--- title: Rees Valuation Rings Fundamentals url: https://www.emergentmind.com/topics/rees-valuation-rings type: topic --- # Rees Valuation Rings Fundamentals Searching arXiv for the cited papers and related work on Rees valuation rings. arXiv search query: "Rees valuation rings 2011.14749 1308.6449 1404.1524 1607.05341 2507.07091" Rees valuation rings are the discrete valuation rings attached to an ideal \(I\) in a Noetherian domain that control the integral closure of all powers of \(I\). In the classical setting, if \(R\) is a Noetherian domain and \(I\subset R\) is nonzero, there exist unique discrete valuations \(V_1,\dots,V_r\) in \(\operatorname{Frac}(R)\) such that \[ \overline{I^n}=\bigcap_{i=1}^r I^nV_i\cap R \qquad \text{for each } n, \] and these valuation rings arise from height-one primes in the normalization of the Rees algebra or extended Rees ring. Modern work places this construction inside a broader valuation theorem for finite type extensions, relates the centers of Rees valuation rings to asymptotic associated primes, studies their behavior for complete ideals in regular local rings, refines them via Itoh’s root constructions, and identifies them with Shilov boundary points in a nonarchimedean setting [2011.14749] [1308.6449]. ## 1. Classical definition and valuation-theoretic control Let \(R\) be a Noetherian integral domain with field of fractions \(K\), let \(I\) be an ideal of \(R\), let \(t\) be an indeterminate, and let \(u=t^{-1}\). The extended Rees ring is \[ \mathscr R = R[u,It]=\bigoplus_{n\in\mathbb Z} I^n t^n, \] and \(\mathscr R'\) denotes its integral closure in the quotient field of \(\mathscr R\). If \(\mathfrak p_1,\dots,\mathfrak p_r\) are the height-one primes of \(\mathscr R'\) that contain \(u\), and \(v_i\) is the valuation associated to the DVR \(\mathscr R'_{\mathfrak p_i}\), then the Rees valuation rings of \(I\) are \[ V_i=\mathscr R'_{\mathfrak p_i}\cap K,\qquad i=1,\dots,r. \] Each \(V_i\) is therefore obtained by restricting a height-one localization of the normalized extended Rees ring back to the ground fraction field [1308.6449]. These valuations control integral dependence through the asymptotic valuation function \[ \overline V_I(x)=\lim_{n\to\infty}\frac{V_I(x^n)}{n}, \] where \(V_I(x)\) is the largest positive integer such that \(x\in I^n\). If \(e_i=v_i(u)\), then \[ \overline V_I(x)=\min_{i=1,\dots,r}\frac{v_i(x)}{e_i}, \] and for every positive integer \(k\), \[ \overline V_I(x)\ge k \Longleftrightarrow x\in \overline{I^k}. \] Equivalently, since \(I^nV_i\) is principal and integrally closed in the DVR \(V_i\), \[ \overline{I^n}=\bigcap_{i=1}^r I^nV_i\cap R \qquad (n>0). \] This is the characteristic valuation-theoretic property of Rees valuation rings: they are the finite minimal family of DVRs that determines the integral closure of every power of \(I\) [1308.6449]. A recurrent simplification is that Rees valuations may be viewed as the divisorial valuations arising from the exceptional locus of the normalized blowup \(\operatorname{Proj}(R[It])\). The papers in the record above present this as a compatible characterization rather than an independent definition. ## 2. Centers, asymptotic associated primes, and localization The centers of Rees valuation rings encode asymptotic prime-theoretic data. Ratliff’s stabilization result gives an increasing chain \[ \operatorname{Ass}_R R/\overline I\subseteq \operatorname{Ass}_R R/\overline{I^2}\subseteq \operatorname{Ass}_R R/\overline{I^3}\subseteq \cdots \] with stable value denoted \(A^*(I)\). McAdam’s \(A^{**}(I)\) agrees with this stable set, and a valuation-theoretic reformulation identifies it by means of centers of Rees valuation rings [1308.6449]. For a Noetherian ring \(R\), the set \[ B^*(I)=\{\mathfrak p\in \operatorname{Spec}R \mid I\subseteq \mathfrak p \text{ and there exists } z\in \operatorname{Min}R \text{ with } z\subseteq \mathfrak p \text{ and } \mathfrak p/z \text{ the