2.1 Integral closure of ideals and Huneke’s uniform theorems

The integral closure of an ideal $I\subseteq $ is the ideal $\overline{I}$ consisting of elements $x\in R$ that satisfy an equation of the form $x^t+a_1x^{t-1}+\cdots + a_{t-1}x + a_t=0$ , such that $a_i\in I^i$ for all $1\leqslant i\leqslant t$ ; see [Reference Swanson and HunekeSH06, Chapter 1] for an introduction to the integral closures of ideals.

Our work relies on the Uniform Artin–Rees and Uniform Briançon–Skoda Theorems of [Reference HunekeHun92]. We recall the Uniform Artin–Rees and Briançon–Skoda Properties here for convenience.

Huneke conjectured that any excellent Noetherian ring R of finite Krull dimension possesses the Uniform Artin–Rees Property [Reference HunekeHun92, Conjecture 1.3], and that any excellent Noetherian reduced ring of finite Krull dimension exhibits the Uniform Briançon–Skoda Property [Reference HunekeHun92, Conjecture 1.4]. Strong support for these conjectures arises from the same paper, where they are proven under certain additional mild hypotheses.

2.2 Rees valuations

Continue to assume R is an excellent reduced ring and $I\subseteq R$ is an ideal. Introduce a variable T; the extended Rees algebra of I is the $\mathbb{Z}$ -graded ring

\[R[IT, T^{-1}] = \cdots \oplus RT^{-2} \oplus RT^{-1} \oplus R \oplus IT \oplus I^2T^2 \oplus \cdots.\]

The nth-degree component of $R[IT,T^{-1}]$ is $I^nT^n$ for $n \gt 0$ , coinciding with R in non-positive degrees. In particular,

\[\frac{R[IT, T^{-1}]}{T^{-1}R[IT,T^{-1}]} \cong \frac{R}{I}\oplus \frac{I}{I^2}T \oplus \frac{I^2}{I^3}T^2\oplus \cdots\]

is the associated graded ring of I and denoted by $\mathrm{Gr}_I(R)$ . If $I=(f_1,f_2,\ldots,f_t)$ , then the homogeneous localizations of $R[IT,T^{-1}]$ at the elements $f_iT$ give the Laurent polynomial ring over the affine charts of the blowup:

\[R[(f_1,f_2,\ldots,f_t)T,T^{-1}]_{f_iT} =R\left[\frac{f_1}{f_i},\frac{f_2}{f_i},\ldots,\frac{f_t}{f_i}\right]\![T,T^{-1}].\]

Let $\overline{R[IT, T^{-1}]}$ denote the integral closure of $R[IT, T^{-1}]$ in its total ring of fractions K(T). If $n\geqslant 1$ , we have $T^{-n}\overline{R[IT, T^{-1}]}\cap R = \overline{I^n\overline{R}}\cap R=\overline{I^n}$ , and $\overline{R[IT,T^{-1}]}$ coincides with $\overline{R}$ in non-positive degrees. Notably, if $R=\overline{R}$ , i.e. R is normal, then

\[\overline{R[IT, T^{-1}]} = \cdots \oplus RT^{-2} \oplus RT^{-1} \oplus R \oplus \overline{I}T \oplus \overline{I^2}T^2 \oplus \cdots.\]

Homogeneous localizations of $\overline{R[IT,T^{-1}]}$ at the elements $f_iT$ give the Laurent polynomial ring of the affine charts of the normalized blowup:

\[v\overline{R[(f_1,f_2,\ldots,f_t)T,T^{-1}]}_{f_iT} = \overline{R\left[\frac{f_1}{f_i},\frac{f_2}{f_i},\ldots,\frac{f_t}{f_i}\right]}[T,T^{-1}].\]

The Rees valuations of I, denoted $\mathcal{R}_I$ , are the discrete valuation rings obtained through homogeneous localization of the associated primes of $T^{-1}\overline{R[IT,T^{-1}]}$ .Footnote ¹ If $\nu\in\mathcal{R}_I$ , then we typically denote the corresponding minimal primes of $T^{-1}\overline{R[IT,T^{-1}]}$ as $Q_\nu$ . Minimal primes of $T^{-1}\overline{R[IT,T^{-1}]}$ are the exceptional primes of $\overline{R[IT,T^{-1}]}$ .

Example 2.3. Let $R=\mathbb{C}[x_1,x_2,x_3]/(x_1x_2+x_3^3)$ and $\mathfrak{m}=(x_1,x_2,x_3)$ . Then R is a normal domain with an isolated rational double-point singularity at $\mathfrak{m}$ . It is not difficult to show that $\mathfrak{m}^n=\overline{\mathfrak{m}^n}$ for all $n\in\mathbb{N}$ , and therefore

\[ R[\mathfrak{m} T,T^{-1}]=\overline{R[\mathfrak{m} T,T^{-1}]}\cong \frac{\mathbb{C}[y_1,y_2,y_3,T^{-1}]}{(y_1y_2+T^{-1}y_3^3)}. \]

The injective $\mathbb{C}$ -algebra map $R\to R[\mathfrak{m} T,T^{-1}]$ is defined by $x_i\mapsto T^{-1}y_i$ for all $1\leqslant i\leqslant 3$ . Observe that $T^{-1}\overline{R[\mathfrak{m} T,T^{-1}]}=(T^{-1},y_1)\cap (T^{-1},y_2)$ . Therefore, $\mathcal{R}_{\mathfrak{m}}$ is a 2-element set with Rees valuation rings $(\mathbb{C}[y_1,y_2,y_3]/(y_1y_2+T^{-1}y_3^3))_{(T^{-1},y_1)}$ and $(\mathbb{C}[y_1,y_2,y_3]/(y_1y_2+T^{-1}y_3^3))_{(T^{-1},y_2)}$ .

The following theorem is a list of known properties of Rees valuations used throughout this article.

Theorem 2.6 (Properties of Rees valuations). Let R be an excellent Noetherian reduced ring and $I\subseteq R$ an ideal not contained in a minimal prime of R. Let $\mathcal{R}_I$ denote the set of Rees valuations of I. For each $\nu\in\mathcal{R}_I$ , let $Q_\nu$ be the corresponding exceptional prime of $\overline{R[IT,T^{-1}]}$ and $\nu(I)$ the unique natural number such that

\[ T^{-1}\overline{R[IT,T^{-1}]}=\bigcap_{\nu\in\mathcal{R}_I} Q_\nu^{(\nu(I))}. \]

1. (a) Let $f\in R$ . Then $f\in \overline{I^n}$ if and only if $\nu(f)\geqslant n\nu(I)$ for all $\nu\in\mathcal{R}_I$ [Reference ReesRee56].Footnote ² Consequently:

   • – if W is a multiplicative set, then the Rees valuations of $IR_W$ are the Rees valuations of I whose centers do not intersect W [Reference Swanson and HunekeSH06, Proposition 10.4.1];

   • – $\mathcal{R}_I=\mathcal{R}_{\overline{I}}$ ;

   • – if $t\in\mathbb{N}$ , then $\mathcal{R}_I=\mathcal{R}_{I^t}$ .

2. (b) Let $\mathfrak{p}\in\mathrm{Spec}(R)$ . Then $\mathfrak{p}$ is an associated prime of $\overline{I^n}$ for some n if and only if $\mathfrak{p}$ is a center of a Rees valuation of I. If $\mathfrak{p}$ is an associated prime of $\overline{I^{n_0}}$ , then $\mathfrak{p}$ is an associated prime of $\overline{I^n}$ for all $n\geqslant n_0$ [Reference RatliffRat84, Theorems 2.4 and 2.7].

3. (c) If $\mathfrak{p}\in\mathrm{Spec}(R)$ is of height h, then $\mathfrak{p}$ is a center of a Rees valuation of I if and only if $IR_{\mathfrak{p}}$ has analytic spread h [Reference McAdamMcA80, Theorem 3].

4. (d) If $(R,\mathfrak{m},k)$ is local and $a\in I$ has the property that $\nu(a)=\nu(I)$ for every $\nu\in\mathcal{R}_I$ , then $\mathcal{R}_I$ is the set of all valuation domains of the associated primes of the principal ideal generated by a in the affine chart $\overline{R({I}/{a})}$ of the normalized blowup of I. Such an element a exists if R has infinite residue fields [Reference Swanson and HunekeSH06, Proposition 10.2.5].

5. (e) If $(R,\mathfrak{m},k)$ is local with infinite residue field and $I\subseteq R$ is $\mathfrak{m}$ -primary, then $f\in I$ is part of a minimal reduction of I if and only if $\nu(f)=\nu(I)$ for all $\nu\in\mathcal{R}_I$ [Reference SallySal89, pp. 437–438].

2.3 Gaussian extensions of valuations

Suppose that X is a variable and consider $R\to R[X]$ . The fraction field of R[X] is K(X). If $\nu$ is a K-valuation then $\nu$ extends to a K(X)-valuation $\nu'$ via the Gaussian extension of $\nu$ ; if $f\in K[x]$ , $f=a_0+a_1x +\cdots +a_nx^n$ with $a_i\in K$ , then $\nu'(f)=\min\{\nu(a_0),\nu(a_1),\ldots,\nu(a_n)\}$ ; see [Reference Swanson and HunekeSH06, Remark 6.1.3].

