Syzygies of associated graded modules
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910864096886784 |
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| author | Ananthnarayan, H. Batavia, Manav Javadekar, Omkar |
| author_facet | Ananthnarayan, H. Batavia, Manav Javadekar, Omkar |
| contents | Given a finitely generated module $M$ over a Noetherian local ring $R$, we give a characterization for the first syzygy of the associated graded module $G_{\mathfrak{m}}(M)$ to be equigenerated. As an application of this, we identify a complex of free $G_{\mathfrak{m}}(R)$-modules, arising from given free resolution of $M$ over $R$, which is a resolution of $G_{\mathfrak{m}}(M)$ if and only if $G_{\mathfrak{m}}(M)$ is a pure $G_{\mathfrak{m}}(R)$-module. We also give several applications of the purity of $G_{\mathfrak{m}}(M)$. Our results demonstrate that while not all algebraic properties of a module carry over to its associated graded module, the purity of the minimal free resolution of $G_{\mathfrak{m}}(M)$ ensures that several important invariants are inherited. In addition, we provide sufficient conditions for Cohen-Macaulayness and purity of $G_{\mathfrak{m}}(M)$, and provide a local version of the Herzog-Kühl equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_00654 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Syzygies of associated graded modules Ananthnarayan, H. Batavia, Manav Javadekar, Omkar Commutative Algebra Given a finitely generated module $M$ over a Noetherian local ring $R$, we give a characterization for the first syzygy of the associated graded module $G_{\mathfrak{m}}(M)$ to be equigenerated. As an application of this, we identify a complex of free $G_{\mathfrak{m}}(R)$-modules, arising from given free resolution of $M$ over $R$, which is a resolution of $G_{\mathfrak{m}}(M)$ if and only if $G_{\mathfrak{m}}(M)$ is a pure $G_{\mathfrak{m}}(R)$-module. We also give several applications of the purity of $G_{\mathfrak{m}}(M)$. Our results demonstrate that while not all algebraic properties of a module carry over to its associated graded module, the purity of the minimal free resolution of $G_{\mathfrak{m}}(M)$ ensures that several important invariants are inherited. In addition, we provide sufficient conditions for Cohen-Macaulayness and purity of $G_{\mathfrak{m}}(M)$, and provide a local version of the Herzog-Kühl equations. |
| title | Syzygies of associated graded modules |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2308.00654 |