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The module of vector-valued modular forms is Cohen-Macaulay
Czechoslovak Mathematical Journal, Tome 70 (2020) no. 4, pp. 1211-1218
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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 $\dim \rho .$
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 $\dim \rho .$
DOI : 10.21136/CMJ.2020.0476-19
Classification : 11F03, 13C14
Keywords: vector-valued modular form; Cohen-Macaulay module 
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Gottesman, Richard. The module of vector-valued modular forms is Cohen-Macaulay. Czechoslovak Mathematical Journal, Tome 70 (2020) no. 4, pp. 1211-1218. doi: 10.21136/CMJ.2020.0476-19

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