Rees Valuation Rings Fundamentals

• Rees valuation rings are discrete valuation rings attached to an ideal in a Noetherian domain that uniquely determine the integral closure of all powers of the ideal.
• They are constructed from height-one primes in the normalization of the extended Rees ring and relate closely to asymptotic associated primes, complete ideals, and finite type birational extensions.
• Recent advances connect these rings with nonarchimedean geometry, identifying them with Shilov boundary points in Tate rings and linking them to Itoh’s (e)-valuation rings.

Searching arXiv for the cited papers and related work on Rees valuation rings. arXiv search query: "Rees valuation rings (Rangachev, 2020, Jahandoust et al., 2013, Heinzer et al., 2014, Kim et al., 2016, Dine, 9 Jul 2025)" 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 (Rangachev, 2020, Jahandoust et al., 2013).

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 $I$0, let $I$1 be an indeterminate, and let $I$2. The extended Rees ring is

$I$3

and $I$4 denotes its integral closure in the quotient field of $I$5. If $I$6 are the height-one primes of $I$7 that contain $I$8, and $I$9 is the valuation associated to the DVR $R$0, then the Rees valuation rings of $R$1 are

$R$2

Each $R$3 is therefore obtained by restricting a height-one localization of the normalized extended Rees ring back to the ground fraction field (Jahandoust et al., 2013).

These valuations control integral dependence through the asymptotic valuation function

$R$4

where $R$5 is the largest positive integer such that $R$6. If $R$7, then

$R$8

and for every positive integer $R$9,

$I\subset R$0

Equivalently, since $I\subset R$1 is principal and integrally closed in the DVR $I\subset R$2,

$I\subset R$3

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\subset R$4 (Jahandoust et al., 2013).

A recurrent simplification is that Rees valuations may be viewed as the divisorial valuations arising from the exceptional locus of the normalized blowup $I\subset R$5. 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

$I\subset R$6

with stable value denoted $I\subset R$7. McAdam’s $I\subset R$8 agrees with this stable set, and a valuation-theoretic reformulation identifies it by means of centers of Rees valuation rings (Jahandoust et al., 2013).

For a Noetherian ring $I\subset R$9, the set

$V_1,\dots,V_r$0

satisfies

$V_1,\dots,V_r$1

In the domain case this becomes especially transparent: $V_1,\dots,V_r$2 Thus the primes that persist as associated primes of $V_1,\dots,V_r$3 for all large $V_1,\dots,V_r$4 are exactly the centers of the valuations governing $V_1,\dots,V_r$5 (Jahandoust et al., 2013).

This description has a localization consequence. If $V_1,\dots,V_r$6 is multiplicatively closed and avoids every prime in $V_1,\dots,V_r$7, then

$V_1,\dots,V_r$8

The mechanism is purely valuative: elements of $V_1,\dots,V_r$9 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 $\operatorname{Frac}(R)$0.

3. Generalization from ideals to finite type birational extensions

A major extension of the Rees valuation framework replaces the special inclusion $\operatorname{Frac}(R)$1 by a general finite type inclusion of domains $\operatorname{Frac}(R)$2. If $\operatorname{Frac}(R)$3 is Noetherian, $\operatorname{Frac}(R)$4 is a finitely generated $\operatorname{Frac}(R)$5-algebra, and $\operatorname{Frac}(R)$6 denotes the integral closure of $\operatorname{Frac}(R)$7 in $\operatorname{Frac}(R)$8, then one has a valuation decomposition

$\operatorname{Frac}(R)$9

where the $$\overline{I^n}=\bigcap_{i=1}^r I^nV_i\cap R \qquad \text{for each } n,$$0 are unique discrete valuation rings in $$\overline{I^n}=\bigcap_{i=1}^r I^nV_i\cap R \qquad \text{for each } n,$$1, and the decomposition is minimal in the sense that no $$\overline{I^n}=\bigcap_{i=1}^r I^nV_i\cap R \qquad \text{for each } n,$$2 can be omitted (Rangachev, 2020).

The construction is explicit. If $$\overline{I^n}=\bigcap_{i=1}^r I^nV_i\cap R \qquad \text{for each } n,$$3 are the nonzero associated primes of the $$\overline{I^n}=\bigcap_{i=1}^r I^nV_i\cap R \qquad \text{for each } n,$$4-module $$\overline{I^n}=\bigcap_{i=1}^r I^nV_i\cap R \qquad \text{for each } n,$$5, then each localization $$\overline{I^n}=\bigcap_{i=1}^r I^nV_i\cap R \qquad \text{for each } n,$$6 is a DVR, and

$$\overline{I^n}=\bigcap_{i=1}^r I^nV_i\cap R \qquad \text{for each } n,$$7

These are precisely the DVRs appearing in the decomposition, so the valuations are localizations of $$\overline{I^n}=\bigcap_{i=1}^r I^nV_i\cap R \qquad \text{for each } n,$$8 at height-one primes in $$\overline{I^n}=\bigcap_{i=1}^r I^nV_i\cap R \qquad \text{for each } n,$$9. Under the additional hypothesis that $R$0 is locally formally equidimensional, each $R$1 is a divisorial valuation ring with respect to a Noetherian subring of $R$2 (Rangachev, 2020).

