Non‐integrality of some Steinberg modules

We prove that the Steinberg module of the special linear group of a quadratic imaginary number ring which is not Euclidean is not generated by integral apartment classes. Assuming the generalized Riemann hypothesis, this shows that the Steinberg module of a number ring is generated by integral apart...

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Bibliographic Details
Published in:Journal of topology 2020-06, Vol.13 (2), p.441-459
Main Authors: Miller, Jeremy, Patzt, Peter, Wilson, Jennifer C. H., Yasaki, Dan
Format: Article
Language:English
Subjects:
11F75
55N25 (primary)
55R35
55U10 (secondary)
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Summary:We prove that the Steinberg module of the special linear group of a quadratic imaginary number ring which is not Euclidean is not generated by integral apartment classes. Assuming the generalized Riemann hypothesis, this shows that the Steinberg module of a number ring is generated by integral apartment classes if and only if the ring is Euclidean. We also construct new cohomology classes in the top‐dimensional cohomology group of the special linear group of some quadratic imaginary number rings.
ISSN:1753-8416
1753-8424
DOI:10.1112/topo.12132