  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
$\textrm{dim } \rho.$
