  We build on the results of [6] to show that the homology groups
$\mathrm{H}_{r_1+r_2}(Y_0(\mathcal{N}_\Sigma),\mathcal{O})_{\mathfrak{m}_\Sigma}$
of arithmetic manifolds are free over certain deformation rings $R_\Sigma$,
when there are enough geometric characteristic 0 representations. Hitherto we
had proved that the homology group has a nonzero free $R_\Sigma$-direct
summand. The new ingredient is a commutative algebra argument involving
congruence modules defined in higher codimension in [6].
