Rees' theorem for filtrations, multiplicity function and reduction   criteria

Parangama Sarkar

arXiv:1704.07643·math.AC·May 11, 2021

Rees' theorem for filtrations, multiplicity function and reduction criteria

Parangama Sarkar

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TL;DR

This paper extends Rees' theorem to broader classes of filtrations in local rings, linking polynomial degrees of length and multiplicity functions to reduction criteria, with explicit degree characterizations in special cases.

Contribution

It generalizes Rees' theorem to Noetherian and non-Noetherian filtrations, providing new criteria for reductions based on polynomial degrees of length and multiplicity functions.

Findings

01

Degree of length polynomial indicates reduction status.

02

Explicit degree formulas for multiplicity functions in certain cases.

03

Characterization of reductions via polynomial degrees for ideals with analytic deviation one.

Abstract

Let $J \subset I$ be ideals in a formally equidimensional local ring with $λ (I / J) < \infty.$ Rees proved that for all $n ≫ 0$ , $λ (I^{n} / J^{n})$ is a polynomial $P (I / J) (X)$ in $n$ of degree at most dim $R$ and $J$ is a reduction of $I$ if and only if deg $P (I / J) (X) \leq$ dim $R - 1.$ We extend this result for all Noetherian filtrations of ideals in a formally equidimensional local ring and for (not necessarily Noetherian) filtrations of ideals in analytically irreducible rings. We provide certain classes of ideals such that deg $P (I / J)$ achieves its maximal degree. On the other hand, for ideals $J \subset I$ in a formally equidimensional local ring, we consider the multiplicity function $e (I^{n} / J^{n})$ which is a polynomial in $n$ for all large $n .$ We explicitly determine the deg $e (I^{n} / J^{n})$ in some special cases. For an ideal $J$ of analytic deviation one, we give characterization of…

Equations152

P_{I} (n) = e_{0} (I) (d n + d - 1) - e_{1} (I) (d - 1 n + d - 2) + \dots + (- 1)^{d} e_{d} (I)

P_{I} (n) = e_{0} (I) (d n + d - 1) - e_{1} (I) (d - 1 n + d - 2) + \dots + (- 1)^{d} e_{d} (I)

R_{1} = m_{1}, \dots, m_{r} \geq 0 ⨁ I (1)_{m_{1}} \dots I (r)_{m_{r}} \mbox an d R_{2} = m_{1}, \dots, m_{r} \geq 0 ⨁ J (1)_{m_{1}} \dots J (r)_{m_{r}} .

R_{1} = m_{1}, \dots, m_{r} \geq 0 ⨁ I (1)_{m_{1}} \dots I (r)_{m_{r}} \mbox an d R_{2} = m_{1}, \dots, m_{r} \geq 0 ⨁ J (1)_{m_{1}} \dots J (r)_{m_{r}} .

m \to \infty lim λ (I (1)_{m n_{1}} \dots I (r)_{m n_{r}} / J (1)_{m n_{1}} \dots J (r)_{m n_{r}}) / m^{d} = 0.

m \to \infty lim λ (I (1)_{m n_{1}} \dots I (r)_{m n_{r}} / J (1)_{m n_{1}} \dots J (r)_{m n_{r}}) / m^{d} = 0.

m^{c i} \cap I (1)_{i n_{1}} \dots I (r)_{i n_{r}} = m^{c i} \cap J (1)_{i n_{1}} \dots J (r)_{i n_{r}} .

m^{c i} \cap I (1)_{i n_{1}} \dots I (r)_{i n_{r}} = m^{c i} \cap J (1)_{i n_{1}} \dots J (r)_{i n_{r}} .

m \to \infty lim λ (I (1)_{m n_{1}} \dots I (r)_{m n_{r}} / J (1)_{m n_{1}} \dots J (r)_{m n_{r}}) / m^{d} = 0.

m \to \infty lim λ (I (1)_{m n_{1}} \dots I (r)_{m n_{r}} / J (1)_{m n_{1}} \dots J (r)_{m n_{r}}) / m^{d} = 0.

J_{1} I_{1}^{n} (I_{2} \dots I_{s})^{n + 1} \supset (J_{1} I_{1}^{n}) \dots (J_{s} I_{s}^{n}) = I_{1} I_{1}^{n} (I_{2} \dots I_{s})^{n + 1} \supset J_{1} I_{1}^{n} (I_{2} \dots I_{s})^{n + 1} .

J_{1} I_{1}^{n} (I_{2} \dots I_{s})^{n + 1} \supset (J_{1} I_{1}^{n}) \dots (J_{s} I_{s}^{n}) = I_{1} I_{1}^{n} (I_{2} \dots I_{s})^{n + 1} \supset J_{1} I_{1}^{n} (I_{2} \dots I_{s})^{n + 1} .

R_{1} = m_{1}, \dots, m_{r} \geq 0 ⨁ I (1)_{m_{1}} \dots I (r)_{m_{r}} \mbox an d R_{2} = m_{1}, \dots, m_{r} \geq 0 ⨁ J (1)_{m_{1}} \dots J (r)_{m_{r}} .

R_{1} = m_{1}, \dots, m_{r} \geq 0 ⨁ I (1)_{m_{1}} \dots I (r)_{m_{r}} \mbox an d R_{2} = m_{1}, \dots, m_{r} \geq 0 ⨁ J (1)_{m_{1}} \dots J (r)_{m_{r}} .

m \to \infty lim λ (I (1)_{m n_{1}} \dots I (r)_{m n_{r}} / J (1)_{m n_{1}} \dots J (r)_{m n_{r}}) / m^{d} = 0.

m \to \infty lim λ (I (1)_{m n_{1}} \dots I (r)_{m n_{r}} / J (1)_{m n_{1}} \dots J (r)_{m n_{r}}) / m^{d} = 0.

G^{(b_{1}, \dots, b_{r})} = m_{1}, \dots, m_{r} \geq 0 ⨁ I (1)_{α m_{1} + b_{1}} \dots I (r)_{α m_{r} + b_{r}} / J (1)_{α m_{1} + b_{1}} \dots J (r)_{α m_{r} + b_{r}}

G^{(b_{1}, \dots, b_{r})} = m_{1}, \dots, m_{r} \geq 0 ⨁ I (1)_{α m_{1} + b_{1}} \dots I (r)_{α m_{r} + b_{r}} / J (1)_{α m_{1} + b_{1}} \dots J (r)_{α m_{r} + b_{r}}

G_{m_{1}, \dots, m_{r}}^{(b_{1}, \dots, b_{r})} = I (1)_{α m_{1} + b_{1}} \dots I (r)_{α m_{r} + b_{r}} / J (1)_{α m_{1} + b_{1}} \dots J (r)_{α m_{r} + b_{r}}

G_{m_{1}, \dots, m_{r}}^{(b_{1}, \dots, b_{r})} = I (1)_{α m_{1} + b_{1}} \dots I (r)_{α m_{r} + b_{r}} / J (1)_{α m_{1} + b_{1}} \dots J (r)_{α m_{r} + b_{r}}

P_{(b_{1}, \dots, b_{r})} (m_{1}, \dots, m_{r}) = λ (G_{m_{1}, \dots, m_{r}}^{(b_{1}, \dots, b_{r})}) .

P_{(b_{1}, \dots, b_{r})} (m_{1}, \dots, m_{r}) = λ (G_{m_{1}, \dots, m_{r}}^{(b_{1}, \dots, b_{r})}) .

dim Supp_{++} T / m^{t} T \leq dim Supp_{++} T / m T = l (J (1)_{α} \dots J (r)_{α}) - 1 < d

dim Supp_{++} T / m^{t} T \leq dim Supp_{++} T / m T = l (J (1)_{α} \dots J (r)_{α}) - 1 < d

P_{(b_{1}, \dots, b_{r})} (X_{1}, \dots, X_{r}) = i_{1} + \dots + i_{r} < d \sum z_{i_{1}, \dots, i_{r}} (b_{1}, \dots, b_{r}) X_{1}^{i_{1}} X_{2}^{i_{2}} \dots X_{r}^{i_{r}}

P_{(b_{1}, \dots, b_{r})} (X_{1}, \dots, X_{r}) = i_{1} + \dots + i_{r} < d \sum z_{i_{1}, \dots, i_{r}} (b_{1}, \dots, b_{r}) X_{1}^{i_{1}} X_{2}^{i_{2}} \dots X_{r}^{i_{r}}

\begin{array}[]{lll}&&C:=\max\{|z_{i_{1},\ldots,i_{r}}(b_{1},\ldots,b_{r})|(n_{1}\cdots n_{r})^{d},\lambda(I(1)_{mn_{1}}\cdots I(r)_{mn_{r}}/J(1)_{mn_{1}}\cdots J(r)_{mn_{r}})\\ &&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ :0\leq b_{1},\ldots,b_{r}\leq\alpha-1,\sum\limits_{j=1}^{r}i_{j}<d\mbox{ and }0\leq m\leq\alpha(f+1)\}.\end{array}

\begin{array}[]{lll}&&C:=\max\{|z_{i_{1},\ldots,i_{r}}(b_{1},\ldots,b_{r})|(n_{1}\cdots n_{r})^{d},\lambda(I(1)_{mn_{1}}\cdots I(r)_{mn_{r}}/J(1)_{mn_{1}}\cdots J(r)_{mn_{r}})\\ &&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ :0\leq b_{1},\ldots,b_{r}\leq\alpha-1,\sum\limits_{j=1}^{r}i_{j}<d\mbox{ and }0\leq m\leq\alpha(f+1)\}.\end{array}

λ (I (1)_{m n_{1}} \dots I (r)_{m n_{r}} / J (1)_{m n_{1}} \dots J (r)_{m n_{r}}) / m^{d} \leq C / m < C .

λ (I (1)_{m n_{1}} \dots I (r)_{m n_{r}} / J (1)_{m n_{1}} \dots J (r)_{m n_{r}}) / m^{d} \leq C / m < C .

\begin{array}[]{lll}0&\leq&\liminf_{m\to\infty}\lambda(I(1)_{mn_{1}}\cdots I(r)_{mn_{r}}/J(1)_{mn_{1}}\cdots J(r)_{mn_{r}})/m^{d}\\ &\leq&\limsup_{m\to\infty}\lambda(I(1)_{mn_{1}}\cdots I(r)_{mn_{r}}/J(1)_{mn_{1}}\cdots J(r)_{mn_{r}})/m^{d}=0\end{array}

\begin{array}[]{lll}0&\leq&\liminf_{m\to\infty}\lambda(I(1)_{mn_{1}}\cdots I(r)_{mn_{r}}/J(1)_{mn_{1}}\cdots J(r)_{mn_{r}})/m^{d}\\ &\leq&\limsup_{m\to\infty}\lambda(I(1)_{mn_{1}}\cdots I(r)_{mn_{r}}/J(1)_{mn_{1}}\cdots J(r)_{mn_{r}})/m^{d}=0\end{array}

m \to \infty lim λ (I (1)_{m n_{1}} \dots I (r)_{m n_{r}} / J (1)_{m n_{1}} \dots J (r)_{m n_{r}}) / m^{d} = 0.

m \to \infty lim λ (I (1)_{m n_{1}} \dots I (r)_{m n_{r}} / J (1)_{m n_{1}} \dots J (r)_{m n_{r}}) / m^{d} = 0.

