Mixed multiplicities and the multiplicity of rees modules of reductions (published in j. Algebra appl.)

TL;DR

This paper establishes that mixed multiplicities and the multiplicity of Rees modules are equal for good filtrations and their reductions, leading to new insights in algebraic multiplicity theory.

Contribution

It proves the equality of mixed multiplicities and Rees module multiplicities for good filtrations and their reductions, a novel result in algebraic multiplicity theory.

Findings

Mixed multiplicities equal the multiplicity of Rees modules for good filtrations.

Results apply to reductions of filtrations, simplifying multiplicity calculations.

Provides new theorems on the structure of Rees modules and their multiplicities.

Abstract

This paper shows that mixed multiplicities and the multiplicity of Rees modules of good filtrations and that of their reductions are the same. As an application of this result, we obtain interesting results on mixed multiplicities and the multiplicity of Rees modules.

Equations

$$\mathbf{n}=(n_1,\dots ,n_s);\mathbf{k}=(k_1,\dots ,k_s);0=(0,\dots ,0);1=(1,\dots ,1);$$

$${\mathbf{n}}^{\mathbf{k}}=n_1^{k_1}\cdots n_s^{k_s};\mathbf{k}!=k_1!\cdots k_s!;|\mathbf{k}|=k_1+\cdots +k_s;$$

$$\mathbf{I}=I_1,\dots ,I_s;{\mathbf{I}}^{[\mathbf{k}]}=I_1^{[k_1]},\dots ,I_s^{[k_s]};{\mathbb{I}}^{\mathbf{n}}=I_1^{n_1}\cdots I_s^{n_s};$$

$$\mathbf{F}=F_1,\dots ,F_s;{\mathbf{F}}^{[\mathbf{k}]}=F_1^{[k_1]},\dots ,F_s^{[k_s]};{\mathbb{F}}_{\mathbf{n}}=(F_1{)}_{n_1}\cdots (F_s{)}_{n_s}.$$

$$\mathfrak{R}(\mathbf{I};A)=\underset{\mathbf{n}⩾0}{⨁}{\mathbb{I}}^{\mathbf{n}}{T}^{\mathbf{n}}\text{ and }\mathfrak{R}(\mathbf{I};M)=\underset{\mathbf{n}⩾0}{⨁}{\mathbb{I}}^{\mathbf{n}}M{T}^{\mathbf{n}}$$

$$e((\mathfrak{I},\mathfrak{R}(\mathbf{F};A{)}_{+});\mathfrak{R}(\mathbf{F};M))=e((\mathfrak{I},\mathfrak{R}(\mathbf{I};A{)}_{+});\mathfrak{R}(\mathbf{I};M)).$$

$$A=(F{)}_0(F{)}_1\supseteq \cdots \supseteq (F{)}_n\cdots$$

$$F_1=\{(F_1{)}_n{\}}_{n⩾0},\dots ,F_s=\{(F_s{)}_n{\}}_{n⩾0}$$

$$\mathbf{n}=(n_1,\dots ,n_s);\mathbf{k}=(k_1,\dots ,k_s);0=(0,\dots ,0);1=(1,\dots ,1)\in {\mathbb{N}}^s;$$

$${\mathbf{n}}^{\mathbf{k}}=n_1^{k_1}\cdots n_s^{k_s};\mathbf{k}!=k_1!\cdots k_s!;|\mathbf{k}|=k_1+\cdots +k_s;$$

$$\mathbf{I}=I_1,\dots ,I_s;{\mathbf{I}}^{[\mathbf{k}]}=I_1^{[k_1]},\dots ,I_s^{[k_s]};{\mathbb{I}}^{\mathbf{n}}=I_1^{n_1}\cdots I_s^{n_s};$$

$$\mathbf{F}=F_1,\dots ,F_s;{\mathbf{F}}^{[\mathbf{k}]}=F_1^{[k_1]},\dots ,F_s^{[k_s]};{\mathbb{F}}_{\mathbf{n}}=(F_1{)}_{n_1}\cdots (F_s{)}_{n_s}.$$

$$\mathfrak{R}(\mathbf{I};A)=\underset{\mathbf{n}⩾0}{⨁}{\mathbb{I}}^{\mathbf{n}}{T}^{\mathbf{n}}\text{ and }\mathfrak{R}(\mathbf{I};M)=\underset{\mathbf{n}⩾0}{⨁}{\mathbb{I}}^{\mathbf{n}}M{T}^{\mathbf{n}}$$

$$\mathfrak{R}(\mathbf{F};A)=\underset{\mathbf{n}⩾0}{⨁}{\mathbb{F}}_{\mathbf{n}}{T}^{\mathbf{n}}\text{ and }\mathfrak{R}(\mathbf{F};M)=\underset{\mathbf{n}⩾0}{⨁}{\mathbb{F}}_{\mathbf{n}}M{T}^{\mathbf{n}}$$

$$\mathcal{S}=\underset{n_0⩾0,\mathbf{n}⩾0}{⨁}\frac{(F{)}_{n_0}{\mathbb{F}}_{\mathbf{n}}}{(F{)}_{n_0+1}{\mathbb{F}}_{\mathbf{n}}}$$

$$B(n_{0},{\bf n};F,\mathbf{F};M)=\ell_{A}\Big{(}\frac{(F)_{n_{0}}\mathbb{F}_{\bf n}M}{(F)_{n_{0}+1}\mathbb{F}_{\bf n}M}\Big{)}.$$

$$\frac{(F{)}_{n_0+c}{\mathbb{F}}_{\mathbf{n}+c1}M}{(F{)}_{n_0+1+c}{\mathbb{F}}_{\mathbf{n}+c1}M}=\frac{J^{n_0}{\mathbb{I}}^{\mathbf{n}}{\mathcal{I}}_{c}M}{J^{n_0+1}{\mathbb{I}}^{\mathbf{n}}{\mathcal{I}}_{c}M},$$

$$B(n_0+c,\mathbf{n}+c1;F,\mathbf{F};M)=B(n_0,\mathbf{n};J,\mathbf{I};{\mathcal{I}}_{c}M)$$

$$\dim {\mathcal{I}}_{c}M/0_{{\mathcal{I}}_{c}M}:I^{\infty }$$

$$=\dim A/(0_M:I^{\infty }):M=\dim M/0_M:I^{\infty }=q.$$

$$\sum _{k_0+|\mathbf{k}|=q-1}e(F^{[k_0+1]},{\mathbf{F}}^{[\mathbf{k}]};M)\frac{n_0^{k_0}{\mathbf{n}}^{\mathbf{k}}}{k_0!\mathbf{k}!},$$

