Discrete valuation overrings of a graded Noetherian domain
In this paper, as usual a valuation ring is an intregral domain \(V\) such that, for every \(x\) in the fraction field of \(V\), \(x\) or \(x^{-1}\) belongs to \(V\). Now, a discrete valuation ring (in short DVR) is a valuation ring such that every primary ideal is a power of its radical. The authors do not make the connection between this definition and the usual definition of a DVR (nor they talk about valuations). This paper is mainly about rings and ideals. The purpose of this paper is to study a graded domain analogue of a result about rings which is called the Krull-Akizuki theorem. In the following, \(R\) is a graded integral domain, \(H\) is the set of nonzero homogeneous elements of \(R\), and the homogeneous quotient field of \(R\) is the graded ring \(R_H\) such that the nonzero homogeneous elements of \(R_H\) are the quotients of the nonzero homogeneous elements of \(R\). A homogeneous graded overring of \(R\) is a graded subring of \(R_H\) (i.e.\ generated by homogeneous elements) which contains \(R\). A homogeneous ideal of \(R\) is an ideal which is generated by homogeneous elements. A homogeneous maximal ideal is an ideal which is maximal among proper homogeneous ideals. If \(R\) satisfies the ascending chain condition on homogeneous ideals, then \(R\) is said to be Noetherian; this equivalent to saying that every homogeneous ideal is finitely generated. The set of homogeneous prime ideals of \(R\) is denoted by h-Spec\((R)\). For \(P\) in h-Spec\((R)\), let h-ht\((P)\) be the supremum of the lengths of chains of homogeneous prime ideals descending from \(P\). The h-dimension h-dim\((R)\) of \(R\) is the supremum of the set \(\{\mathrm{h-ht}(R)\mid P\in \mathrm{h-Spec}(R)\}\). Let \(S(H)=R\backslash \bigcup_{P\in \mathrm{h-Spec}(R)}P\). The first result of this paper states that \(R\) is a graded Noetherian domain with h-dimension \(1\) if, and only if, \(R_{S(H)}\) is a one dimensional Noetherian domain. Furthermore, if this holds, then the set of maximal ideals of \(R_{S(H)}\) is \(\{PR_{S(H)}\mid P\in \mathrm{h-Spec}(R)\backslash \{(0)\}\}\). They deduce some corollaries from this result. Now, we denote by \(V\) a homogeneous graded overring of \(R\) and by \(M\) a homogeneous maximal ideal of \(V\). We say that \(V\) is a graded valuation ring if, for every homogeneous element \(x\) of the homogeneous quotient field of \(V\), \(x\) or \(x^{-1}\) belongs to \(V\). Then, \(V\) contains a unique homogeneous maximal ideal. We say that \(V\) is a discrete graded valuation ring if in addition every homogeneous primary ideal is a power of its radical. For \(f\in R_H\), the fractionary ideal of \(R\) generated by the homogeneous components of \(f\) is denoted by \(C(f)\). Let \(\widehat{V}=\{f/g\mid f\in R,\; g\in R\backslash \{0\},\; C(f)V\subseteq C(g)V\}\). The authors prove that \(\widehat{V}\) is a valuation overring of \(R\), \(\widehat{V}\cap R_H=V\), \(\widehat{V}=V_M\) and \(\mathrm{dim}(\widehat{V})=\mathrm{h-dim}(V)\). Furthermore, \(\widehat{V}\) is a DVR if, and only if, \(V\) is a discrete graded valuation ring. Assume in addition that \(R\) is Noetherian and let \((0)=P_0\subsetneq P_1\subsetneq \cdots \subsetneq P_n\) be a saturated chain of homogeneous prime ideals of \(R\). Then, there exists a homogeneous graded valuation overring \(V\) of \(R\) which contains a chain \(\{Q_{\alpha}\}_{\alpha\in\Lambda}\) such that\(\{Q_{\alpha}\cap R\}_{\alpha\in\Lambda}= \{P_i\}_{i=0}^n\). In this case, \(\{Q_{\alpha}\}_{\alpha\in\Lambda}=\{(0)=Q_0\subsetneq Q_1\subsetneq \cdots \subsetneq Q_n \}\). Furthermore, \(V_{M\backslash Q_n}\) is a discrete graded valuation ring of h-dimension \(n\), and the homogeneous quotient field of \(V/Q_i\) (\(1\leq i\leq n\)) is finite over the homogeneous quotient field of \(R/P_i\).
scientific article; zbMATH DE number 6875413
scientific article