Let $V$ be a representation of the modular group $\Gamma$ of dimension $p$. We show that the $\mathbb{ Z}$-graded space $\mathcal{ H}(V)$ of holomorphic vector-valued modular forms associated to $V$ is a free module of rank $p$ over the algebra $\mathcal{ M}$ of classical holomorphic modular forms. We study the nature of $\mathcal{ H}$ considered as a functor from $\Gamma$-modules to graded $\mathcal{ M}$-lattices and give some applications, including the calculation of the Hilbert-Poincar\'{ e} of $\mathcal{ H}(V)$ in some cases.
Categories: math.NT math.AC