$$e((\mathfrak{I},\mathfrak{R}(\mathbf{F};A{)}_{+});\mathfrak{R}(\mathbf{F};M))=e((\mathfrak{I},\mathfrak{R}(\mathbf{I};A{)}_{+});\mathfrak{R}(\mathbf{I};M)).$$

$$B(n_0+c,\mathbf{n}+c1;F,\mathbf{F};M)=B(n_0,\mathbf{n};J,\mathbf{I};{\mathcal{I}}_{c}M)$$

$$e(F^{[k_0+1]},{\mathbf{F}}^{[\mathbf{k}]};M)=e(J^{[k_0+1]},{\mathbf{I}}^{[\mathbf{k}]};{\mathcal{I}}_{c}M).$$

$$\overline{M}=M/0_M:I^{\infty }.$$

$$e(J^{[k_0+1]},{\mathbf{I}}^{[\mathbf{k}]};{\mathcal{I}}_{c}M)=e(J^{[k_0+1]},{\mathbf{I}}^{[\mathbf{k}]};{\mathcal{I}}_c\overline{M}).$$

$$0\to {\mathcal{I}}_c\overline{M}\to \overline{M}\to \overline{M}/{\mathcal{I}}_c\overline{M}\to 0.$$

$$\dim \overline{M}/{\mathcal{I}}_c\overline{M}<\dim \overline{M}.$$

$$e(J^{[k_0+1]},{\mathbf{I}}^{[\mathbf{k}]};{\mathcal{I}}_c\overline{M})=e(J^{[k_0+1]},{\mathbf{I}}^{[\mathbf{k}]};\overline{M}).$$

$$e(J^{[k_0+1]},{\mathbf{I}}^{[\mathbf{k}]};\overline{M})=e(J^{[k_0+1]},{\mathbf{I}}^{[\mathbf{k}]};M).$$

$$e(F^{[k_0+1]},{\mathbf{F}}^{[\mathbf{k}]};M)=e(J^{[k_0+1]},{\mathbf{I}}^{[\mathbf{k}]};M).$$

$$I_c=(F_1{)}_c\cdots (F_s{)}_c.$$

$$0={\mathcal{M}}_0\subseteq {\mathcal{M}}_1\subseteq {\mathcal{M}}_2\subseteq \cdots \subseteq {\mathcal{M}}_u=\frac{I_c\mathfrak{R}(\mathbf{F};M)}{{\mathfrak{V}}^n{I}_c\mathfrak{R}(\mathbf{F};M)}$$

Full text

MIXED MULTIPLICITIES AND THE MULTIPLICITY

OF REES MODULES OF REDUCTIONS

(Published in J. Algebra Appl.)

Truong Thi Hong Thanh and Duong Quoc Viet

Department of Mathematics, Hanoi National University of Education

136 Xuan Thuy Street, Hanoi, Vietnam

Email: thanhtth@gmail.com, vduong99@gmail.com

ABSTRACT: This paper shows that mixed multiplicities and the multiplicity of Rees modules of good filtrations and that of their reductions are the same. As an application of this result, we obtain interesting results on mixed multiplicities and the multiplicity of Rees modules.

1 Introduction

In past years, mixed multiplicities and the multiplicity of Rees modules have attracted much attention (see e.g. 2, 6$-$9, 13$-$31). However up to now, results on the relationship between mixed multiplicities (multiplicities of Rees modules) and mixed multiplicities (multiplicities of Rees modules) of reductions, even in the case of ideals, are still restricted. In this paper, wishing to have broader applications, we consider this problem in the context of good filtrations.

00footnotetext:

Mathematics Subject Classification (2010): Primary 13H15. Secondary 13C15, 13D40, 14C17.

Key words and phrases: Multiplicity, mixed multiplicity, good filtration, Rees ring.

Let $(A,\mathfrak{m})$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$; $M$ be a finitely generated $A$-module. A filtration of ideals $F=\{(F)_{n}\}_{n\geqslant 0}$ in $A$ is a decreasing chain of ideals $A=(F)_{0}\varsupsetneq(F)_{1}\supseteq\cdots\supseteq(F)_{n}\cdots$ such that $(F)_{m}(F)_{n}\subseteq(F)_{m+n}$ for all $m,n\geqslant 0.$ For any ideal $I$ of $A,$ the filtration $\{I^{n}\}_{n\geqslant 0}$ determined by the powers of $I$ is called the $I$-adic filtration. Let $I_{1},\ldots,I_{s}$ be ideals and $F_{1}=\{(F_{1})_{n}\}_{n\geqslant 0},\ldots,$ $F_{s}=\{(F_{s})_{n}\}_{n\geqslant 0}$ be filtrations in $A$. Put

[TABLE]

Set $T^{\bf n}=t_{1}^{n_{1}}\cdots t_{s}^{n_{s}},$ here $t_{i}$ is a variable over $A$ for all $i=1,\ldots,s.$ Denote by

[TABLE]

the multi-Rees algebra and the multi-Rees module of the ideals $\mathbf{I}$ with respect to $M$, respectively; and by $\mathfrak{R}(\mathrm{\bf F};A)=\bigoplus_{\bf n\geqslant 0}\mathbb{F}_{\mathrm{\bf n}}T^{\bf n}\;\text{ and }\;\mathfrak{R}(\mathrm{\bf F};M)=\bigoplus_{\bf n\geqslant 0}\mathbb{F}_{\mathrm{\bf n}}MT^{\bf n}$ the multi-Rees algebra and the multi-Rees module with respect to $M$ of the filtrations $\mathbf{F}$, respectively. Set $\mathfrak{R}(\mathrm{\bf I};A)_{+}=\bigoplus_{|{\bf n}|>0}\mathbb{I}^{\mathrm{\bf n}}T^{\bf n}$ and $\mathfrak{R}(\mathrm{\bf F};A)_{+}=\bigoplus_{|{\bf n}|>0}\mathbb{F}_{\mathrm{\bf n}}T^{\bf n}$.

