The dualizing module and top-dimensional cohomology group of GL_n(O)

2019/09/30 by Andrew Putman, Daniel Studenmund · 5 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Cohomological dimension #Cohomology #Combinatorics #Dimension (graph theory) #Duality (order theory) #Geometry #Group (periodic table) #Homotopy and Cohomology in Algebraic Topology #Mathematics #Orientation (vector space) #Physics #Pure mathematics #math.AT #math.GR #math.GT #math.NT

paper · pdf · doi:10.1007/s00209-021-02769-9

published in Mathematische Zeitschrift 300(1), 1-31 (Springer Science+Business Media) · 34 pages, minor revision. To appear in Math. Z

arxiv created 2021/04/23 · openalex publication_date 2021/05/24 · openalex created_date 2021/06/07 · arxiv updated 2022/03/08 · openalex updated_date 2026/07/23

Abstract

For a number ring O, Borel and Serre proved that SL_n(O) is a virtual duality group whose dualizing module is the Steinberg module. They also proved that GL_n(O) is a virtual duality group. In contrast to SL_n(O), we prove that the dualizing module of GL_n(O) is sometimes the Steinberg module, but sometimes instead is a variant that takes into account a sort of orientation. Using this, we obtain vanishing and nonvanishing theorems for the cohomology of GL_n(O) in its virtual cohomological dimension.

Citations

• Algebraic Number Theory