Stability theorems for positively graded domains and a question of Lindel

Given a commutative Noetherian graded domain $R={⨁}_{i\ge 0}{R}_i$ of dimension $d\ge 2$ with $\dim (R_0)\ge 1$, we prove that any unimodular row of length $d+1$ in $R$ can be completed to the first row of an invertible matrix $\alpha$ such that $\alpha$ is homotopic to the identity matrix. Utilizing this result we establish that if $I\subset R$ is an ideal satisfying $\mu (I\mathrm{/}{I}^2)=\text{ht}(I)=d$, then any set of generators of $I\mathrm{/}{I}^2$ lifts to a set of generators of $I$, where $\mu (-)$ denotes the minimal number of generators. Consequently, any projective $R$-module of rank $d$ with trivial determinant splits into a free factor of rank one. This provides an affirmative answer to an old question of Lindel. Finally, we prove that for any projective $R$-module $P$ of rank $d$, if the Quillen ideal of $P$ is non-zero, then $P$ is cancellative.