A filtration $F$ is called an $I$-good filtration if $I(F)_{n}\subseteq(F)_{n+1}$ for all $n\geqslant 0$ and $(F)_{n+1}=I(F)_{n}$ for all large $n$. In this case, $I$ is also called a reduction of $F$. The filtration $F$ is called a good filtration if it is an $I$-good filtration for some ideal $I.$ A filtration $F$ is called $\mathfrak{m}$-primary if $(F)_{1}$ is an $\mathfrak{m}$-primary ideal. Let $F$ be an $\mathfrak{m}$-primary good filtration and $\mathbf{F}$ be good filtrations of ideals in $A$ such that $I=(F_{1})_{1}\cdots(F_{s})_{1}$ is not contained in $\sqrt{\mathrm{Ann}M}$. Set $q=\dim M/0_{M}:I^{\infty}$. Then the paper shows that \ell_{A}\Big{(}\frac{(F)_{n_{0}}\mathbb{F}_{\bf n}M}{(F)_{n_{0}+1}\mathbb{F}_{\bf n}M}\Big{)} is a polynomial of total degree $q-1$ in $n_{0},\bf n$ for all large $n_{0},\bf n$ (see Proposition 2.2). Now if we write the terms of total degree $q-1$ in this polynomial in the form $\sum_{k_{0}+|\mathbf{k}|=q-1}e(F^{[k_{0}+1]},\mathbf{F}^{[\mathbf{k}]};M)\frac{n_{0}^{k_{0}}\mathbf{n}^{\mathbf{k}}}{k_{0}!\mathbf{k}!},$ then $e(F^{[k_{0}+1]},\mathbf{F}^{[\mathbf{k}]};M)$ are non-negative integers not all zero. And $e(F^{[k_{0}+1]},\mathbf{F}^{[\mathbf{k}]};M)$ is called the mixed multiplicity of $M$ with respect to the good filtrations $F,\mathbf{F}$ of the type $(k_{0}+1,\mathbf{k})$. Remember that in the case where $F$ is a $J$-adic filtration and $F_{i}$ is an $I_{i}$-adic filtration for all $i=1,\ldots,s$, $e(F^{[k_{0}+1]},\mathbf{F}^{[\mathbf{k}]};M)$ is denoted by $e(J^{[k_{0}+1]},\mathbf{I}^{[\mathbf{k}]};M)$ and called the mixed multiplicity of $M$ with respect to the ideals $J,\mathbf{I}$ of the type $(k_{0}+1,\mathbf{k})$ (see e.g. [9, 17, 24]). And as one might expect, we obtain the following result which is also the main theorem of this paper.

Theorem 1.1** (Theorem 2.3).**

Let $F$ be an $\mathfrak{m}$-primary good filtration and $\mathbf{F}$ be good filtrations of ideals in $A$ such that $I=(F_{1})_{1}\cdots(F_{s})_{1}$ is not contained in $\sqrt{\mathrm{Ann}M}$. Let $J,I_{1},\ldots,I_{s}$ be reductions of $F,F_{1},\ldots,F_{s}$, respectively. Let $\mathfrak{I}$ be an $\mathfrak{m}$-primary ideal of $A$. Then the following statements hold.

1. (i)

$e(F^{[k_{0}+1]},\mathbf{F}^{[\mathbf{k}]};M)=e(J^{[k_{0}+1]},\mathbf{I}^{[\mathbf{k}]};M)$. 2. (ii)

Assume that $\mathrm{ht}\dfrac{I+\mathrm{Ann}M}{\mathrm{Ann}M}>0.$ Then we have

[TABLE]

Now, if $\overline{\mathfrak{R}(I;A)}$ denotes the integral closure of the Rees algebra $\mathfrak{R}(I;A)=A[It]$ in the polynomial ring $A[t],$ then applying Theorem 1.1, we obtain the following.

Corollary 1.2** (Corollary 2.4).**

Let $A$ be an analytically unramified ring and $I$ an ideal of positive height of $A.$ Then $e(\overline{\mathfrak{R}(I;A)})=e(\mathfrak{R}(I;A)).$

Moreover, using Theorem 1.1, we easily obtain results for multiplicities of good filtrations: expressing mixed multiplicities of modules in terms of the multiplicity of joint reductions (Corollary 2.5); the multiplicity of Rees modules as the sum of the mixed multiplicities (Corollary 3.1); the additivity on exact sequences (Corollary 3.2) and the additivity and reduction formula (Corollary 3.3) for mixed multiplicities and the multiplicity of Rees modules; the relationship between mixed multiplicities of modules and mixed multiplicities of rings via the rank of modules (Corollary 3.4).

This paper is divided into three sections. In Section 2, first we define mixed multiplicities of good filtrations (see Proposition 2.2); next we prove the main theorem (Theorem 2.3) and give Corollary 2.4 and Corollary 2.5. Section 3 discusses some applications of Theorem 2.3 to formulas for mixed multiplicities and multiplicities of Rees modules (see Corol. 3.1, Corol. 3.2, Corol. 3.3, Corol. 3.4).

2 Mixed multiplicities and multiplicities of Rees modules

In this section, first we give Proposition 2.2 which is used for defining mixed multiplicities of good filtrations, and prove the main theorem (Theorem 2.3) and give some applications of this theorem to multiplicities of good filtrations.

$(A,\mathfrak{m})$ denotes a Noetherian local ring with maximal ideal $\mathfrak{m}.$ Recall that a filtration of ideals $F=\{(F)_{n}\}_{n\geqslant 0}$ in $A$ is a decreasing chain of ideals

[TABLE]

such that $(F)_{m}(F)_{n}\subseteq(F)_{m+n}$ for all $m,n\geqslant 0.$ For any ideal $I$ of $A,$ the filtration $\{I^{n}\}_{n\geqslant 0}$ determined by the powers of $I$ is called the $I$-adic filtration. Let $I$ be an ideal of $A$ and let $F=\{(F)_{n}\}_{n\geqslant 0}$ be a filtration of ideals in $A$. Then we say that $F$ is an $I$-good filtration if $I(F)_{n}\subseteq(F)_{n+1}$ for all $n\geqslant 0$ and $(F)_{n+1}=I(F)_{n}$ for all sufficiently large $n$. In this case, $I\subseteq(F)_{1}$. If $F$ is an $I$-good filtration, then $I$ is called a reduction of $F$. The filtration $F$ is called a good filtration if it is an $I$-good filtration for some ideal $I$ of $A.$ It is easily seen that $F$ is a good filtration if and only if $F$ is an $(F)_{1}$-good filtration and in this case, $(F)_{1}$ is a reduction of $F$. There are numerous examples of non-ideal-adic good filtrations: for an ideal $I$ of $A$ containing a non-zero-divisor, $\{\widetilde{I^{n}}\}_{n\geqslant 0}$ is a good filtration, where $\widetilde{I^{n}}$ is the Ratliff-Rush closure of $I^{n}$ (see [10, Theorem 2.1]). Furthermore, if $A$ is an analytically unramified ring, then $\{\overline{I^{n}}\}_{n\geqslant 0}$ is an $I$-good filtration (see [12]) and if $A$ is an analytically unramified ring containing a field, then $\{(I^{n})^{*}\}_{n\geqslant 0}$ is an $I$-good filtration, here $\overline{I^{n}}$ and $(I^{n})^{*}$ denote the integral closure and tight closure of $I^{n}$, respectively (see [5, Ch. 13]).