For each divisorial valuation $\nu$ of R, let $\nu'$ be the Gaussian extension of $\nu$ to R’. Then the collection of valuations $\{\nu'\mid \nu\in \mathcal{R}_I\}$ will form the Rees valuations of I’ and has the described properties of the lemma. Moreover, if $\nu,\nu', Q_\nu,$ and $Q_{\nu'}$ are as in the statement of (g), then $\overline{R'[I'T,T^{-1}]}\cong \overline{R[I T,T^{-1}]}[X]$ and hence $Q_{\nu'} = Q_{\nu}\overline{R'[I'T,T^{-1}]}$ .

2.4 Multiplicity and valuations

Let R be an excellent reduced ring. If M is a finite-length R-module, then $\mathcal{L}(M)$ is the length of R. If $(R,\mathfrak{m},k)$ is local, M a nonzero and finitely generated R-module of Krull dimension d, and $I\subseteq R$ an $\mathfrak{m}$ -primary ideal, then the (Hilbert–Samuel) multiplicity of M with respect to I is $e_I(M):=\lim_{n\to\infty}{d!\mathcal{L}(M/I^nM)}/{n^{d}}$ . If $(R,\mathfrak{m},k)$ is local, then e(M) is the Hilbert–Samuel multiplicity of M with respect to the maximal ideal. If R is $Z-graded, I\subseteq R$ a homogeneous ideal such that $R/I$ is of finite length and an Artin local ring in degree 0, and $M=\oplus_{n\in\mathbb{Z}} M_n$ a finitely generated graded R-module of dimension d, then the multiplicity of M with respect to I is $e_I(M)=\lim_{n\to \infty}{(d-1)!\ell(M_n/IM\cap M_n)}/{n^{d-1}}$ . In particular, if $(R,\mathfrak{m},k)$ is local and $I\subseteq R$ an $\mathfrak{m}$ -primary ideal, then

\[e_I(R)=e_{(IT,T^{-1})}\bigg(\frac{R[I T,T^{-1}]}{T^{-1}R[IT,T^{-1}]}\bigg).\]

If R is excellent and equidimensional, then Hilbert–Samuel multiplicity defines an upper-semi-continuous function $\mathrm{Spec}(R)\to \mathbb{N}$ by $\mathfrak{p}\mapsto e(R_{\mathfrak{p}})$ [Reference BennettBen70, Theorem 4]. We implicitly use this result throughout this article when we assert either of the following consequences of Nagata’s criteria for openness [Reference MatsumuraMat89, Theorem 24.2] and quasi-compactness of $\mathrm{Spec}(R)$ .

• – If $\mathfrak{p}\subseteq \mathfrak{q}$ are prime ideals, then $e(R_{\mathfrak{p}})\leqslant e(R_{\mathfrak{q}})$ .

• – There exists an upper bound e for the Hilbert–Samuel multiplicity of each localization of R at a prime ideal, i.e. if $\mathfrak{p}\in\mathrm{Spec}(R)$ , then $e(R_{\mathfrak{p}})\leqslant e$ .

The following theorem is Rees’ Order Ideal Theorem, which plays a crucial role in the comparison of divisorial valuations with differing centers in this article. We augment the Order Ideal Theorem statement with insights not explicitly stated in Rees’ original statements. Instead, the additional insights can be derived from Rees’ proof. We provide a streamlined and somewhat novel proof of Rees’ result. Doing so eliminates extensive terminology translation and justifications that exceed the following presentation, resulting in a more concise treatment of the necessary materials.

Theorem 2.8 [Reference ReesRee61, Rees’ Order Ideal Theorem]. Let $(R,\mathfrak{m},k)$ be an equidimensional local ring and of Krull dimension d. Suppose that R is analytically reduced and I an $\mathfrak{m}$ -primary ideal. For each Rees valuation $\nu\in\mathcal{R}_I$ , let $Q_\nu$ be the corresponding exceptional prime of the normalized extended Rees algebra $\overline{R[IT,T^{-1}]}$ . If f is an element of $\mathfrak{m}$ avoiding all minimal primes of R, then

\[e_{{(I,f)}/{(f)}}\bigg(\frac{R}{fR}\bigg) = \sum_{\nu \in \mathcal{R}_I} \nu(f)e\bigg(\frac{\overline{R[IT,T^{-1}]}}{Q_\nu}\bigg).\]

Proof. Let $\mathfrak{a}= \bigcap_{\nu \in \mathcal{R}_I}Q_\nu^{(\nu(f))}\subseteq \overline{R[IT,T^{-1}]}$ . By the associativity formula for multiplicity [Reference MatsumuraMat89, Theorem 14.7],

\[ e\bigg(\frac{\overline{R[IT,T^{-1}]}}{\mathfrak{a}}\bigg) = \sum_{\nu \in \mathcal{R}_I} \nu(f)e\bigg(\frac{\overline{R[IT,T^{-1}]}}{Q_\nu}\bigg). \]

The degree-n piece of $\mathfrak{a}$ , denoted by $\mathfrak{a}_n$ , is

\begin{align*} \mathfrak{a}_n &= (T^{-n}\mathfrak{a} )\cap R \\ &= \bigg(\bigcap_{\nu\in\mathcal{R}_I}Q_\nu^{(\nu(I)n + \nu(f))}\bigg)\cap R \\ & = \{x\in R \mid \nu(x)\geqslant n\nu(I) + \nu(f), \forall \nu\in\mathcal{R}_I\}. \end{align*}

By Rees’ valuation criteria for containment in integral closure, $\bigcap_{\nu \mathcal{R}_I}I_{\nu\geqslant n\nu(I)}=\overline{I^n}$ . Therefore, $g\in \overline{I^n}$ if and only if, for all $\nu\in\mathcal{R}_I$ , $\nu(gf)=\nu(g)+\nu(f)\geqslant n\nu(I)+\nu(f)$ . Hence, $\overline{I^n} = (\mathfrak{a}_n:_Rf)$ .

If $h\geqslant \nu_i(f)$ for all $1\leqslant i\leqslant t$ , then $\overline{I^{n+h}}\subseteq \mathfrak{a}_n \subseteq \overline{I^n}$ for all n, i.e. the chains of ideals $\{\mathfrak{a}_n\}$ , $\{\overline{I^n}\}$ are cofinal. We are assuming R is analytically reduced, therefore, $\{\mathfrak{a}_n\}$ is also cofinal with $\{I^n\}$ . In conclusion,

\begin{align*} e\bigg(\frac{\overline{R[IT,T^{-1}]}}{\mathfrak{a}}\bigg) &= \lim_{n\to \infty}\frac{(d-1)!}{n^{d-1}}\mathcal{L}\bigg(\frac{\overline{I^n}}{\mathfrak{a}_n}\bigg) \\ & = \lim_{n\to \infty}\frac{(d-1)!}{n^{d-1}}\mathcal{L}\bigg(\frac{(\mathfrak{a}_n:f)}{\mathfrak{a}_n}\bigg) \\ &= \lim_{n\to \infty}\frac{(d-1)!}{n^{d-1}}\bigg(\mathcal{L}\bigg(\frac{R}{\mathfrak{a}_n}\bigg)-\mathcal{L}\bigg(\frac{R}{(\mathfrak{a}_n:f)}\bigg)\bigg)\\ &= \lim_{n\to \infty}\frac{(d-1)!}{n^{d-1}}\mathcal{L}\bigg(\frac{R}{(\mathfrak{a}_n,f)}\bigg) \\ &= \lim_{n\to \infty}\frac{(d-1)!}{n^{d-1}}\mathcal{L}\bigg(\frac{R}{(I^n,f)}\bigg) \\ &= e_{{(I,f)}/{(f)}}\bigg(\frac{R}{fR}\bigg).\\[-36pt] \end{align*}

Corollary 2.9 (Corollary of Rees’ Order Ideal Theorem). Let $(R,\mathfrak{m},k)$ be an equidimensional local ring and of Krull dimension d. Suppose that R is analytically reduced and I an $\mathfrak{m}$ -primary ideal. For each Rees valuation $\nu\in\mathcal{R}_I$ , let $Q_\nu$ be the corresponding exceptional prime of the normalized extended Rees algebra $\overline{R[IT,T^{-1}]}$ . Then

\[e_I(R) = \sum_{\nu\in\mathcal{R}_I} \nu(I)e\bigg(\frac{\overline{R[IT,T^{-1}]}}{Q_\nu}\bigg).\]

In particular, if $\nu\in\mathcal{R}_I$ , then $\nu(I)\leqslant e_I(R)$ .

