Monic Inversion Principle and Complete intersection

Abstract

Let $A$ be a regular ring of dimension $d$ essentially of finite type over an infinite field $k$ of characteristic $\neq 2$. Let $P$ be a projective $A$-module of rank $n$ with $2n\geq d+3$. Let $I$ be an ideal of $A[T]$ of height $n$ and $\phi:P[T]\twoheadrightarrow I/I^2$ be a surjection. If $\phi\otimes A(T)$ has a surjective lift $\theta :P[T]\otimes A(T)\twoheadrightarrow IA(T)$, then $\phi$ has a surjective lift $\Phi:P[T]\twoheadrightarrow I$.