A filtration $F=\{(F)_{n}\}_{n\geqslant 0}$ is $\mathfrak{m}$-primary if $(F)_{1}$ is an $\mathfrak{m}$-primary ideal. Note that $\mathfrak{m}$-primary filtrations are sometimes called Hilbert filtrations (see e.g. [4]).

Let $M$ be a finitely generated $A$-module and $I_{1},\ldots,I_{s}$ be ideals of $A.$ Let

[TABLE]

be filtrations in $A$ $(s>0)$. Set

[TABLE]

Put $T^{\bf n}=t_{1}^{n_{1}}\cdots t_{s}^{n_{s}},$ here $t_{i}$ is a variable over $A$ for all $i=1,\ldots,s.$ Denote by

[TABLE]

the multi-Rees algebra and the multi-Rees module with respect to $M$ of the ideals $\mathbf{I}$, respectively; and by

[TABLE]

the multi-Rees algebra and the multi-Rees module of the filtrations $\mathbf{F}$ with respect to $M$, respectively. Set $\mathfrak{R}(\mathrm{\bf I};A)_{+}=\bigoplus_{|{\bf n}|>0}\mathbb{I}^{\mathrm{\bf n}}T^{\bf n}$ and $\mathfrak{R}(\mathrm{\bf F};A)_{+}=\bigoplus_{|{\bf n}|>0}\mathbb{F}_{\mathrm{\bf n}}T^{\bf n}$.

Let $\mathbf{F}:F_{1}=\{(F_{1})_{n}\}_{n\geqslant 0},\ldots,F_{s}=\{(F_{s})_{n}\}_{n\geqslant 0}$ be good filtrations of ideals in $A$ such that $I=(F_{1})_{1}\cdots(F_{s})_{1}$ is not contained in $\sqrt{\mathrm{Ann}M}$ and $F=\{(F)_{n}\}_{n\geqslant 0}$ be an $\mathfrak{m}$-primary good filtration. Now, we consider the $\mathbb{N}^{s+1}$-graded algebra:

[TABLE]

and the $\mathbb{N}^{s+1}$-graded $\mathcal{S}$-module: $\mathcal{M}=\bigoplus_{n_{0}\geqslant 0,{\bf n}\geqslant{\bf 0}}\frac{(F)_{n_{0}}\mathbb{F}_{\bf n}M}{(F)_{n_{0}+1}\mathbb{F}_{\bf n}M}.$ Then we get the Bhattacharya function of $\mathcal{M}$ (see [1])

[TABLE]

Note that $\mathcal{S}$ is not a standard graded algebra. This is an obstruction for proving $B(n_{0},{\bf n};F,\mathbf{F};M)$ is a polynomial for all large $n_{0},\bf n$. So we need the following note which plays an important role in the approach of this paper.

Note 2.1**.**

For $n\geqslant 1$, set $\mathcal{I}_{n}=(F)_{n}(F_{1})_{n}\cdots(F_{s})_{n}$. Assume that $J,I_{1},\ldots,I_{s}$ are reductions of $F,F_{1},\ldots,F_{s}$, respectively. Then there exists a large enough positive integer $c$ such that $(F)_{n_{0}+c}=J^{n_{0}}(F)_{c}$, $(F_{i})_{n_{i}+c}=I_{i}^{n_{i}}(F_{i})_{c}$ for all $n_{0},n_{i}$ and all $i=1,\ldots,s$. Thus $(F)_{n_{0}+c}\mathbb{F}_{{\bf n}+c{\bf 1}}=J^{n_{0}}\mathbb{I}^{\mathbf{n}}\mathcal{I}_{c}$ for all $n_{0},{\bf n}.$ So

[TABLE]

and hence

[TABLE]

for all $n_{0},{\bf n},$ here B(n_{0},{\bf n};J,\mathbf{I};\mathcal{I}_{c}M)=\ell_{A}\Big{(}\dfrac{J^{n_{0}}\mathbb{I}^{\mathbf{n}}\mathcal{I}_{c}M}{J^{n_{0}+1}\mathbb{I}^{\mathbf{n}}\mathcal{I}_{c}M}\Big{)}.

Using Note 2.1, we prove the following proposition.

Proposition 2.2**.**

Let $F$ be an $\mathfrak{m}$-primary good filtration and $\mathbf{F}$ be good filtrations such that $I=(F_{1})_{1}\cdots(F_{s})_{1}$ is not contained in $\sqrt{\mathrm{Ann}M}$. Set $q=\dim M/0_{M}:I^{\infty}$. Then $B(n_{0},{\bf n};F,\mathbf{F};M)$ is a polynomial of degree $q-1$ in $n_{0},\bf n$ for all large $n_{0},\bf n$.

Proof.

Let $J,I_{1},\ldots,I_{s}$ be reductions of $F,F_{1},\ldots,F_{s}$, respectively. Then it is easily to see that $0_{M}:(I_{1}\cdots I_{s})^{\infty}=0_{M}:I^{\infty}$. By Note 2.1, there exists a positive integer $c$ such that

[TABLE]

for all $n_{0},{\bf n}.$ Since $B(n_{0},{\bf n};J,\mathbf{I};\mathcal{I}_{c}M)$ is a polynomial in $n_{0},\bf n$ for all large $n_{0},\bf n$ by [3, Theorem 4.1] and this polynomial has degree: $\dim\mathcal{I}_{c}M/0_{\mathcal{I}_{c}M}:I^{\infty}-1$ by [17, Proposition 3.1 (i)] (see [9]), it is easily seen that $B(n_{0},{\bf n};F,\mathbf{F};M)$ is a polynomial of degree $\dim\mathcal{I}_{c}M/0_{\mathcal{I}_{c}M}:I^{\infty}-1$ for all large $n_{0},\bf n.$ Note that $(0_{M}:I^{\infty}):\mathcal{I}_{c}=0_{M}:I^{\infty}$. Therefore

[TABLE]

Hence $B(n_{0},{\bf n};F,\mathbf{F};M)$ is a polynomial of degree $q-1$ for all large $n_{0},\bf n$. ∎

With assumptions as in Proposition 2.2, $B(n_{0},{\bf n};F,\mathbf{F};M)$ is a polynomial of degree $q-1$ for all large $n_{0},\bf n.$ Denote by $P(n_{0},{\bf n};F,\mathbf{F};M)$ this polynomial. Write the terms of total degree $q-1$ in the polynomial $P(n_{0},{\bf n};F,\mathbf{F};M)$ in the form

[TABLE]

then it is easily seen that $e(F^{[k_{0}+1]},\mathbf{F}^{[\mathbf{k}]};M)$ are non-negative integers not all zero. We call $e(F^{[k_{0}+1]},\mathbf{F}^{[\mathbf{k}]};M)$ the mixed multiplicity of $M$ with respect to the good filtrations $F,\mathbf{F}$ of the type $(k_{0}+1,\mathbf{k})$.