R_{1}^{(α n_{1}, \dots, α n_{r})} = \oplus_{m_{1}, \dots, m_{r} \geq 0} I (1)_{α n_{1} m_{1}} \dots I (r)_{α n_{r} m_{r}} = \oplus_{m_{1}, \dots, m_{r} \geq 0} I (1)_{α n_{1}}^{m_{1}} \dots I (r)_{α n_{r}}^{m_{r}}

R_{1}^{(α n_{1}, \dots, α n_{r})} = \oplus_{m_{1}, \dots, m_{r} \geq 0} I (1)_{α n_{1} m_{1}} \dots I (r)_{α n_{r} m_{r}} = \oplus_{m_{1}, \dots, m_{r} \geq 0} I (1)_{α n_{1}}^{m_{1}} \dots I (r)_{α n_{r}}^{m_{r}}

R_{2}^{(α n_{1}, \dots, α n_{r})} = \oplus_{m_{1}, \dots, m_{r} \geq 0} J (1)_{α n_{1} m_{1}} \dots J (r)_{α n_{r} m_{r}} = \oplus_{m_{1}, \dots, m_{r} \geq 0} J (1)_{α n_{1}}^{m_{1}} \dots J (r)_{α n_{r}}^{m_{r}} .

R_{2}^{(α n_{1}, \dots, α n_{r})} = \oplus_{m_{1}, \dots, m_{r} \geq 0} J (1)_{α n_{1} m_{1}} \dots J (r)_{α n_{r} m_{r}} = \oplus_{m_{1}, \dots, m_{r} \geq 0} J (1)_{α n_{1}}^{m_{1}} \dots J (r)_{α n_{r}}^{m_{r}} .

R (I (1)_{α n_{1}} \dots I (r)_{α n_{r}}) = \oplus_{m \geq 0} (I (1)_{α n_{1}} \dots I (r)_{α n_{r}})^{m} = \oplus_{m \geq 0} I (1)_{α n_{1} m} \dots I (r)_{α n_{r} m}

R (I (1)_{α n_{1}} \dots I (r)_{α n_{r}}) = \oplus_{m \geq 0} (I (1)_{α n_{1}} \dots I (r)_{α n_{r}})^{m} = \oplus_{m \geq 0} I (1)_{α n_{1} m} \dots I (r)_{α n_{r} m}

R (J (1)_{α n_{1}} \dots J (r)_{α n_{r}}) = \oplus_{m \geq 0} (J (1)_{α n_{1}} \dots J (r)_{α n_{r}})^{m} = \oplus_{m \geq 0} J (1)_{α n_{1} m} \dots J (r)_{α n_{r} m} .

R (J (1)_{α n_{1}} \dots J (r)_{α n_{r}}) = \oplus_{m \geq 0} (J (1)_{α n_{1}} \dots J (r)_{α n_{r}})^{m} = \oplus_{m \geq 0} J (1)_{α n_{1} m} \dots J (r)_{α n_{r} m} .

K_{μ} = {f \in Q (R) ∣ ν (f) \geq μ},

K_{μ} = {f \in Q (R) ∣ ν (f) \geq μ},

K_{μ}^{+} = {f \in Q (R) ∣ ν (f) > μ} .

K_{μ}^{+} = {f \in Q (R) ∣ ν (f) > μ} .

T = {T_{m} := T (1)_{m n_{1}} \dots T (r)_{m n_{r}}} \mbox an d T (a) = {T (a)_{m} := T_{a} (1)_{m n_{1}} \dots T_{a} (r)_{m n_{r}}} .

T = {T_{m} := T (1)_{m n_{1}} \dots T (r)_{m n_{r}}} \mbox an d T (a) = {T (a)_{m} := T_{a} (1)_{m n_{1}} \dots T_{a} (r)_{m n_{r}}} .

T (a)_{n} \subset F (a)_{n} \subset T_{n} .

T (a)_{n} \subset F (a)_{n} \subset T_{n} .

\begin{array}[]{lll}&&\lim_{a\to\infty}\big{(}{\lim_{m\to\infty}\lambda({\mathcal{T}(a)_{m}/\mathcal{T}(a)_{m}\cap K_{\alpha mb}})}\big{)}/m^{d}=\lim_{m\to\infty}\lambda({\mathcal{T}_{m}/\mathcal{T}_{m}\cap K_{\alpha mb})/m^{d}}\\ &=&\lim_{a\to\infty}\big{(}{\lim_{m\to\infty}\lambda({\mathcal{F}(a)_{m}/\mathcal{F}(a)_{m}\cap K_{\alpha mb}})}\big{)}/m^{d}.\end{array}

\begin{array}[]{lll}&&\lim_{a\to\infty}\big{(}{\lim_{m\to\infty}\lambda({\mathcal{T}(a)_{m}/\mathcal{T}(a)_{m}\cap K_{\alpha mb}})}\big{)}/m^{d}=\lim_{m\to\infty}\lambda({\mathcal{T}_{m}/\mathcal{T}_{m}\cap K_{\alpha mb})/m^{d}}\\ &=&\lim_{a\to\infty}\big{(}{\lim_{m\to\infty}\lambda({\mathcal{F}(a)_{m}/\mathcal{F}(a)_{m}\cap K_{\alpha mb}})}\big{)}/m^{d}.\end{array}

\begin{array}[]{lll}\Gamma^{\prime(t)}&=&\{(m_{1},\ldots,m_{d},i)\in\mathbb{N}^{d+1}\mid\operatorname{dim}_{k}\mathcal{T}_{i}\cap K_{m_{1}\lambda_{1}+\cdots+m_{d}\lambda_{d}}/\mathcal{T}_{i}\cap K^{+}_{m_{1}\lambda_{1}+\cdots+m_{d}\lambda_{d}}\geq t\\ &&\mbox{ and }m_{1}+\cdots+m_{d}\leq\alpha b^{\prime}i\},\end{array}

\begin{array}[]{lll}\Gamma^{\prime(t)}&=&\{(m_{1},\ldots,m_{d},i)\in\mathbb{N}^{d+1}\mid\operatorname{dim}_{k}\mathcal{T}_{i}\cap K_{m_{1}\lambda_{1}+\cdots+m_{d}\lambda_{d}}/\mathcal{T}_{i}\cap K^{+}_{m_{1}\lambda_{1}+\cdots+m_{d}\lambda_{d}}\geq t\\ &&\mbox{ and }m_{1}+\cdots+m_{d}\leq\alpha b^{\prime}i\},\end{array}

\begin{array}[]{lll}\Gamma(a)^{\prime(t)}&=&\{(m_{1},\ldots,m_{d},i)\in\mathbb{N}^{d+1}\mid\\ &&\operatorname{dim}_{k}\mathcal{T}(a)_{i}\cap K_{m_{1}\lambda_{1}+\cdots+m_{d}\lambda_{d}}/\mathcal{T}(a)_{i}\cap K^{+}_{m_{1}\lambda_{1}+\cdots+m_{d}\lambda_{d}}\geq t\\ &&\mbox{ and }m_{1}+\cdots+m_{d}\leq\alpha b^{\prime}i\}\end{array}

\begin{array}[]{lll}\Gamma(a)^{\prime(t)}&=&\{(m_{1},\ldots,m_{d},i)\in\mathbb{N}^{d+1}\mid\\ &&\operatorname{dim}_{k}\mathcal{T}(a)_{i}\cap K_{m_{1}\lambda_{1}+\cdots+m_{d}\lambda_{d}}/\mathcal{T}(a)_{i}\cap K^{+}_{m_{1}\lambda_{1}+\cdots+m_{d}\lambda_{d}}\geq t\\ &&\mbox{ and }m_{1}+\cdots+m_{d}\leq\alpha b^{\prime}i\}\end{array}

\begin{array}[]{lll}\Gamma(\mathcal{F}(a))^{\prime(t)}&=&\{(m_{1},\ldots,m_{d},i)\in\mathbb{N}^{d+1}\mid\\ &&\operatorname{dim}_{k}\mathcal{F}(a)_{i}\cap K_{m_{1}\lambda_{1}+\cdots+m_{d}\lambda_{d}}/\mathcal{F}(a)_{i}\cap K^{+}_{m_{1}\lambda_{1}+\cdots+m_{d}\lambda_{d}}\geq t\\ &&\mbox{ and }m_{1}+\cdots+m_{d}\leq\alpha b^{\prime}i\}.\end{array}

\begin{array}[]{lll}\Gamma(\mathcal{F}(a))^{\prime(t)}&=&\{(m_{1},\ldots,m_{d},i)\in\mathbb{N}^{d+1}\mid\\ &&\operatorname{dim}_{k}\mathcal{F}(a)_{i}\cap K_{m_{1}\lambda_{1}+\cdots+m_{d}\lambda_{d}}/\mathcal{F}(a)_{i}\cap K^{+}_{m_{1}\lambda_{1}+\cdots+m_{d}\lambda_{d}}\geq t\\ &&\mbox{ and }m_{1}+\cdots+m_{d}\leq\alpha b^{\prime}i\}.\end{array}

\begin{array}[]{lll}\Gamma^{(t)}&=&\{(m_{1},\ldots,m_{d},i)\in\mathbb{N}^{d+1}\mid\operatorname{dim}_{k}\mathcal{T}_{i}\cap K_{m_{1}\lambda_{1}+\cdots+m_{d}\lambda_{d}}/\mathcal{T}_{i}\cap K^{+}_{m_{1}\lambda_{1}+\cdots+m_{d}\lambda_{d}}\geq t\\ &&\mbox{ and }m_{1}+\cdots+m_{d}\leq\alpha bi\},\end{array}

\begin{array}[]{lll}\Gamma^{(t)}&=&\{(m_{1},\ldots,m_{d},i)\in\mathbb{N}^{d+1}\mid\operatorname{dim}_{k}\mathcal{T}_{i}\cap K_{m_{1}\lambda_{1}+\cdots+m_{d}\lambda_{d}}/\mathcal{T}_{i}\cap K^{+}_{m_{1}\lambda_{1}+\cdots+m_{d}\lambda_{d}}\geq t\\ &&\mbox{ and }m_{1}+\cdots+m_{d}\leq\alpha bi\},\end{array}

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Full text

Rees’ theorem for filtrations, multiplicity function and reduction criteria

Parangama Sarkar

chennai mathematical institute, h1, sipcot it park, siruseri, chennai, india-603103

[email protected], [email protected]

Abstract.

Let $J\subset I$ be ideals in a formally equidimensional local ring with ${\lambda}(I/J)<\infty.$ Rees proved that for all $n\gg 0$ , ${\lambda}(I^{n}/J^{n})$ is a polynomial $P(I/J)(X)$ in $n$ of degree at most $\operatorname{dim}R$ and $J$ is a reduction of $I$ if and only if $\deg P(I/J)(X)\leq\operatorname{dim}R-1.$ We extend this result for all Noetherian filtrations of ideals in a formally equidimensional local ring and for (not necessarily Noetherian) filtrations of ideals in analytically irreducible rings. We provide certain classes of ideals such that $\deg P(I/J)$ achieves its maximal degree. On the other hand, for ideals $J\subset I$ in a formally equidimensional local ring, we consider the multiplicity function $e(I^{n}/J^{n})$ which is a polynomial in $n$ for all large $n.$ We explicitly determine the $\deg e(I^{n}/J^{n})$ in some special cases. For an ideal $J$ of analytic deviation one, we give characterization of reductions in terms of $\deg e(I^{n}/J^{n})$ under some additional conditions.

2010 Mathematics Subject Classification:

Primary 13D40, 13A30, 13D45, 13E05

*Keywords: Filtrations, reduction of an ideal, analytic spread, associated graded ring, $d$ -sequence, Rees algebra, analytic deviation, unmixed ideal. *

The author is supported by Department of Science & Technology (DST); INSPIRE Faculty Fellowship.

