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تسجيل مستخدم جديدThe dualizing module and top-dimensional cohomology group of $text{GL}_n(mathcal{O})$
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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.
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