Splitting criteria for projective modules over polynomial algebras

TL;DR

This paper studies the splitting problem for projective modules over polynomial algebras, addresses an open question on monic inversion principles, and provides new results for affine algebras over algebraically closed fields.

Contribution

It proves the monic inversion principle for finitely generated rings and affine algebras over algebraically closed fields, advancing understanding of projective modules and their splitting criteria.

Findings

Proved the monic inversion principle for finitely generated rings.

Established the principle for affine algebras over algebraically closed fields.

Presented applications of the theoretical results.

Abstract

This article investigates the splitting problem for finitely generated projective modules $P$ over affine algebras over algebraically closed fields and their polynomial extensions. We then address an open question due to M. Roitman on monic inversion principle for projective modules and prove it in the affirmative for finitely generated rings. For affine algebras over $\overline{\mathbb{F}}_p$, we prove a monic inversion principle for ideals. We also exhibit some applications.