Integrality in the Steinberg module and the top-dimensional cohomology of SLn OK

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 n (SL n O K ; Q), where O K is the ring of integers in a number field and n 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 n (SL n O K ; V ) with twisted coefficients V .