In the case that $F_{i}$ is an $I_{i}$-adic filtration for all $i=1,\ldots,s$ and $F$ is a $J$-adic filtration, where $J$ is an $\mathfrak{m}$-primary ideal and $I_{1}\cdots I_{s}$ is not contained in $\sqrt{\mathrm{Ann}M}$, then $e(F^{[k_{0}+1]},\mathbf{F}^{[\mathbf{k}]};M)$ is denoted by $e(J^{[k_{0}+1]},\mathbf{I}^{[\mathbf{k}]};M)$ and called the mixed multiplicity of $M$ with respect to the ideals $J,\mathbf{I}$ of the type $(k_{0}+1,\mathbf{k})$ (see e.g. [9, 17, 24]).

Then the main result of this paper is the following theorem.

Theorem 2.3**.**

Let $F$ be an $\mathfrak{m}$-primary good filtration and $\mathbf{F}$ be good filtrations of ideals in $A$ such that $I=(F_{1})_{1}\cdots(F_{s})_{1}$ is not contained in $\sqrt{\mathrm{Ann}M}$. Let $J,I_{1},\ldots,I_{s}$ be reductions of $F,F_{1},\ldots,F_{s}$, respectively. Let $\mathfrak{I}$ be an $\mathfrak{m}$-primary ideal of $A$. Then the following statements hold.

1. (i)

$e(F^{[k_{0}+1]},\mathbf{F}^{[\mathbf{k}]};M)=e(J^{[k_{0}+1]},\mathbf{I}^{[\mathbf{k}]};M)$. 2. (ii)

Assume that $\mathrm{ht}\dfrac{I+\mathrm{Ann}M}{\mathrm{Ann}M}>0.$ Then we have

[TABLE]

Proof.

The proof of (i): From Note 2.1, there exists a positive integer $c$ such that

[TABLE]

for all $n_{0},{\bf n},$ here $\mathcal{I}_{c}=(F)_{c}(F_{1})_{c}\cdots(F_{s})_{c}$. Hence by the definition of the mixed multiplicities, we get

[TABLE]

Note that $0_{M}:(I_{1}\cdots I_{s})^{\infty}=0_{M}:I^{\infty}$. Set

[TABLE]

By [9, Proposition 3.1 (ii)] (see [17, Proposition 3.1 (ii)]), we have

[TABLE]

Consider the short exact sequence of $A$-modules:

[TABLE]

Since $\mathrm{Ann}\overline{M}:\mathcal{I}_{c}=\mathrm{Ann}\overline{M}$, it follows that $\mathrm{ht}\dfrac{\mathcal{I}_{c}+\mathrm{Ann}\overline{M}}{\mathrm{Ann}\overline{M}}>0$. Hence

[TABLE]

So by the additivity of mixed multiplicities [29, Corollary 3.9 (ii)(a)], we get

[TABLE]

Also by [9, Proposition 3.1 (ii)] (see [17, Proposition 3.1 (ii)], we obtain

[TABLE]

Therefore, by (1), (2), (3) and (4), we have

[TABLE]

The proof of (ii): Set $\mathfrak{U}=(\mathfrak{I},\mathfrak{R}(\mathrm{\bf I};A)_{+})$ and $\mathfrak{V}=(\mathfrak{I},\mathfrak{R}(\mathrm{\bf F};A)_{+})$. At first as in Note 2.1, there exists a positive integer $c$ such that $\mathbb{F}_{{\bf n}+c{\bf 1}}=\mathbb{I}^{\mathbf{n}}{I}_{c}$ for all ${\bf n}$, here

[TABLE]

So it can be verified that $I_{c}\mathfrak{R}(\mathbf{F};M)=\mathfrak{R}(\mathbf{I};I_{c}M)$ and $\mathfrak{V}^{n}I_{c}\mathfrak{R}(\mathbf{F};M)=\mathfrak{U}^{n}\mathfrak{R}(\mathbf{I};I_{c}M)$ for all $n\geq 1$. Hence $\dfrac{\mathfrak{R}(\mathbf{I};{I}_{c}M)}{\mathfrak{U}^{n}\mathfrak{R}(\mathbf{I};{I}_{c}M)}=\dfrac{{I}_{c}\mathfrak{R}(\mathbf{F};M)}{\mathfrak{V}^{n}{I}_{c}\mathfrak{R}(\mathbf{F};M)}$ for all $n\geq 1$. Now, for any $n\geq 1,$ assume that \ell_{\mathfrak{R}(\mathrm{\bf F};A)}\bigg{(}\dfrac{{I}_{c}\mathfrak{R}(\mathbf{F};M)}{\mathfrak{V}^{n}{I}_{c}\mathfrak{R}(\mathbf{F};M)}\bigg{)}=u. Then there exists a composition series

