On Regular Sequences in the Form Module with Applications to Local B\'ezout Inequalities

TL;DR

This paper explores regular sequences in form modules of Noetherian local rings to improve inequalities relating multiplicities and lengths, with applications to classical local Bézout inequalities in algebraic geometry.

Contribution

It introduces new conditions for regular sequences in form modules to refine existing inequalities involving multiplicities and lengths.

Findings

Improved inequality relating multiplicities and lengths in Noetherian local rings.

Characterization of regular sequences in form modules via Koszul complex homology.

Application to classical local Bézout inequality in affine space.

Abstract

Let $\mathfrak{q}$ denote an ideal in a Noetherian local ring $(A,\mathfrak{m})$. Let $\underline{a}=a_1,\ldots,a_d \subset \mathfrak{q}$ denote a system of parameters in a finitely generated $A$-module $M$. This note investigate an improvement of the inequality $c_1\cdot \ldots \cdot c_d \cdot e_0(\mathfrak{q};M) \leq \ell_A(M/\underline{a}\,M)$, where $c_i$ denote the initial degrees of $a_i$ in the form ring $G_A(\mathfrak{q})$. To this end, there is an investigation of regular sequences in the form module $G_M(\mathfrak{q})$ by homology of a factor complex of the Koszul complex. In a particular case, there is a discussion of classical local B\'ezout inequality in the affine $d$-space $\mathbb{A}^d_k$.

Equations

$$\sum _{P\in C\cap D}\mu (P;C,D)=\mathrm{deg}C\cdot \mathrm{deg}D,$$

$$e_0(f,g;A)\ge c\cdot d\cdot e_0(\mathfrak{m};A)=c\cdot d,$$

$$e_0(f,g;A)\ge c\cdot d+t,$$

$${\ell }_A(M/\underline{a}M)\ge c\cdot e_0(\mathfrak{q};M)+\mathfrak{x}$$

$$L_i(\underline{a},\mathfrak{q},M;n)=0\text{ for all }i>t,\text{ for all }n.$$

$$L_{d-t}(\underline{a},\mathfrak{q},M;n)\cong {\cap }_{i=1}^d(\underline{b},{\mathfrak{q}}^{n+\beta -\overline{c_i}})M{:}_M{a}_i/(\underline{b},{\mathfrak{q}}^{n+\beta -\overline{c}})M.$$

$${\ell }_A(M/{\mathfrak{q}}^{n}M)=\sum _{i=0}^d{e}_i(\mathfrak{q};M)\left(\genfrac{}{}{0ex}{}{n+d-i-1}{d-i}\right),$$

$$R_A(\mathfrak{q})={\oplus }_{n\ge 0}{\mathfrak{q}}^n{T}^n\subseteq A[T]\text{ and }{G}_A(\mathfrak{q})={\oplus }_{n\ge 0}{\mathfrak{q}}^n/{\mathfrak{q}}^{n+1},$$

$$R_M(\mathfrak{q})={\oplus }_{n\ge 0}{\mathfrak{q}}^{n}M{T}^n\subseteq M[T]\text{ and }{G}_M(\mathfrak{q})={\oplus }_{n\ge 0}{\mathfrak{q}}^{n}M/{\mathfrak{q}}^{n+1}M.$$

$$d_{j_1\dots j_i}:e_{j_1}\wedge \dots \wedge e_{j_i}\mapsto \sum _{k=1}^i(-1{)}^{k+1}{a}_{j_k}{e}_{j_1}\wedge \dots \wedge \widehat{e_{j_k}}\wedge \dots \wedge e_{j_i}$$

$$0\to K_{\bullet }(\underline{a},\mathfrak{q},M;n)\to K_{\bullet }(\underline{a};M)\to {\mathcal{L}}_{\bullet }(\underline{a},\mathfrak{q},M;n)\to 0.$$

$$L_1(a_1,\mathfrak{q},M;n)={\mathfrak{q}}^{n}M{:}_A{a}_1/{\mathfrak{q}}^{n-c_1}M,$$

$$L_1(\underline{a},\mathfrak{q},M;n)\cong 0{:}_{L_0({\underline{a}}^{\prime },\mathfrak{q},M;n-c_d)}{a}_d.$$

$$L_1({\underline{a}}^{\prime },\mathfrak{q},M;n)=a_d{L}_1({\underline{a}}^{\prime },\mathfrak{q},M;n-c_d),$$

$$0=L_1(\underline{a},\mathfrak{q},M;n)\cong ({\underline{a}}^{\prime },{\mathfrak{q}}^n)M{:}_A{a}_d/({\underline{a}}^{\prime },{\mathfrak{q}}^{n-c_d})M$$

$$H_{d-t}(\underline{a};M)\cong (b_1,\dots ,b_t)M{:}_A\underline{a}/(b_1,\dots ,b_t)M,$$

$$0\to {\mathcal{L}}_{\bullet }(\underline{a},\mathfrak{q},M;n-\beta )\stackrel{b}{\to }{\mathcal{L}}_{\bullet }(\underline{a},\mathfrak{q},M;n)\to {\mathcal{L}}_{\bullet }(\underline{a},\mathfrak{q},M/bM;n)\to 0.$$

$$\dots \to L_i(\underline{a},\mathfrak{q},M;n-\beta )\stackrel{b}{\to }{L}_i(\underline{a},\mathfrak{q},M;n)\to L_i(\underline{a},\mathfrak{q},M/bM;n)\to \dots .$$

$$(\sum _{i=1}^t{b}_i{\mathfrak{q}}^{n-{\beta }_i}+{\mathfrak{q}}^{n+1})/{\mathfrak{q}}^{n+1}\subseteq (\sum _{j=1}^d{a}_j{\mathfrak{q}}^{n-c_j}+{\mathfrak{q}}^{n+1})/{\mathfrak{q}}^{n+1}$$

$$L_i(\underline{a},\mathfrak{q},M;n)=0\text{ for all }i>d-t,\text{ for all }n.$$

$$L_{d-t}(\underline{a},\mathfrak{q},M;n)\cong {\cap }_{i=1}^d(\underline{b},{\mathfrak{q}}^{n+\beta -\overline{c_i}})M{:}_M{a}_i/(\underline{b},{\mathfrak{q}}^{n+\beta -\overline{c}})M,$$

$$L_{d-t+1}(\underline{a},\mathfrak{q},M/b_{1}M;n)\cong {\cap }_{i=1}^d(\underline{b},{\mathfrak{q}}^{n+{\sum }_{j=2}^t{\beta }_j-\overline{c_i}})M{:}_M{a}_i/(\underline{b},{\mathfrak{q}}^{n+{\sum }_{j=2}^t{\beta }_j-\overline{c}})M$$