Theorem 2.8 is a generalization of the observation that, if $(R,\mathfrak{m},k)$ is a regular local ring, then for all $0\not=f\in \mathfrak{m}$ , $e(R/fR)=\mathrm{ord}_{\mathfrak{m}}(f)$ . Indeed, if R is regular then the associated graded ring $R[\mathfrak{m} T,T^{-1}]/T^{-1}R[\mathfrak{m} T,T^{-1}]$ is a polynomial ring over k in $\dim(R)$ variables. It follows that $R[\mathfrak{m} T,T^{-1}]$ is a normal domain and $T^{-1}$ is a prime element. Therefore, the collection of Rees valuations of $\mathfrak{m}$ is the 1-element set $\mathcal{R}_{\mathfrak{m}}=\{\omega\}$ and for all $0\not = f\in \mathfrak{m}$ , $\omega(f)=\mathrm{ord}_{\mathfrak{m}}(f)$ . By Theorem 2.8, $e(R/fR)=\mathrm{ord}_{\mathfrak{m}}(f)e({R[\mathfrak{m} T,T^{-1}]}/{T^{-1}R[\mathfrak{m} T^{-1}]})=\mathrm{ord}_{\mathfrak{m}}(f)$ .

Regular local rings are not the only class of local rings whose maximal ideal admits a single Rees valuation determined by $\mathfrak{m}$ -adic order. The following proposition, likely known by experts, points out that the localization of a standard graded normal domain, at the unique homogeneous maximal ideal, produces a local ring $(R,\mathfrak{m},k)$ whose maximal ideal admits a single Rees valuation that agrees with $\mathfrak{m}$ -adic order.

2.5 Rees’ Order Ideal Theorem and the Izumi–Rees Theorem

Let R be an excellent normal domain, $\mathfrak{p}\in\mathrm{Spec}(R)$ , and $0\not=f\in\mathfrak{p}$ . Intersection properties of exceptional components of normalized blowups, described by the below-stated Izumi–Rees Theorem, will be utilized in parallel with semi-continuity of multiplicity, the Rees Order Ideal Theorem, and the Uniform Brainçon–Skoda Theorem in §3 to compare the values of $e(R_{\mathfrak{p}}/fR_{\mathfrak{p}})$ and $\mathrm{ord}_{\mathfrak{p}}(f)$ .

Theorem 2.11 [Reference IzumiIzu85, Reference ReesRee89, Reference Hübl and SwansonHS01, Izumi–Rees Theorem]. Let R be an excellent Noetherian normal domain and $\mathfrak{p}\subseteq R$ a prime ideal. If $\mathcal{R}=\{\nu_1,\nu_2,\ldots,\nu_t\}$ are divisorial valuations of R centered on $\mathfrak{p}$ , e.g. $\mathcal{R}$ is the collection of Rees valuations of the maximal ideal of $R_{\mathfrak{p}}$ , then there is a constant E, depending on the collection of divisorial valuations $\mathcal{R}$ , such that for all $f\in R$ , for all $1\leqslant i,j,\leqslant t$ ,

\[ \nu_i(f)\leqslant E\nu_j(f). \]

The constant E is an Izumi–Rees bound of the collection of valuations $\mathcal{R}$ .

Proof. Suppose that C is a constant such that, if $0\not=f\in\mathfrak{m}$ , then $e_{\mathfrak{m}}(R/fR)\leqslant C\mathrm{ord}_{\mathfrak{m}}(f)$ and let $\nu_1,\nu_2\in\mathcal{R}_{\mathfrak{m}}$ be Rees valuations of the maximal ideal of R. There exists $t\geqslant 1$ such that

\[ t\nu_2(\mathfrak{m})\leqslant \nu_2(f)\lt (t+1)\nu_2(\mathfrak{m}). \]

Then $f\not\in \overline{\mathfrak{m}^{t+1}}$ and hence $\mathrm{ord}_{\mathfrak{m}}(f)\leqslant t$ . By Rees’ Order Ideal Theorem and by assumption,

\[ \nu_1(f)+\nu_2(f)\leqslant e_{\mathfrak{m}}(R/fR)\leqslant Ct\leqslant \frac{C\nu_2(f)}{\nu_2(\mathfrak{m})}. \]

In particular,

\[ \nu_1(f)\leqslant (C-1)\nu_2(f). \]

Conversely, suppose $E\in\mathbb{N}$ has the property that, for all $\nu_1,\nu_2\in\mathcal{R}_{\mathfrak{m}}$ and $0\not=f\in R$ , $\nu_1(f)\leqslant E\nu_2(f)$ . There exists a Rees valuation $\omega\in \mathcal{R}_{\mathfrak{m}}$ such that

By assumption, if $\nu\in\mathcal{R}_{\mathfrak{m}}$ then $\nu(f)\leqslant E\omega(f)\leqslant 2BEe_{\mathfrak{m}}(R)\mathrm{ord}_{\mathfrak{m}}(f)$ . Therefore,

Proposition 2.13 and the Uniform Briançon–Skoda Property provide an equivalent characterization of the Uniform Izumi–Rees Property introduced in the statement of Main Theorem 2.

2.6 Equimultiplicity

In proofs to come, we require a comparison of multiplicities of the form $e(R_{\mathfrak{p}})$ and $e_{\mathfrak{q}}(R_{\mathfrak{q}}/xR_{\mathfrak{q}})$ , where $\mathfrak{p}\subseteq \mathfrak{q}\in\mathrm{Spec}(R)$ and $\mathfrak{q} R_{\mathfrak{q}} = (\mathfrak{p},x)R_{\mathfrak{q}}$ . Central to our comparisons of multiplicity is the notion of analytic spread. If $(R,\mathfrak{m},k)$ is a local ring and $I\subseteq R$ is an ideal, then the analytic spread of I is the Krull dimension of the standard graded k-algebra $\mathrm{Gr}_{I}(R)\otimes_R R/\mathfrak{m} \cong \bigoplus_{n\geqslant 0}{I^n}/{\mathfrak{m} I^{n+1}}$ and denoted by $\ell(I)$ . The following Theorem 2.15 is an application of the theory of equimultiple ideals. Theorem 2.15 and the lemma that follows provide sufficient conditions for $e(R_{\mathfrak{p}}) = e_{\mathfrak{q}}(R_{\mathfrak{q}}/xR_{\mathfrak{q}})$ whenever $\mathfrak{p}\subseteq \mathfrak{q}$ are prime ideals such that $\mathfrak{q} R_{\mathfrak{q}} = (\mathfrak{p},x)R_{\mathfrak{q}}$ for some $x\in R$ .

Consider the associated graded ring of the normalized extended Rees algebra of (I,x),

\[ \overline{\mathrm{Gr}}_{(I,x)}(R):=\frac{\overline{R[(I,x)T,T^{-1}]}}{T^{-1}\overline{R[(I,x)T,T^{-1}]}} = \bigoplus_{n\geqslant 0}\frac{\overline{(I,x)^n}}{\overline{(I,x)^{n+1}}} T^n. \]

The equality of ideals $(\overline{(I,x)^n}:_Rx)=\overline{(I,x)^{n-1}}$ implies that the degree-1 element xT is a nonzero divisor of $\overline{\mathrm{Gr}}_{(I,x)}(R)$ . Equivalently, the degree-1 element xT of the normalized extended Rees algebra $\overline{R[(I,x)T,T^{-1}]}$ avoids all exceptional primes of $\overline{R[(I,x)T,T^{-1}]}$ . If W is the complement of the union of the exceptional primes of $\overline{R[(I,x)T,T^{-1}]}$ , then xT belongs to W and there are maps of homogeneous localizations

\[ \overline{R[(I,x)T,T^{-1}]}_{xT}\cong \overline{R\left[\frac{(I,x)}{(x)}\right]}[T,T^{-1}] \to \overline{R[(I,x)T,T^{-1}]}_W. \]

Even further, $x = xTT^{-1}$ and so $\nu(x) = \nu(xT)+\nu(T^{-1})=0+\nu((I,x))$ . Therefore, $\nu(x)=\nu((I,x))$ for all $\nu\in\mathcal{R}_{(I,x)}$ . The third claim of the lemma is an application of Theorem 2.6(d).

Theorem 2.18 (An application of equimultiplicity theory). Let R be an excellent Noetherian normal domain and $\mathfrak{p}\subsetneq \mathfrak{q}\in\mathrm{Spec}(R)$ . Assume that $\mathfrak{q} R_{\mathfrak{q}} = (\mathfrak{p}, x)R_{\mathfrak{p}}$ . The following are equivalent:

1. (a) $\mathfrak{q} R_{\mathfrak{q}} \not \in \bigcup_{n\geqslant 1}\mathrm{Ass}_R(R_{\mathfrak{q}}/\overline{\mathfrak{p}^n}R_{\mathfrak{q}})$ ;

2. (b) $\ell(\mathfrak{p} R_{\mathfrak{q}})=\mathrm{ht}(\mathfrak{p})$ ;

3. (c) $e(R_{\mathfrak{p}}) = e(R_{\mathfrak{q}})$ .

