Discrete valuation overrings of a graded Noetherian domain

Abstract

Let $R = \bigoplus _{\alpha \in \Gamma } R_{\alpha }$ be an integral domain graded by an arbitrary torsionless grading monoid $\Gamma $, $M$ a homogeneous maximal ideal of $R$ and $S(H) = R \setminus \bigcup _{P \in \text {h-}Spec (R)}P$. We show that $R$ is a graded Noetherian domain with $\text {h-}\dim (R) = 1$ if and only if $R_{S(H)}$ is a one-dimensional Noetherian domain. We then use this result to prove a graded Noetherian domain analogue of the Krull-Akizuki theorem. We prove that, if $R$ is a gr-valuation ring, then $R_M$ is a valuation domain, $\dim (R_M) = \text {h-}\dim (R)$ and $R_M$ is a discrete valuation ring if and only if $R$ is discrete as a gr-valuation ring. We also prove that, if $\{P_i\}$ is a chain of homogeneous prime ideals of a graded Noetherian domain $R$, then there exists a discrete valuation overring of $R$ which has a chain of prime ideals lying over $\{P_i\}$.