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

## TL;DR

This paper proves that the module of vector-valued modular forms is Cohen-Macaulay over the modular forms ring, providing new structural insights and conditions for freeness when the ring is polynomial.

## Contribution

It establishes the Cohen-Macaulay property of the module of vector-valued modular forms, a significant advancement in understanding their algebraic structure.

## Key findings

- $M(
ho)$ is Cohen-Macaulay as an $M(H)$-module.
- If $M(H)$ is polynomial, then $M(
ho)$ is a free module of rank equal to $ ext{dim }
ho$.
- The result clarifies the module's structure and potential freeness under specific conditions.

## Abstract

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 $\textrm{dim } \rho.$