English

Abstract

Let H denote a finite index subgroup of the modular group Γ and let ϱ denote a finite-dimensional complex representation of H. Let M(ϱ) denote the collection of holomorphic vector-valued modular forms for ϱ and let M(H) denote the collection of modular forms on H. Then M(ϱ) is a ℤ-graded M(H)-module. It has been proven that M(ϱ) may not be projective as a M(H)-module. We prove that M(ϱ) 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(ϱ) is a free M(H)-module of rank dim ϱ.