$$L_{d-t+1}(\underline{a},\mathfrak{q},M/b_{1}M;n)\cong L_{d-t}(\underline{a},\mathfrak{q},M;n-{\beta }_1).$$

$$0\to {\mathfrak{q}}^{n}M/{\mathfrak{q}}^{n+1}M\to M/{\mathfrak{q}}^{n+1}M\to M/{\mathfrak{q}}^{n}M\to 0,$$

$$0\to K_{\bullet }({\underline{a}}^{\star };G_M(\mathfrak{q}){)}_n\to {\mathcal{L}}_{\bullet }(\underline{a},\mathfrak{q},M;n+1)\to {\mathcal{L}}_{\bullet }(\underline{a},\mathfrak{q},M;n)\to 0$$

$$\sum _{i=0}^d(-1{)}^i\sum _{1\le j_1<\dots <j_i\le d}{\ell }_A(M/{\mathfrak{q}}^{n-c_{j_1}-\dots -c_{j_i}}M),$$

$${\chi }_A({\mathcal{L}}_{\bullet }(\underline{a},\mathfrak{q},M;n))=\sum _{i\ge 0}(-1{)}^i{\ell }_A(L_i(\underline{a},\mathfrak{q},M;n))\text{ for all }n\gg 0,$$

$${\ell }_A(L_1(\underline{a},\mathfrak{q},M;n))={\ell }_A([{\underline{b}}^{\star }{G}_M(\mathfrak{q}):{\underline{a}}^{\star }/({\underline{b}}^{\star }{G}_M(\mathfrak{q})){]}_{n+\beta -\overline{c}-1})+{\ell }_n,$$

$${\ell }_n={\ell }_A({\cap }_{i=1}^d(\underline{b},{\mathfrak{q}}^{n+\beta -{\overline{c}}_i})M{:}_M{a}_i/({\cap }_{i=1}^d(\underline{b},{\mathfrak{q}}^{n+\beta -{\overline{c}}_i})M{:}_M{a}_i)\cap (\underline{b},{\mathfrak{q}}^{n+\beta -\overline{c}-1})M),$$

$$[{\underline{b}}^{\star }{G}_M(\mathfrak{q}):{\underline{a}}^{\star }/{\underline{b}}^{\star }{G}_M(\mathfrak{q}){]}_n\cong ({\cap }_{i=1}^d(\underline{b},{\mathfrak{q}}^{n+c_i+1})M{:}_M{a}_i)\cap (\underline{b},{\mathfrak{q}}^n)M/(\underline{b},{\mathfrak{q}}^{n+1})M.$$

Full text

The author is grateful to DAAD and HEC, Pakistan for the support of his PhD research under grant number 91524811 and 112-21480-2PS1-015 (50021731) respectively.

1. Introduction

The importance of an improvement of the inequality $\ell_{A}(M/\underline{a}M)\geq c_{1}\cdot\ldots\cdot c_{d}\cdot e_{0}(\mathfrak{q};M)$ has to do with Bézout’s Theorem in the projective plane. Let $C=V(F),D=V(G)\subset\mathbb{P}^{2}_{\Bbbk},\Bbbk=\overline{\Bbbk,}$ be two curves in the projective plane without a common component. Then

[TABLE]

where $\mu(P;C,D)$ denotes the local intersection multiplicity of $P$ in $C\cap D$. In a particular case when $P$ is the origin, it follows that $\mu(P;C,D)=\ell_{A}(A/(f,g)A)$, where $A=\Bbbk[x,y]_{(x,y)}$ and $f,g$ denote the equations in $A$. Note that $\ell_{A}(A/(f,g)A)=e_{0}(f,g;A)$ as $A$ is a regular local ring. Since $C,D$ have no component in common, $\{f,g\}$ forms a system of parameters in $A$. Then

[TABLE]

since $e_{0}(\mathfrak{m};A)=1$, called the local Bézout inequality in the affine plane $\mathbb{A}_{\Bbbk}^{2}$. Here $c,d$ denote the initial degree of $f,g$ respectively. This estimate is well-known (see for instance [3] or [6]) and proved by resultants or Puiseux expansions. Moreover, equality holds if and only if $C,D$ intersect transversally at the origin. In other words $f^{\star},g^{\star}$, the initial forms of $f,g$ in the form ring $G_{A}(\mathfrak{m})\cong\Bbbk[X,Y]$, is a homogeneous system of parameters.

First Bydz̆ovský [5] and most recently Bod̆a-Schenzel [2] presented an improvement of the local Bézout inequality. More precisely,

[TABLE]

where $t$ is the number of common tangents of $f,g$ at origin when counted with multiplicities. We generalized their result to an arbitrary situation. To this end, let $\mathfrak{q}$ denote an ideal in a Noetherian local ring $(A,\mathfrak{m},\Bbbk)$ such that $\ell_{A}(M/\mathfrak{q}M)$ is finite for a finitely generated $A$-module $M$. Let $\underline{a}=a_{1},\ldots,a_{d}\subseteq\mathfrak{q}$ denote a system of parameters of $M$ such that $a_{i}\in\mathfrak{q}^{c_{i}}\setminus\mathfrak{q}^{c_{i}+1}$, $c_{i}>0$, for $i=1,\ldots,d$. Then we have the following result.

Theorem 1.1**.**

(Cor. 5.3) With the previous notations, if $\underline{a}^{\star}G_{A}(\mathfrak{q})$ contains a $G_{M}(\mathfrak{q})$-regular sequence $\underline{b}^{\star}=b_{1}^{\star},\ldots,b_{d-1}^{\star}$ and we choose $b_{i}$ for $i=1,\ldots,d-1$ as in Lemma 4.2. Then

[TABLE]

where $c=c_{1}\cdot\ldots\cdot c_{d}$ and $\mathfrak{x}=\ell_{A}([\operatorname{Ext}^{d-1}_{G_{A}(\mathfrak{q})}(G_{A}(\mathfrak{q})/\underline{a}^{\star}G_{A}(\mathfrak{q}),G_{M}(\mathfrak{q}))]_{n-\overline{c}-1})$ is a constant for all $n\gg 0$ and $\overline{c}=c_{1}+\ldots+c_{d}.$

There are also few applications of the previous theorem. We refer Section 5 and 6.

Another motivation for the author was a recent preprint [8]. In this preprint, the authors define a generalized Koszul complex $\mathcal{L}_{\bullet}(\underline{a},\mathfrak{q},M;n)$ which is factor complex of Koszul complex (see definition 2.3). There are criteria concerning regular sequences in a finitely generated $A$-module $M$, which deal the vanishing and rigidity of the Koszul homology (see [4] and [9]). We present the similar criteria concerning regular sequences in the form module $G_{M}(\mathfrak{q})$ in terms of the homology modules $L_{i}(\underline{a},\mathfrak{q},M;n)$ of the complex $\mathcal{L}_{\bullet}(\underline{a},\mathfrak{q},M;n)$. More precisely, let $\underline{a}=a_{1},\ldots,a_{d}$ and $\underline{b}=b_{1},\ldots,b_{t}$ denote two systems of elements of $A$. There is a following theorem.