1. introduction

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d$ and $I$ be an ideal in $R.$ If $I$ is $\mathfrak{m}$ -primary then Samuel [24] showed that for all large $n,$ the Hilbert-Samuel function of $I,$ $\displaystyle H_{I}(n)=\lambda({R}/{I^{n}})$ (here $\lambda(M)$ denotes the length of an $R$ -module $M$ ) coincides with a polynomial

[TABLE]

of degree $d,$ called the Hilbert-Samuel polynomial of $I.$ The coefficient $e(I):=e_{0}(I)$ is called the multiplicity of $I.$ Rees used this numerical invariant $e(\_)$ to study the numerical characterization of reductions. He showed that if $R$ is a formally equidimensional local ring (i.e. $R$ is a Noetherian local ring and its completion in the topology defined by the maximal ideal is equidimensional) and $J\subset I$ are $\mathfrak{m}$ -primary ideals in $R$ then $J$ is a reduction of $I$ if and only if $e(I)=e(J)$ [20]. In literature this result has been generalized in several directions. Amao [1], considered more general case where $I,J$ are not necessarily $\mathfrak{m}$ -primary ideals and proved that if $M$ is a finitely generated $R$ -module and $J\subset I$ are ideals of $R$ such that $IM/JM$ has finite length then $\mu(n):={\lambda}(I^{n}M/J^{n}M)$ is a polynomial $P(IM/JM)$ in $n$ for $n\gg 0.$ In [22], Rees showed that in a Noetherian local ring $(R,\mathfrak{m}),$ the degree of $P(I/J)$ is at most $\operatorname{dim}R$ and he further proved the following theorem.

Theorem 1.1.

(Rees, [22, Theorem 2.1])* Let $R$ be a formally equidimensional Noetherian local ring and $J\subset I$ be ideals in $R$ with $\lambda(I/J)<\infty.$ Then $J$ is a reduction of $I$ if and only if $\deg P(I/J)$ is at most $\operatorname{dim}R-1.$ *

Since for some $n,$ $J^{n}$ is a reduction of $I^{n}$ implies $J$ is a reduction of $I,$ we can interpret Rees’ result as follows:

Theorem 1.2.

Let $R$ be a formally equidimensional Noetherian local ring of dimension $d$ and $J\subset I$ be ideals in $R$ with $\lambda(I/J)<\infty.$ Then the following are true.

(1)

The function $F(n)=\lim_{m\to\infty}\lambda(I^{mn}/J^{mn})/m^{d}$ is a polynomial in $n$ of degree at most $d.$ 2. (2)

*Fix $n\in\mathbb{Z}_{+},$ then $J$ is a reduction of $I$ if and only if * $F(n)=0.$

In [6, Theorem 6.1], Cutkosky proved the following result.

Theorem 1.3.

(Cutkosky, [6, Theorem 6.1])* Let $R$ be an analytically unramified local ring of dimension $d>0.$ Suppose $\{I_{i}\}$ and $\{J_{i}\}$ are two graded families of ideals such that for all $i,$ $J_{i}\subset I_{i}$ and there exists $c\in\mathbb{Z}_{+},$ such that $\mathfrak{m}^{ci}\cap I_{i}=\mathfrak{m}^{ci}\cap J_{i}$ for all $i.$ Assume that if $P$ is a minimal prime of $R$ then $I_{1}\subset P$ implies $I_{i}\subset P$ for all $i>0.$ Then the limit $\lim_{m\to\infty}\lambda(I_{m}/J_{m})/m^{d}$ exists.*

Therefore it is natural to ask whether Rees’ result (Theorem 1.2) holds true for two (not necessarily Noetherian) filtrations of ideals $\{I_{i}\}$ and $\{J_{i}\}$ in an analytically unramified local ring $R$ of dimension $d.$ In this direction we first prove Rees’ theorem for Noetherian filtrations of ideals.

Theorem 1.4.

(=Theorem 2.3)* Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ and for $1\leq l\leq r,$ $\mathcal{I}(l)=\{I(l)_{n}\},$ $\mathcal{J}(l)=\{J(l)_{n}\}$ be Noetherian filtrations of ideals in $R$ such that $J(l)_{n}\subset I(l)_{n}$ and $\lambda(I(l)_{n}/J(l)_{n})<\infty$ for all $n\in\mathbb{N}$ and $1\leq l\leq r.$ Let*

[TABLE]

Fix $n_{1},\ldots,n_{r}\in\mathbb{Z}_{+}.$ If $R_{1}$ is integral over $R_{2}$ then

[TABLE]

Converse holds if $R$ is formally equidimensional and $\operatorname{grade}(J(1)_{1}\cdots J(r)_{1})\geq 1.$

We prove that the “only if” part of Rees’ theorem 1.2 $(2)$ holds true for (not necessarily Noetherian) filtrations of ideals in analytically irreducible local rings (i.e. $R$ is a Noetherian local ring and its completion in the topology defined by the maximal ideal is domain). We also give an example to show that the “if” part of Rees’ theorem 1.2 $(2)$ is not true for non-Noetherian filtrations of ideals in general (See example 2.7).

Theorem 1.5.

(=Theorem 2.5)* Let $(R,\mathfrak{m})$ be an analytically irreducible local ring of dimension $d>0$ and for $1\leq l\leq r,$ $\mathcal{I}(l)=\{I(l)_{n}\},$ $\mathcal{J}(l)=\{J(l)_{n}\}$ be (not necessarily Noetherian) filtrations of ideals in $R$ such that $J(l)_{n}\subset I(l)_{n}$ for all $n\in\mathbb{N}$ and $1\leq l\leq r.$ Fix $n_{1},\ldots,n_{r}\in\mathbb{Z}_{+}.$ Suppose there exists an integer $c\in\mathbb{Z}_{+}$ such that for all $i\in\mathbb{N},$ *

[TABLE]

Let $R_{1}=\bigoplus_{m_{1},\ldots,m_{r}\geq 0}I(1)_{m_{1}}\cdots I(r)_{m_{r}}$ be integral over $R_{2}=\bigoplus_{m_{1},\ldots,m_{r}\geq 0}J(1)_{m_{1}}\cdots J(r)_{m_{r}}.$ Then

[TABLE]

Rees’ theorem [22, Theorem 2.1] shows that $J$ is a reduction of $I$ implies $P(I/J)$ is a polynomial of degree at most $\operatorname{dim}R-1.$ In this direction one can ask that if $J$ is a reduction of $I$ when the degree of the polynomial $P(I/J)$ attains the maximal. We use the notation $(I,J)$ to denote $J$ is a reduction of $I.$ We produce a lower bound of degree of $P(I/J)$ and show that if $J$ is a complete intersection ideal then $\deg P(I/J)=l(J)-1$ where $l(J)$ is the analytic spread of $J$ (See Theorem 3.7). We also improve the lower bound of $\deg P(I/J)$ for the following class of ideals.

Theorem 1.6.

(=Theorem 3.9)* Let $(R,\mathfrak{m})$ be a Noetherian local ring, $J\subsetneq I$ be ideals in $R$ such that ${\lambda}(I/J)<\infty$ and $\operatorname{grade}(J)=s.$ Let $a_{1},\ldots,a_{s},b_{1},\ldots,b_{t}$ be a $d$ -sequence in $R$ which minimally generates the ideal $J$ and $t\geq 1.$ Suppose $I\cap(J^{\prime}:b_{1})=J^{\prime}$ where $J^{\prime}=(a_{1},\ldots,a_{s})$ $(J^{\prime}=(0)$ if $s=0)$ . Then the following are true.*

(1)

$\operatorname{grade}(J)\leq\deg P(I/J).$ ** 2. (2)

If $J$ is a reduction of $I$ then $\operatorname{grade}(J)\leq\deg P(I/J)\leq\l(J)-1.$

This helps us to provide a large class of ideals $(I,J)$ which attain the maximal degree of $P(I/J),$ i.e. $l(J)-1.$

Corollary 1.7.

(=Corollary 3.11)* Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\geq 2$ with infinite residue field, $I$ be an ideal of $R$ and $J$ be any minimal reduction of $I$ with ${\lambda}(I/J)<\infty.$ Suppose $I$ satisfies $G_{l(I)}$ and $AN_{l(I)-2}^{-}$ conditions. Then for any ideal $K$ with $J\subsetneq K\subseteq I,$ $\deg P(K/J)=l(J)-1.$ *

In the other direction, Herzog, Puthenpurakal and Verma [12] proved that, for ideals $J\subset I$ in a formally equidimensional local ring, $e(I^{n}/J^{n})$ is eventually a polynomial function. In the same vein, in a recent paper, Ciupercă [2], showed that $e(I^{n}/J^{n})$ is a polynomial function of degree at most $\operatorname{dim}R-t$ where $t$ is a constant equal to $\operatorname{dim}Q$ where $Q=R/J^{n}:I^{n}$ for all $n\gg 0$ and in [3], he remarked that if $J$ is reduction of $I$ then $\deg e(I^{n}/J^{n})\leq l(J)-1.$ He also studied the function $e(I^{n}/J^{n})$ where $J$ is an equimultiple ideal (i.e. $\operatorname{ht}J=l(J)$ ) and proved the following characterization of reductions for equimultiple ideals in terms of $\deg e(I^{n}/J^{n})$ [3, Theorem 2.6].

Theorem 1.8.

(Ciupercă, [3, Theorem 2.6])* Let $(R,\mathfrak{m})$ be a formally equidimensional local ring and $J\subseteq I$ proper ideals of $R$ with $J$ equimultiple. Let $f(n)=e(I^{n}/J^{n}).$ The following are true.*

(1)

If $J\subseteq I$ is a reduction then $\deg f(n)\leq l(J)-1.$ 2. (2)

If $J\subseteq I$ is not a reduction then $\deg f(n)=l(J).$

In this situation one may ask if $J$ has analytic deviation one (i.e. $l(J)=\operatorname{ht}(J)+1$ ) can we give characterization of reduction in terms of $\deg e(I^{n}/J^{n})?$

Motivated by the result of Ciupercă (Theorem 1.8), we provide upper and lower bounds of $\deg e(I^{n}/J^{n})$ and show that if $J$ is a complete intersection ideal then $J$ is reduction of $I$ if and only if $\deg e(I^{n}/J^{n})=l(J)-1$ (See Proposition 4.1). In the case $J$ has analytic deviation one, we characterize when $J$ is a reduction of $I$ in terms of $\deg e(I^{n}/J^{n})$ under some additional hypotheses.

Theorem 1.9.

(=Theorem 4.4)* Let $(R,\mathfrak{m})$ be a formally equidimensional local ring of dimension $d\geq 2,$ $J\subsetneq I$ be ideals in $R$ and $J$ has analytic deviation one. Suppose $l(J_{p})<l(J)$ for all prime ideals $p$ in $R$ such that $\operatorname{ht}p=l(J).$ Then the following are true.*

(1)

If $J$ is not a reduction of $I$ then $\deg e(I^{n}/J^{n})=l(J)-1.$ 2. (2)

*If $l(J)=d-1,$ $\operatorname{depth}(R/J)>0$ and for all $n\geq 1,$ $\sqrt{J:I}=\sqrt{J^{n}:I^{n}}$ then *

$J$ * is a reduction of $I$ if and only if $\deg e(I^{n}/J^{n})\leq l(J)-2.$ *

We can not omit the condition $\operatorname{depth}(R/J)>0$ from Theorem 4.4 $(2)$ (see Example 4.5). We give sufficient conditions on the ideal $J$ for the equality $\sqrt{J:I}=\sqrt{J^{n}:I^{n}}$ for all $n\geq 1$ (see Proposition 4.3).

2. Rees’ theorem for filtrations

In this section we prove Rees’ theorem for (not necessarily Noetherian) filtrations of ideals. A family $\mathcal{I}=\{I_{n}\}_{n\in\mathbb{N}}$ of ideals in $R$ is called filtration of ideals if $I_{0}=R,$ $I_{m}\subseteq I_{n}$ for all $m\geq n$ and $I_{m}I_{n}\subseteq I_{m+n}$ for all $m,n\in\mathbb{N}.$ A filtration $\mathcal{I}=\{I_{n}\}_{n\in\mathbb{N}}$ of ideals in $R$ is said to be Noetherian filtration if $\oplus_{n\geq 0}I_{n}$ is a finitely generated $R$ -algebra. We first prove Rees’ result for Noetherian filtrations of ideals and then using iterated Noetherian filtrations, we prove the “only if” part for (not necessarily Noetherian) filtrations of ideals in analytically irreducible rings. We conclude this section with an example which shows that for non-Noetherian filtrations the “if” part of Rees’ theorem is not true in general.

