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

TL;DR

This paper proves that projective modules over certain graded subalgebras of Laurent polynomial rings are cancellative and split off free summands, extending known results to a broader class of rings.

Contribution

It establishes the cancellative property and splitting off free summands for projective modules over graded subalgebras of Laurent polynomial rings, generalizing previous results.

Findings

Projective modules are cancellative over these subalgebras.

Modules split off a free summand of rank one.

Results extend known cases for polynomial and Laurent polynomial rings.

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.