Theorem 1.2**.**

(1)(Theorem 3.1) With the previous notations, the following are equivalent:

• (a)

$\underline{a}^{\star}=a_{1}^{\star},\ldots,a_{d}^{\star}$* is $G_{M}(\mathfrak{q})$-regular sequence.*

• (b)

$L_{1}(\underline{a},\mathfrak{q},M;n)=0$* for all $n$.*

• (c)

$L_{i}(\underline{a},\mathfrak{q},M;n)=0$* for all $i>0$, for all $n$.*

(2)(Theorem 4.3, Prop. 4.4)With the previous notations, if $\underline{a}^{\star}G_{A}(\mathfrak{q})$ contains a $G_{M}(\mathfrak{q})$-regular sequence $\underline{b}^{\star}=b_{1}^{\star},\ldots,b_{t}^{\star}$, then

[TABLE]

The converse is also true. Moreover, given $\underline{b}^{\star}=b_{1}^{\star},\ldots,b_{t}^{\star}$ we choose $b_{i}$ for $i=1,\ldots,t$ as in Lemma 4.2, there is an isomorphism

[TABLE]

We refer Section 3 and 4 for the detail discussion about above theorem. In [2] and [8], authors posed the problem to study the Euler characteristic $\chi_{A}(\underline{a},\mathfrak{q},M)$ of the complex $K_{\bullet}(\underline{a},\mathfrak{q},M;n)$, see def. 2.3, independently of its value which is equal to $e_{0}(\underline{a};M)-c_{1}\cdot\ldots\cdot c_{d}\cdot e_{0}(\mathfrak{q};M)\geq 0$ for $n\gg 0$, cf. [2]. In section 6, we discuss a few properties of this Euler characteristic. A further investigation of the geometric meanings of the length involved in Theorem 1.1 in affine space $\mathbb{A}_{\Bbbk}^{d}$ when $d\geq 3$ is in progress.

As a source for basic notions in Commutative Algebra, we refer to [1] and [9]. For results on Homological Algebra, we refer to [10] and [14].

2. Preliminaries

In this section, we present the basic notations, which we are going to use in upcoming sections. For more detail, we refer the text book [9] and lecture notes [11].

Notation 2.1**.**

(1) Let $(A,\mathfrak{m})$ be a local Noetherian ring, $\mathfrak{q}$ be an ideal in $A$ and $M$ be a finitely generated $A$-module. Then $\mathfrak{q}$ is said to be an ideal of definition with respect to $M$ if the length $\ell_{A}(M/\mathfrak{q}M)$ of $A$-module $M/\mathfrak{q}M$ is finite. Now, it is easily seen that the length of $A$-modules $M/\mathfrak{q}^{n}M$ is also finite for all $n\in\mathbb{N}$.

For $n$ large enough, $\ell_{A}(M/\mathfrak{q}^{n}M)$ becomes a polynomial, which is written as

[TABLE]

where degree $d$ is equal to $\dim M$ (see [9]). Here, $e_{i}(\mathfrak{q};M)$ are called the Hilbert-Samuel local multiplicities of $M$ with respect to $\mathfrak{q}$. The first $e_{0}(\mathfrak{q};M)$ of them is our main ingredient for the rest of the note, and we call it just the multiplicity of $M$ with respect to $\mathfrak{q}$.

(2) The Rees and form rings of $A$ with respect to $\mathfrak{q}$ are defined by

[TABLE]

where $T$ denotes an indeterminate over $A$. The Rees and form modules are defined in the corresponding way by

[TABLE]

(3) Assume that $m\in M$ such that $m\in\mathfrak{q}^{c}M\setminus\mathfrak{q}^{c+1}M$. We define $m^{\star}:=m+\mathfrak{q}^{c+1}M\in[G_{M}(\mathfrak{q})]_{c}$. If $m\in\cap_{n\geq 1}\mathfrak{q}^{n}M$, then we write $m^{\star}=0$. Here $m^{\star}$ and $c$ are called the initial form and initial degree of $m$ in $G_{M}(\mathfrak{q})$ respectively. We refer [12] for more detail.

For basic results about multiplicities, we refer [4] and [9]. Another tool for the investigation is use of Koszul complex.

Remark 2.2**.**

(Koszul Complex) Let $\underline{a}=a_{1},\ldots,a_{d}$ denote a system of elements of the ring $A$. The Koszul complex $K_{\bullet}(\underline{a};A)$ is defined as follows: Assume that $F$ is a free $A$-module with basis $e_{1},\ldots,e_{d}$. Then $K_{i}(\underline{a};A)=\bigwedge^{i}F$ for $i=1,\ldots,d$. A basis of $K_{i}(\underline{a};A)$ is given by the wedge products $e_{j_{1}}\wedge\ldots\wedge e_{j_{i}}$ for $1\leq j_{1}<\ldots<j_{i}\leq d$. The boundary homomorphism $K_{i}(\underline{a};A)\to K_{i-1}(\underline{a};A)$ is defined by

[TABLE]

on the free generators $e_{j_{1}}\wedge\ldots\wedge e_{j_{i}}$. Also $K_{\bullet}(\underline{a};M)\cong K_{\bullet}(\underline{a};A)\otimes_{A}M$. We write $H_{i}(\underline{a};M),i\in\mathbb{Z},$ for the $i$-th homology of $K_{\bullet}(\underline{a};M)$.

For more detail about Koszul homology, we refer [4] and [9]. The following are main ingredients for the investigation, see [8] for reference.

Notation 2.3**.**

(Khadam-Schenzel [8]) (1) Assume that $a_{i}\in\mathfrak{q}^{c_{i}}$ for $i=1,\ldots,d$ and $n$ is a non-negative integer. For $n<0$, we assume that $\mathfrak{q}^{n}M=0$. We define a complex $K_{\bullet}(\underline{a},\mathfrak{q},M;n)$ in the following way:

• (i)

The $i$-th term $K_{i}(\underline{a},\mathfrak{q},M;n):=\oplus_{1\leq j_{1}<\ldots<j_{i}\leq d}\mathfrak{q}^{n-c_{j_{1}}-\ldots-c_{j_{i}}}M$ for $0\leq i\leq d$ and $K_{i}(\underline{a},\mathfrak{q},M;n)=0$ otherwise.

