Projective modules over overrings of polynomial rings

Let A be a commutative Noetherian ring of dimension d and let P be a projective R = A[X-1,...,X-l, Y-1,...,Y-m, 1/f(1)...f(m)]-module of rank r >= max{2, dim A + 1}, where f(i) is an element of A[Y-i]. Then (i) The natural map Phi(r) : GL(r)(R)/ELr1(R) -> K-1(R) is surjective (3.8). (ii) Assume f(i) is a monic polynomial. Then Phi(r+1) is an isomorphism (3.8). (iii) EL1(R circle plus P) acts transitively on Um(R circle plus 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. f(i) = Y-i), (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]. (C) 2009 Elsevier Inc. All rights reserved.