[TABLE]

of the $\mathfrak{R}(\mathrm{\bf F};A)$-module $\dfrac{{I}_{c}\mathfrak{R}(\mathbf{F};M)}{\mathfrak{V}^{n}{I}_{c}\mathfrak{R}(\mathbf{F};M)},$ i.e., ${\cal M}_{i+1}/{\cal M}_{i}\cong\mathfrak{R}(\mathbf{F};A)/(\mathfrak{m},\mathfrak{R}(\mathrm{\bf F};A)_{+})$ for all $0\leq i\leq u-1.$ Since $\mathfrak{R}(\mathbf{I};A)/(\mathfrak{m},\mathfrak{R}(\mathrm{\bf I};A)_{+})\cong A/\mathfrak{m}\cong\mathfrak{R}(\mathbf{F};A)/(\mathfrak{m},\mathfrak{R}(\mathrm{\bf F};A)_{+})$ and $\dfrac{\mathfrak{R}(\mathbf{I};{I}_{c}M)}{\mathfrak{U}^{n}\mathfrak{R}(\mathbf{I};{I}_{c}M)}=\dfrac{{I}_{c}\mathfrak{R}(\mathbf{F};M)}{\mathfrak{V}^{n}{I}_{c}\mathfrak{R}(\mathbf{F};M)}$ is also an $\mathfrak{R}(\mathbf{I};A)$-module, it follows that the composition series $(5)$ is also a composition series of the $\mathfrak{R}(\mathrm{\bf I};A)$-module $\dfrac{\mathfrak{R}(\mathbf{I};{I}_{c}M)}{\mathfrak{U}^{n}\mathfrak{R}(\mathbf{I};{I}_{c}M)}.$ Consequently \ell_{\mathfrak{R}(\mathrm{\bf I};A)}\bigg{(}\dfrac{\mathfrak{R}(\mathbf{I};{I}_{c}M)}{\mathfrak{U}^{n}\mathfrak{R}(\mathbf{I};{I}_{c}M)}\bigg{)}=u. From this it follows that

[TABLE]

for all $n$. Therefore

[TABLE]

By the assumption $\mathrm{ht}\dfrac{I+\mathrm{Ann}M}{\mathrm{Ann}M}>0,$ it implies that $\mathrm{ht}\dfrac{{I}_{c}+\mathrm{Ann}M}{\mathrm{Ann}M}>0.$ So

[TABLE]

Hence from the short exact sequence of $A$-modules: $0\rightarrow{I}_{c}M\rightarrow M\rightarrow M/{I}_{c}M\rightarrow 0,$ we get

[TABLE]

by [29, Corollary 3.9 (ii)(b)] on the additivity of multiplicities of Rees modules. Since $\mathrm{ht}\dfrac{I_{c}+\mathrm{Ann}M}{\mathrm{Ann}M}>0,$ we have $\dim\mathfrak{R}(\mathbf{F};M)/{I}_{c}\mathfrak{R}(\mathbf{F};M)<\dim\mathfrak{R}(\mathbf{F};M).$ Therefore from the short exact sequence of $\mathfrak{R}(\mathrm{\bf F};A)$-modules:

[TABLE]

it follows that

[TABLE]

(see e.g. [5, Theorem 11.2.3]). Consequently, by (6), (7) and (8), we obtain

[TABLE]

∎

A version of Theorem 2.3 (i) for the case of reductions of ideals in Noetherian local rings was proved by Viet in [20, Theorem 4.1] by a different approach. Theorem 2.3 is an important key which help us to obtain the following results for mixed multiplicities and the multiplicity of Rees modules by short arguments.

Let $I$ be an ideal of $A$ and $t$ a variable over $A,$ and $\mathfrak{R}(I;A)=A[It]$ the Rees algebra of $I.$ Denote by $\overline{\mathfrak{R}(I;A)}$ the integral closure of $A[It]$ in the polynomial ring $A[t].$ Then $\overline{\mathfrak{R}(I;A)}=\bigoplus_{n\geq 0}\overline{I^{n}}t^{n}$ (see [5, Proposition 5.2.1]). Moreover, if $A$ is an analytically unramified ring, then $\{\overline{I^{n}}\}_{n\geqslant 0}$ is an $I$-good filtration (see [12]). In this case, $I$ is a reduction of $\{\overline{I^{n}}\}_{n\geqslant 0}.$ Hence if we assume further that $\mathrm{ht}I>0$, then $e(\overline{\mathfrak{R}(I;A)})=e(\mathfrak{R}(I;A))$ by Theorem 2.3 (ii). We get the following result.

Corollary 2.4**.**

Let $A$ be an analytically unramified ring and $I$ an ideal of positive height of $A.$ Then $e(\overline{\mathfrak{R}(I;A)})=e(\mathfrak{R}(I;A)).$

Let $\mathfrak{I}_{1}$ be a sequence consisting $k_{1}$ elements of $(F_{1})_{1},\ldots,$ $\mathfrak{I}_{s}$ be a sequence consisting $k_{s}$ elements of $(F_{s})_{1}$ with $k_{1},\ldots,k_{s}\geq 0.$ Put $(\emptyset)=0_{A}$ and $\mathfrak{R}=\mathfrak{I}_{1},\ldots,\mathfrak{I}_{s}.$ For any $1\leq i\leq s,$ set $\varepsilon_{i}=(1,\ldots,1,0,1,\ldots,1)\in\mathbb{N}^{s}$ (the $i$th coordinate is 0). Then $\mathfrak{R}$ is called a joint reduction of filtrations $\mathbf{F}$ with respect to $M$ of the type $\mathbf{k}=(k_{1},\ldots,k_{s})$ if $\mathbb{F}_{\mathbf{n}+\mathbf{1}}M=\sum_{i=1}^{s}(\mathfrak{I}_{i})\mathbb{F}_{\mathbf{n}+\mathbf{\varepsilon}_{i}}M$ for all large $\bf n.$ Recall that the concept of joint reductions of $\mathfrak{m}$-primary ideals was given by Rees in 1984 [13]. This concept was extended to the set of arbitrary ideals by [11, 19, 20, 25, 31].

We obtain the following corollary on expressing mixed multiplicities of modules with respect to filtrations in terms of the multiplicity of their joint reductions, which is an extension of [25, Theorem 3.1] and [13, Theorem 2.4].

Corollary 2.5**.**

Let $F$ be an $\mathfrak{m}$-primary good filtration and $\mathbf{F}$ be good filtrations in $A$. Set $\dim M=d$ and $I=(F_{1})_{1}\cdots(F_{s})_{1}.$ Assume that \mathrm{ht}\Big{(}\dfrac{I+\mathrm{Ann}_{A}M}{\mathrm{Ann}_{A}M}\Big{)}=h>0 and $k_{0}\in\mathbb{N}$, ${\bf k}=(k_{1},\ldots,k_{s})\in\mathbb{N}^{s}$ such that $k_{0}+|\mathbf{k}|=d-1$ and $|\mathbf{k}|<h.$ Let $\mathfrak{R}=\mathfrak{I}_{0},\mathfrak{I}_{1},\ldots,\mathfrak{I}_{s}$ be a joint reduction of $F,\mathbf{F}$ with respect to $M$ of the type $(k_{0}+1,{\bf k})$ such that $\mathfrak{R}$ is a system of parameters for $M.$ Then

[TABLE]

Proof.

