Freeness of Hecke modules at non-minimal levels

TL;DR

This paper proves that certain homology groups of arithmetic manifolds are free over deformation rings at non-minimal levels, extending previous results by employing advanced commutative algebra techniques.

Contribution

It establishes the freeness of homology groups over deformation rings at non-minimal levels, using a novel commutative algebra argument involving higher codimension congruence modules.

Findings

Homology groups are free over deformation rings at non-minimal levels.

Extension of previous results to more general levels.

Application of higher codimension congruence modules in the proof.

Abstract

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].