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

Abstract:

Let $(R, \mathfrak {m})$ 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 $I$.