Subrings of polynomial rings and the conjectures of Eisenbud and Evans

TL;DR

This paper investigates the validity of Eisenbud and Evans's conjectures on polynomial rings over certain Noetherian subrings, extending known results to a broader class of rings including Rees algebras and symbolic Rees algebras.

Contribution

It introduces a class of Noetherian subrings of polynomial rings and proves that Eisenbud and Evans's conjectures hold for these rings, expanding the scope of previous results.

Findings

Conjectures hold for polynomial rings and Rees algebras.

Established validity over Noetherian subrings of dimension d+1.

Includes symbolic Rees algebras in the class where conjectures are valid.

Abstract

Let $R$ be a commutative Noetherian ring of dimension $d$. In 1973, Eisenbud and Evans proposed three conjectures on the polynomial ring $R[T]$. These conjectures were settled in the affirmative by Sathaye, Mohan Kumar and Plumstead. One of the primary objectives of this article is to investigate the validity of these conjectures over Noetherian subrings of $R[T]$ of dimension $d+1$, containing $R$. We formulate a class of such rings, which includes polynomial rings, Rees algebras, Rees-like algebras and Noetherian symbolic Rees algebras, and exhibit that all three conjectures hold for rings belonging to this class.