Moreover, if there exists an $n_0$ such that $\mathfrak{q}\not \in \bigcup_{n\geqslant n_0}\mathrm{Ass}_R(R_{\mathfrak{q}}/\mathfrak{p}^nR_{\mathfrak{q}})$ , then

\[ e(R_{\mathfrak{p}})= e(R_{\mathfrak{q}}) = e\bigg(\frac{R_{\mathfrak{q}}}{xR_{\mathfrak{q}}}\bigg). \]

If $\mathfrak{q} R_{\mathfrak{q}}$ is not among the elements of $\bigcup_{n\geqslant n_0}\mathrm{Ass}_R(R_{\mathfrak{q}}/\mathfrak{p}^nR_{\mathfrak{q}})$ , then $\mathfrak{q} R_{\mathfrak{q}}\not \in \bigcup_{n\geqslant 1}\mathrm{Ass}_R(R_{\mathfrak{q}}/\overline{\mathfrak{p}^n}R_{\mathfrak{q}})$ , see Theorem 2.6(b), and hence $e(R_{\mathfrak{p}})=e(R_{\mathfrak{q}})$ by the above. By Lemma 2.16 and Remark 2.17, if $\nu\in\mathcal{R}_{\mathfrak{q} R_{\mathfrak{q}}}$ then $\nu(x)=\nu(\mathfrak{q})$ . By Theorem 2.8 and Corollary 2.9, $e(R_{\mathfrak{q}}) = e(R_{\mathfrak{q}}/xR_{\mathfrak{q}})$ .

The following specific corollary of Theorem 2.18 is used in the proof of Theorem 3.3.

Corollary 2.19. Let k be a field and $R=k[x_1,\ldots,x_n]/P$ an affine normal domain, where P is a prime ideal. Endow the polynomial ring $k[x_1,\ldots,x_n]$ with the standard grading (we are not assuming R is a graded ring). Consider $f_t,\ldots,f_{t+c}\in k[x_1,\ldots,x_n]$ such that $f_i$ is either 0 or homogeneous of degree i. Assume that the image of $f:=f_t+f_{t+1}+\cdots+f_{t+c}$ in R is nonzero. Let $T^{-1}$ be a variable of degree $-1$ and $k[x_1,\ldots,x_n,T^{-1}]$ a $\mathbb{Z}$ -graded polynomial ring.

Define $f'=f_t+T^{-1}f_{t+1}+\cdots + T^{-c}f_{t+c}$ in the $\mathbb{Z}$ -graded polynomial ring $k[x_1,\ldots,x_n,T^{-1}]$ . By changing the expansion of f’ around $T^{-1}=0$ to an expansion around $T^{-1}=1$ in the polynomial ring $k[x_1,\ldots,x_n,T^{-1}]$ , we obtain polynomial functions $g_t,g_{t+1},\ldots,g_{t+c}\in k[x_1,\ldots,x_n]$ such that

\[ f' = (f_t+ f_{t+1}+\cdots+f_{t+c}) +(T^{-1}-1)g_{t}+\cdots + (T^{-1}-1)^cg_{t+c}. \]

Abuse notation and let $f_i, g_i$ denote the images of $f_i,g_i$ in the quotient ring R for each $t\leqslant i\leqslant t+c$ . For any Rees valuation $\nu\in\mathcal{R}_{\mathfrak{m}}$ of the maximal ideal $\mathfrak{m}=(x_1,\ldots,x_n)$ of R,

\begin{align*} \min\{\nu(f_t(\underline{x})+f_{t+1}(\underline{x})+\cdots + f_{t+c}(\underline{x})), \nu(g_t(\underline{x})),\ldots,\nu(g_{t+c}(\underline{x}))\} &= \nu(f_t(\underline{x})+f_{t+1}(\underline{x})+\cdots \\&\quad + f_{t+c}(\underline{x})). \end{align*}

Let $f'(\underline{y})$ be the homogeneous degree-t element of $k[y_1,\ldots,y_n,T^{-1}]$ obtained by substituting $y_i$ for $x_i$ . For each $n\in\mathbb{N}$ , the ideal $((\mathfrak{m} T)^n,f')\subseteq R[\mathfrak{m} T,T^{-1}]$ is homogeneous and admits a homogeneous primary decomposition. Therefore, the non-homogeneous element $T^{-1}-1$ avoids such components. By Theorem 2.18,

\[ e\bigg(\bigg(\frac{R[\mathfrak{m} T,T^{-1}]}{(f'(\underline{y}))}\bigg)_{\!\mathfrak{m} T}\bigg)=e\bigg(\bigg(\frac{R[\mathfrak{m} T,T^{-1}]}{(f'(\underline{y}),T^{-1}-1)}\bigg)_{\!(\mathfrak{m} T,T^{-1}-1)}\bigg). \]

Observe that

\[ \frac{R[\mathfrak{m} T,T^{-1}]}{(f'(\underline{y}),T^{-1}-1)}\cong \frac{R}{(f_t+f_{t+1}+\cdots+f_{t+c})R}. \]

Therefore,

(2.1) \begin{align} e\bigg(\bigg(\frac{R[\mathfrak{m} T,T^{-1}]}{(f'(\underline{y}))}\bigg)_{\!\mathfrak{m} T}\bigg)&=e\bigg(\bigg(\frac{R[\mathfrak{m} T,T^{-1}]}{(f'(\underline{y}),T^{-1}-1)}\bigg)_{\!(\mathfrak{m} T,T^{-1}-1)}\bigg) \end{align}

(2.2) \begin{align} &= e\bigg(\bigg(\frac{R}{(f_t+f_{t+1}+\cdots+f_{t+c})R}\bigg)_{\!\mathfrak{m}}\bigg). \end{align}

As an element of the polynomial ring $k[y_1,\ldots,y_n,T^{-1}]$ ,

\[ f'(\underline{y})=(f_t(\underline{y})+f_{t+1}(\underline{y})+\cdots + f_{t+c}(\underline{y})) + (T^{-1}-1)g_t(\underline{y})+\cdots+(T^{-1}-1)^cg_{t+c}(\underline{y}). \]

The Rees valuations of $\mathcal{R}_{(\mathfrak{m} T)_{(\mathfrak{m} T,T^{-1}-1)}}$ are the Gaussian extensions of the Rees valuations of $\mathcal{R}_{\mathfrak{m}}$ . If $\nu'\in \mathcal{R}_{(\mathfrak{m} T)_{(\mathfrak{m} T,T^{-1}-1)}}$ , we let $\nu\in \mathcal{R}_{\mathfrak{m}}$ be the corresponding Rees valuation of $\mathfrak{m}\subseteq R$ . For every $\nu'\in \mathcal{R}_{(\mathfrak{m} T)_{(\mathfrak{m} T,T^{-1}-1)}}$ ,

(2.3) \begin{align} \nu'(f') = \min\{\nu(f_t+f_{t+1}+\cdots + f_{t+c}), \nu(g_t),\ldots,\nu(g_{t+c})\}. \end{align}

By Theorem 2.8, for each $\nu'\in \mathcal{R}_{(\mathfrak{m} T)_{(\mathfrak{m} T,T^{-1}-1)}}$ there are natural numbers $d_{\nu'}\geqslant 1$ such that

\[ e\bigg(\bigg(\frac{R[\mathfrak{m} T,T^{-1}]}{(f')}\bigg)_{\!\mathfrak{m} T}\bigg) = \sum_{\nu'\in \mathcal{R}_{(\mathfrak{m} T)_{(\mathfrak{m} T,T^{-1}-1)}}}\nu'(f')d_{\nu'} \]

and

\[ e\bigg(\bigg(\frac{R[\mathfrak{m} T,T^{-1}]}{(f_t+f_{t+1}+\cdots+f_{t+c})}\bigg)_{\!\mathfrak{m} T}\bigg) = \sum_{\nu'\in \mathcal{R}_{(\mathfrak{m} T)_{(\mathfrak{m} T,T^{-1}-1)}}}\nu(f_t+f_{t+1}+\cdots+f_{t+c})d_{\nu'}. \]

By (2.1) and (2.2),

(2.4) \begin{align} \sum_{\nu'\in \mathcal{R}_{(\mathfrak{m} T)_{(\mathfrak{m} T,T^{-1}-1)}}}\nu'(f')d_{\nu'} = \sum_{\nu'\in \mathcal{R}_{(\mathfrak{m} T)_{(\mathfrak{m} T,T^{-1}-1)}}}\nu(f_t+f_{t+1}+\cdots+f_{t+c})d_{\nu'}. \end{align}

By (2.3) and (2.4),

\begin{align*} \min\{\nu(f_t(\underline{y})+f_{t+1}(\underline{y})+\cdots + f_{t+c}(\underline{y})), \nu(g_t(\underline{y})),\ldots,\nu(g_{t+c}(\underline{y}))\} = \nu(f_t+f_{t+1}+\cdots+f_{t+c}).\end{align*}

2.7 Homogenization and projective closures

Let k be a field and $R={k[x_1,x_2,\ldots,x_n]}/{P}$ an affine domain, W a multiplicative set, and $R_W=({k[x_1,x_2,\ldots,x_n]}/{P})_W$ . The homogenization of R, with respect to the presentation $R = {k[x_1,x_2,\ldots,x_n]}/{P}$ , produces a standard graded domain S that is the coordinate ring of a choice of projective closure of $\mathrm{Spec}(R_W)$ . The graded ring S can be derived as follows: Let $X_0,X_1,\ldots,X_n$ be variables, $x_i={X_i}/{X_0}$ ,

\[ ^{h}{P}=\bigg(F\in k[X_0,X_1,\ldots,X_n]\mid F\mbox{ is homogeneous and } \frac{F}{X_0^{\deg(F)}}\in P\bigg), \]

and $S = k[X_0,X_1,X_2,\ldots,X_n]/^{h}{P}$ . The homogeneous maximal ideal $\mathcal{M}=(X_0,X_1,\ldots,X_n)$ is the irrelevant homogeneous ideal of S. Given a homogeneous element $F\in S$ , we can lift F to a homogeneous element of the polynomial ring $k[X_0,X_1,\ldots,X_n]$ and define $^{a}{}{F}$ to be the image of ${F}/{X_0^{\deg(F)}}$ in R. The operation $^{a}{-}$ is well-defined as R is a domain. Given an ideal $I\subseteq R$ , $^{h}{}{I}$ is the ideal-generated homogeneous element F such that $^{a}{}{F}\in I$ . The operation $^{h}{}{-}$ is not well-defined on elements of R, only the ideals of R. The affine variety $\mathrm{Spec}(R)$ is the open subset of the affine piece $D(X_0)$ of $\mathrm{Proj}_k(S)$ .

