A generalization of Rao's theorem to graded $R$-subalgebras of $R[t]$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909664315179008 |
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| author | Garg, Diksha Gupta, Anjan |
| author_facet | Garg, Diksha Gupta, Anjan |
| contents | Let $R$ be a Noetherian local ring of Krull dimension $d$ such that $(d!)R = R$, and let $A$ be a graded $R$-subalgebra of the polynomial algebra $R[t]$. We prove that every unimodular row of length $d + 1$ over $A$ can be completed to an invertible matrix. This is a generalization of a classical result by Rao, who proved that in the same setting, every unimodular row of length $d + 1$ over $R[t]$ admits a completion to an invertible matrix. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_18836 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A generalization of Rao's theorem to graded $R$-subalgebras of $R[t]$ Garg, Diksha Gupta, Anjan Commutative Algebra K-Theory and Homology 13A02, 13B25, 13C10, 19A13, 19B10, 19B14 Let $R$ be a Noetherian local ring of Krull dimension $d$ such that $(d!)R = R$, and let $A$ be a graded $R$-subalgebra of the polynomial algebra $R[t]$. We prove that every unimodular row of length $d + 1$ over $A$ can be completed to an invertible matrix. This is a generalization of a classical result by Rao, who proved that in the same setting, every unimodular row of length $d + 1$ over $R[t]$ admits a completion to an invertible matrix. |
| title | A generalization of Rao's theorem to graded $R$-subalgebras of $R[t]$ |
| topic | Commutative Algebra K-Theory and Homology 13A02, 13B25, 13C10, 19A13, 19B10, 19B14 |
| url | https://arxiv.org/abs/2506.18836 |