• (ii)

The boundary homomorphism $K_{i}(\underline{a},\mathfrak{q},M;n)\to K_{i-1}(\underline{a},\mathfrak{q},M;n)$ is defined by homomorphisms on each of the direct summands $\mathfrak{q}^{n-c_{j_{1}}-\ldots-c_{j_{i}}}M$. On $\mathfrak{q}^{n-c_{j_{1}}-\ldots-c_{j_{i}}}M$, it is the map given by $d_{j_{1}\ldots j_{i}}\otimes 1_{M}$ restricted to $\mathfrak{q}^{n-c_{j_{1}}-\ldots-c_{j_{i}}}M$, where $d_{j_{1}\ldots j_{i}}$ denotes the homomorphism as defined in 2.2.

It is clear that $K_{\bullet}(\underline{a},\mathfrak{q},M;n)$ is a complex. Moreover, by construction, $K_{\bullet}(\underline{a},\mathfrak{q},M;n)$ is a sub complex of the Koszul complex $K_{\bullet}(\underline{a};M)$. We write $H_{i}(\underline{a},\mathfrak{q},M;n)$,$i\in\mathbb{Z}$, for the $i$-th homology of the complex $K_{\bullet}(\underline{a},\mathfrak{q},M;n)$. Note that $[K_{\bullet}(\underline{aT^{c}};R_{M}(\mathfrak{q}))]_{n}\cong K_{\bullet}(\underline{a},\mathfrak{q},M;n)$ for $n\in\mathbb{N}$.

(2) We define $\mathcal{L}_{\bullet}(\underline{a},\mathfrak{q},M;n)$ as the quotient of the embedding $K_{\bullet}(\underline{a},\mathfrak{q},M;n)\to K_{\bullet}(\underline{a};M)$. That is, there is a short exact sequence of complexes

[TABLE]

Note that $\mathcal{L}_{i}(\underline{a},\mathfrak{q},M;n)\cong\oplus_{1\leq j_{1}<\ldots<j_{i}\leq d}M/\mathfrak{q}^{n-c_{j_{1}}-\ldots-c_{j_{i}}}M$. The boundary homomorphisms are those induced by the Koszul complex. We write $L_{i}(\underline{a},\mathfrak{q},M;n)$, $i\in\mathbb{Z}$, for the $i$-th homology of the complex $\mathcal{L}_{\bullet}(\underline{a},\mathfrak{q},M;n)$.

For more detail about complexes $K_{\bullet}(\underline{a},\mathfrak{q},M;n)$ and $\mathcal{L}_{\bullet}(\underline{a},\mathfrak{q},M;n)$, and their relationship with the local cohomology module, we refer Khadam-Schenzel [8].

3. Regular Sequences in the Form and Rees Modules

There is a criterion in terms of Koszul homology which ensures whether a sequence of elements in $A$ is $M$-regular or not. More precisely, the sequence $\underline{a}=a_{1},\ldots,a_{d}$ is $M$-regular if and only if $H_{i}(\underline{a};M)=0$ for all $i>0$ (see [9]). In this section, we present a similar criterion in the form module $G_{M}(\mathfrak{q})$ in terms of the homology modules $L_{i}(\underline{a},\mathfrak{q},M;n)$.

Let $\underline{a}^{\star}=a_{1}^{\star},\ldots,a_{d}^{\star}$ denote a sequence of initial forms in the form ring $G_{A}(\mathfrak{q})$ with $\deg a_{i}^{\star}=c_{i}$ for $i=1,\ldots,d$. We start with the main result of the section.

Theorem 3.1**.**

With the previous notations, the following are equivalent:

• (1)

$\underline{a}^{\star}=a_{1}^{\star},\ldots,a_{d}^{\star}$* is $G_{M}(\mathfrak{q})$-regular sequence.*

• (2)

$L_{1}(\underline{a},\mathfrak{q},M;n)=0$* for all $n$.*

• (3)

$L_{i}(\underline{a},\mathfrak{q},M;n)=0$* for all $i>0$, for all $n$.*

Proof.

For (1) $\Rightarrow$ (3), we use induction on $d$. If $d=1$, then

[TABLE]

which is equal to zero for all $n$ if and only if $a_{1}^{\star}$ is $G_{M}(\mathfrak{q})$-regular. Now, by virtue of long exact homology sequence coming from the mapping cone construction of the complex $\mathcal{L}_{\bullet}(\underline{a},\mathfrak{q},M;n)$ (see [8]) and by inductive step, $L_{i}(\underline{a},\mathfrak{q},M;n)=0$ for all $i>1$, for all $n$, and

[TABLE]

The latter is isomorphic to $(\underline{a}^{\prime},\mathfrak{q}^{n})M:_{A}a_{d}/(\underline{a}^{\prime},\mathfrak{q}^{n-c_{d}})M$, which is equal to zero for all $n$, see [13]. It is obvious that (3) $\Rightarrow$ (2).

For (2) $\Rightarrow$ (1), we apply induction on $d$ once again. The case $d=1$ is clear, see above. Again, from the mapping cone construction and by assumption,

[TABLE]

and therefore, by virtue of Nakayama lemma $L_{1}(\underline{a}^{\prime},\mathfrak{q},M;n)=0$ for all $n$. Hence $\underline{a}^{\prime\star}=a_{1}^{\star},\ldots,a_{d-1}^{\star}$ is a $G_{M}(\mathfrak{q})$-regular sequence by induction. Moreover,

[TABLE]

for all $n$, and hence $a_{d}^{\star}$ is $G_{M}(\mathfrak{q})/(\underline{a}^{\prime\star})G_{M}(\mathfrak{q})$-regular. This finishes the argument. ∎

The following is a consequence of the previous theorem.

Corollary 3.2**.**

With the previous notations, $\underline{a}^{\star}=a_{1}^{\star},\ldots,a_{d}^{\star}$ is a $G_{M}(\mathfrak{q})$-regular sequence implies that $\underline{aT^{c}}=a_{1}T^{c_{1}},\ldots,a_{d}T^{c_{d}}$ is an $R_{M}(\mathfrak{q})$-regular sequence.

Proof.

