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arXiv · 2506.18826

On Projective modules over graded $R$-subalgebras of $R[X,1/X]$

Diksha Garg·Anjan Gupta
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Abstract

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 $\tau$ such that $\tau (\v) = (1, 0)$. 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.

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Diksha Garg, Anjan Gupta. 2025-06-23. On Projective modules over graded $R$-subalgebras of $R[X,1/X]$. https://arxiv.org/abs/2506.18826

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