Splitting criteria for projective modules over polynomial algebras

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.