290 research outputs found
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,\ldots,X_l,Y_1,\ldots,Y_m,\frac {1}{f_1\ldots f_m}]-module
of rank r\geq max {2,dim A+1, where f_i\in A[Y_i]. Then
(i) \EL^1(R\op P) acts transitively on Um(R\oplus P). In particular, P is
cancellative.
(ii) If A is an affine algebra over a field, then P has a unimodular element.
(iii) The natural map \Phi_r : GL_r(R)/EL^1_r(R) \ra K_1(R) is surjective.
(iv) Assume f_i is a monic polynomial. Then \Phi_{r+1} is an isomorphism.
In the case of Laurent polynomial ring (i.e. f_i=Y_i), (i) is due to Lindel,
(ii) is due to Bhatwadekar, Lindel and Rao and (iii, iv) is due to Suslin
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