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 2 n ≥ d + 2. Moreover, if A is smooth, we associate to a unimodular row v of length n + 1 its Euler class e (v) ∈ 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). [ABSTRACT FROM AUTHOR]