Efficient generation, unimodular element in a geometric subring of a polynomial ring

TL;DR

This paper introduces the concept of geometric subrings of polynomial rings over Noetherian rings and proves new results on ideal generation and projective module splitting, extending classical bounds.

Contribution

It defines geometric subrings of polynomial rings and proves that locally complete intersection ideals are complete, improving bounds for ideal generation and projective modules.

Findings

Every locally complete intersection ideal of height d+1 is a complete intersection.

Finitely generated projective modules of rank d+1 split off a free summand.

Applications to set-theoretic generation of ideals in geometric rings.

Abstract

Let $R$ be a commutative Noetherian ring of dimension $d$. First, we define the "geometric subring" $A$ of a polynomial ring $R[T]$ of dimension $d+1$ (the definition of geometric subring is more general, see (1.2)). Then we prove that every locally complete intersection ideal of height $d+1$ is a complete intersection ideal. Thus improving the general bound of Mohan Kumar \cite{NMK78} for an arbitrary ring of dimension $d+1$. Afterward, we deduce that every finitely generated projective $A$-module of rank $d+1$ splits off a free summand of rank one. This improves the general bound of Serre \cite{Serre58} for an arbitrary ring. Finally, applications are given to a set-theoretic generation of an ideal in the geometric ring $A$ and its polynomial extension $A[X]$.