Which series are Hilbert series of graded modules over polynomial rings?
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arXiv
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| Format: | Preprint |
| Published: |
2016
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| _version_ | 1866912189944692736 |
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| author | Katthän, Lukas Moyano-Fernández, Julio José Uliczka, Jan |
| author_facet | Katthän, Lukas Moyano-Fernández, Julio José Uliczka, Jan |
| contents | Let $S$ be a multigraded polynomial ring such that the degree of each variable is a unit vector; so $S$ is the homogeneous coordinate ring of a product of projective spaces. In this setting, we characterize the formal Laurent series which arise as Hilbert series of finitely generated $S$-modules.
Also we provide necessary conditions for a formal Laurent series to be the Hilbert series of a finitely generated module with a given depth. In the bigraded case (corresponding to the product of two projective spaces), we completely classify the Hilbert series of finitely generated modules of positive depth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1603_06215 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Which series are Hilbert series of graded modules over polynomial rings? Katthän, Lukas Moyano-Fernández, Julio José Uliczka, Jan Commutative Algebra Combinatorics Primary: 13C05, Secondary: 05E40, 13A02 Let $S$ be a multigraded polynomial ring such that the degree of each variable is a unit vector; so $S$ is the homogeneous coordinate ring of a product of projective spaces. In this setting, we characterize the formal Laurent series which arise as Hilbert series of finitely generated $S$-modules. Also we provide necessary conditions for a formal Laurent series to be the Hilbert series of a finitely generated module with a given depth. In the bigraded case (corresponding to the product of two projective spaces), we completely classify the Hilbert series of finitely generated modules of positive depth. |
| title | Which series are Hilbert series of graded modules over polynomial rings? |
| topic | Commutative Algebra Combinatorics Primary: 13C05, Secondary: 05E40, 13A02 |
| url | https://arxiv.org/abs/1603.06215 |