analyze the torsion to get π(THH(A)p).

topological Hochschild homology, maximal orders, division algebras, simple algebras
Mathematical Subject Classification 2010
Primary: 16E40, 19D55
Secondary: 16H10, 55T99
Received: 24 June 2015
Revised: 4 February 2018
Accepted: 5 August 2018
Published: 6 February 2019
Henry Yi-Wei Chan
Department of Mathematics
University of Chicago
Chicago, IL
United States
Ayelet Lindenstrauss
Department of Mathematics
Indiana University
Bloomington, IN
United States