Stability theorems for positively graded domains and a question of Lindel

TL;DR

This paper proves stability theorems for positively graded domains, showing that certain unimodular rows can be completed to invertible matrices homotopic to the identity, and applies these results to projective modules, answering Lindel's question.

Contribution

It establishes new stability results for graded domains and projective modules, providing an affirmative answer to Lindel's longstanding question.

Findings

Unimodular rows of length d+1 can be completed to invertible matrices homotopic to identity.

Ideals with minimal generators equal to height have their generators lifted from quotient.

Projective modules of rank d with non-zero Quillen ideal are cancellative.

Abstract

Given a commutative Noetherian graded domain $R = \bigoplus_{i\ge 0} R_i$ of dimension $d\geq 2$ with $\dim(R_0) \geq 1$, 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 $\mu(I/I^2) = \text{ht}(I) = d$, then any set of generators of $I/I^2$ 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.