center of a Rees valuation ring of } I(R/z)\} \] satisfies \[ B^*(I)=A^{**}(I)=A^*(I). \] In the domain case this becomes especially transparent: \[ A^*(I)=\{\mathfrak p\in \operatorname{Spec}R \mid I\subseteq \mathfrak p,\ \mathfrak p \text{ is the center on }R\text{ of some Rees valuation ring of }I\}. \] Thus the primes that persist as associated primes of \(R/\overline{I^n}\) for all large \(n\) are exactly the centers of the valuations governing \(\overline{I^n}\) [1308.6449]. This description has a localization consequence. If \(S\subseteq R\) is multiplicatively closed and avoids every prime in \(A^{**}(I)\), then \[ \overline{I^n}R_S\cap R=\overline{I^n}\qquad \text{for all }n>0. \] The mechanism is purely valuative: elements of \(S\) avoid the centers of all Rees valuation rings, hence become units in every such valuation ring, and unit multiplication does not change membership in the valuation ideals \(I^nV\). ## 3. Generalization from ideals to finite type birational extensions A major extension of the Rees valuation framework replaces the special inclusion \(R[It]\subset R[t]\) by a general finite type inclusion of domains \(A\subset B\). If \(A\) is Noetherian, \(B\) is a finitely generated \(A\)-algebra, and \(\overline A\) denotes the integral closure of \(A\) in \(B\), then one has a valuation decomposition \[ \overline A=\bigcap_{i=1}^r V_i\cap B, \] where the \(V_i\) are unique discrete valuation rings in \(\operatorname{Frac}(A)\), and the decomposition is minimal in the sense that no \(V_i\) can be omitted [2011.14749]. The construction is explicit. If \(\mathfrak q_1,\dots,\mathfrak q_r\) are the nonzero associated primes of the \(A\)-module \(B/A\), then each localization \(A_{\mathfrak q_i}\) is a DVR, and \[ V_i=A_{\mathfrak q_i}. \] These are precisely the DVRs appearing in the decomposition, so the valuations are localizations of \(A\) at height-one primes in \(\operatorname{Ass}_-(B/A)\). Under the additional hypothesis that \(A\) is locally formally equidimensional, each \(V_i\) is a divisorial valuation ring with respect to a Noetherian subring of \(A\) [2011.14749]. The classical Rees theorem is recovered by taking \(A=R[It]\) and \(B=R[t]\). In that graded setting, the degree-\(n\) piece of the integral closure yields \[ \overline{I^n}=\bigcap_{i=1}^r I^nV_i\cap R, \] so the discrete valuation rings furnished by the general theorem specialize exactly to the Rees valuation rings of the ideal \(I\) [2011.14749]. A common misconception is suggested by the general fact that an integrally closed domain is an intersection of all valuation rings of its fraction field containing it. The finite type theorem isolates a much sharper statement: under finite generation of \(B\) over \(A\), one may replace that infinite family by finitely many uniquely determined DVRs, each coming from a height-one localization of \(A\). ## 4. Complete ideals in regular local rings In a regular local ring, Rees valuation rings interact with complete ideals, infinitely near points, and local quadratic transforms. In dimension two, Zariski’s theory gives a particularly rigid picture: every complete \(\mathfrak m\)-primary ideal factors uniquely as a product of powers of simple complete ideals, and each simple complete factor has a unique Rees valuation. Distinct simple factors correspond to distinct Rees valuation rings [1404.1524]. In higher dimension, the two-dimensional picture fails in two ways recorded explicitly in the literature. First, a simple complete ideal can have more than one Rees valuation. Second, a complete \(\mathfrak m\)-primary ideal may have finitely many or infinitely many base points. For finitely supported complete ideals, Lipman’s factorization expresses the ideal as a \(*\)-product of special \(*\)-simple complete ideals, possibly with negative exponents. Here \(I*J=\overline{IJ}\), and the special \(*\)-simple ideal \(P_{RT}\) is attached to a pair of infinitely near points \(R