If $R_W$ is normal, normalization commutes with localization, allowing us to adjust the presentation of R such that R is also normal. Moreover, by embedding $\overline{\mathrm{Proj}_k(S)}$ into projective space, we can refine the presentation of R further to assume that the homogenization S of R is a standard graded normal domain. Consequently, every normal domain essentially of finite type over a field can be viewed as an open subset of an arithmetically normal projective variety.

The following proposition points out a well-known relationship between an affine domain $R = k[x_1,x_2,\ldots,x_n]/P$ and the homogeneous ring $S = k[X_0,X_1,X_2,\ldots,X_n]/^{h}{}{P}$ .

Proposition 2.20. Let k be a field, $R = k[x_1,x_2,\ldots,x_n]/P$ an affine domain, $X_0,X_1,\ldots,X_n$ homogeneous variables of degree 1 such that ${X_i}/{X_0}= x_i$ , and S the standard graded domain $k[X_0,X_1,\ldots,X_n]/^{h}{}{P}$ . Abuse notation by letting $X_i$ denote the image $X_i$ in S. Then, as subsets of the fraction field of S,

\[ S\left[\frac{X_1}{X_0},\ldots,\frac{X_n}{X_0}\right] \cong R[X_0]. \]

Proof. The algebra $S[{X_1}/{X_0},\ldots,{X_n}/{X_0}]$ is an affine chart of the blowup of the homogeneous maximal ideal of S. Therefore, $\dim(S[{X_1}/{X_0},\ldots,{X_n}/{X_0}]) = \dim(R)+1$ . Moreover, $S[{X_1}/{X_0},\ldots,{X_n}/{X_0}]$ is the homomorphic image of the polynomial algebra $k[X_0,{X_1}/{X_0},\ldots, {X_n}/{X_0}]$ and the kernel $\mathfrak{p}$ of $k[X_0,{X_1}/{X_0},\ldots, {X_n}/{X_0}]\to S[{X_1}/{X_0},\ldots,{X_n}/{X_0}]$ contains the extension $^{h}{}{P}$ to $k[X_0,{X_1}/{X_0},\ldots, {X_n}/{X_0}]$ . The element $X_0$ is a nonzero divisor of $S[{X_1}/{X_0},\ldots, {X_n}/{X_0}]$ , therefore

\[ \bigg({^{h}}{P}k\left[X_0,\frac{X_1}{X_0},\ldots, \frac{X_n}{X_0}\right]:_{k[X_0,{X_1}/{X_0},\ldots, {X_n}/{X_0}]} X_0^{\infty}\bigg)\subseteq \mathfrak{p}. \]

Recall that $^{h}{}{P} = (^{h}{}{f}\mid f\in P)$ . Therefore, $Pk[X_0,{X_1}/{X_0},\ldots, {X_n}/{X_0}]\subseteq \mathfrak{p}$ . But $Pk[X_0,{X_1}/{X_0}, \ldots, {X_n}/{X_0}]$ is a prime, so $\dim({k[X_0,{X_1}/{X_0},\ldots, {X_n}/{X_0}]}/{Pk[X_0,{X_1}/{X_0},\ldots, {X_n}/{X_0}]}) =\dim(R)+1$ . Therefore, $\mathfrak{p} = Pk[X_0,{X_1}/{X_0},\ldots, {X_n}/{X_0}]$ and $S[{X_1}/{X_0},\ldots,{X_n}/{X_0}] \cong R[X_0]$ , as claimed.

3.1 Quasi-projective varieties over an algebraically closed field

Main Theorem 1 is Corollary 3.6 of the following theorem.

Theorem 3.3. Let k be an algebraically closed field and R a normal domain essentially of finite type over k. Let S be a coordinate ring of an arithmetically normal projective closure of $\mathrm{Spec}(R)$ and $\mathcal{M}$ the homogeneous maximal ideal of S. Then for all prime ideals $\mathfrak{p}\in\mathrm{Spec}(R)$ and all $0\not=f\in\mathfrak{p}$ ,

\[ e\bigg(\frac{R_{\mathfrak{p}}}{fR_{\mathfrak{p}}}\bigg)\leqslant e(S_{\mathcal{M}})\mathrm{ord}_{\mathfrak{p}}(f). \]

Proof. Suppose that $R\cong (k[x_1,\ldots,x_n]/P)_W$ , $X_0,X_1,\ldots, X_n$ are variables such that $x_i={X_i}/{X_0}$ , and $S={k[X_0,X_1,\ldots,X_n]}/{^{h}{}{P}}$ is a standard graded normal domain with homogeneous maximal ideal $\mathcal{M}=(X_0,X_1,\ldots,X_n)$ . We may assume that R is the affine normal domain $k[x_1,\ldots,x_n]/P$ . Let $\mathfrak{p} \in \mathrm{Spec}(R)$ . We abuse notation by letting $X_0,X_1,\ldots,X_n$ denote the images of the graded variables in S and $x_1,\ldots,x_n$ the images of the variables in R. By [Reference Eisenbud and HochsterEH79, Theorem], there is an open subset of maximal ideals $U\subseteq V(\mathfrak{p})\cap \mathrm{Max}(R)$ such that

\[ \mathfrak{p}^{(\mathrm{ord}_{\mathfrak{p}}(f))} = \bigcap_{\mathfrak{m} \in U} \mathfrak{m}^{\mathrm{ord}_{\mathfrak{p}}(f)} \]

and a dense open subset of maximal ideals $V\subseteq V(\mathfrak{p})\cap \mathrm{Max}(R)$ such that

\[ \mathfrak{p}^{(\mathrm{ord}_{\mathfrak{p}}(f))} = \bigcap_{\mathfrak{m} \in V} \mathfrak{m}^{\mathrm{ord}_{\mathfrak{p}}(f)}+1. \]

Therefore, there exists a maximal ideal $\mathfrak{m}\in \mathrm{Max}(R)$ such that $\mathrm{ord}_{\mathfrak{p}}(f) = \mathrm{ord}_{\mathfrak{m}}(f)$ . By semi-continuity of multiplicity, we can assume $\mathfrak{p}=\mathfrak{m}$ is a maximal ideal of $\mathrm{Spec}(R)$ . By Zariski’s Nullstellensatz, every maximal ideal of R has the form $(x_1-a_1,\ldots,x_n-a_n)$ for some $(a_1,\ldots,a_n)\in V(P)\subseteq \mathbb{A}^n_k$ . After a linear change of coordinates, a process that does not change the homogeneous coordinate ring $S=k[X_0,X_1,\ldots,X_n]/^{h}{}{P}$ ,Footnote ³ we may assume $\mathfrak{m}=(x_1,\ldots,x_n)$ .

Suppose that $f\in \mathfrak{m}$ and $\mathrm{ord}_{\mathfrak{m}}(f)=t$ . There exists a lift of f in the polynomial ring $k[x_1,\ldots,x_n]=k[{X_1}/{X_0},\ldots,{X_n}/{X_0}]$ , $f=f_t+f_{t+1}+\cdots + f_{t+c}$ , such that each element $f_i$ , if a nonzero element in R, is a homogeneous polynomial of degree i with respect to the maximal ideal $(x_1,\ldots, x_n)\subseteq k[x_1,\ldots,x_n]$ . For each $t\leqslant i \leqslant t+c$ , let $F_i$ be the corresponding homogeneous polynomial of $k[X_1,\ldots,X_n]\subseteq k[X_0,X_1,\ldots,X_n]$ and $F=F_t+\cdots +F_{t+c}$ .

The ring S is a standard graded normal domain. By Proposition 2.10, the (non-homogeneous) local ring $S_{\mathcal{M}}$ enjoys the property that the collection of Rees valuations of the maximal ideal $\mathcal{M} S_{\mathcal{M}}$ is a 1-element set, $\mathcal{R}_{\mathcal{M} S_{\mathcal{M}}}=\{\omega\}$ , $\omega(g)=\max\{n\in \mathbb{N}\mid g\in \mathcal{M}^nS_{\mathcal{M}}\}$ , and $e(S_{\mathcal{M}}/gS_{\mathcal{M}}) = e(S)\omega(g)$ for all $0\not =g\in \mathcal{M} S_{\mathcal{M}}$ .

