On Projective modules over graded $R$-subalgebras of $R[X,1/X]$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915356137750528 |
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| author | Garg, Diksha Gupta, Anjan |
| author_facet | Garg, Diksha Gupta, Anjan |
| contents | Let $R$ be a Noetherian ring of dimension $d$ and $A$ be a graded $R$-subalgebra of $R[X,1/X]$. Let $P$ be a projective module over $A$ of rank $r \geq \max\{d+1,2\}$ and $\v=(a,p)$ be a unimodular element of $A \oplus P$. We find an elementary automorphism $τ$ such that $τ(\v) = (1, 0)$. Consequently, we obtain the cancellative property of $P$. We show that $P$ splits off a free summand of rank one. When $A = R[X]$ or $R[X, 1/ X]$, the results are well-known due to the contributions by various authors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18826 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Projective modules over graded $R$-subalgebras of $R[X,1/X]$ Garg, Diksha Gupta, Anjan Commutative Algebra K-Theory and Homology 13A02, 13B25, 13C10, 19A13 Let $R$ be a Noetherian ring of dimension $d$ and $A$ be a graded $R$-subalgebra of $R[X,1/X]$. Let $P$ be a projective module over $A$ of rank $r \geq \max\{d+1,2\}$ and $\v=(a,p)$ be a unimodular element of $A \oplus P$. We find an elementary automorphism $τ$ such that $τ(\v) = (1, 0)$. Consequently, we obtain the cancellative property of $P$. We show that $P$ splits off a free summand of rank one. When $A = R[X]$ or $R[X, 1/ X]$, the results are well-known due to the contributions by various authors. |
| title | On Projective modules over graded $R$-subalgebras of $R[X,1/X]$ |
| topic | Commutative Algebra K-Theory and Homology 13A02, 13B25, 13C10, 19A13 |
| url | https://arxiv.org/abs/2506.18826 |