Unimodular rows over monoid extensions of overrings of polynomial rings

Abstract

Let R be a commutative Noetherian ring of dimension d and M a commutative cancellative torsion-free seminormal monoid. Then: (1) Let A be a ring of type R[d,m,n] and P be a projective A[M]-module of rank r≥max {2,d+1}. Then the action of E(A[M]⊕P) on (A[M]⊕P) is transitive and (2) Assume (R,m,K) is a regular local ring containing a field k such that either char k=0 or char k=p and tr-deg K∕𝔽p≥1. Let A be a ring of type R[d,m,n]∗ and f∈R be a regular parameter. Then all finitely generated projective modules over A[M], A[M]f and A[M]⊗RR(T) are free. When M is free both results are due to Keshari and Lokhande (2014).