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.

中文翻译：

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某些Steinberg模块的非完整性

我们证明，不是整数单元的二次虚数环的特殊线性群的Steinberg模不是由整数单元类生成的。假设广义Riemann假设，这表明，当且仅当环为欧几里得时，数字环的Steinberg模块才由整数单元类生成。我们还在一些二次虚数环的特殊线性组的高维同调组中构造了新的同调类。