Subrings of polynomial rings and the conjectures of Eisenbud-Evans

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. Furthermore, for a graded subring $B$ of $R[T]$ containing $R$, we improve some existing stability theorems for projective modules over $B$, generalizing results due to Bass, Serre and Vaser{\v{s}}te{\u{\i}}n.Comment: 25 pages. New results in section 6. Comments are welcom