Lemma 2.1.

Let $(R,\mathfrak{m})$ be a local ring of dimension $d>0$ and $A=\bigoplus_{m_{1},\ldots,m_{r}\geq 0}A_{m_{1},\ldots,m_{r}}$ be a $\mathbb{N}^{r}$ -graded finitely generated $R$ -algebra where $A_{0,\ldots,0}=R.$ Let $u_{1},\ldots,u_{r}\in\mathbb{Z}_{+}.$ Consider the grdaed subring $A^{(u_{1},\ldots,u_{r})}=\bigoplus_{m_{1},\ldots,m_{r}\geq 0}A_{u_{1}m_{1},\ldots,u_{r}m_{r}}$ of $A.$ Then $A$ is finitely generated $A^{(u_{1},\ldots,u_{r})}$ -module.

Proof.

Let $\{a_{1},\ldots,a_{p}\}$ be a generating set of $A$ as $R$ algebra. Define $\mathcal{S}=\{a_{1}^{r_{1}}\cdots a_{p}^{r_{p}}\mid 0\leq r_{i}<u_{1}\cdots u_{r}\}.$ Let $h_{1},\ldots,h_{p}\in\mathbb{N}$ such that $h_{i}=u_{1}\cdots u_{r}q_{i}+t_{i}$ where $0\leq t_{i}<u_{1}\cdots u_{r}$ and $1\leq i\leq p.$ Then $a_{1}^{h_{1}}\cdots a_{p}^{h_{p}}=(a_{1}^{q_{1}}\cdots a_{p}^{q_{p}})^{u_{1}\cdots u_{r}}a_{1}^{t_{1}}\cdots a_{p}^{t_{p}}.$ Thus $\mathcal{S}$ is a generating set of $A$ as $A^{(u_{1},\ldots,u_{r})}$ -module. ∎

Lemma 2.2.

Let $(R,\mathfrak{m})$ be a Noetherian local ring and for $i=1,\ldots,s,$ $J_{i}\subset I_{i}$ be ideals in $R$ and $\operatorname{grade}(J_{1}\cdots J_{s})\geq 1.$ Suppose $J_{i}$ is not a reduction of $I_{i}$ for some $i\in\{1,\ldots,s\}.$ Then $J_{1}\cdots J_{s}$ is not a reduction of $I_{1}\cdots I_{s}.$

Proof.

Without loss of generality, assume that $J_{1}$ is not a reduction of $I_{1}.$ Suppose for some $n\geq 0,$ $(J_{1}\cdots J_{s})(I_{1}\cdots I_{s})^{n}=(I_{1}\cdots I_{s})^{n+1}.$ Then

[TABLE]

Hence $J_{1}M=I_{1}M$ where $M=I^{n}_{1}(I_{2}\cdots I_{s})^{n+1}.$ Therefore by [21, Lemma 1.5], $J_{1}$ is a reduction of $I_{1}$ which is a contradiction. ∎

Theorem 2.3.

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ and for $1\leq l\leq r,$ $\mathcal{I}(l)=\{I(l)_{n}\},$ $\mathcal{J}(l)=\{J(l)_{n}\}$ be Noetherian filtrations of ideals in $R$ such that $J(l)_{n}\subset I(l)_{n}$ and $\lambda(I(l)_{n}/J(l)_{n})<\infty$ for all $n\in\mathbb{N}$ and $1\leq l\leq r.$ Let

[TABLE]

Fix $n_{1},\ldots,n_{r}\in\mathbb{Z}_{+}.$ If $R_{1}$ is integral over $R_{2}$ then

[TABLE]

Converse holds if $R$ is formally equidimensional and $\operatorname{grade}(J(1)_{1}\cdots J(r)_{1})\geq 1.$

Proof.

Since for all $1\leq l\leq r,$ $\mathcal{J}(l)=\{J(l)_{n}\}$ are Noetherian filtrations of ideals in $R,$ there exists an integer $\alpha\in\mathbb{Z}_{+},$ such that $G_{l}^{\alpha}=\oplus_{n\geq 0}J(l)_{\alpha n}$ are Noetherian standard graded $R$ -algebras for all $1\leq l\leq r.$ Hence $T=\bigoplus_{m_{1},\ldots,m_{r}\geq 0}J(1)_{\alpha m_{1}}\cdots J(r)_{\alpha m_{r}}$ is a Noetherian standard graded $R$ -algebra and graded subring of $R_{2}$ .

Using Lemma 2.1 for $u_{1}=\ldots=u_{r}=\alpha,$ we get $R_{2}$ is finitely generated $T$ -module. Since $R_{1}$ is finitely generated $R_{2}$ -module, we have $R_{1}$ is finitely generated $T$ -module. Now for all integers $0\leq b_{j}\leq\alpha-1$ with $1\leq j\leq r,$ $S^{(\alpha,b_{1},\ldots,b_{r})}=\bigoplus_{m_{1},\ldots,m_{r}\geq 0}I(1)_{\alpha m_{1}+b_{1}}\cdots I(r)_{\alpha m_{r}+b_{r}}$ are $T$ -submodules of $R_{1}.$ Therefore $S^{(\alpha,b_{1},\ldots,b_{r})}$ are finitely generated $T$ -modules for all integers $0\leq b_{j}\leq\alpha-1$ with $1\leq j\leq r$ and hence

[TABLE]

are finitely generated $T$ -modules for all integers $0\leq b_{j}\leq\alpha-1$ with $1\leq j\leq r$ where we consider the grading

[TABLE]

and $T_{m_{1},\ldots,m_{r}}=J(1)_{\alpha m_{1}}\cdots J(r)_{\alpha m_{r}}.$ Since $\lambda(I(l)_{n}/J(l)_{n})<\infty$ for all $n\in\mathbb{N}$ and $1\leq l\leq r,$ there exists an integer $t\in\mathbb{Z}_{+},$ such that $G^{(b_{1},\ldots,b_{r})}$ is finitely generated $T/\mathfrak{m}^{t}T$ -module for all $0\leq b_{1},\ldots,b_{r}\leq\alpha-1.$

By [11, Theorem 4.1], for all $1\leq j\leq r$ and integers $0\leq b_{j}\leq\alpha-1,$ there exist polynomials $P_{(b_{1},\ldots,b_{r})}(X_{1},\ldots,X_{r})\in\mathbb{Q}[X_{1},\ldots,X_{r}]$ of total degree at most $\operatorname{dim}\operatorname{Supp}_{++}T/\mathfrak{m}^{t}T$ and an integer $f\in\mathbb{Z}_{+}$ such that for all $m_{1},\ldots,m_{r}\geq f,$

[TABLE]

Since $J(1)_{\alpha m_{1}}\cdots J(r)_{\alpha m_{r}}=J(1)_{\alpha}^{m_{1}}\cdots J(r)_{\alpha}^{m_{r}},$ by [16], [10, Corollary 3.3 and Corollary 5.3], we get

[TABLE]

where $l(J(1)_{\alpha}\cdots J(r)_{\alpha})$ is the analytic spread of $J(1)_{\alpha}\cdots J(r)_{\alpha}.$

Let $i_{1},\ldots,i_{r}\in\mathbb{N}$ with $i_{1}+\cdots+i_{r}<d$ and

[TABLE]

where $z_{i_{1},\ldots,i_{r}}(b_{1},\ldots,b_{r})\in\mathbb{Q}.$ Let

[TABLE]

Then for any $m\in\mathbb{Z}_{+},$ by equation (2.3.1), we get

[TABLE]

Therefore

[TABLE]

implies

[TABLE]

Now we prove the converse. Let $R$ be a formally equidimensional local ring,

$\operatorname{grade}(J(1)_{1}\cdots J(r)_{1})\geq 1$ and $\lim_{m\to\infty}\lambda(I(1)_{mn_{1}}\cdots I(r)_{mn_{r}}/J(1)_{mn_{1}}\cdots J(r)_{mn_{r}})/m^{d}=0.$

Suppose $R_{1}$ is not integral over $R_{2}.$

Claim : For any $u_{1},\ldots,u_{r}\in\mathbb{Z}_{+},$ $R_{1}^{(u_{1},\ldots,u_{r})}=\oplus_{m_{1},\ldots,m_{r}\geq 0}I(1)_{u_{1}m_{1}}\cdots I(r)_{u_{r}m_{r}}$ is not integral over $R_{2}^{(u_{1},\ldots,u_{r})}=\oplus_{m_{1},\ldots,m_{r}\geq 0}J(1)_{u_{1}m_{1}}\cdots J(r)_{u_{r}m_{r}}.$

Proof of the claim: Suppose there exist $u_{1},\ldots,u_{r}\in\mathbb{Z}_{+}$ such that $R_{1}^{(u_{1},\ldots,u_{r})}$ is integral over $R_{2}^{(u_{1},\ldots,u_{r})}.$ By Lemma 2.1, $R_{1}$ is finitely generated $R_{1}^{(u_{1},\ldots,u_{r})}$ -module. Hence $R_{1}$ is integral over $R_{2}^{(u_{1},\ldots,u_{r})}$ as well as over $R_{2}$ which is a contradiction.

Since for all $1\leq l\leq r,$ $\mathcal{I}(l)=\{I(l)_{n}\},$ $\mathcal{J}(l)=\{J(l)_{n}\}$ are Noetherian filtrations of ideals in $R,$ there exists an integer $\alpha\in\mathbb{Z}_{+},$ such that $\oplus_{n\geq 0}I(l)_{\alpha n},$ $\oplus_{n\geq 0}J(l)_{\alpha n}$ are Noetherian standard graded $R$ -algebras for all $1\leq l\leq r.$ Now by the claim,

[TABLE]

is not integral over

[TABLE]

Therefore there exists $t\in\{1,\ldots,r\}$ such that $\oplus_{m_{t}\geq 0}I(t)_{\alpha n_{t}}^{m_{t}}$ is not finitely generated over $\oplus_{m_{t}\geq 0}J(t)_{\alpha n_{t}}^{m_{t}},$ i.e., $J(t)_{\alpha n_{t}}$ is not a reduction of $I(t)_{\alpha n_{t}}.$

Let $n=\max\{n_{1},\ldots,n_{r}\}.$ Since $(J(1)_{1}\cdots J(r)_{1})^{n\alpha}\subset J(1)_{\alpha n_{1}}\cdots J(r)_{\alpha n_{r}},$ by Lemma 2.2, $J(1)_{\alpha n_{1}}\cdots J(r)_{\alpha n_{r}}$ is not a reduction of $I(1)_{\alpha n_{1}}\cdots I(r)_{\alpha n_{r}}.$ Therefore

[TABLE]

is not finitely generated module over

[TABLE]

Thus by [22, Theorem 2.1], $\lim_{m\to\infty}\lambda(I(1)_{\alpha mn_{1}}\cdots I(r)_{\alpha mn_{r}}/J(1)_{\alpha mn_{1}}\cdots J(r)_{\alpha mn_{r}})/m^{d}\neq 0,$ which is a contradiction. ∎

Now we prove the “only if” part of Rees’ theorem 1.2 for (not necessarily Noetherian) filtrations. Let $(R,\mathfrak{m})$ be a complete local domain of dimension $d>0.$ Then by [7, Lemma 4.2], [6], there exists a regular local ring $S$ of dimension $d$ which birationally dominates $R$ . Let ${\lambda}_{1},\ldots,{\lambda}_{d}\in\mathbb{R}$ be rationally independent real numbers such that $\lambda_{i}\geq 1$ for all $i$ and $y_{1},\ldots,y_{d}\in S$ be a regular system of parameters in $S.$ We consider a valuation $\nu$ on the quotient field of $R$ which dominates $S$ as mentioned in [6], i.e., $\nu(y_{1}^{a_{1}}\cdots y_{d}^{a_{d}})=a_{1}\lambda_{1}+\cdots+a_{d}\lambda_{d}$ for $a_{1},\ldots,a_{d}\in\mathbb{N}$ and $\nu(\gamma)=0$ if $\gamma\in S$ is a unit. Let $k=R/m_{R}$ and $k^{\prime}=S/m_{S}$ . Then by [6] (Page 10), we get $k^{\prime}=S/\mathfrak{m}_{S}=V_{\nu}/\mathfrak{m}_{\nu}$ where $V_{\nu}$ is the valuation ring of $\nu$ and $[k^{\prime}:k]<\infty.$

For $\mu\in\mathbb{R}_{>0}$ , define ideals $K_{\mu}$ and $K_{\mu}^{+}$ in the valuation ring $V_{\nu}$ by

[TABLE]

Let $\mathcal{T}(m)=\{T(m)_{n}\}$ be filtrations of ideals in $R$ for all $1\leq m\leq r$ and $a\in\mathbb{Z}_{+}.$ We recall the definition of $a$ th truncated filtration of ideals defined in [7]. The $a$ -th truncated filtration of ideals $\mathcal{T}_{a}(m)=\{T_{a}(m)_{n}\}$ of $\mathcal{T}(m)$ is defined by $T_{a}(m)_{n}=T(m)_{n}$ if $n\leq a$ and if $n>a$ , then $T_{a}(m)_{n}=\sum T_{a}(m)_{i}T_{a}(m)_{j}$ where the sum is over $i,j>0$ such that $i+j=n$ .