For each $i=1,\ldots,s$, set $I_{i}=(F_{i})_{1}$ and $I_{0}=(F)_{1}$. Since $I_{0},I_{1},\ldots,I_{s}$ are reductions of $F,\mathbf{F}$, there exists a large enough positive integer $c$ such that

[TABLE]

for all $n_{0},\mathbf{n}$, here $\mathcal{I}_{c}=(F)_{c}(F_{1})_{c}\cdots(F_{s})_{c}$ (see Note 2.1). Then by the assumption that $\mathfrak{R}$ is a joint reduction of $F,\mathbf{F}$ with respect to $M$, we have

[TABLE]

for all large $n_{0},\bf n$, here $\varepsilon_{0}=\mathbf{1}\in\mathbb{N}^{s},$ $\varepsilon_{i}=(1,\ldots,1,0,1,\ldots,1)\in\mathbb{N}^{s}$ (the $i$th coordinate is 0) and $u_{0}=0,$ $u_{i}=1$ for all $1\leq i\leq s.$ Thus

[TABLE]

for all large $n_{0},\bf n.$ Hence $\mathfrak{R}$ is a joint reduction of $I_{0},\mathbf{I}$ with respect to $\mathcal{I}_{c}M$. Recall that $I=(F_{1})_{1}\cdots(F_{s})_{1},$ $\mathcal{I}_{c}=(F)_{c}(F_{1})_{c}\cdots(F_{s})_{c},$ $(F)_{c}$ is $\mathfrak{m}$-primary, and $\mathrm{ht}\dfrac{I+\mathrm{Ann}M}{\mathrm{Ann}M}=h>0.$ Hence $\dim M=\dim\mathcal{I}_{c}M$ and $\mathrm{ht}\dfrac{I+\mathrm{Ann}\mathcal{I}_{c}M}{\mathrm{Ann}\mathcal{I}_{c}M}=h.$ So by [25, Theorem 3.1], we get $e(I_{0}^{[k_{0}+1]},\mathbf{I}^{[\mathbf{k}]};\mathcal{I}_{c}M)=e_{A}(\mathfrak{R};\mathcal{I}_{c}M).$ Moreover, by $(1)$ in the proof of Theorem 2.3 (i), we have $e(F^{[k_{0}+1]},\mathbf{F}^{[\mathbf{k}]};M)=e(I_{0}^{[k_{0}+1]},\mathbf{I}^{[\mathbf{k}]};\mathcal{I}_{c}M).$ Therefore

[TABLE]

Since $\mathrm{ht}\dfrac{I+\mathrm{Ann}M}{\mathrm{Ann}M}>0$, $\mathrm{ht}\dfrac{\mathcal{I}_{c}+\mathrm{Ann}M}{\mathrm{Ann}M}>0.$ So $\dim M/\mathcal{I}_{c}M<\dim M.$ Hence from the short exact sequence of $A$-modules: $0\rightarrow\mathcal{I}_{c}M\rightarrow M\rightarrow M/\mathcal{I}_{c}M\rightarrow 0,$ it follows that $e_{A}(\mathfrak{R};\mathcal{I}_{c}M)=e_{A}(\mathfrak{R};M)$ (see e.g. [5, Theorem 11.2.3]). Consequently we obtain $e(F^{[k_{0}+1]},\mathbf{F}^{[\mathbf{k}]};M)=e_{A}(\mathfrak{R};M).$ ∎

3 On some formulas for multiplicities

Continuing to apply the main theorem (Theorem 2.3), in this section, we give some formulas for transforming mixed multiplicities and multiplicities of Rees modules of good filtrations.

Keep the notations in Theorem 2.3. Let $J$ be a reduction of $F$. We choose the reductions $I_{1}=(F_{1})_{1},\ldots,I_{s}=(F_{s})_{1}$ of filtrations $F_{1},\ldots,F_{s}$, respectively. Set $\dim M=d$ and $I=(F_{1})_{1}\cdots(F_{s})_{1}.$ If $\mathrm{ht}\dfrac{I+\mathrm{Ann}M}{\mathrm{Ann}M}>0$, then by [3, Theorem 4.4] which is a generalized version of [16, Theorem 1.4] (see [29]), we have

[TABLE]

On one hand, by Theorem 2.3 (ii), we get

[TABLE]

On the other hand, by Theorem 2.3 (i), it follows that

[TABLE]

Hence we obtain the following result.

Corollary 3.1**.**

Let $F$ be an $\mathfrak{m}$-primary good filtration and $\mathbf{F}$ be good filtrations. Let $J$ be a reduction of $F$. Set $\dim M=d$ and $I=(F_{1})_{1}\cdots(F_{s})_{1}.$ Assume that $\mathrm{ht}\dfrac{I+\mathrm{Ann}M}{\mathrm{Ann}M}>0$. Then

[TABLE]

The next corollary is the additivity on exact sequences of mixed multiplicities and the multiplicity of Rees modules of filtrations. Let $W_{1},W_{2},W_{3}$ be finitely generated $A$-modules and $0\longrightarrow W_{1}\longrightarrow W_{3}\longrightarrow W_{2}\longrightarrow 0$ be a short exact sequence of $A$-modules. For any $i=1,2,3,$ set $\overline{W}_{i}=\dfrac{W_{i}}{0_{W_{i}}:I^{\infty}}$. Suppose that $I=(F_{1})_{1}\cdots(F_{s})_{1}$ is not contained in $\sqrt{\mathrm{Ann}{W_{i}}}$  for all $i=1,2,3.$

From Theorem 2.3 and [29, Corollary 3.9] and Corollary 3.1, we get the following result on the additivity on exact sequences of multiplicities of filtrations.

Corollary 3.2**.**

Keep the above notations. Let $\mathfrak{I}$ be an $\mathfrak{m}$-primary ideal of $A$. Set $\mathfrak{J}=(\mathfrak{I},\mathfrak{R}(\mathrm{\bf F};A)_{+})$. Assume that $k_{0}\in\mathbb{N}$, ${\bf k}\in\mathbb{N}^{s}$ such that $k_{0}+1+|\mathbf{k}|=\dim\overline{W}_{3}.$ Then the following statements hold.

• $\mathrm{(i)}$

If $\dim\overline{W}_{1}=\dim\overline{W}_{2}=\dim\overline{W}_{3}$, then

[TABLE]

• $\mathrm{(ii)}$

If $h\neq k=1,2$ and $\dim\overline{W}_{3}>\dim\overline{W}_{h}$, then

[TABLE]

Proof.

