The dualizing module and top-dimensional cohomology group of ${\text{GL}}_n(\mathcal{O})$

Abstract

For a number ring $\mathcal{O}$, Borel and Serre proved that ${\text{SL}}_n(\mathcal{O})$ is a virtual duality group whose dualizing module is the Steinberg module. They also proved that ${\text{GL}}_n(\mathcal{O})$ is a virtual duality group. In contrast to ${\text{SL}}_n(\mathcal{O})$, we prove that the dualizing module of ${\text{GL}}_n(\mathcal{O})$ 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 ${\text{GL}}_n(\mathcal{O})$ in its virtual cohomological dimension.

34 pages, minor revision. To appear in Math. Z

References in corpus (1)

• Non-integrality of some Steinberg modules

Cited by in corpus (4)

• On the codimension-two cohomology of ${\mathrm{SL}}_n(\mathbb{Z})$
• The Steinberg representation is irreducible
• Unstable cohomology of ${\mathsf{GL}}_{2n}(\mathbb{Z})$ and the odd commutative graph complex
• Cohen--Macaulay Complexes, Duality Groups, and the dualizing module of $\mathrm{Out}(F_N)$