The module of vector-valued modular forms is Cohen-Macaulay

Let $H$ denote a finite index subgroup of the modular group $\Gamma$ and let $\rho$ denote a finite-dimensional complex representation of $H.$ Let $M(\rho)$ denote the collection of holomorphic vector-valued modular forms for $\rho$ and let $M(H)$ denote the collection of modular forms on $H$. Then $M(\rho)$ is a $\textbf{Z}$-graded $M(H)$-module. It has been proven that $M(\rho)$ may not be projective as a $M(H)$-module. We prove that $M(\rho)$ is Cohen-Macaulay as a $M(H)$-module. We also explain how to apply this result to prove that if $M(H)$ is a polynomial ring then $M(\rho)$ is a free $M(H)$-module of rank $\textrm{dim } \rho.$