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

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 $r\ge \max \{d+1,2\}$ and $\check{=}(a,p)$ be a unimodular element of $A\oplus P$. We find an elementary automorphism $\tau$ such that $\tau (\check{)}=(1,0)$. 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.

This work is part of the first author's Ph.D. dissertation