On triviality of the Euler class group of a deleted neighbourhood of a smooth local scheme (Q2846969)

The author addresses a question whether can a given basis of \(I/I^2\) be lifted to a minimal set of generators of \(I\)? This paper has four sections. In the introduction, the author formulates the main question and provides a scheme to prove it. In the second section, he states some preliminary results and definitions. In the third section, he provides a result analogous to Gabber's theorem [\textit{O. Gabber}, in: Groupe de Brauer, Semin., Les Plans-sur-Bex 1980, Lect. Notes Math. 844, 129--209 (1981; Zbl 0472.14013)]. In the fourth section, the author provides the main result.NEWLINENEWLINE Theorem. Let \((R, m)\) be a regular local ring of dimension \(d\) which is essentially of finite type over a field \(k\) such that the residue field of \(R\) is infinite. Let \( f \in \mathfrak{m} \smallsetminus \mathfrak{m}^2\) be a regular parameter and \(n\) be an integer such that \(2n \geq d+1\). Let \(I \subset R_f\) be an ideal of height n such that \(I/I^2\) is generated by \(n\) elements. It is proved that any given set of \(n\) generators of \(I/I^2\) can be lifted to a set of \(n\) generators of \(I\).