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

2025/06/23 by Garg, Diksha, Gupta, Anjan
#13A02 #13B25 #13C10 #19A13 #19B10 #19B14 #Commutative Algebra (math.AC) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2506.18836

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.

Citations

• On Projective modules over graded R-subalgebras of R[X,1/X]
• Stability theorems for positively graded domains and a question of Lindel
• Unimodular rows over monoid rings
• A nice group structure on the orbit space of unimodular rows-II
• Euler class group of a Noetherian ring

Related

• On completion of unimodular rows over polynomial extension of finitely\n generated rings over \ℤ
• Stability theorems for positively graded domains and a question of Lindel
• Graded r-Submodules
• Ratliff-Rush Filtration, regularity and depth of Higher Associated graded modules: Part II
• On Projective modules over graded R-subalgebras of R[X,1/X]