Top-degree rational cohomology in the symplectic group of a number ring

Abstract Let K be a number field with ring of integers $$R = \mathcal {O}_K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> </mml:math> . We show that if R is not a principal ideal domain, then the symplectic group $$\operatorname {Sp}_{2n}(R)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>Sp</mml:mo> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> has non-trivial rational cohomology in its virtual cohomological dimension. This demonstrates a sharp contrast to the situation where R is Euclidean. To prove our result, we study the symplectic Steinberg module, i.e. the top-dimensional homology group of the spherical building associated to $$\operatorname {Sp}_{2n}(K)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>Sp</mml:mo> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We show that this module is not generated by integral apartment classes. Both of these results follow from a vanishing theorem for homology with Steinberg coefficients.