If $\underline{a}^{\star}=a_{1}^{\star},\ldots,a_{d}^{\star}$ is a $G_{M}(\mathfrak{q})$-regular sequence, then $\underline{a}=a_{1},\ldots,a_{d}$ is an $M$-regular sequence, see [13]. Hence $H_{i}(\underline{a};M)=0$ for all $i>0$. Therefore, from the long exact sequence of homology coming from the short exact sequence of 2.3, $H_{i}(\underline{a},\mathfrak{q},M;n)=0$ for all $i>0$ and for all $n$ (see 3.1). That is $H_{i}(\underline{aT^{c}};R_{M}(\mathfrak{q}))=0$ for all $i>0$. Hence, by virtue of Koszul criterion, $\underline{aT^{c}}=a_{1}T^{c_{1}},\ldots,a_{d}T^{c_{d}}$ is an $R_{M}(\mathfrak{q})$-regular sequence. ∎

4. A Formula for Homology

There is a classical result concerning the length of an $M$-sequence inside the ideal $(a_{1},\ldots,a_{d})$ and vanishing of the Koszul homology. More precisely, if $b_{1},\ldots,b_{t}$ is an $M$-sequence contained in the ideal $(a_{1},\ldots,a_{d})$, then $H_{i}(\underline{a};M)=0$ for all $i>d-t$, and there is a formula

[TABLE]

see [4]. In this section, we present the similar result for the homology modules $L_{i}(\underline{a},\mathfrak{q},M;n)$.

We begin with a lemma.

Lemma 4.1**.**

With the previous notations, let $b^{\star}$ be a $G_{M}(\mathfrak{q})$-regular element of degree $\beta$, then there is a short exact sequence of complexes

[TABLE]

In particular, there is the long exact homology sequence

[TABLE]

Proof.

The kernel of the map $\mathcal{L}_{\bullet}(\underline{a},\mathfrak{q},M;n-\beta)\stackrel{{\scriptstyle b}}{{\to}}\mathcal{L}_{\bullet}(\underline{a},\mathfrak{q},M;n)$ is zero since $b^{\star}$ is $G_{M}(\mathfrak{q})$-regular. Also, it is easy to see that $\operatorname{Coker}(m_{b})=\mathcal{L}_{\bullet}(\underline{a},\mathfrak{q},M/bM;n)$. This provides the short exact sequence of complexes. By taking homology it yields the long exact sequence. ∎

Let $\underline{b}=b_{1},\ldots,b_{t}$ denote a sequence of elements in $A$, and $\underline{b}^{\star}=b_{1}^{\star},\ldots,b_{t}^{\star}$ denote a sequence of initial forms in $G_{A}(\mathfrak{q})$ with $\deg b_{i}^{\star}=\beta_{i}$. There is another technical lemma.

Lemma 4.2**.**

With the previous notations, assume that $\underline{b}^{\star}G_{A}(\mathfrak{q})\subseteq\underline{a}^{\star}G_{A}(\mathfrak{q})$. Then there are elements $b_{1}^{\prime},\cdots,b_{t}^{\prime}\in A$ such that

• (i)

$b_{i}^{\star}=b_{i}^{\prime\star}$* for $i=1,\cdots,t$.*

• (ii)

$(b_{1}^{\prime},\cdots,b_{t}^{\prime})A\subseteq(a_{1},\cdots,a_{d})A$.

Proof.

The containment relation of the assumption restricted to degree $n\in\mathbb{Z}$ provides

[TABLE]

for all $n$, and hence $\sum_{i=1}^{t}b_{i}\mathfrak{q}^{n-\beta_{i}}\subseteq\sum_{j=1}^{d}a_{j}\mathfrak{q}^{n-c_{j}}+\mathfrak{q}^{n+1}$ for all $n$. Now choose $n=\beta_{k}$ and therefore $b_{k}\in\sum_{j=1}^{d}a_{j}\mathfrak{q}^{\beta_{k}-c_{j}}+\mathfrak{q}^{\beta_{k}+1}$. Whence there exist $r_{jk}\in\mathfrak{q}^{\beta_{k}-c_{j}}$ for $j=1,\cdots,d,$ such that $b_{k}-\sum_{j=1}^{d}a_{j}r_{jk}\in\mathfrak{q}^{\beta_{k}+1}$. Note that $\sum_{j=1}^{d}a_{j}r_{jk}\in\mathfrak{q}^{\beta_{k}}\setminus\mathfrak{q}^{\beta_{k}+1}$. We choose $b_{k}^{\prime}=\sum_{j=1}^{d}a_{j}r_{jk}$ for $k=1,\cdots,t$, and this finishes the proof. ∎

Now, we present the main result of the section.

Theorem 4.3**.**

With the previous notations, if $\underline{a}^{\star}G_{A}(\mathfrak{q})$ contains a $G_{M}(\mathfrak{q})$-regular sequence $\underline{b}^{\star}=b_{1}^{\star},\ldots,b_{t}^{\star}$, then

[TABLE]

Moreover, given $\underline{b}^{\star}=b_{1}^{\star},\ldots,b_{t}^{\star}$ we choose $b_{i}$ for $i=1,\ldots,t$ as in Lemma 4.2, there is an isomorphism

[TABLE]

where $c:=c_{1}\cdot\ldots\cdot c_{d},\overline{c}:=\sum_{i=1}^{d}c_{i},\overline{c_{i}}:=c_{1}+\ldots+c_{i-1}+c_{i+1}+\ldots+c_{d},$ and $\beta:=\sum_{j=1}^{t}\beta_{j}$.

Proof.

We proceed by induction on $t$. The vanishing $L_{i}(\underline{a},\mathfrak{q},M;n)=0$ for $i>d$ is trivial, and it is easily seen that $L_{d}(\underline{a},\mathfrak{q},M;n)\cong\cap_{i=1}^{d}\mathfrak{q}^{n-\overline{c_{i}}}M:_{M}a_{i}/\mathfrak{q}^{n-\overline{c}}M$. Now assume that $t>0$ and $\underline{b}A\subseteq\underline{a}A$ by lemma 4.2. As by virtue of Valla-Valabrega [13], $b_{2}^{\star},\ldots,b_{t}^{\star}$ is a $G_{M/b_{1}M}(\mathfrak{q})$-regular sequence, therefore by induction $L_{i}(\underline{a},\mathfrak{q},M/b_{1}M;n)=0$ for all $i>d-t+1$ and for all $n$. Hence by lemma 4.1, $L_{i}(\underline{a},\mathfrak{q},M;n)=0$ for all $i>d-t+1$ and for all $n$, and $L_{d-t+1}(\underline{a},\mathfrak{q},M;n-\beta_{1})=0$ for all $n$. Note that $\underline{b}L_{i}(\underline{a},\mathfrak{q},M;n)=0$ for all $i$, for all $n$, see [8, Theorem 3.5(b)].

Again by induction

[TABLE]

and hence by using 4.1 and $b_{1}L_{i}(\underline{a},\mathfrak{q},M;n)=0$ for all $i,n$, we get

[TABLE]

This finishes the inductive argument. ∎

There is a converse of the previous theorem.

Proposition 4.4**.**

With the previous notations, assume that $L_{i}(\underline{a},\mathfrak{q},M;n)=0$ for all $i>d-t$, for all $n$, then $\underline{a}^{\star}G_{A}(\mathfrak{q})$ contains a $G_{M}(\mathfrak{q})$-regular sequence $\underline{b}^{\star}=b_{1}^{\star},\ldots,b_{t}^{\star}$.

