Monic Inversion Principle and Complete intersection

TL;DR

This paper proves a monic inversion principle for projective modules over regular rings, showing that certain surjections can be lifted from the localization at the polynomial variable to the entire polynomial ring.

Contribution

It establishes a new lifting result for projective modules over regular rings, extending the monic inversion principle to ideals of polynomial extensions.

Findings

Surjective lifts exist under specified conditions.

The result applies to regular rings over infinite fields with characteristic not 2.

Provides a method to lift surjections from localized to global settings.

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 $\Phi:P[T]\twoheadrightarrow I$.