Fix $n_{1},\ldots,n_{r}\in\mathbb{Z}_{+}.$ Define filtrations of ideals

[TABLE]

By [5, Lemma 4.3], there exists $\alpha\in\mathbb{Z}_{+}$ such that $K_{\alpha m}\cap R\subset\mathfrak{m}^{m}$ for all $m\in\mathbb{N}.$

Note that $\mathfrak{m}^{\alpha bmd}\mathcal{T}_{m}\subset\mathcal{T}_{m}\cap K_{\alpha mb}$ and $\mathfrak{m}^{\alpha bmd}\mathcal{T}(a)_{m}\subset\mathcal{T}(a)_{m}\cap K_{\alpha mb}$ for all $m\in\mathbb{N}$ and $b\in\mathbb{Z}_{+}.$

Proposition 2.4.

Suppose $(R,\mathfrak{m}),$ $\mathcal{T},$ $\mathcal{T}(a),$ $\alpha$ and $K_{\mu}$ as above. Fix $b^{\prime}\in\mathbb{Z}_{+}.$ For each $a\in\mathbb{Z}_{+},$ let $\mathcal{F}(a)=\{\mathcal{F}(a)_{n}\}$ be a filtration of ideals such that for all $n,$

[TABLE]

Then there exists an integer $b$ such that $b\geq b^{\prime}$ and

[TABLE]

Proof.

We follow the argument given in [6], [7, Proposition 4.3]. For $t\geq 1$ , define the semigroups

[TABLE]

By [6, Lemma 4.5], there exists an integer $b\geq b^{\prime}$ such that the semigroups

[TABLE]

satisfy equations $(5)$ and $(6)$ of [6].

Define $\Gamma_{m}^{(t)}=\Gamma^{(t)}\cap(\mathbb{N}^{d}\times\{m\}),$ $\Gamma(a)^{(t)}_{m}=\Gamma(a)^{(t)}\cap(\mathbb{N}^{d}\times\{m\})$ and $\Gamma(\mathcal{F}(a))^{(t)}_{m}=\Gamma(\mathcal{F}(a))^{(t)}\cap(\mathbb{N}^{d}\times\{m\})$ for $m\in\mathbb{N}$ . For a (strongly nonnegative) sub semigroup $S$ of $\mathbb{Z}^{d}\times\mathbb{N},$ ${\rm con}(S)$ is defined as the closed convex cone which is the closure of the set of all linear combinations $\sum\lambda_{i}s_{i}$ with $s_{i}\in S$ and $\lambda_{i}$ a nonnegative real number and the Newton-Okounkov body is defined as

[TABLE]

This theory is developed in [19], [17] and [15] and is summarized in [6, Section 3]. By [6, Lemma 4.5] and [6, Theorem 3.2],

[TABLE]

Therefore

[TABLE]

Similarly

[TABLE]

and

[TABLE]

Let $\hat{a}=\lfloor a/\max\{n_{1},\ldots,n_{r}\}\rfloor$ where $\lfloor x\rfloor$ be the largest integer smaller than or equal to the real number $x.$ Now

[TABLE]

Hence

[TABLE]

By [6, Theorem 3.3] and since $\hat{a}\to\infty$ as $a\to\infty,$ given $\epsilon>0$ , there exists $a_{0}>0$ such that for all $a\geq a_{0},$ we have

[TABLE]

and

[TABLE]

By equations (2.4.4)-(2.4.9), we get the desired result. ∎

Theorem 2.5.

Let $(R,\mathfrak{m})$ be an analytically irreducible local ring of dimension $d>0$ and for $1\leq l\leq r,$ $\mathcal{I}(l)=\{I(l)_{n}\},$ $\mathcal{J}(l)=\{J(l)_{n}\}$ be (not necessarily Noetherian) filtrations of ideals in $R$ such that $J(l)_{n}\subset I(l)_{n}$ for all $n\in\mathbb{N}$ and $1\leq l\leq r.$ Fix $n_{1},\ldots,n_{r}\in\mathbb{Z}_{+}.$ Suppose there exists an integer $c\in\mathbb{Z}_{+}$ such that for all $i\in\mathbb{N},$

[TABLE]

Let $R_{1}=\oplus_{m_{1},\ldots,m_{r}\geq 0}I(1)_{m_{1}}\cdots I(r)_{m_{r}}$ be integral over $R_{2}=\oplus_{m_{1},\ldots,m_{r}\geq 0}J(1)_{m_{1}}\cdots J(r)_{m_{r}}.$ Then

[TABLE]

Proof.

Let $\hat{R}$ denote the $\mathfrak{m}$ -adic completion of $R.$ Without loss of generality, we may replace $R$ by $\hat{R}$ and $I(l)_{n},J(l)_{n}$ by $I(l)_{n}{\hat{R}},J(l)_{n}{\hat{R}}$ respectively for all $n$ and $1\leq l\leq r.$

For $a\in\mathbb{Z}_{+},$ we define filtrations of ideals

[TABLE]

and

[TABLE]

Now there exists an integer $\underline{a}\in\mathbb{Z}_{+},$ such that $a\leq\underline{a}$ and every element of $\oplus_{i\geq 0}\mathcal{I}(a)_{i}$ is integral over $\oplus_{i\geq 0}\mathcal{J}(\underline{a})_{i}$ where

[TABLE]

Define the following filtrations of ideals

[TABLE]

and

[TABLE]

Note that $\oplus_{n\geq 0}\mathcal{A}_{a,n}$ is finitely generated $\oplus_{n\geq 0}\mathcal{B}_{\underline{a},n}$ -module. Thus by Theorem 2.3,

[TABLE]

Now for all $n,$

[TABLE]

Let $\alpha\in\mathbb{Z}_{+}$ be the integer such that $K_{\alpha n}\cap R\subset\mathfrak{m}^{n}$ for all $n\in\mathbb{N}$ [5, Lemma 4.3]. Therefore

[TABLE]

and hence $\mathcal{A}_{a,n}\cap K_{\alpha cn}=\mathcal{B}_{\underline{a},n}\cap K_{\alpha cn}$ for all $n.$

Now for all $n\in\mathbb{N},$

[TABLE]

Choose an integer $c^{\prime}\geq c$ such that Proposition 2.4 holds for the above two equations. Also note that $i\in\mathbb{N},$

[TABLE]

Replace $c$ by $c^{\prime}$ . Since $\underline{a}\to\infty$ as $a\to\infty,$ by equation (2.5.1) and Proposition 2.4, we get

[TABLE]

∎

It is natural to ask the following.

Question 2.6.

Does Theorem 2.5 hold true for any analytically unramified local ring $?$

The author believes the answer is positive but is unable to prove.

The following example shows that the “if” part of Rees’ theorem 1.2 $(2)$ is not true for non-Noetherian filtrations of ideals in general.

Example 2.7.

Let $R=k[|x,y|]$ be a formal power series ring of dimension two over a field $k.$ Let $\mathcal{I}=\{I_{m}:=x^{m}\}$ and $\mathcal{J}=\{J_{m}:=(x^{m+1},x^{m}y)\}.$ Note that $I_{m}\cap\mathfrak{m}^{2m}=J_{m}\cap\mathfrak{m}^{2m}$ for all $m\geq 0$ and $\lim_{m\to\infty}\lambda(I_{mn}/J_{mn})/m^{2}=0$ but $\oplus_{m\geq 0}I_{m}$ is not integral over $\oplus_{m\geq 0}J_{m}.$

Let $r\geq 2.$ We define a multigraded filtration $\mathcal{I}=\{I_{{\bf{n}}}\}$ where ${\bf{n}}=(n_{1},\ldots,n_{r})\in\mathbb{N}^{r}$ of ideals in a ring $R$ to be a collection of ideals of $R$ such that $R=I_{\bf 0}$ , $I_{{\bf{n}}+{\bf e_{i}}}\subset I_{{\bf{n}}}$ for all $i=1,\ldots,r$ where $e_{i}=(0,\ldots,1,\ldots,0)$ with $1$ at $i$ th position and $I_{{\bf{n}}}I_{{\bf{m}}}\subset I_{{\bf{n}}+{\bf{m}}}$ whenever ${\bf{n}},{\bf{m}}\in\mathbb{N}^{r}$ . As a generalization of Rees’ theorem 1.2, we pose the following question for multigraded filtration of ideals.

Question 2.8.

Let $\{I_{{\bf{n}}}\}$ and $\{J_{{\bf{n}}}\}$ be multigraded filtrations of ideals in a Noetherian local ring such that $J_{{\bf{n}}}\subset I_{{\bf{n}}}$ and ${\lambda}(I_{{\bf{n}}}/J_{{\bf{n}}})<\infty$ for all ${\bf{n}}\in\mathbb{N}^{r}.$ Fix $n_{1},\ldots,n_{r}\in\mathbb{Z}_{+}.$

(1)

Is it true that if $\bigoplus\limits_{{\bf{n}}\in\mathbb{N}^{r}}I_{{\bf{n}}}$ is integral over $\bigoplus\limits_{{\bf{n}}\in\mathbb{N}^{r}}J_{{\bf{n}}}$ then

[TABLE] 2. (2)

Does the converse of $(1)$ hold true if $\bigoplus\limits_{{\bf{n}}\in\mathbb{N}^{r}}I_{{\bf{n}}}$ and $\bigoplus\limits_{{\bf{n}}\in\mathbb{N}^{r}}J_{{\bf{n}}}$ both are Noetherian $?$

3. the degree of the rees polynomial

In this section, we provide bounds for the degree of the function ${\lambda}(I^{n}/J^{n})$ where $J\subsetneq I$ are ideals in a Noetherian local ring with ${\lambda}(I/J)<\infty$ and $J$ is reduction of $I.$ We concentrate on the ideals generated by $d$ -sequences. The notion of $d$ -sequence was introduced by Huneke [13]. Let ${\underline{x}}=x_{1},\ldots,x_{n}$ be a sequence of elements in $R.$ Then ${\underline{x}}$ is called a $d$ -sequence if

(1)

$x_{i}\notin(x_{1},\ldots,{\widehat{x_{i}}},\ldots,x_{n})$ for all $i=1,\ldots,n,$ 2. (2)

for all $k\geq i+1$ and all $i\geq 0,$ $(x_{0}=0)$

[TABLE]

Every regular sequence is a $d$ -sequence but the converse is not true. Every system of parameters in a Buchsbaum local ring is a $d$ -sequence. Examples of $d$ -sequences are abundant. Ideals generated by $d$ -sequences have nice properties, e.g. they are of linear type [13], Castelnuovo-Mumford regularity of Rees algebras of such ideals is zero [25]. For our convenience we omit the condition $(1)$ in the definition of $d$ -sequence. We say, a sequence of elements ${\underline{x}}=x_{1},\ldots,x_{n}$ in $R$ is a $d$ -sequence if condition $(2)$ is satisfied.

