Geodesic
The module of vector-valued modular forms is Cohen-Macaulay
Czechoslovak Mathematical Journal, Tome 70 (2020) no. 4, pp. 1211-1218 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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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     author = {Gottesman, Richard},
     title = {The module of vector-valued modular forms is {Cohen-Macaulay}},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1211--1218},
     year = {2020},
     volume = {70},
     number = {4},
     doi = {10.21136/CMJ.2020.0476-19},
     mrnumber = {4181810},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2020.0476-19/}
}
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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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