Projective modules over overrings of polynomial rings

Let A be a commutative Noetherian ring of dimension d and let P be a projective -module of rank r⩾max{2,dimA+1}, where fi∈A[Yi]. Then(i)The natural map is surjective (3.8).(ii)Assume fi is a monic polynomial. Then Φr+1 is an isomorphism (3.8).(iii)EL1(R⊕P) acts transitively on Um(R⊕P). In particular, P is cancellative (3.12).(iv)If A is an affine algebra over a field, then P has a unimodular element (3.13). In the case of Laurent polynomial ring (i.e. fi=Yi), (i), (ii) are due to Suslin (1977) [12], , (iii) is due to Lindel (1995) [4], and (iv) is due to Bhatwadekar, Lindel and Rao (1985) [2].