Proof.

Note that from the short exact sequence

[TABLE]

there is the following short exact sequence of complexes

[TABLE]

for $n\in\mathbb{N}$, where $K_{\bullet}(\underline{a}^{\star};G_{M}(\mathfrak{q}))_{n}$ denotes the $n$th component of the Koszul complex of $G_{M}(\mathfrak{q})$ w.r.t $\underline{a}^{\star}=a_{1}^{\star},\ldots,a_{d}^{\star}$. From here, by view of long homology exact sequence $H_{i}(\underline{a}^{\star};G_{M}(\mathfrak{q}))_{n}=0$ for all $i>d-t$, for all $n$, and hence $H_{i}(\underline{a}^{\star};G_{M}(\mathfrak{q}))=0$ for all $i>d-t$. Now the result follows by virtue of Koszul homology, see [9]. ∎

5. Applications

Let $(A,\mathfrak{m})$ denote a local Noetherian ring and $M$ be a finitely generated $A$-module with $\dim M=d$. Let $\underline{a}=a_{1},\ldots,a_{d}$ denote a system of parameters of $M$ such that $\underline{a}\subset\mathfrak{q}$. We present the main result of the section.

Theorem 5.1**.**

With the previous notations, if $\underline{a}^{\star}G_{A}(\mathfrak{q})$ contains a $G_{M}(\mathfrak{q})$-regular sequence of length $d-1$, then

• (1)

$\ell_{A}(L_{1}(\underline{a},\mathfrak{q},M;n))$* is a constant for all $n\gg 0$.*

• (2)

$\ell_{A}(M/\underline{a}\,M)=c\cdot e_{0}(\mathfrak{q};M)+\ell_{A}(L_{1}(\underline{a},\mathfrak{q},M;n))$* for all $n\gg 0$, where $c=c_{1}\cdot\ldots\cdot c_{d}$.*

Proof.

Note that the alternating sum of the lengths of modules in the complex $\mathcal{L}_{\bullet}(\underline{a},\mathfrak{q},M;n)$ is

[TABLE]

which is a weighted $d$-fold difference operator of Hilbert-Samuel polynomial for all $n\gg 0$ and hence is a constant $c\cdot e_{0}(\mathfrak{q};M)$. Also, it coincides with the Euler characteristic

[TABLE]

see [8] for more detail. As $L_{0}(\underline{a},\mathfrak{q},M;n)\cong M/(\underline{a}M,\mathfrak{q}^{n}M)=M/\underline{a}M$ for all $n\gg 0$ since $\mathfrak{q}^{n}M\subseteq\underline{a}M$, therefore (2) follows from theorem 4.3. Also, (1) follows from (2). This completes the argument. ∎

Now, we describe the length $\ell_{A}(L_{1}(\underline{a},\mathfrak{q},M;n))$.

Proposition 5.2**.**

With the previous notations, if $\underline{a}^{\star}G_{A}(\mathfrak{q})$ contains a $G_{M}(\mathfrak{q})$-regular sequence $\underline{b}^{\star}=b_{1}^{\star},\ldots,b_{d-1}^{\star}$ such that $\deg b_{i}^{\star}=\beta_{i}$ and we choose $b_{i}$ for $i=1,\ldots,d-1$ as in Lemma 4.2. Then $\ell_{A}(L_{1}(\underline{a},\mathfrak{q},M;n))$ might be broken into two pieces. That is,

[TABLE]

where

[TABLE]

with $c=c_{1}\ldots c_{d},\overline{c}=\sum_{i=1}^{d}c_{i},\overline{c_{i}}=c_{1}+\ldots+c_{i-1}+c_{i+1}+\ldots+c_{d}$, and $\beta=\sum_{j=1}^{d-1}\beta_{j}$.

Moreover, for $n\gg 0$, all of the lengths involved here are constants and independent of the choice of $\underline{b}^{\star}$. We write $\mathfrak{x}=\ell_{A}([\underline{b}^{\star}G_{M}(\mathfrak{q}):\underline{a}^{\star}/(\underline{b}^{\star}G_{M}(\mathfrak{q}))]_{n})$ and $\ell=\ell_{n}$ for $n\gg 0$.

Proof.

As $\underline{b}^{\star}$ is a $G_{M}(\mathfrak{q})$-regular sequence, hence $G_{M}(\mathfrak{q})/(\underline{b}^{\star})G_{M}(\mathfrak{q})\cong G_{M/\underline{b}M}(\mathfrak{q})$, see [13]. Therefore, it is easily seen that

[TABLE]

Now, we have the following short exact sequence

[TABLE]

see theorem 4.3. By counting the lengths, it provides the first equality of the statement. The length of the module in the middle is constant for $n\gg 0$, see 5.1. Also, the length of the module in the left is constant for $n\gg 0$ since it is of dimension 1. By comparing the Hilbert polynomials, this proves that all the lengths are constants for all $n\gg 0$.

Note that

[TABLE]

Therefore, we conclude that $\ell_{A}([\underline{b}^{\star}G_{M}(\mathfrak{q}):\underline{a}^{\star}/\underline{b}^{\star}G_{M}(\mathfrak{q})]_{n})$ is independent of the choice of $\underline{b}^{\star}$, and consequently, $\ell$ is also independent of the choice of $\underline{b}^{\star}$. This completes the proof. ∎

Now, we have the main result of the section, which is also the consequence of previous two results.

Corollary 5.3**.**

With the previous notations, if $\underline{a}^{\star}G_{A}(\mathfrak{q})$ contains a $G_{M}(\mathfrak{q})$-regular sequence $\underline{b}^{\star}=b_{1}^{\star},\ldots,b_{d-1}^{\star}$ and we choose $b_{i}$ for $i=1,\ldots,d-1$ as in Lemma 4.2. Then

[TABLE]

where $c=c_{1}\cdot\ldots\cdot c_{d}$ and $\mathfrak{x}=\ell_{A}([\operatorname{Ext}^{d-1}_{G_{A}(\mathfrak{q})}(G_{A}(\mathfrak{q})/\underline{a}^{\star}G_{A}(\mathfrak{q}),G_{M}(\mathfrak{q}))]_{n-\overline{c}-1})$ is a constant for all $n\gg 0$ and $\overline{c}=c_{1}+\ldots+c_{d}.$

We mention a geometric application to local Bézout inequality in the affine plane $\mathbb{A}_{k}^{2}$.

