Integrality in the Steinberg module and the top-dimensional cohomology of GLnOK

We prove a new structural result for the spherical Tits building attached to GLnK for many number fields K, and more generally for the fraction fields of many Dedekind domains O: the Steinberg module Stn(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 is Cohen–Macaulay. We apply this to prove new vanishing and nonvanishing results for H(GLnOK ;Q), where OK is the ring of integers in a number field and νn is the virtual cohomological dimension of GLnOK . The (non)vanishing depends on the (non)triviality of the class group of OK . We also obtain a vanishing theorem for the cohomology H(GLnOK ;V ) with twisted coefficients V . The same results hold for SLnOK as well.