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

• Benjamin Brück ^[2] ; Zachary Himes ^[1]
  1. [1] University of Michigan–Ann Arbor

     University of Michigan–Ann Arbor

     City of Ann Arbor, Estados Unidos

     
  2. [2] Münster University, Münster, Germany
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 3, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01051-8
• Enlaces
  • Texto completo
• Resumen

  • Let K be a number field with ring of integers R = OK . We show that if R is not a principal ideal domain, then the symplectic group Sp2n(R) 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 Sp2n(K). 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

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