The classical Rees theorem is recovered by taking $R$3 and $R$4. In that graded setting, the degree-$R$5 piece of the integral closure yields

$R$6

so the discrete valuation rings furnished by the general theorem specialize exactly to the Rees valuation rings of the ideal $R$7 (Rangachev, 2020).

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 $R$8 over $R$9, one may replace that infinite family by finitely many uniquely determined DVRs, each coming from a height-one localization of $K$0.

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 $K$1-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 (Heinzer et al., 2014).

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 $K$2-primary ideal may have finitely many or infinitely many base points. For finitely supported complete ideals, Lipman’s factorization expresses the ideal as a $K$3-product of special $K$4-simple complete ideals, possibly with negative exponents. Here $K$5, and the special $K$6-simple ideal $K$7 is attached to a pair of infinitely near points $K$8 of the same dimension (Heinzer et al., 2014).

The central higher-dimensional criterion concerns change of direction in the unique quadratic sequence

$K$9

Assume $I$0 and $I$1. Then the special $I$2-simple complete ideal $I$3 has the order valuation ring of $I$4 as its unique Rees valuation if and only if either $I$5, or there is no change of direction in the sequence from $I$6 to $I$7. In the paper’s terminology, “no change of direction” means that there exists an element $I$8 such that $I$9 is part of a minimal generating set for $I$00 (Heinzer et al., 2014).

The examples exhibited in dimension three show the range of possibilities:

Quadratic-sequence pattern	Point basis	Rees valuations
No change of direction	$I$01	$I$02
One change of direction	$I$03	$I$04
Two changes of direction	$I$05	$I$06

These examples make precise that “simple” does not imply “one-fibered” in higher dimension. The same source also proves that every special $I$07-simple complete ideal is projectively full, so its projective equivalence class is especially rigid (Heinzer et al., 2014).

5. Rees integers and Itoh $I$08-valuation rings

The valuation-theoretic data attached to a Rees valuation ring includes its Rees integer. If $I$09 is a Rees valuation ring of a regular proper ideal $I$10 in a Noetherian ring, then $I$11 for a unique positive integer $I$12; these exponents measure the multiplicity of $I$13 along the corresponding valuations (Kim et al., 2016).

Itoh’s construction adjoins roots of the Rees parameter. With $I$14 and $I$15, define

$I$16

The Itoh $I$17-valuation rings of $I$18 are the Rees valuation rings of the principal ideal $I$19. Equivalently, they are the rings

$I$20

where $I$21 ranges over the height-one associated primes of $I$22 and $I$23 is the unique minimal prime of $I$24 contained in $I$25 (Kim et al., 2016).

A structural theorem gives a one-to-one correspondence between the Itoh $I$26-valuation rings $I$27 of $I$28 and the Rees valuation rings $I$29 of the ideal $I$30. If $I$31 is the quotient field of $I$32, then $I$33 is the integral closure of $I$34 in $I$35. Moreover, if $I$36, $I$37, and $I$38, then

$I$39

When $I$40 is a multiple of $I$41, there exists a unit $I$42 such that

$I$43

$I$44 is a finite free integral extension domain of $I$45, $I$46, $I$47, and $I$48. If all Rees integers of $I$49 are equal to $I$50, then this simplifies to

$I$51

The radicality criterion is equally precise: $I$52 is radical if and only if $I$53 is a common multiple of the Rees integers of $I$54 (Kim et al., 2016).

6. Nonarchimedean reinterpretation and current directions

A recent development identifies Rees valuation rings with Shilov boundary points in nonarchimedean geometry. If $I$55 is a Tate ring with Noetherian ring of definition $I$56 and pseudo-uniformizer $I$57, then the Shilov boundary of $I$58 naturally coincides with the set of Rees valuation rings of the principal ideal $I$59. For affinoid algebras in the sense of Tate whose underlying rings are integral domains, this recovers a well-known result of Berkovich (Dine, 9 Jul 2025).

The analytic side is organized by the spectral seminorm, while the algebraic side is organized by integral closure of powers of $I$60. The identification shows that, in the principal Noetherian setting, the finite family of valuations determining $I$61 is exactly the finite family of rank-one seminorms on which the spectral seminorm attains its maximum. The paper further characterizes the Shilov boundary for a wide class of Tate rings by means of minimal open prime ideals in the subring of power-bounded elements, and proves stability of this characterization under certain completed integral extensions (Dine, 9 Jul 2025).

One consequence concerns mixed characteristic. For every mixed-characteristic Noetherian domain $I$62, the Tate ring

$I$63

admits a Shilov boundary description of this type, where $I$64 is the $I$65-adic completion of the absolute integral closure of $I$66. This places Rees valuation rings in a setting where blowup-theoretic, valuation-theoretic, and analytic boundary constructions coincide (Dine, 9 Jul 2025).

Taken together, these developments show that Rees valuation rings are not merely auxiliary devices for the study of $I$67. They form a finite and unique valuation package controlling integral closure, asymptotic associated primes, birational finite type extensions, the geometry of complete ideals in regular local rings, root constructions governed by Rees integers, and, in the principal case, the Shilov boundary of a Tate ring.