Stability theorems for positively graded domains

Abstract

We consider a positively (and non-trivially) graded integral domain $R=\bigoplus_{i\ge 0}R_i$ of dimension $d\geq 2$, where $\dim(R_0)\geq 1$. Let $A=S^{-1}R$ be the localization of $R$ with respect to a multiplicatively closed set $S\subset R$. In this article, we establish that any unimodular row of length $d+1$ in $A$ can be completed to the first row of an invertible matrix $\alpha$ such that $\alpha$ is homotopic to the identity matrix. We also prove that the injective stability of $\text{K}_1(R)$ is $d+1$. Furthermore, we show that if $I\subset A$ is an ideal such that $\mu(I/I^2)=d$ and $\text{ht}(I)=\text{dim}(A)$, then any set of generators of $I/I^2$ lifts to a set of generators of $I$, where $\mu(-)$ represents the minimal number of generators. As a consequence, every projective $A$-module of rank $d$ with trivial determinant splits off a free summand of rank one. Finally, for a projective $R$-module $P$ of rank $d$, we establish that if the Quillen ideal $J(R_0,P)$ of $P$ is non-zero, then $P$ is cancellative.Comment: 21 pages. Comments are welcom