Integrality in the Steinberg module and the top-dimensional cohomology of SL_n(O_K)

TL;DR

This paper establishes a link between the structure of the Steinberg module for SL_n over number fields and the triviality of the class group, leading to new results on the cohomology of these groups.

Contribution

It proves that the Steinberg module is generated by integral apartments iff the class group is trivial, and derives new cohomology vanishing and nonvanishing results based on this.

Findings

Steinberg module generated by integral apartments iff class group is trivial

New vanishing results for cohomology of SL_n(O_K)

Nonvanishing results depending on class group triviality

Abstract

We prove a new structural result for the spherical Tits building attached to SL_n(K) for many number fields K, and more generally for the fraction fields of many Dedekind domains O: the Steinberg module St_n(K) is generated by integral apartments if and only if the ideal class group cl(O) is trivial. We deduce this integrality by proving that the complex of partial bases of O^n is Cohen-Macaulay. We apply this to prove new vanishing and nonvanishing results for H^{vcd}(SL_n(O_K); Q), where O_K is the ring of integers in a number field and vcd is the virtual cohomological dimension of SL_n(O_K). The (non)vanishing depends on the (non)triviality of the class group of O_K. We also obtain a vanishing theorem for the cohomology H^{vcd}(SL_n(O_K); V) with twisted coefficients V.