The proof of (i)(a) and (ii)(a): We choose the reductions

[TABLE]

of filtrations $F,F_{1},\ldots,F_{s}$, respectively. Then by Theorem 2.3 (i), we obtain

[TABLE]

for all $1\leq i\leq 3.$ Consequently, the part (i)(a) and the part (ii)(a) follow immediately from [29, Corollary 3.9 (i)(a) and (ii)(a)], respectively.

The proof of (i)(b): Note that since $\mathrm{ht}\dfrac{I+\mathrm{Ann}W_{i}}{\mathrm{Ann}W_{i}}>0,$ $\dim\overline{W}_{i}=\dim{W}_{i}.$ Hence $\dim{W}_{1}=\dim{W}_{2}=\dim{W}_{3}.$ Set $\dim W_{i}=d$ for $i=1,2,3$ and $\mathcal{F}=\{\mathfrak{I}^{n}\}_{n\geqslant 0}$ the $\mathfrak{I}$-adic filtration. On one hand, by Corollary 3.1, we have

[TABLE]

for $i=1,2,3$. On the other hand by (i)(a), it follows that

[TABLE]

Hence by (9), we obtain e\big{(}\mathfrak{J};\mathfrak{R}(\mathrm{\bf F};{W}_{3})\big{)}=e\big{(}\mathfrak{J};\mathfrak{R}(\mathrm{\bf F};{W}_{1})\big{)}+e\big{(}\mathfrak{J};\mathfrak{R}(\mathrm{\bf F};{W}_{2})\big{)}. The proof of (ii)(b): Similarly to the proof of (i)(b), by (9) and (ii)(a), we get (ii)(b). ∎

We also easily prove the following additivity and reduction formulas for mixed multiplicities and the multiplicity of Rees modules of filtrations which are generalizations of [29, Theorem 3.2] and [5, Theorem 17.4.8].

Corollary 3.3**.**

Let $F$ be an $\mathfrak{m}$-primary good filtration and $\mathbf{F}$ be good filtrations in $A$ such that $I=(F_{1})_{1}\cdots(F_{s})_{1}$ is not contained in $\sqrt{\mathrm{Ann}M}$. Set $\overline{M}=\dfrac{M}{0_{M}:I^{\infty}}.$ Denote by $\Pi$ the set of all prime ideals $\mathfrak{p}$ of $A$ such that $\mathfrak{p}\in\mathrm{Min}(A/\mathrm{Ann}\overline{M})$ and $\dim A/\mathfrak{p}=\dim\overline{M}.$ Let $\mathfrak{I}$ be an $\mathfrak{m}$-primary ideal of $A$. Then we have

1. (i)

$e(F^{[k_{0}+1]},\mathrm{\bf F}^{[\mathrm{\bf k}]};M)=\sum_{\mathfrak{p}\in\Pi}\ell({M}_{\mathfrak{p}})e(F^{[k_{0}+1]},\mathrm{\bf F}^{[\mathrm{\bf k}]};A/\mathfrak{p}).$** 2. (ii)

If $\mathrm{ht}\dfrac{I+\mathrm{Ann}M}{\mathrm{Ann}M}>0$, then

[TABLE]

Proof. The proof of (i): Choose the reductions $J=(F)_{1},I_{1}=(F_{1})_{1},\ldots,I_{s}=(F_{s})_{1}$ of filtrations $F,F_{1},\ldots,F_{s}$, respectively. Then on one hand by Theorem 2.3 (i), we get $e(F^{[k_{0}+1]},\mathrm{\bf F}^{[\mathrm{\bf k}]};M)=e(J^{[k_{0}+1]},\mathrm{\bf I}^{[\mathrm{\bf k}]};M)$ and $e(F^{[k_{0}+1]},\mathrm{\bf F}^{[\mathrm{\bf k}]};A/\mathfrak{p})=e(J^{[k_{0}+1]},\mathrm{\bf I}^{[\mathrm{\bf k}]};A/\mathfrak{p})$ for all $\mathfrak{p}\in\Pi.$ On the other hand by [29, Theorem 3.2], we have

[TABLE]

So $e(F^{[k_{0}+1]},\mathrm{\bf F}^{[\mathrm{\bf k}]};M)=\sum_{\mathfrak{p}\in\Pi}\ell({M}_{\mathfrak{p}})e(F^{[k_{0}+1]},\mathrm{\bf F}^{[\mathrm{\bf k}]};A/\mathfrak{p}).$

The proof of (ii): Set $\dim M=d$ and $\mathcal{F}=\{\mathfrak{I}^{n}\}_{n\geqslant 0}$ the $\mathfrak{I}$-adic filtration. By Corollary 3.1 and (i) we have

[TABLE]

Denote by $U$ the set of all non-zero divisors of $A.$ Recall that an $A$-module $N$ has rank $r$ if $U^{-1}N$ (the localization of $N$ with respect to $U$) is a free $U^{-1}A$-module of rank $r.$

Now, using Theorem 2.3 (i) and [30, Theorem 3.4] we give the following.

Corollary 3.4**.**

Let $F$ be an $\mathfrak{m}$-primary good filtration and $\mathbf{F}$ be good filtrations of ideals in $A$ such that $I=(F_{1})_{1}\cdots(F_{s})_{1}$ is not contained in $\sqrt{\mathrm{Ann}M}$. Suppose that $M$ has rank $r>0$. Then e\big{(}{F}^{[k_{0}+1]},{\mathrm{\bf F}}^{[\mathrm{\bf k}]};M\big{)}=e\big{(}F^{[k_{0}+1]},\mathrm{\bf F}^{[\mathrm{\bf k}]};A\big{)}\mathrm{rank}_{A}M.

Proof.

Choose the reductions $J=(F)_{1},I_{1}=(F_{1})_{1},\ldots,I_{s}=(F_{s})_{1}$ of filtrations $F,F_{1},\ldots,F_{s}$, respectively. Then by Theorem 2.3 (i), we have

[TABLE]

and $e(F^{[k_{0}+1]},\mathrm{\bf F}^{[\mathrm{\bf k}]};M)=e(J^{[k_{0}+1]},\mathrm{\bf I}^{[\mathrm{\bf k}]};M).$ By [30, Theorem 3.4], we obtain

[TABLE]

So e\big{(}{F}^{[k_{0}+1]},{\mathrm{\bf F}}^{[\mathrm{\bf k}]};M\big{)}=e\big{(}F^{[k_{0}+1]},\mathrm{\bf F}^{[\mathrm{\bf k}]};A\big{)}\mathrm{rank}_{A}M. ∎

Acknowledgement: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04.2015.01.