On a question of Nori: Obstructions, improvements, and applications

This article concerns a question asked by M. V. Nori on homotopy of sections of projective modules defined on the polynomial algebra over a smooth affine domain R. While this question has an affirmative answer, it is known that the assertion does not hold if: (1) dim(R) = 2; or (2) d >= 3 but R is not smooth. We first prove that an affirmative answer can be given for dim(R) = 2 when R is an F-p-algebra. Next, for d >= 3 we find the precise obstruction for the failure in the singular case. Further, we improve a result of Mandal (related to Nori's question) in the case when the ring A is an affine F-p-algebra of dimension d. We apply this improvement to define the n-th Euler class group E-n(A), where 2n >= d + 2. Moreover, if A is smooth, we associate to a unimodular row v of length n +1 its Euler class e(v) is an element of E-n(A) and show that the corresponding stably free module, say, P(v) has a unimodular element if and only if e(v) vanishes in E-n(A).