Homotopy and sections of algebraic vector bundles over real affine algebras
In this paper the author gives an algebraic proof (due to Nori) of the fact that the tangent bundle of the real 2-sphere does not have a unimodular element. Let \(A\) be a regular affine domain of dimension \(n\), over a field \(k\) of characteristic zero and let \(K_A\) be the canonical line bundle of \(A\). Let \(P\) be a projective \(A\)-module of rank \(n\) and \(i:K_A@>\sim>>\bigwedge^nP\) be an isomorphism. The center piece of the proof of the above result is to define an invariant for such pairs \((P,i)\). This invariant was defined by Nori which is explained in complete detail in this paper. The invariant takes values in a certain quotient of the Witt group of the ground field. The author also uses this invariant to get many interesting results about projective modules over regular rings of dimension two.
scientific article; zbMATH DE number 949853