Topological Hochschild homology of maximal orders in simple ℚ–algebras

Chan, Henry; Lindenstrauss, Ayelet

doi:10.2140/agt.2019.19.31

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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

DOI: 10.2140/agt.2019.19.31

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 $ℚ_{p}$ –algebras. We show that the topological Hochschild homology of $A ∕ (p)$ splits as the tensor product of its Hochschild homology with ${THH}_{*} (F_{p})$ . We use this result in Brun’s spectral sequence to calculate ${THH}_{*} (A, A ∕ (p))$ , and then we analyze the torsion to get $π_{*} (THH {(A)}_{p}^{\land})$ .

Keywords

topological Hochschild homology, maximal orders, division algebras, simple algebras

Mathematical Subject Classification 2010

Primary: 16E40, 19D55

Secondary: 16H10, 55T99

References

Publication

Received: 24 June 2015

Revised: 4 February 2018

Accepted: 5 August 2018

Published: 6 February 2019

Authors

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

Volume 19, issue 1 (2019)