*Remark 3.1**.*

Let $J\subsetneq I$ be ideals in a Noetherian local ring $(R,\mathfrak{m})$ with the property that ${\lambda}(I/J)<\infty.$ Suppose $J$ is a reduction of $I$ and $JI^{m}=I^{m+1}$ for all $m\geq l.$ Consider the Veronese subring $A=\bigoplus_{n\geq 0}J^{nl}$ of $\mathcal{R}(J).$ Then $M=\bigoplus_{n\geq 0}I^{nl}/J^{nl}$ is a finite $A$ -module. Therefore there exists an integer $t\in\mathbb{Z}_{+},$ such that $\mathfrak{m}^{t}I^{nl}\subseteq J^{nl}$ for all $n\geq 1.$ Hence $M$ is $A/\mathfrak{m}^{t}A$ -module. Thus $\operatorname{dim}M\leq\operatorname{dim}A/\mathfrak{m}A=\operatorname{dim}\mathcal{R}(J)/\mathfrak{m}\mathcal{R}(J)=l(J).$ Therefore ${\lambda}(I^{nl}/J^{nl})$ is a polynomial type function of degree at most $l(J)-1.$ Hence ${\lambda}(I^{n}/J^{n})$ is a polynomial type function of degree at most $l(J)-1.$

Let $J\subset I$ be ideals in a Noetherian local ring $(R,\mathfrak{m})$ and $a_{1},\ldots,a_{s}$ be a sequence of elements in $J.$ Throughout this section we use the following notation: $R_{i}=R/(a_{0},a_{1},\ldots,a_{i})$ for all $i=0,\ldots,s$ where $a_{0}=0$ and $I_{i},J_{i}$ denote the images of $I,J$ in $R_{i}$ respectively for all $i=0,\ldots,s.$ The next two results are requied to prove our main results.

Lemma 3.2.

Let $(R,\mathfrak{m})$ be a Noetherian local ring, $J\subsetneq I$ be ideals in $R$ and ${\lambda}(I/J)<\infty.$ Let $a_{1},\ldots,a_{s}$ be a sequence of elements in $J.$ Suppose for some $i\in\{0,1,\ldots,s-1\}$ and all $n\gg 0,$ $(J^{n+1}_{i}:(a^{(i)}_{i+1}))\cap I^{n}_{i}=J^{n}_{i}$ where $a^{(i)}_{i+1}$ denotes the image of $a_{i+1}$ in $R_{i}.$ Then for all $n\gg 0,$

[TABLE]

Proof.

For all $n\gg 0,$ consider the following exact sequence

[TABLE]

Since

[TABLE]

and $a^{(i)}_{i+1}I^{n}_{i}+J^{n+1}_{i}\subseteq(a^{(i)}_{i+1})\cap I^{n+1}_{i}+J^{n+1}_{i},$ we have the required inequality. ∎

Proposition 3.3.

Let $x_{1},\ldots,x_{n}$ be a $d$ -sequence in a Noetherian local ring $(R,\mathfrak{m})$ and $J=(x_{1},\ldots,x_{n}).$ Then $J^{n}\cap(x_{1})=x_{1}J^{n-1}$ for all $n\geq 1.$

Proof.

Note that images of $x_{2},\ldots,x_{n}$ is a $d$ -sequence in $R/(x_{1}).$ Let $x_{1}r\in J^{n}\cap(x_{1}).$ Then $x_{1}r=x_{1}a+b$ for some $a\in J^{n-1}$ and $b\in J^{n}_{1}$ where $J_{1}=(x_{2},\ldots,x_{n}).$ By [13, Theorem 2.1], for all $n\geq 1,$

[TABLE]

Therefore $x_{1}r\in x_{1}J^{n-1}.$ ∎

*Remark 3.4**.*

Note that if $J=(x_{1},\ldots,x_{n})$ where $x_{1},\ldots,x_{n}$ is a $d$ -sequence and $\operatorname{grade}(J)=r$ then $x_{1},\ldots,x_{r}$ is a regular sequence.

Next we state some well-known properties of $d$ -sequence. Here $G(J)=\oplus_{n\geq 0}J^{n}/J^{n+1}$ denotes the graded associated ring of $J.$

Proposition 3.5.

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $J$ an ideal in $R$ with $\operatorname{grade}(J)=s.$ Suppose $J$ is generated by a $d$ -sequence $a_{1},\ldots,a_{s},\ldots,a_{t}.$ Then $\operatorname{grade}G(J)_{+}=\operatorname{grade}(J).$

Proposition 3.6.

Let $x_{1},\ldots,x_{s}$ be a $d$ -sequence in a Noetherian local ring $(R,\mathfrak{m})$ and $J$ be an ideal minimally generated by $x_{1},\ldots,x_{s}.$ Suppose $x_{1}$ is a nonzerodivisor on $R.$ Then $l(J/(x_{1}))=l(J)-1.$

The following theorem gives a lower bound for $\deg P(I/J)$ and using the lower bound of $\deg P(I/J)$ we show that if $J$ is a complete intersection ideal then $\deg P(I/J)=l(J)-1.$

Theorem 3.7.

Let $(R,\mathfrak{m})$ be a Noetherian local ring, $J\subsetneq I$ be ideals in $R$ such that ${\lambda}(I/J)<\infty.$ Then the following are true.

(1)

$\operatorname{grade}G(J)_{+}-1\leq\deg P(I/J).$ ** 2. (2)

If $J$ is a reduction of $I$ then $\operatorname{grade}G(J)_{+}-1\leq\deg P(I/J)\leq l(J)-1.$ 3. (3)

If $J$ is a complete intersection ideal and reduction of $I$ then $\deg P(I/J)=l(J)-1.$

Proof.

$(1)$ Let $S=R[X]_{\mathfrak{m}[x]}.$ Then $S$ is faithfully flat extension of $R,$ $S$ has infinite residue field, ${\lambda}_{S}(I^{n}S/J^{n}S)={\lambda}_{R}(I^{n}/J^{n}){\lambda}_{S}(S/\mathfrak{m}S)={\lambda}_{R}(I^{n}/J^{n})$ whenever ${\lambda}_{R}(I^{n}/J^{n})<\infty$ and $\operatorname{grade}G(JS)_{+}\geq\operatorname{grade}G(J)_{+}.$ Therefore without loss of generality we assume that $R$ has infinite residue field. Let $\operatorname{grade}G(J)_{+}=s.$ We may assume $s\geq 1.$ Then there exist $a^{*}_{1},\ldots,a^{*}_{s}\in G(J)_{1}$ such that $a^{*}_{1},\ldots,a^{*}_{s}$ is $G(J)$ -sequence. Therefore for all $i=0,\ldots,s-1$ and $n\geq 0,$ we have $(J^{n+1}_{i}:(a^{(i)}_{i+1}))=J^{n}_{i}$ and hence $(J^{n+1}_{i}:(a^{(i)}_{i+1}))\cap I^{n}_{i}=J^{n}_{i}$ where $a^{(i)}_{i+1}$ denotes the image of $a_{i+1}$ in $R_{i}$ ( $R_{i},I_{i},J_{i}$ are defined before Lemma 3.2). Thus by Lemma 3.2, for all $i=0,\ldots,s-2$ and $n\gg 0,$ we have

[TABLE]

Suppose $\deg P(I/J)<s-1.$ Then using the inequality (3.7.1), for all $i=0,\ldots,s-2,$ we get that ${\lambda}(I^{n}_{s-1}/J^{n}_{s-1})$ is a polynomial type function of degree less than zero and hence there exists an integer $k$ such that $I^{n}_{s-1}=J^{n}_{s-1}$ for all $n\geq k.$

We show that if $I^{m}_{s-1}=J^{m}_{s-1}$ for some integer $m$ then $I^{m-1}_{s-1}=J^{m-1}_{s-1}.$ Let $x^{\prime}\in I^{m-1}_{s-1}$ where $x^{\prime}$ denotes the image of $x$ in $R_{s-1}.$ Then $x^{\prime}a^{(s-1)}_{s}\in I^{m}_{s-1}=J^{m}_{s-1}.$ Thus $x^{\prime}\in J^{m}_{s-1}:a^{(s-1)}_{s}=J^{m-1}_{s-1}.$ Using this technique $m-1$ times, we get $I_{s-1}=J_{s-1}.$ This implies $I=J$ which is a contradiction. Therefore $\operatorname{grade}G(J)_{+}-1\leq\deg P(I/J).$

$(2)$ This follows from part $(1)$ and Remark 3.1.

$(3)$ since $J$ is a complete intersection ideal, $J$ is generated by a regular sequence and by Proposition 3.5, we have $\operatorname{grade}G(J)_{+}=\operatorname{grade}(J)=\operatorname{ht}J=l(J).$ Hence the result follows from part $(2).$ ∎

Example 3.8.

Let $R=K[X,Y]_{(X,Y)}$ where $K$ is a field. Let $J=(XY^{2},X^{4})$ and $I=(X^{4},XY^{2},X^{3}Y).$ Then ${\lambda}(I/J)<\infty$ and $J$ is a reduction of $I.$ Now $\operatorname{grade}G(J)_{+}\geq 1,$ $J$ is generated by a $d$ -sequence. For all $n\gg 0,$

[TABLE]

where $J^{n}=(X^{4n},X^{4n-3}Y^{2},X^{4n-6}Y^{4},\ldots,X^{4n-3n}Y^{2n}).$ Then ${\lambda}(I^{n}/J^{n})$ is a polynomial in $n$ of degree one.**

The following theorem provides a better lower bound for $\deg P(I/J)$ and as a consequence of that we obtain a class of ideals for which the polynomial $P(I/J)(X)$ attains the maximal degree.

Theorem 3.9.

Let $(R,\mathfrak{m})$ be a Noetherian local ring, $J\subsetneq I$ be ideals in $R$ such that ${\lambda}(I/J)<\infty$ and $\operatorname{grade}(J)=s.$ Let $a_{1},\ldots,a_{s},b_{1},\ldots,b_{t}$ be a $d$ -sequence in $R$ which minimally generates the ideal $J$ and $t\geq 1.$ Suppose $I\cap(J^{\prime}:b_{1})=J^{\prime}$ where $J^{\prime}=(a_{1},\ldots,a_{s})$ $(J^{\prime}=(0)$ if $s=0)$ . Then the following are true.

(1)

$\operatorname{grade}(J)\leq\deg P(I/J).$ ** 2. (2)

If $J$ is a reduction of $I$ then $\operatorname{grade}(J)\leq\deg P(I/J)\leq\l(J)-1.$

Proof.

$(1)$ ( $R_{i},I_{i},J_{i}$ are defined before Lemma 3.2).

