Resolutions of the Steinberg module for GL(n)

We give several resolutions of the Steinberg representation Stn for the general linear group over a principal ideal domain, in particular over Z. We compare them, and use these results to prove that the computations in Avner Ash et al. (2011) [AGM11], are definitive. In particular, in Avner Ash et al. (2011) [AGM11] we use two complexes to compute certain cohomology groups of congruence subgroups of SL(4,Z). One complex is based on Voronoiʼs polyhedral decomposition of the symmetric space for SL(n,R), whereas the other is a larger complex that has an action of the Hecke operators. We prove that both complexes allow us to compute the relevant cohomology groups, and that the use of the Voronoi complex does not introduce any spurious Hecke eigenclasses.