American Journal of Mathematics

abstract:

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