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

TL;DR

This paper extends dimension theory to non-Noetherian graded rings via Hilbert-Serre rings, generalizing Krull's theorem and applying results to initial algebras with explicit examples.

Contribution

It introduces Hilbert-Serre rings to generalize classical dimension theorems and applies these to initial algebras, showing dimension equalities and strict inequalities.

Findings

Established inequalities relating dimension, GK-dimension, and pole order.

Proved that dimensions coincide for monomial algebras.

Provided examples where inequalities are strict, even for integral domains.

Abstract

This paper explores the dimension theory of non-Noetherian graded rings by introducing the class of Hilbert-Serre rings. We generalize Krull's dimension theorem and Smoke's dimension theorem by establishing the fundamental inequalities $\dim(R) \le \operatorname{GKdim}_k(R) \le d(R)$ for any Hilbert-Serre ring $R$, where $d(R)$ is the pole order of its Poincar\'e series at $t=1$. Furthermore, we apply these results to initial algebras, proving that all these dimensions, including the transcendence degree, coincide for monomial algebras. Finally, we provide explicit examples demonstrating that these inequalities can be strict in general, even for integral domains.