On Krull's Dimension Theorem for Certain Graded Rings and Its Applications

Abstract

This paper explores the dimension theory of non-Noetherian graded rings by introducing the class of Hilbert--Serre rings. Motivated by Krull's dimension theorem and Smoke's dimension theorem, we establish the fundamental inequalities $\operatorname{dim_{gr}}(R) \leq \dim(R) \leq \operatorname{GKdim}_k(R) \leq d(R)$ for any Hilbert--Serre ring $R$, where $d(R)$ is the pole order of its Poincar\'e series at $t=1$. We then prove that all these dimensions, including the transcendence degree in the domain case, coincide for monomial algebras. As an application, let $S$ be a finitely generated homogeneous subalgebra of a polynomial ring over a field $k$, and let $\mathrm{in}_{<}(S)$ be its initial algebra with respect to a monomial order $<$. Even though $\mathrm{in}_{<}(S)$ need not be Noetherian, we prove that taking initial algebras preserves all the dimensions considered in this paper; in particular, $\dim(S)=\dim(\mathrm{in}_{<}(S))$. Finally, we provide explicit examples demonstrating that these inequalities can be strict in general, even for Hilbert--Serre domains.