A note on cancellation of projective modules

We prove the following results. (i) Let A be an affine algebra of dimension d≥4 over (with p≥d). Then all projective A-modules of rank d−1 are cancellative.(ii) Let A be a ring of dimension d such that Ed+1(R) acts transitively on Umd+1(R) for every finite extension R of A. Then for any projective A-module P of rank d, E(A⊕P) acts transitively on Um(A⊕P).