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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.07248 |
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| _version_ | 1866915292397961216 |
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| author | Van Kien, Do Nguyen, Hop D. |
| author_facet | Van Kien, Do Nguyen, Hop D. |
| contents | The linearity defect is a measure for the non-linearity of minimal free resolutions of modules over noetherian local rings. A tantalizing open question due to Herzog and Iyengar asks whether a noetherian local ring $(R,\mathfrak{m})$ is Koszul if its residue field $R/\mathfrak{m}$ has a finite linearity defect. We provide a positive answer to this question when $R$ is a Cohen-Macaulay local ring of almost minimal multiplicity with the residue field of characteristic zero. The proof depends on the study of noetherian local rings $(R,\mathfrak{m})$ such that $\mathfrak{m}^2$ is a principal ideal, which we call $g$-$stretched$ local rings. The class of $g$-stretched local rings subsumes stretched artinian local rings studied by Sally, and generic artinian reductions of Cohen-Macaulay local rings of almost minimal multiplicity. An essential part in the proof of our main result is a complete characterization of one-dimensional complete $g$-stretched local rings. Beside partial progress on Herzog-Iyengar's question, another consequence of our study is a numerical characterization of all $g$-stretched Koszul rings, strengthening previous work of Avramov, Iyengar, and Şega. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07248 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Koszul property and finite linearity defect over $g$-stretched local rings Van Kien, Do Nguyen, Hop D. Commutative Algebra 13F20, 14N05, 13A02 The linearity defect is a measure for the non-linearity of minimal free resolutions of modules over noetherian local rings. A tantalizing open question due to Herzog and Iyengar asks whether a noetherian local ring $(R,\mathfrak{m})$ is Koszul if its residue field $R/\mathfrak{m}$ has a finite linearity defect. We provide a positive answer to this question when $R$ is a Cohen-Macaulay local ring of almost minimal multiplicity with the residue field of characteristic zero. The proof depends on the study of noetherian local rings $(R,\mathfrak{m})$ such that $\mathfrak{m}^2$ is a principal ideal, which we call $g$-$stretched$ local rings. The class of $g$-stretched local rings subsumes stretched artinian local rings studied by Sally, and generic artinian reductions of Cohen-Macaulay local rings of almost minimal multiplicity. An essential part in the proof of our main result is a complete characterization of one-dimensional complete $g$-stretched local rings. Beside partial progress on Herzog-Iyengar's question, another consequence of our study is a numerical characterization of all $g$-stretched Koszul rings, strengthening previous work of Avramov, Iyengar, and Şega. |
| title | Koszul property and finite linearity defect over $g$-stretched local rings |
| topic | Commutative Algebra 13F20, 14N05, 13A02 |
| url | https://arxiv.org/abs/2505.07248 |