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

Richard Gottesman — 2020

Czechoslovak Mathematical Journal

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 -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...

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