Freeness of Hecke modules at non-minimal levels

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