Remark 5.4**.**

Let $\Bbbk$ be an algebraically close field and $A=\Bbbk[x,y]_{(x,y)}$ be a local ring. Also, let $f,g$ denote a system of parameters in $A$ and $\mathfrak{m}$ denote the maximal ideal of $A$. Then $B:=\Bbbk[X,Y]\cong G_{A}(\mathfrak{m})$ and $1=e_{0}(\mathfrak{m};A)$. Then the above two results imply that

[TABLE]

where $t$ denotes the number of common tangents to $f,g$ at origin when counted with multiplicities. Note that $\ell_{A}([f^{\star}B:_{B}g^{\star}/f^{\star}B]_{n})=t$ for all $n\gg 0$ (see [2]).

Problem 5.5**.**

Let $M=A=k[x_{1},\ldots,x_{d}]_{(x_{1},\ldots,x_{d})}$ be the local ring and $\mathfrak{q}=\mathfrak{m}=(x_{1},\ldots,x_{d})A$, where $d\geq 3$. Let $\underline{a}^{\prime\star}=a_{1}^{\star},\ldots,a_{d-1}^{\star}$, then the author does not know the geometric interpretation of

[TABLE]

for all $n\gg 0$. This problem can be related to the homological terms as in case of $d=2$.

In the next, we present another consequence. More precisely, there is an upper bound to $\ell_{A}(M/\underline{a}\,M)-e_{0}(\underline{a};M)\geq 0.$

Corollary 5.6**.**

With the previous notations, if $\underline{a}^{\star}G_{A}(\mathfrak{q})$ contains a $G_{M}(\mathfrak{q})$-regular sequence $\underline{b}^{\star}=b_{1}^{\star},\ldots,b_{d-1}^{\star}$, then

• (1)

$\ell_{A}(M/\underline{a}\,M)-e_{0}(\underline{a};M)\leq\ell_{A}(L_{1}(\underline{a},\mathfrak{q},M;n))$* for all $n\gg 0$.*

• (2)

Equality occurs when $\underline{a}^{\star}$ is a system of parameters of $G_{M}(\mathfrak{q})$. The converse is not true in general.

Proof.

Since $c\cdot e_{0}(\mathfrak{q};M)\leq e(\underline{a};M)$ (see [2]). Therefore claim in (1) follows from previous theorem 5.1. Note that

[TABLE]

(see theorem 5.1). Now, the claim in (2) follows by [2, Theorem 5.1]. ∎

6. An Euler Characteristic

With the notations of the previous section, we have the following lemma.

Lemma 6.1**.**

With the previous notations, if $\underline{a}^{\prime\star}=a_{1}^{\star},\ldots,a_{d-1}^{\star}$ is a $G_{M}(\mathfrak{q})$-regular sequence, then

[TABLE]

for all $n\gg 0$, where $c=c_{1}\cdot\ldots\cdot c_{d}$.

Proof.

Note $H_{i}(\underline{a};M)=0$ for all $i>1$ since $a_{1},\ldots,a_{d-1}$ is $M$-regular sequence, cf. [13]. Also, $L_{i}(\underline{a},\mathfrak{q},M;n)=0$ for all $i>1$ and for all $n$, see 4.3. Moreover,

[TABLE]

since $\mathfrak{q}^{n}M\subseteq\underline{a}M$ for $n\gg 0$. Therefore, from the long exact homology sequence coming from the short exact sequence in 2.3, we get the following exact sequence

[TABLE]

for all $n\gg 0$. Note that $H_{1}(\underline{a};M)\cong\underline{a}^{\prime}M:_{M}a_{d}/\underline{a}^{\prime}M$, where $\underline{a}^{\prime}=a_{1},\ldots,a_{d-1}$. By Theorem 5.1, we get

[TABLE]

for all $n\gg 0$. Finally, note that

[TABLE]

and

[TABLE]

since $a_{1}T^{c_{1}},\ldots,a_{d-1}T^{c_{d-1}}$ is an $R_{M}(\mathfrak{q})$-regular sequence, see 3.2. This finishes the proof. ∎

Let $\chi_{A}(\underline{a},\mathfrak{q},M;n)$ denote the Euler characteristic of the complex $K_{\bullet}(\underline{a},\mathfrak{q},M;n)$. With the assumption of previous lemma,

[TABLE]

which is a constant. Even in a more general situation, we have

[TABLE]

see [2] or [8]. Moreover, the authors mentioned a problem of giving an interpretation to $\chi_{A}(\underline{a},\mathfrak{q},M;n)$ for $n\gg 0$. In case of $M=A$, Bod̆a-Schenzel [2] proved that $\chi_{A}(\underline{a},\mathfrak{q},M;n)\geq 0$ for all $n\gg 0$. This can be generalized for an $A-$module $M$ with slight modification. In the following, in a particular case, we bound this Euler characteristic from the upper side, and also discuss the equality in terms of Cohen-Macauleyness of $M$.

Corollary 6.2**.**

With the previous notations, if $\underline{a}^{\prime\star}=a_{1}^{\star},\ldots,a_{d-1}^{\star}$ is a $G_{M}(\mathfrak{q})$-regular sequence, then

• (1)

$\chi_{A}(\underline{a},\mathfrak{q},M;n)\leq\ell_{A}(L_{1}(\underline{a},\mathfrak{q},M;n))$* for all $n\gg 0$.*

• (2)

Equality occurs if and only if $M$ is Cohen-Macauley.

Proof.

Since $\ell_{A}(M/\underline{a}\,M)\geq e_{0}(\underline{a};M)$, therefore claim in (1) follows from previous theorem 5.1. Note that

[TABLE]

see theorem 5.1. The latter is equivalent to the fact that $M$ is Cohen-Macaulay, see [4]. This finishes (2). ∎

In the following, we discuss a few more properties of Euler characteristic $\chi_{A}(\underline{a},\mathfrak{q},M;n)$. We need the following lemma.

Lemma 6.3**.**

With the previous notations, assume that $a\in\mathfrak{q}^{c}\setminus\mathfrak{q}^{c+1}$ such that $\dim M/aM=d-1$. Then the following holds.

• (1)

If $\dim 0:_{M}a\leq d-2$, then $c\cdot e_{0}(\mathfrak{q};M)\leq e_{0}(\mathfrak{q};M/aM)$. Moreover, equality occurs if and only if $\deg\ell_{A}(\mathfrak{q}^{n}M:a/(\mathfrak{q}^{n-c}M+0:_{M}a))\leq d-2$ for all $n\gg 0$.

• (2)

If $\dim 0:_{M}a=d-1$, then $c\cdot e_{0}(\mathfrak{q};M)+e_{0}(\mathfrak{q};0:_{M}a)\leq e_{0}(\mathfrak{q};M/aM)$. Moreover, equality occurs if and only if $\deg\ell_{A}(\mathfrak{q}^{n}M:a/(\mathfrak{q}^{n-c}M+0:_{M}a))\leq d-2$ for all $n\gg 0$.

Proof.