First we show that if $s=0$ then $H^{0}_{\mathcal{R}(J)_{+}}(\mathcal{R}(I))_{1}=0.$ Let $at\in H^{0}_{\mathcal{R}(J)_{+}}(\mathcal{R}(I))_{1}.$ Then for some $m>0,$ we have $({b^{m}_{1}}t^{m})(at)=0$ in $\mathcal{R}(I).$ Hence $ab^{m}_{1}=0.$ If $m\geq 2,$ since $b_{1},\ldots,b_{t}$ is a $d$ -sequence in $R,$ we have

[TABLE]

Thus $ab^{m-1}_{1}=0.$ Using the same method $m-1$ -times, we get $ab_{1}=0.$ Therefore $a\in I\cap((0):b_{1})=(0).$

Note that images of $a_{i},\ldots,a_{s},b_{1},\ldots,b_{t}$ in $R_{i-1}$ is a $d$ -sequence for all $i=1,\ldots,s.$ Now $\operatorname{grade}(J)=s$ implies $a_{1},\ldots,a_{s}$ is a regular sequence in $R.$ Thus by Proposition 3.3, for all $i=0,\ldots,s-1$ and $n\gg 0,$ we have

[TABLE]

Hence by Lemma 3.2, for all $i=0,\ldots,s-1$ and $n\gg 0,$ we have

[TABLE]

Suppose $\deg P(I/J)\leq s-1.$ Then using the inequality (3.9.1), for all $i=0,\ldots,s-1,$ we get that ${\lambda}(I^{n}_{s}/J^{n}_{s})$ is a polynomial type function of degree less than zero and hence there exists an integer $k$ such that $I^{n}_{s}=J^{n}_{s}$ for all $n\geq k.$

Now replace $R$ by $R_{s}$ and $I,J$ by $I_{s},J_{s}$ respectively. Hence $J\subsetneq I,$ $J=(c_{1},\ldots,c_{t})$ where $c_{1},\ldots,c_{t}$ is a $d$ -sequence in $R$ $(c_{1},\ldots,c_{t}\mbox{ are images of }b_{1},\ldots,b_{t}\mbox{ in }R_{s}\mbox{ respectively}),$ $\operatorname{grade}(J)=0,$ $J^{\prime}=(0)$ and $I\cap((0):c_{1})=(0).$ Hence $H^{0}_{\mathcal{R}(J)_{+}}(\mathcal{R}(I))_{1}=0.$

Consider the exact sequence of $\mathcal{R}(J)$ -modules

[TABLE]

which induces a long exact sequence of local cohomology modules whose $n$ -graded component is

[TABLE]

Consider the case $i=0$ and $n=1.$ Since $I^{n}=J^{n}$ for all $n\gg 0,$ $\mathcal{R}(I)/\mathcal{R}(J)$ is $\mathcal{R}(J)_{+}$ -torsion. Thus $H^{0}_{\mathcal{R}(J)_{+}}(\mathcal{R}(I)/\mathcal{R}(J))_{1}=I/J.$ Since $J$ is generated by $d$ -sequence, by Theorem [25, Corollary 5.2], $\operatorname{reg}(\mathcal{R}(J))=0$ and hence $H^{1}_{\mathcal{R}(J)_{+}}(\mathcal{R}(J))_{1}=0.$ Therefore from the long exact sequence, for $i=0$ and $n=1,$ we get $I=J$ which is a contradiction. Hence $s\leq\deg P(I/J).$

$(2)$ Suppose $J$ is a reduction of $I.$ Then by part $(1)$ and Remark 3.1, we get the required result. ∎

Example 3.10.

Let $R=K[X,Y,Z,W]_{(X,Y,Z,W)}$ where $K$ is a field. Let

[TABLE]

Then $J$ is a reduction of $I$ and ${\lambda}(I/J)<\infty.$ By computations on Macaulay2 [9], we get $\operatorname{grade}(J)=2,$ $I\cap((XZ,YW):(XW+YZ))=(XZ,YW)$ and $XZ,YW,XW+YZ$ is a $d$ -sequence. Consider the ideals $\overline{J}=J/(XZ,YW)$ and $\overline{I}=I/(XZ,YW)$ in $\overline{R}=R/(XZ,YW).$ Let $x,y,z,w$ denote images of $X,Y,Z,W$ in $\overline{R}.$ Then $\overline{J}^{n}=(x^{n}w^{n}+y^{n}z^{n})$ and $\overline{I}^{n}=(x^{n}w^{n},y^{n}z^{n}).$ Hence ${\lambda}(\overline{I}^{n}/\overline{J}^{n})$ is nonzero constant for all $n\gg 0.$ Thus $2\leq\deg P(I/J).$ Since $l(J)=l(I)=\operatorname{dim}K[XZ,XW,YZ,YW]=3,$ we get $\deg P(I/J)=2.$ **

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\geq 1.$ We say an ideal $I$ satisfies $G_{s}$ if $\mu(I_{p})\leq\operatorname{ht}p$ for every $p\in V(I)$ such that $\operatorname{ht}(p)<s.$ A proper ideal $J$ is an $s$ -residual intersection of $I$ if there exist $s$ elements $x_{1},\ldots,x_{s}$ of $I$ such that $J=(x_{1},\ldots,x_{s}):I$ and $\operatorname{ht}J\geq s.$ We say $J$ is a geometric $s$ -residual intersection of $I$ if in addition we have $\operatorname{ht}I+J>s.$ The ideal $I$ is said to satisfy the Artin-Nagata property $AN_{s}^{-}$ if the ring $R/K$ is Cohen-Macaulay for every $0\leq i\leq s$ and for every geometric $i$ -residual intersection $K$ of $I.$

Corollary 3.11.

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\geq 2$ with infinite residue field, $I$ be an ideal of $R$ and $J$ be any minimal reduction of $I$ with ${\lambda}(I/J)<\infty.$ Suppose $I$ satisfies $G_{l(I)}$ and $AN_{l(I)-2}^{-}$ conditions. Then for any ideal $K$ with $J\subsetneq K\subseteq I,$ $\deg P(K/J)=l(J)-1.$

Proof.

By [26] and [14], $\operatorname{grade}(J)\geq l(I)-1,$ $x_{1},\ldots,x_{l(I)}$ is a $d$ -sequence and

[TABLE]

If $\operatorname{grade}(J)=l(I)$ then by Theorem 3.7, $\deg P(K/J)=l(J)-1.$ Suppose $\operatorname{grade}(J)=l(I)-1.$ Then by Theorem 3.9, we obtain $\deg P(K/J)=l(J)-1.$ ∎

4. degree of the multiplicity function $e(I^{n}/J^{n})$

Let $(R,\mathfrak{m})$ be a formally equidimensional local ring and $J\subsetneq I$ be ideals in $R.$ Then by [2, Lemma 2.2], the chain of ideals

[TABLE]

stabilizes. Choose an integer $r\geq 1$ such that $K:=\sqrt{J^{n}:I^{n}}=\sqrt{J^{n+1}:I^{n+1}}$ for all $n\geq r.$ Put $\operatorname{dim}R/K=t.$ Then using associativity formula, for all $n\geq r,$ we have

[TABLE]

By [2, Proposition 2.4], $e(I^{n}/J^{n})$ is a function of polynomial type of degree at most $\operatorname{dim}R-t.$

Throughout this section we use the following notation: $K=\sqrt{J^{n}:I^{n}}=\sqrt{J^{n+1}:I^{n+1}}$ for all $n\geq r,$ $t=\operatorname{dim}R/K$ and $S=\{p\in\operatorname{Spec}R:K\subseteq p,\operatorname{dim}R/p=t\}.$

Next we show that if $J$ is a complete intersection ideal we can explicitly detect the $\deg e(I^{n}/J^{n}).$

Proposition 4.1.

Let $(R,\mathfrak{m})$ be a formally equidimensional local ring, $J\subsetneq I$ be ideals in $R.$

(1)

Then $\operatorname{grade}G(J)_{+}-1\leq\deg e(I^{n}/J^{n})\leq l(J).$ 2. (2)

If $J$ is a reduction of $I$ then $\operatorname{grade}G(J)_{+}-1\leq\deg e(I^{n}/J^{n})\leq l(J)-1.$ 3. (3)

If $J$ is a complete intersection ideal then

$(a)$

$J$ * is a reduction of $I$ implies $\deg e(I^{n}/J^{n})=l(J)-1.$ * 2. $(b)$

$J$ * is not a reduction of $I$ implies $\deg e(I^{n}/J^{n})=l(J).$ *

Proof.

$(1)$ ( $K,S$ are defined at the beginning of the Section $3$ ). Suppose $J_{p}$ is reduction of $I_{p}$ for all $p\in S$ then by Theorem 3.7, for all $p\in S,$

[TABLE]

Hence from the equation (4.0.1), $\operatorname{grade}G(J)_{+}-1\leq\deg e(I^{n}/J^{n})\leq l(J)-1.$

Suppose $J_{q}$ is not a reduction of $I_{q}$ for some $q\in S.$ Therefore $J$ is not a reduction of $I$ and by [8, Proposition 3.6.3], for all $n\gg 0,$ we have $\operatorname{ht}K\leq\operatorname{ht}{\sqrt{JI^{n-1}:I^{n}}}\leq l(J).$ Then by [22, Theorem 2.1], $\operatorname{ht}J\leq\deg P(I_{q}/J_{q})=\operatorname{ht}q\leq l(J).$ Therefore the result follows from the equation (4.0.1).

$(2)$ If $J$ is a reduction of $I$ then $J_{p}$ is reduction of $I_{p}$ for all $p\in S$ and the result follows from the argument in $(1).$

$(3)$ Since $J$ is a complete intersection ideal, we have $\operatorname{grade}(J)=l(J).$ Therefore by Proposition 3.5 and part $(2),$ we get that $J$ is a reduction of $I$ implies $\deg e(I^{n}/J^{n})=l(J)-1.$ If $J$ is not a reduction of $I,$ then the result follows from [3, Theorem 2.6]. ∎

Lemma 4.2.

Let $(R,\mathfrak{m})$ be a formally equidimensional local ring, $J\subsetneq I$ be ideals in $R.$ Suppose there exists a prime ideal $Q\supseteq J$ such that $\operatorname{dim}R/J=\operatorname{dim}R/Q$ and $I\nsubseteq Q.$ Then $\deg e(I^{n}/J^{n})=\operatorname{ht}J.$ In particular, if $J$ is unmixed (i.e. all associated primes of $J$ have same height) and $\sqrt{J}\subsetneq\sqrt{I},$ then $\deg e(I^{n}/J^{n})=\operatorname{ht}J.$

Proof.

Let $\mbox{Min}(R/K)=\{Q_{1},\ldots,Q_{k}\}$ ( $K$ is defined at the beginning of the Section $3$ ). Put $T=Q_{1}\cdots Q_{k}.$ Since $K=\sqrt{J^{r}:I^{r}},$ there exists an integer $c>0$ such that $T^{c}I^{r}\subseteq J^{r}.$ Let $J=\bigcap\limits_{P\in Ass(R/J)}q(P)$ be an irredundant primary decomposition of $J.$ Then

[TABLE]

Since $I\nsubseteq Q,$ we have $T\subseteq Q.$ Therefore $Q_{j}\subseteq Q$ for some $j$ and hence $Q=Q_{j}.$ Thus $J_{Q_{j}}$ is not a reduction of $I_{Q_{j}}=R_{Q_{j}}$ and $\operatorname{dim}R/Q=\operatorname{dim}R/K.$ Hence by [22, Theorem 2.1] and the equation (4.0.1), we get $\deg e(I^{n}/J^{n})=\deg{\lambda}(I^{n}_{Q_{j}}/J^{n}_{Q_{j}})=\operatorname{dim}R_{Q_{j}}=\operatorname{ht}J.$ ∎

We provide some sufficient conditions on $J$ such that for any $J\subsetneq I,$ we have $\sqrt{J:I}=\sqrt{J^{n}:I^{n}}$ for all $n\geq 1.$

Proposition 4.3.

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $J\subsetneq I$ be ideals in $R$ such that one of the following conditions hold,

(1)

the residue field of $R$ is infinite and $\operatorname{grade}G(J)_{+}\geq 1,$ 2. (2)

$\operatorname{grade}(J)\geq 1$ * and $J$ is generated by $d$ -sequence,* 3. (3)

$J^{k+1}:J=J^{k}$ * for all * $k\geq 0.$

Then $\sqrt{J:I}=\sqrt{J^{n}:I^{n}}$ for all $n\geq 1.$

Proof.

