A generalization of Rao's theorem to graded $R$-subalgebras of $R[t]$

TL;DR

This paper extends Rao's theorem, showing that over certain graded subalgebras of polynomial rings, unimodular rows of length d+1 can be completed to invertible matrices, generalizing classical results in algebra.

Contribution

It generalizes Rao's theorem from polynomial rings to graded subalgebras under specific conditions, broadening the scope of unimodular row completion.

Findings

Unimodular rows of length d+1 over the subalgebra can be completed to invertible matrices.

The result applies to graded R-subalgebras of R[t] with certain factorial conditions.

Generalizes classical Rao's theorem to a wider algebraic setting.

Abstract

Let $R$ be a Noetherian local ring of Krull dimension $d$ such that $(d!)R = R$, and let $A$ be a graded $R$-subalgebra of the polynomial algebra $R[t]$. We prove that every unimodular row of length $d + 1$ over $A$ can be completed to an invertible matrix. This is a generalization of a classical result by Rao, who proved that in the same setting, every unimodular row of length $d + 1$ over $R[t]$ admits a completion to an invertible matrix.