Subrings of polynomial rings and the conjectures of Eisenbud and Evans (Q6187301)

Let \(R\) be a commutative Noetherian ring of dimension \(d\). Eisenbud and Evans formulated in 1973 three conjectures on the polynomial ring \(R[T]\), later proved by Sathaye, Mohan Kumar and Plumstead. In this paper, the author shows that these conjectures are valid on a certain class of subrings of \(R[T]\), which includes polynomial rings, Rees algebras, Rees-like algebras and Noetherian symbolic Rees algebras. Let \(f\in R[T]\setminus R\) be a non-zero divisor and \(n \in Z\). A Noetherian subring \(A\) of \(R[T, f^n]\) containing \(R\), is called a geometric subring of \(R[T, f^n]\), if there exists a non-zero divisor \(s\in R\) such that \(A_s=R_s[T, f^n]\) and \(\dim(A)=d+1\). The main results of the paper read as follows. Let \(A\) be a geometric subring of \(R[T]\), and let \(M\) be an \(A\)-module such that \(\mu_\mathfrak{p}(M) \ge \dim(A/\mathfrak{p})\) for all minimal prime ideal \(\mathfrak{p} \in \mathrm{Spec}(A)\); then \(M\) has a basic element. Let \(f\in R[T] \setminus R\) be a non-zero divisor, let \(A\) be a geometric subring of \(R[T,f^n]\), where \(n \in Z\), and let \(P\) be a projective \(A\)-module such that \(\mathrm{rank}(P)\ge d +1\); then the projective module \(P\) is cancellative. Let \(A\) be a geometric subring of \(R[T]\); then any \(A\)-module \(M\) is generated by \(e(M)\) elements, where \(e(M) = \sup \{\mu_\mathfrak{p}(M) + \dim(R[T]/\mathfrak{p} \mid \mathfrak{p} \in \mathrm{Spec}(R[T]) \textrm{ such that } \dim(R[T]/\mathfrak{p} \le d\}\).