If $0\leqslant i\leqslant t-1$ , then $F\in \mathcal{M}^i$ and represents the 0-element of $\mathcal{M}^i/\mathcal{M}^{i+1}$ . The elements F and $F_t$ represent the same element of $\mathcal{M}^t/\mathcal{M}^{t+1}$ . Note that ${F_t}/{X_0^t}=f_t$ is a nonzero element of R. Therefore, $F_t$ is a nonzero element of $S\cong \mathrm{Gr}_{\mathcal{M}}(S)$ . Hence, the image of F in $\mathcal{M}^t/\mathcal{M}^{t+1}\subseteq S$ is nonzero, $\mathrm{ord}_{\mathcal{M}}(F)=t$ , and

(3.1) \begin{equation} e\bigg(\bigg(\frac{S}{FS}\bigg)_{\!\mathcal{M}}\bigg) = e(S)t=e(S_{\mathcal{M}})\mathrm{ord}_{\mathfrak{m}}(f). \end{equation}

Note that $S/(X_1,\ldots,X_n)\cong k[X_0]$ ; in particular, the ideal $(X_1,\ldots,X_n)$ is a prime ideal. By semi-continuity of multiplicity,

(3.2) \begin{equation} e\bigg(\bigg(\frac{S}{FS}\bigg)_{\!(X_1,\ldots,X_n)}\bigg)\leqslant e\bigg(\bigg(\frac{S}{FS}\bigg)_{\!\mathcal{M}}\bigg). \end{equation}

The element $X_0$ avoids the prime ideal $(X_1,\ldots,X_n)S$ . Therefore, as subsets of the fraction field of S,

\[ S\left[\frac{X_1}{X_0},\ldots,\frac{X_n}{X_0}\right]_{(\frac{X_1}{X_0},\ldots,\frac{X_n}{X_0})} = S_{(X_1,\ldots,X_n)}. \]

By Proposition 2.20, $S[{X_1}/{X_0},\ldots,{X_n}/{X_0}]= R[X_0]$ . Let

\[ F'=\frac{F}{X_{0}^t} =f_t+X_0f_t+\cdots + X_0^cf_{t+c}. \]

By the above,

(3.3) \begin{equation} e\bigg(\bigg(\frac{S}{FS}\bigg)_{\!(X_1,\ldots,X_n)}\bigg) = e\bigg(\bigg(\frac{R[X_0]}{F'R[X_0]}\bigg)_{\!(x_1,\ldots,x_n)R[X_0]}\bigg). \end{equation}

Claim 3.4. Let $A={R[X_0]}/{F'R[X_0]}$ . Then

\[ e((A)_{\mathfrak{m} A})= e\bigg(\bigg(\frac{A}{(X_0-1)A}\bigg)_{\!(\mathfrak{m},X_0-1)A}\bigg) = e\bigg(\bigg(\frac{R}{fR}\bigg)_{\!\mathfrak{m}}\bigg). \]

Proof of Claim. Observe that

\[ \frac{A}{(X_0-1)A} \cong \frac{R}{(f_t+f_{t+1}+\cdots + f_{t+c})R} \]

and consequently

(3.4) \begin{equation} e\bigg(\bigg(\frac{A}{(X_0-1)A}\bigg)_{\!(x_1,\ldots,x_n,X_0-1)}\bigg) = e\bigg(\bigg(\frac{R}{(f_t+f_{t+1}+\cdots + f_{t+c})R}\bigg)_{\!\mathfrak{m}}\bigg) = e\bigg(\bigg(\frac{R}{fR}\bigg)_{\!\mathfrak{m}}\bigg). \end{equation}

It remains to show $e((A)_{\mathfrak{m} A})= e(({A}/{(X_0-1)A})_{(\mathfrak{m},X_0-1)A})$ .

Recall that $F'=f_t+X_0f_{t+1}+\cdots + X_0^cf_{t+c}$ . We change the expansion of F’ around $X_0=0$ to an expansion around $X_0=1$ . Then, if $g_t,\ldots,g_{t+c}$ are the polynomials described by Corollary 2.19,

\[ F' = (f_t+\cdots + f_{t+c}) + (X_0-1)g_{t+1}+\cdots+(X_0-1)^cg_{t+c}. \]

Hence if $\nu''\in \mathcal{R}_{\mathfrak{m} R[X_0]}$ is a Gaussian extension of a valuation $\nu\in \mathcal{R}_{\mathfrak{m}}$ to $R[X_0]=R[X_0-1]$ then

\[ \nu''(F') = \min\{\nu(f_t+f_{t+1}+\cdots + f_{t+c}), \nu(g_t),\ldots,\nu(g_{t+c})\}. \]

By Corollary 2.19,

\[ \nu''(F') =\min\{\nu(f_t+f_{t+1}+\cdots + f_{t+c}), \nu(g_t),\ldots,\nu(g_{t+c})\} = \nu(f_t+f_{t+1}+\cdots + f_{t+c}). \]

Theorem 2.8 and Corollary 2.19 imply

This completes the proof of the claim.

We continue with the proof of the theorem and let A be as in Claim 3.4, such that

\[e\bigg(\bigg(\frac{A}{(X_0-1)A}\bigg)_{\!(\mathfrak{m},X_0-1)A}v\bigg)= e\bigg(\bigg(\frac{R}{fR}\bigg)_{\!\mathfrak{m}}\bigg)\]

and then, by (3.3),

\[e\bigg(\bigg(\frac{R}{fR}\bigg)_{\!\mathfrak{m}}\bigg) =e\bigg(\bigg(\frac{S}{FS}\bigg)_{\!(X_1,\ldots,X_n)}\bigg).\]

By (3.2),

\[e\bigg(\bigg(\frac{R}{fR}\bigg)_{\!\mathfrak{m}}\bigg)\leqslant e\bigg(\bigg(\frac{S}{FS}\bigg)_{\!\mathcal{M}}\bigg).\]

Finally, by (3.1),

\begin{align*}e\bigg(\bigg(\frac{R}{fR}\bigg)_{\!\mathfrak{m}}\bigg) \leqslant e(S_{\mathcal{M}})\mathrm{ord}_{\mathfrak{m}}(f).\\[-36pt]\end{align*}

Corollary 3.5 (Main Theorem 1). Let k be an algebraically closed field and R a normal domain essentially of finite type over k. Let S be a coordinate ring of an arithmetically normal projective closure of $\mathrm{Spec}(R)$ , $\mathcal{M}$ the homogeneous maximal ideal of S, and e(S) the multiplicity of S with respect to $\mathcal{M}$ . Then, for all prime ideals $\mathfrak{p}\subseteq \mathfrak{q}\in\mathrm{Spec}(R)$ and for all $n\geqslant 1$ ,

\[ \mathfrak{p}^{(2e(S)n)}\subseteq \mathfrak{p}^{(e(S)n+1)}\subseteq \mathfrak{q}^{(n)}. \]

Proof. It is clear that $\mathfrak{p}^{(2e(S)n)}\subseteq \mathfrak{p}^{(e(S)n+1)}$ . If $f\in \mathfrak{p}^{(e(S)n+1)}$ , then by semi-continuity of multiplicity

\[ e(S)n+1\leqslant e\bigg(\frac{R_{\mathfrak{p}}}{fR_{\mathfrak{p}}}\bigg)\leqslant e\bigg(\frac{R_{\mathfrak{q}}}{fR_{\mathfrak{q}}}\bigg). \]

By Theorem 3.3, $f\in\mathfrak{q}^{(n)}$ .

Corollary 3.6. Let k be an algebraically closed field and R a normal domain essentially of finite type over k. Let S be a coordinate ring of an arithmetically normal projective closure of $\mathrm{Spec}(R)$ , $\mathcal{M}$ the homogeneous maximal ideal of S, and e(S) the multiplicity of S with respect to $\mathcal{M}$ . Then, for every $\mathfrak{p}\in\mathrm{Spec}(R)$ , for every pair of distinct Rees valuations $\nu_1,\nu_2\in \mathcal{R}_{\mathfrak{p} R_{\mathfrak{p}}}$ of the maximal ideal of $R_{\mathfrak{p}}$ , and $0\not = f\in\mathfrak{p}$ ,

\[ \nu_1(f)\leqslant (e(S)-1)\nu_2(f). \]

3.2 Reduction to an algebraically closed field

The valuation criteria of the Uniform Izumi–Rees Property (see Definition 3.1), Lemmas 3.8 and 3.10, are utilized to reduce the Uniform Izumi–Rees Property for an algebra essentially of finite type over a field to the Uniform Izumi–Rees Property for an algebra essentially of finite type over an algebraically closed field.