By [2, Lemma 2.2], we know $\sqrt{J:I}\subseteq\sqrt{J^{n}:I^{n}}$ for all $n\geq 1.$ Suppose $x\in\sqrt{J^{n}:I^{n}}$ for any $n>1.$ Then $x^{r}I^{n}\subseteq J^{n}$ for some $r\geq 1.$

First consider the case that $\operatorname{grade}G(J)_{+}\geq 1$ and the residue field of $R$ is infinite. Then there exists an element $y\in J$ such that $J^{n}:(y)=J^{n-1}$ for all $n\geq 1.$ Then $x^{r}yI^{n-1}\subseteq x^{r}II^{n-1}\subseteq J^{n}$ implies $x^{r}I^{n-1}\subseteq J^{n}:(y)=J^{n-1}.$ Using this technique $n-1$ times we get $x^{r}I\subseteq J.$

Note that in condition $(1),$ we require infinite residue field just to get a $G(J)$ -regular element of degree $1.$ If $\operatorname{grade}(J)\geq 1$ and $J$ is generated by a $d$ -sequence, say $J=(a_{1},\ldots,a_{s}),$ by Propositions 3.3 and 3.5, we get a $G(J)$ -regular element of degree $1.$ Hence the result follows from the previous paragraph.

If $(3)$ holds then for all $n\geq 1,$ $x^{r}JI^{n-1}\subseteq x^{r}II^{n-1}\subseteq J^{n}$ implies $x^{r}I^{n-1}\subseteq J^{n}:J=J^{n-1}.$ ∎

The following theorem gives characterization of reduction in terms of $\deg e(I^{n}/J^{n}).$

Theorem 4.4.

Let $(R,\mathfrak{m})$ be a formally equidimensional local ring of dimension $d\geq 2,$ $J\subsetneq I$ be ideals in $R$ and $J$ has analytic deviation one. Suppose $l(J_{p})<l(J)$ for all prime ideals $p$ in $R$ such that $\operatorname{ht}p=l(J).$ Then the following are true.

(1)

If $J$ is not a reduction of $I$ then $\deg e(I^{n}/J^{n})=l(J)-1.$ 2. (2)

*If $l(J)=d-1,$ $\operatorname{depth}(R/J)>0$ and for all $n\geq 1,$ $\sqrt{J:I}=\sqrt{J^{n}:I^{n}}$ then *

$J$ * is a reduction of $I$ if and only if $\deg e(I^{n}/J^{n})\leq l(J)-2.$ *

Proof.

(Note that the condition $l(J_{p})<l(J)$ for all prime ideals $p$ in $R$ such that $\operatorname{ht}p=l(J)$ implies $l(J)\leq d-1.$ ) First we show that if $\operatorname{ht}K=l(J)$ then $J$ is a reduction of $I$ (where $K=\sqrt{J^{r}:I^{r}},$ defined at the beginning of the section $3$ ). Without loss of generality we may asumme that $\operatorname{Ass}(R/{\overline{J^{n}}})=\operatorname{Ass}(R/{\overline{J^{n+1}}})$ for all $n\geq r.$ Let $q\in$ Ass $(R/{\overline{J^{r}}}).$ We show that $(J^{r}:I^{r})\nsubseteq q.$ If possible suppose $(J^{r}:I^{r})\subseteq q.$ Then either $\operatorname{ht}q>l(J)\geq l(J_{q})=l(J^{r}_{q})$ or $\operatorname{ht}q=l(J)>l(J_{q})=l(J^{r}_{q}).$ Therefore by [18], [22, Lemma 3.1], $q\notin\operatorname{Ass}(R/\overline{J^{nr}})=\operatorname{Ass}(R/\overline{J^{r}})$ for any $n\geq 1,$ which is a contradiction. Thus by prime avoidance lemma, we have an element

[TABLE]

Therefore $x^{\prime}$ is a nonzerodivisor in $R/\overline{J^{r}}$ where $\prime$ denotes the image in $R/\overline{J^{r}},$ i.e. $\overline{J^{r}}:(x)=\overline{J^{r}}.$ This implies $I^{r}\subseteq J^{r}:(x)\subseteq\overline{J^{r}}:(x)=\overline{J^{r}}.$ Thus $J^{r}$ is a reduction of $I^{r}$ and hence $J$ is a reduction of $I.$

$(1)$ Since $J$ is not a reduction of $I,$ by [8, Proposition 3.6.3] and the previous paragraph, $\operatorname{ht}K=\operatorname{ht}J$ (where $K=\sqrt{J^{r}:I^{r}},$ defined at the beginning of the section $3$ ). Now $J$ is not a reduction of $I$ implies $J^{r}$ is not a reduction of $I^{r}.$ Therefore, by [18],[23],[27, Theorem 8.21], we have $J^{r}_{p}$ is not a reduction of $I^{r}_{p}$ for some prime ideal $p$ such that $K\subset p$ and $\operatorname{ht}p=l(J_{p}).$

Let $K\subset p\in\operatorname{Spec}(R)$ such that $\operatorname{ht}p\geq l(J)=\operatorname{ht}K+1.$ Then either $\operatorname{ht}p>l(J)\geq l(J_{p})$ or $\operatorname{ht}p=l(J)>l(J_{p}).$ Therefore $J^{r}_{p}$ is not a reduction of $I^{r}_{p}$ for some prime ideal $p\in S$ ( $S$ is defined at the beginning of the section $3$ ). Hence $J_{p}$ is not a reduction of $I_{p}$ for some prime ideal $p\in S$ and thus by [22, Theorem 2.1], $\deg e(I^{n}/J^{n})=\operatorname{ht}p=\operatorname{ht}J=l(J)-1.$

$(2)$ It is enough to show that if $J$ is a reduction of $I$ then $\deg e(I^{n}/J^{n})\leq l(J)-2.$ First we show that if $J$ is a reduction of $I$ then $\operatorname{ht}K\leq l(J).$ If $\operatorname{ht}K>l(J)$ then $K$ is $\mathfrak{m}$ -primary and hence there exists an integer $n>0$ such that $\mathfrak{m}^{n}I\subseteq J.$ Since $\mathfrak{m}\notin\operatorname{Ass}(R/J),$ we have $I\subseteq J$ which is a contradiction. Suppose $\operatorname{ht}K=\operatorname{ht}J=l(J)-1.$ Since $J$ is a reduction of $I$ , $I_{p},J_{p}$ are $pR_{p}$ -primary for all $p\in S$ and hence we get $\deg e(I^{n}/J^{n})\leq l(J)-2.$ Now suppose $\operatorname{ht}K=l(J).$ Since $J$ is a reduction of $I,$ for any $p\in S,$ we have $\operatorname{ht}p=l(J)$ and hence $\deg P(I_{p}/J_{p})\leq l(J_{p})-1\leq l(J)-2.$ Thus $\deg e(I^{n}/J^{n})\leq l(J)-2.$ ∎

Example 4.5.

We can not omit the condition $\operatorname{depth}(R/J)>0$ from Theorem 4.4 $(2)$ .

Let $R=\mathbb{Q}[x,y,z,w]_{(x,y,z,w)}$ and $J=(xz,yw,xw+yz).$ Then $3=l(J)=\operatorname{ht}J+1,$ $\mathfrak{m}\in$ Ass $(R/J)$ and $l(J_{p})\leq l(J)-1$ for all prime ideals $p$ in $R$ such that $\operatorname{ht}p=l(J).$ Let $I=(xz,xw,yz,yw).$ Then $J$ is a reduction of $I$ and $\sqrt{J:I}=(x,y,z,w).$ Since $xz,yw,xw+yz$ is a $d$ -sequence, by Proposition 4.3, $\sqrt{J:I}=\sqrt{J^{n}:I^{n}}$ for all $n\geq 1.$ Then $\deg e(I^{n}/J^{n})=\deg{\lambda}(I^{n}/J^{n})=2.$ **

Example 4.6.

This example shows existence of an ideal $J$ which satisfies conditions of Theorem 4.4. Let $R=\mathbb{Q}[x,y,z,w]_{(x,y,z,w)}$ and $J=(xyw^{2},xyz^{2},xw^{2}+yz^{2}).$ Then by computations on Macaulay2 [9], we get $l(J)=3,$ $J$ is an ideal of linear type (hence basic) and Ass $(R/J)=\{p_{1}=(w,z),p_{2}=(w,y),p_{3}=(x,z),p_{4}=(x,y)\}.$ Since $xyzw\in R\setminus J$ is integral over $J,$ $J$ is not integrally closed. Note that $J_{p_{1}}=(w^{2},z^{2})R_{p_{1}},J_{p_{2}}=(y,w^{2})R_{p_{2}},J_{p_{3}}=(x,z^{2})R_{p_{3}},J_{p_{4}}=(xy,xw^{2}+yz^{2})R_{p_{4}}.$ Hence $J$ is generically complete intersection. Therefore by [4, Lemma 2.5], $\operatorname{grade}G(J)_{+}\geq 1.$ Thus by Proposition 4.3, $\sqrt{J:I}=\sqrt{J^{n}:I^{n}}$ for all $n\geq 1$ and any ideal $I$ containing $J.$ Suppose $p$ is a prime ideal in $R$ of height $3.$ Then either $x\notin p\mbox{ (hence }J_{p}=(yz^{2},w^{2})R_{p})$ or $y\notin p\mbox{ (hence }J_{p}=(xw^{2},z^{2})R_{p})$ or $z\notin p\mbox{ (hence }J_{p}=(xy,xw^{2}+yz^{2})R_{p})$ or $w\notin p\mbox{ (hence }J_{p}=(xy,xw^{2}+yz^{2})R_{p}).$ Thus $l(J_{p})\leq l(J)-1.$ **

Bibliography27

The reference list from the paper itself. Each links out to its DOI / PubMed record.

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TL;DR

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Rees’ theorem for filtrations, multiplicity function and reduction criteria

Abstract.

2010 Mathematics Subject Classification:

1. introduction

Theorem 1.1**.**

Theorem 1.2**.**

Theorem 1.3**.**

Theorem 1.4**.**

Theorem 1.5**.**

Theorem 1.6**.**

Corollary 1.7**.**

Theorem 1.8**.**

Theorem 1.9**.**

2. Rees’ theorem for filtrations

Lemma 2.1**.**

Proof.

Lemma 2.2**.**

Proof.

Theorem 2.3**.**

Proof.

Proposition 2.4**.**

Proof.

Theorem 2.5**.**

Proof.

Question 2.6**.**

Example 2.7**.**

Question 2.8**.**

3. the degree of the rees polynomial

Remark 3.1*.*

Lemma 3.2**.**

Proof.

Proposition 3.3**.**

Proof.

Remark 3.4*.*

Proposition 3.5**.**

Proposition 3.6**.**

Theorem 3.7**.**

Proof.

Example 3.8**.**

Theorem 3.9**.**

Proof.

Example 3.10**.**

Corollary 3.11**.**

Proof.

4. degree of the multiplicity function e(In/Jn)e(I^{n}/J^{n})e(In/Jn)

Proposition 4.1**.**

Proof.

Lemma 4.2**.**

Proof.

Proposition 4.3**.**

Proof.

Theorem 4.4**.**

Proof.

Example 4.5**.**

Example 4.6**.**

Theorem 1.1.

Theorem 1.2.

Theorem 1.3.

Theorem 1.4.

Theorem 1.5.

Theorem 1.6.

Corollary 1.7.

Theorem 1.8.

Theorem 1.9.

Lemma 2.1.

Lemma 2.2.

Theorem 2.3.

Proposition 2.4.

Theorem 2.5.

Question 2.6.

Example 2.7.

Question 2.8.

*Remark 3.1**.*

Lemma 3.2.

Proposition 3.3.

*Remark 3.4**.*

Proposition 3.5.

Proposition 3.6.

Theorem 3.7.

Example 3.8.

Theorem 3.9.

Example 3.10.

Corollary 3.11.

4. degree of the multiplicity function $e(I^{n}/J^{n})$

Proposition 4.1.

Lemma 4.2.

Proposition 4.3.

Theorem 4.4.

Example 4.5.

Example 4.6.