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