If k is a field and $\overline{k}$ an algebraic closure of k, then the extension $k \to \overline{k}$ can be factored as

\[k \longrightarrow k_{\mathrm{sep}} \longrightarrow \overline{k},\]

where $k_{\mathrm{sep}}$ denotes the separable closure of k, and the extension $k_{\mathrm{sep}} \to \overline{k}$ is purely inseparable. To establish the Uniform Izumi Property for a normal domain essentially of finite type over k, we examine the behavior of Rees valuations and the Uniform Izumi–Rees Property (Definition 3.1) separately along separable and inseparable field extensions. We remark that if $k\to k'$ is an algebraic extension, R an algebra essentially of finite type over k, and $R' = R\otimes_kk'$ , then $R\to R'$ is a faithfully flat ring extension. By lying over, the map of spectrum $\mathrm{Spec}(R')\to \mathrm{Spec}(R)$ is surjective, and for all $\mathfrak{p}\in \mathrm{Spec}(R)$ , the minimal primes of the extended ideal $\mathfrak{p} R'$ have the same height as $\mathfrak{p}$ , see [Reference Swanson and HunekeSH06, 2.2] for necessary details.

Now let $k \to k'$ be a separable field extension. If R is a normal domain essentially of finite type over k, then $R \otimes_k k'$ is not necessarily a domain, but instead decomposes as a product of normal domains (see [Reference Swanson and HunekeSH06, Corollary 2.1.13 and Theorem 19.4.3]). However, the study of the Uniform Izumi–Rees Property naturally extends to products of normal domains: Since the Uniform Izumi–Rees Property is local, a product of normal domains satisfies the Uniform Izumi–Rees Property if and only if each factor does.

Consider the map of extended Rees algebras

\[ \varphi: R[\mathfrak{p} T, T^{-1}]\to R'[\mathfrak{p}' T, T^{-1}] \]

and the induced map of normalizations

\[ \overline{\varphi}: \overline{R[\mathfrak{p} T, T^{-1}]}\to \overline{R'[\mathfrak{p}' T, T^{-1}]}. \]

By Remark 3.7,

\[ \overline{R'[\mathfrak{p}' T, T^{-1}]}\cong \overline{R[\mathfrak{p} T, T^{-1}]}\otimes_k k'. \]

Proof of Claim. In general, if S is a k-algebra, $S'=S\otimes_kk'$ , and $\mathfrak{p}\subseteq S$ a prime ideal, then $S/\mathfrak{p}\to S'/\mathfrak{p} S'$ is flat and algebraic, implying that $S'/\mathfrak{p} S'$ injects into $\overline{S/\mathfrak{p}}\otimes_k k'$ , the latter of which is a product of normal domains by [Reference Swanson and HunekeSH06, Theorem 19.4.3] and [Reference Swanson and HunekeSH06, Corollary 2.1.13]. In particular, $\mathfrak{p} S'$ is a reduced ideal whose components have a common height. In the context of extended Rees algebras and Rees valuations, for each Rees valuation $\nu\in \mathcal{R}_{\mathfrak{p}}$ , there are unique prime components of $Q_{\nu'_1},\ldots, Q_{\nu'_t}$ of $T^{-1}\overline{R'[\mathfrak{p}' T, T^{-1}]}$ such that

\[ Q_\nu \overline{R'[\mathfrak{p}' T, T^{-1}]} = Q_{\nu_1'}\cap \cdots \cap Q_{\nu_t'}. \]

Hence, there is a partition of the Rees valuations $\mathfrak{p}'$ , indexed by the Rees valuations of $\mathfrak{p}$ ,

\[ \mathcal{R}_{\mathfrak{p}'} = \bigcup_{\nu \in \mathcal{R}_{\mathfrak{p}}}\Lambda_\nu, \]

with the defining property that if $\nu'\in \mathcal{R}_{\mathfrak{p}'}$ , then $\nu' \in \Lambda_\nu$ if and only if $\nu'(Q_\nu)\geqslant 1$ , if and only if $\nu'(Q_\nu)= 1$ , as the extension of $Q_{\nu}$ to $\overline{R'}[\mathfrak{p}'T,T^{-1}]$ is reduced. This completes part (a) of the claim.

Now suppose that $\nu \in \mathcal{R}_{\mathfrak{p}}$ is a Rees valuation of $\mathfrak{p}$ . Let $\Lambda_\nu=\{\omega_1,\ldots,\omega_t\}$ be the corresponding Rees valuations of $\mathfrak{p}'$ , let W be the complement of the union of the prime ideals $Q_{\omega_i}$ in $\overline{R'[\mathfrak{p}'T, T^{-1}]}$ , and consider the map of localizations

\[ \overline{\varphi}_{Q_{\nu}}: \overline{R[\mathfrak{p} T,T^{-1}]}_{Q_\nu}\to \overline{R'[\mathfrak{p}'T, T^{-1}]}_{W}. \]

Let $f\in R$ and consider the principal ideal

\[ f\overline{R[\mathfrak{p} T,T^{-1}]}_{Q_\nu} = (Q_\nu^{\nu(f)})_{Q_\nu}, \]

and its expansion under $\overline{\varphi}_{Q_\nu}$ ,

\[ f\overline{R'[\mathfrak{p}' T,T^{-1}]}_{W} = \bigg(\bigcap_{i=1}^tQ_{\omega_i}^{\omega_i(f)}\bigg)_{\!W}. \]

Therefore, $\omega_i(f) = \nu(f)\omega_i(Q_\nu) = \nu(f)$ for each $1\leqslant i \leqslant t$ , as claimed.

Only part (3.9) of the claim remains to be proven. Assume that

\[ \mathfrak{p}' = \mathfrak{p} R' = \mathfrak{q}_1 \cap \cdots \cap \mathfrak{q}_t \]

is the minimal decomposition of $\mathfrak{p}'$ as a reduced ideal of R’. Since $R \to R'$ is algebraic, the primes $\{\mathfrak{q}_1, \mathfrak{q}_2, \ldots, \mathfrak{q}_t\}$ are precisely the primes of $\mathrm{Spec}(R')$ lying over $\mathfrak{p} \in \mathrm{Spec}(R)$ . The claim in (3.9) is that for each $1 \leqslant i \leqslant t$ , there exist Rees valuations $\omega_1, \omega_2 \in \mathcal{R}_{\mathfrak{q}_i}$ such that

\[ Q_{\omega_1} \cap \overline{R[\mathfrak{p} T, T^{-1}]} = Q_{\nu_1} \quad \text{and} \quad Q_{\omega_2} \cap \overline{R[\mathfrak{p} T, T^{-1}]} = Q_{\nu_2}. \]

The minimal prime components of $T^{-1}\overline{R[\mathfrak{p} T, T^{-1}]}$ that contract to $\mathfrak{p}$ in R are in bijection with the minimal prime components of

\[ \frac{\overline{R_{\mathfrak{p}}[\mathfrak{p} R_{\mathfrak{p}} T, T^{-1}]}}{T^{-1}\overline{R_{\mathfrak{p}}[\mathfrak{p} R_{\mathfrak{p}} T, T^{-1}]}}. \]

Similarly, the minimal prime components of $T^{-1}\overline{R'[\mathfrak{p} R'T, T^{-1}]}$ lying over a minimal prime of $\mathfrak{p}' = \mathfrak{p} R'$ correspond to the minimal prime components of

\[ \frac{\overline{R'_{\mathfrak{p}}[\mathfrak{p} R'_{\mathfrak{p}} T, T^{-1}]}}{T^{-1}\overline{R'_{\mathfrak{p}}[\mathfrak{p} R'_{\mathfrak{p}} T, T^{-1}]}}. \]

Now, since $\mathfrak{p}' = \mathfrak{p} R' = \mathfrak{q}_1 \cap \cdots \cap \mathfrak{q}_t$ is the minimal decomposition of $\mathfrak{p}'$ as a reduced ideal of R’, and $R \to R'$ is algebraic, we have

By Remark 3.7, it follows that

Since $k \to k'$ is separable and algebraic, we deduce that

is algebraic. Equivalently, for each $1 \leqslant i \leqslant t$ ,

\[ \frac{\overline{R_{\mathfrak{p}}[\mathfrak{p} R_{\mathfrak{p}} T, T^{-1}]}}{T^{-1}\overline{R_{\mathfrak{p}}[\mathfrak{p} R_{\mathfrak{p}} T, T^{-1}]}} \;\;\longrightarrow\;\; \frac{\overline{R'_{\mathfrak{q}_i}[\mathfrak{q}_i R'_{\mathfrak{q}_i} T, T^{-1}]}}{T^{-1}\overline{R'_{\mathfrak{q}_i}[\mathfrak{q}_i R'_{\mathfrak{q}_i} T, T^{-1}]}} \]

is algebraic.

Therefore, the minimal primes $Q_{\nu_1}$ and $Q_{\nu_2}$ of $T^{-1}\overline{R_{\mathfrak{p}}[\mathfrak{p} R_{\mathfrak{p}} T, T^{-1}]}$ are contractions of minimal primes of $T^{-1}\overline{R'_{\mathfrak{q}_i}[\mathfrak{q}_i R'_{\mathfrak{q}_i} T, T^{-1}]}$ . These, in turn, correspond to minimal primes of $T^{-1}\overline{R'[\mathfrak{q}_i R'T, T^{-1}]}$ that contract to $\mathfrak{q}_i$ in R’, completing the proof of the claim.

