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Keywords:
vector-valued modular form; Cohen-Macaulay module
vector-valued modular form; Cohen-Macaulay module
Summary:
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 $\mathbb {Z}$-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 $\dim \rho .$
References:
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