Projective modules over overrings of polynomial rings

Abstract

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 ∈ A [ Y i ] . Then (i) The natural map Φ r : GL r ( R ) / EL r 1 ( R ) → K 1 ( R ) is surjective (3.8). (ii) Assume f i is a monic polynomial. Then Φ r + 1 is an isomorphism (3.8). (iii) EL 1 ( 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. 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].