We continue with the proof of the lemma. Let $\nu_1,\nu_2\in \mathcal{R}_{\mathfrak{p}}$ be Rees valuations of $\mathfrak{p}$ centered on $\mathfrak{p}$ and choose Rees valuations $\nu_1',\nu_2'$ of $\mathfrak{p}'$ belonging to $\Lambda_{\nu_1}$ and $\Lambda_{\nu_2}$ , respectively. By (3.9) of Claim 3.9, we can choose $\nu_1'$ and $\nu_2'$ to be centered on a common minimal prime component of $\mathfrak{p}'$ . If E is a uniform Izumi–Rees bound of R’, then by (3.9) of Claim 3.9,

\[ \nu_1(f) = \nu_1'(f) \leqslant E\nu_2'(f) = E\nu_2(f). \]

Therefore, the Uniform Izumi–Rees Property of R’ descends to R with uniform Izumi–Rees bound E.

Let k be a field of prime characteristic $p\gt0$ and $k\to k'$ an algebraic and purely inseparable field extension of k, i.e. for each $\alpha\in k'$ , there exists $e\in\mathbb{N}$ such that $\alpha^{p^e}\in k$ . Let R be an algebra essentially of finite type over k, $R' = R\otimes_k k'$ . If $\mathfrak{p}\in \mathrm{Spec}(R)$ , then the extended ideal $\mathfrak{p} R'$ is easily checked to be primary to $\sqrt{\mathfrak{p} R'}$ . Therefore, $\mathfrak{p} R'$ omits a unique minimal prime, namely $\sqrt{\mathfrak{p} R'}$ , and the induced map of spectrum $\mathrm{Spec}(R')\to \mathrm{Spec}(R)$ is a bijection.

Unlike a separable base change, if R is a normal domain essentially of finite type over k, and $k\to k'$ purely inseparable, the algebra $R\otimes_k k'$ no longer has to be integrally closed in its total ring of fractions as $R\otimes_k k'$ can be non-reduced. The following lemma reduces the study of the Uniform Izumi–Rees Property of R to a normal domain obtained from $R\otimes_k k'$ , namely the normalization of the integral domain $(R\otimes_k k')/\sqrt{0}$ .

Lemma 3.10. Let R be a normal domain essentially of finite type over a field k of prime characteristic $p\gt0$ . Let $k\to k'$ be a purely inseparable algebraic extension of k, $R'=(R\otimes_k k')/\sqrt{0}$ , and $\overline{R'}$ the normalization of R’. If $I\subseteq R$ is an ideal and $I'=IR'$ , then there is a bijection of Rees valuations $\psi:\mathcal{R}_I\to \mathcal{R}_{I'}$ . Moreover, for all $\nu\in\mathcal{R}_I$ , if $\nu'=\psi(\nu)$ , $\mathfrak{p}_\nu$ the center of $\nu$ in R, then $\sqrt{\mathfrak{p}_\nu R'}=\mathfrak{p}_{\nu'}$ is the center of $\nu'$ in R’, and for all $f\in R$ ,

\[ \frac{\nu(f)}{\nu(I)}=\frac{\nu'(f)}{\nu'(I')}. \]

Let e and e’ be respective uniform upper bounds of the Hilbert–Samuel multiplicities of the localizations of R and $\overline{R'}$ at their prime ideals. If $\overline{R'}$ has the Uniform Izumi–Rees Property with uniform Izumi–Rees bound E, then R has the Uniform Izumi–Rees Property with uniform Izumi–Rees bound Eee’.

Proof. Let K’ be the field of fractions of R’. Then $R\to R'$ and $K\to K'$ are purely inseparable. The extension $R \to \overline{R'}$ is purely inseparable. Indeed, if $f\in K'$ and satisfies a polynomial equation

\[ f^{t}+a_1f^{t-1}+\cdots+a_{t-1}f+a_t=0 \]

with each coefficient $a_i$ belonging to R’, then we can choose $e\gg 0$ such that $f^{p^e}\in K$ and $a_i^{p^e}\in R$ . If we raise the above equation of integral dependence to $p^e$ , then

\[ f^{p^et}+a_1^{p^e}f^{p^e(t-1)}+\cdots+a_{t-1}^{p^e}f^{p^e}+a_t^{p^e}=0. \]

Therefore, $f^{p^e}$ belongs to the normalization of R. The ring R is assumed to be normal, therefore $f^{p^e}\in R$ .Footnote ⁴

Consider the map of extended Rees algebras

\[ \varphi: R[I T, T^{-1}]\to R'[I' T, T^{-1}] \]

and the induced map of normalizations

\[ \overline{\varphi}: \overline{R[I T, T^{-1}]}\to \overline{R'[I' T, T^{-1}]}. \]

The extension $\varphi$ is purely inseparable. Therefore, $\overline{\varphi}$ is a purely inseparable extension by the footnote. Hence, there is a bijection of the components of $T^{-1}\overline{R[I T, T^{-1}]}$ with the components of $T^{-1}\overline{R'[I' T, T^{-1}]}$ . Equivalently, there is a bijection of Rees valuations

\[ \psi: \mathcal{R}_I\to \mathcal{R}_{I'} \]

defined as follows: If $Q_\nu$ is the component of $T^{-1}\overline{R[I T, T^{-1}]}$ corresponding to $\nu$ and $Q_{\nu'}$ is the prime ideal $\sqrt{Q_\nu \overline{R'[I' T, T^{-1}]}}$ , then $\psi(\nu)=\nu'$ .

Let W be the complement of the union of the components of $T^{-1} \overline{R[I T, T^{-1}]}$ . Consider the map $\overline{\varphi}_W$ of localizations

\[ \overline{\varphi}_W:\overline{R[IT, T^{-1}]}_W\to \overline{R'[I' T, T^{-1}]}_{W}. \]

We examine the decomposition of the localized principal ideal

\[ T^{-1}\overline{R[I T, T^{-1}]}_W = \bigg(\bigcap_{\nu\in\mathcal{R}_I} Q_\nu^{\nu(I)}\bigg)_{\!W} \]

and its decomposition under $\overline{\varphi}_W$ ,

\[ T^{-1}\overline{R'[I' T, T^{-1}]} = \bigg(\bigcap_{\nu\in\mathcal{R}_{\mathfrak{p}}} Q_{\nu'}^{\nu'(I')}\bigg)_{\!W}. \]

Then the bijection of components under expansion implies

(3.5) \begin{align} \nu'(I') = \nu(I)\nu'(Q_\nu). \end{align}

Let $f\in R$ and consider the principal ideal

\[ f\overline{R[I T, T^{-1}]}_W = \bigg(\bigcap_{\nu\in\mathcal{R}_I} Q_\nu^{\nu(f)}\bigg)_{\!W}, \]

and its image under $\overline{\varphi}_W$ in $\overline{R'[I' T, T^{-1}]}_{W}$ ,

\[ f\overline{R'[I' T, T^{-1}]}_{W} = \bigg(\bigcap_{I'\in\mathcal{R}_{I'}} Q_{\nu'}^{\nu'(f)}\bigg)_{\!W}. \]

The bijection of exceptional components and (3.5) implies

\[ \nu'(f)=\nu(f)\nu'(Q_\nu)=\nu(f)\frac{\nu'(I)}{\nu(I)}. \]

Therefore,

\[ \frac{\nu(f)}{\nu(I)} = \frac{\nu'(f)}{\nu'(I')}, \]

as claimed.

Suppose that $I=\mathfrak{p}\in\mathrm{Spec}(R)$ . Let $\nu_1,\nu_2\in \mathcal{R}_{\mathfrak{p}}$ be Rees valuations of $\mathfrak{p}$ , both centered on $\mathfrak{p}$ , and $\nu_1',\nu_2'\in \mathcal{R}_{\mathfrak{p}'}$ the corresponding Rees valuations of $\mathfrak{p}'$ , both of which are necessarily centered on $\mathfrak{p}'$ . By the above and the assumption that R’ enjoys the Uniform Izumi–Rees Property with uniform Izumi–Rees bound E,

\[ \frac{\nu_1(f)}{\nu_1(\mathfrak{p})} = \frac{\nu'_1(f)}{\nu'_1(\mathfrak{p}')}\leqslant \frac{E\nu_2'(f)}{\nu_1'(\mathfrak{p}')}=\frac{E\nu_2'(\mathfrak{p}')}{\nu'_1(\mathfrak{p}')}\cdot \frac{\nu_2'(f)}{\nu_2'(\mathfrak{p}')} = \frac{E\nu_2'(\mathfrak{p}')}{\nu_1'(\mathfrak{p})}\cdot \frac{\nu_2(f)}{\nu_2(\mathfrak{p})}\leqslant E\nu_2'(\mathfrak{p}')\nu_2(f). \]

Therefore,

\[ \nu_1(f)\leqslant E\nu_1(\mathfrak{p})\nu_2'(\mathfrak{p}')\nu_2(f). \]

The values $\nu_1(\mathfrak{p})$ and $\nu_2'(\mathfrak{p}')$ are bounded from above by the Hilbert–Samuel multiplicities of $R_{\mathfrak{p}}$ and $R'_{\mathfrak{p}'}$ , respectively by Corollary 2.9, values that are bounded from above by e and e’, respectively. Therefore, R enjoys the Uniform Izumi–Rees Property with uniform Izumi–Rees bound Eee’.