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

Gottesman, Richard. "The module of vector-valued modular forms is Cohen-Macaulay." Czechoslovak Mathematical Journal 70.4 (2020): 1211-1218. <http://eudml.org/doc/297297>.

@article{Gottesman2020,
abstract = {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 $\dim \rho .$},
author = {Gottesman, Richard},
journal = {Czechoslovak Mathematical Journal},
keywords = {vector-valued modular form; Cohen-Macaulay module},
language = {eng},
number = {4},
pages = {1211-1218},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The module of vector-valued modular forms is Cohen-Macaulay},
url = {http://eudml.org/doc/297297},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Gottesman, Richard
TI - The module of vector-valued modular forms is Cohen-Macaulay
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 4
SP - 1211
EP - 1218
AB - 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 $\dim \rho .$
LA - eng
KW - vector-valued modular form; Cohen-Macaulay module
UR - http://eudml.org/doc/297297
ER -