Note that we have the following complex

[TABLE]

and hence

[TABLE]

where $L_{0}(a,\mathfrak{q},M;n)\cong M/(a,\mathfrak{q}^{n})M$ and $L_{1}(a,\mathfrak{q},M;n)\cong\mathfrak{q}^{n}M:a/\mathfrak{q}^{n-c}M$. We break

$\ell_{A}(\mathfrak{q}^{n}M:a/\mathfrak{q}^{n-c}M)$ by using the following short exact sequence:

[TABLE]

Also, by Artin-Rees we have

[TABLE]

for some $l\in\mathbb{N}$ and for all $n\geq l$. That is,

[TABLE]

By using last equation into $(\star)$, we get

[TABLE]

where all lengths involved are polynomials for $n\gg 0$ with $\deg(\ell_{A}(M/\mathfrak{q}^{n}M)-\ell_{A}(M/\mathfrak{q}^{n-c}M))=\deg\ell_{A}(M/(a,\mathfrak{q}^{n})M)=d-1$, $\deg\ell_{A}(\mathfrak{q}^{l}M\cap 0:_{M}a/\mathfrak{q}^{n-c-l}(\mathfrak{q}^{l}M\cap 0:_{M}a))=\dim 0:_{M}a\leq d-1$, $\deg\ell_{A}(\mathfrak{q}^{n}M:a/(\mathfrak{q}^{n-c}M+0:_{M}a))\leq d-1$ and $\ell_{A}(0:_{M}a/0:_{M}a\cap\mathfrak{q}^{l}M)$ is a constant. Also, leading terms of $\ell_{A}(M/\mathfrak{q}^{n}M)-\ell_{A}(M/\mathfrak{q}^{n-c}M)$ and $\ell_{A}(M/(a,\mathfrak{q}^{n})M)$ are $c\cdot e_{0}(\mathfrak{q};M)$ and $e_{0}(\mathfrak{q};M/aM)$ respectively for all $n\gg 0$. Now, in case of (1), we get

[TABLE]

and in case of (2), we get

[TABLE]

Note that leading term of $\ell_{A}(\mathfrak{q}^{l}M\cap 0:_{M}a/\mathfrak{q}^{n-c-l}(\mathfrak{q}^{l}M\cap 0:_{M}a))$ is $e_{0}(\mathfrak{q};\mathfrak{q}^{l}M\cap 0:_{M}a),$ which is equal to $e_{0}(\mathfrak{q};0:_{M}a)$. Indeed, we have the following short exact sequence

[TABLE]

and $\dim 0:_{M}a/(\mathfrak{q}^{l}M\cap 0:_{M}a)=0$ whereas $\dim 0:_{M}a=\dim(\mathfrak{q}^{l}M\cap 0:_{M}a)$. Therefore $e_{0}(\mathfrak{q};0:_{M}a)=e_{0}(\mathfrak{q};\mathfrak{q}^{l}M\cap 0:_{M}a)$, cf. [9, Theorem 13.3]. Finally, equality in both cases occur if and only if $\deg\ell_{A}(\mathfrak{q}^{n}M:a/(\mathfrak{q}^{n-c}M+0:_{M}a))\leq d-2$ for all $n\gg 0$. ∎

There is the following consequence of previous lemma.

Proposition 6.4**.**

With the previous notations, let $\underline{a}=a_{1},\ldots,a_{d}$ be a system of parameters of $M$, then the following holds.

• (1)

If $\dim 0:_{M}a\leq d-2$, then $\chi_{A}(\underline{a},\mathfrak{q},M)\geq\chi_{A}(\underline{a}^{\prime},\mathfrak{q},M/a_{1}M)$, where $\underline{a}^{\prime}=a_{2},\ldots,a_{d}$. Moreover, equality occurs if and only if

[TABLE]

for all $n\gg 0$.

• (2)

If $\dim 0:_{M}a=d-1$, then $\chi_{A}(\underline{a},\mathfrak{q},M)+\chi_{A}(\underline{a}^{\prime},\mathfrak{q},0:_{M}a_{1})\geq\chi_{A}(\underline{a}^{\prime},\mathfrak{q},M/a_{1}M)$, where $\underline{a}^{\prime}=a_{2},\ldots,a_{d}$. Moreover, equality occurs if and only if

[TABLE]

for all $n\gg 0$.

Proof.

Note that $\underline{a}^{\prime}=a_{2},\ldots,a_{d}$ is a system of parameters for both $A$-modules $M/a_{1}M$ and $0:_{M}a_{1}$. Also, it is a well known fact that

[TABLE]

see for example [4]. For (1), we use Lemma 6.3(1) and get $c\cdot e_{0}(\mathfrak{q};M)\leq c_{2}\cdot\ldots\cdot c_{d}\cdot e_{0}(\mathfrak{q};M/a_{1}M)$, where $c=c_{1}\cdot\ldots\cdot c_{d}$. Hence by definition

[TABLE]

where the equality occurs if and only if $\deg\ell_{A}(\mathfrak{q}^{n}M:a/(\mathfrak{q}^{n-c_{1}}M+0:_{M}a_{1}))\leq d-2$ for all $n\gg 0$.

For (2), we use Lemma 6.3(2) and get

[TABLE]

Hence by definition

[TABLE]

where the equality occurs if and only if $\deg\ell_{A}(\mathfrak{q}^{n}M:a/(\mathfrak{q}^{n-c_{1}}M+0:_{M}a_{1}))\leq d-2$ for all $n\gg 0$. ∎

We finish with the following remark.

Remark 6.5**.**

Lemma 6.3, with slight modification, originally proved by Flenner-Vogel [7]. More precisely, they proved the equality in lemma if and only if $a^{\star}$ is a parameter for $G_{M}(\mathfrak{q})$. The author of present note tried to prove directly that $\deg\ell_{A}(\mathfrak{q}^{n}M:a/(\mathfrak{q}^{n-c_{1}}M+0:_{M}a))\leq d-2$ for all $n\gg 0$ if and only if $a^{\star}$ is a parameter for $G_{M}(\mathfrak{q})$. The ”if” part is easy. Indeed, $\deg\ell_{A}(\mathfrak{q}^{n}M:a/(\mathfrak{q}^{n-c_{1}}M+0:_{M}a))\leq d-2$ for all $n\gg 0$ implies that

[TABLE]

where

[TABLE]

But, this is equivalent to $\dim(\ker(G_{M}(\mathfrak{q})/a^{\star}G_{M}(\mathfrak{q})\to G_{M/aM}(\mathfrak{q})))\leq d-1$ which is equivalent to the fact that $a^{\star}$ is a parameter for $G_{M}(\mathfrak{q})$.

Acknowledgement: The author is grateful to the reviewer for comments and suggestions.