A degree map on unimodular rows (Q2907424)

The main aim of the author is to show that the answer to Nori's question is negative in general for \(n\) odd. Let \(k\) be a field and let \((A^n-0)\) be the punctured affine space. He associates to any morphism \(g: (A^n-0)\rightarrow(A^n-0)\) an element in the Witt group \(W(k)\) that is said to be the degree of \(g\). By using this degree map the author gives a negative answer to a question of Nori about unimodular rows: Let \(k\) be a field such that \( \sqrt{-1}\notin k\). Consider the unimodular row \((x_1, x_2, x_3) \in Um_3(S_3)\) and the map \(g: (A^n-0)\rightarrow(A^n-0)\) defined by the algebra homomorphism \(\varphi: k[x_1, x_2, x_3] \to k[x_1, x_2, x_3]\) given by \(\varphi(x_1)=x^2_1-x^2_2\), \(\varphi(x_2)=x_1x_2\) and \(\varphi(x_3)=x_3\). Then \(k[x_1, x_2, x_3]/(x^2_1-x^2_2,x_1x_2, x_3)\) is of length 4, but \((x^2_1-x^2_2,x_1x_2, x_3)\in Um_3(S_3)\) is not completable.