| Title:
|
The module of vector-valued modular forms is Cohen-Macaulay (English) |
| Author:
|
Gottesman, Richard |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
70 |
| Issue:
|
4 |
| Year:
|
2020 |
| Pages:
|
1211-1218 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $\dim \rho .$ (English) |
| Keyword:
|
vector-valued modular form |
| Keyword:
|
Cohen-Macaulay module |
| MSC:
|
11F03 |
| MSC:
|
13C14 |
| idZBL:
|
07285993 |
| idMR:
|
MR4181810 |
| DOI:
|
10.21136/CMJ.2020.0476-19 |
| . |
| Date available:
|
2020-11-18T09:52:45Z |
| Last updated:
|
2023-01-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148425 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| . |
