Upper bound of multiplicity in prime characteristic

TL;DR

This paper establishes an upper bound on the multiplicity of local rings in prime characteristic based on Frobenius test exponents, extending previous results and improving bounds for specific classes like F-nilpotent rings.

Contribution

The paper provides a new upper bound on the multiplicity of local rings in prime characteristic using Frobenius test exponents, extending prior work and refining bounds for F-nilpotent rings.

Findings

Derived an explicit upper bound for multiplicity involving Frobenius test exponent.

Extended and improved bounds for F-nilpotent rings.

Generalized previous results by Huneke, Watanabe, Katzman, and Zhang.

Abstract

Let $(R, \frak m)$ be a local ring of prime characteristic $p$ of dimension $d$ with the embedding dimension $v$. Suppose the Frobenius test exponent for parameter ideals $Fte(R)$ of $R$ is finite, and let $Q = p^{Fte(R)}$. It is shown that $$e(R) \le Q^{v-d} \binom{v}{d}.$$ We also improve our bound for $F$-nilpotent rings. Our result extends the main results of Huneke and Watanabe and of Katzman and Zhang.

Equations

$$e(R)\le Q^{v-d}\left(\genfrac{}{}{0ex}{}{v}{d}\right).$$

$$e(R)\le \left(\genfrac{}{}{0ex}{}{v}{d}\right)$$

$$e(R)\le Q^{v-d}\left(\genfrac{}{}{0ex}{}{v}{d}\right),$$

$$e(R)\le Q^{v-d}\left(\genfrac{}{}{0ex}{}{v-1}{d-1}\right),$$

$$e(R)\le Q^{v-d}\left(\genfrac{}{}{0ex}{}{v}{d}\right),$$

$$HSL(H_{\mathfrak{m}}^i(R))=\min \{e\mid F^e(0_{H_{\mathfrak{m}}^i(R)}^F)=0\}.$$

$$e(R)\le Q^{v-d}\left(\genfrac{}{}{0ex}{}{v-1}{d-1}\right),$$

$$e(R)\le Q^{v-d}\left(\genfrac{}{}{0ex}{}{v}{d}\right),$$

$$x_1^{{\alpha }_1}\cdots x_d^{{\alpha }_d}{y}_1^{{\beta }_{1}Q+{\gamma }_1}\cdots y_1^{{\beta }_{v-d}Q+{\gamma }_{v-d}},$$

$$e(R)=\frac{1}{Q^d}e({\mathfrak{q}}^{[Q]})\le \frac{1}{Q^d}{\ell }_R(R/{\mathfrak{q}}^{[Q]})\le \frac{1}{Q^d}{Q}^v\left(\genfrac{}{}{0ex}{}{v-1}{d-1}\right)=Q^{v-d}\left(\genfrac{}{}{0ex}{}{v-1}{d-1}\right).$$

$$e(R)=4>3=\left(\genfrac{}{}{0ex}{}{v-1}{d-1}\right).$$

Full text

[TABLE]

We also improve the bound for $F$-nilpotent rings. Our result extends the main results of Huneke and Watanabe [6] and of Katzman and Zhang [9].

2010 Mathematics Subject Classification: 13H15, 13A35.

The second author is partially supported by a fund of Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2017.10.

1. Introduction

Throughout this paper, let $(R,\mathfrak{m})$ be a Noetherian commutative local ring of prime characteristic $p>0$ and of dimension $d$. Our work is inspired by the work of Huneke and Watanabe [6] in what they gave an upper bound of the multiplicity $e(R)$ of an $F$-pure ring $R$ in terms of the embedding dimension $v$. Namely, Huneke and Watanabe proved that

[TABLE]

for any $F$-pure ring. If $R$ is $F$-rational, the authors of [6] provided a better bound that $e(R)\leq\binom{v-1}{d-1}$ (cf. [6, Theorem 3.1]). Recently, Katzman and Zhang tried to remove the $F$-pure condition in Huneke-Watanabe’s theorem by using the Hartshorne-Speiser-Lyubeznik number $HSL(R)$. Notice that $HLS(R)=0$ if $R$ is $F$-injective (e.g. $R$ is $F$-pure). If $R$ is Cohen-Macaulay, Katzman and Zhang [9, Theorem 3.1] proved the following inequality

[TABLE]

where $Q=p^{HSL(R)}$. They also constructed examples to show that their bound is asymptotically sharp (cf. [9, Remark 3.2]).

The key ingredient of this paper is the Frobenius test exponent for parameter ideals of $R$. Recall that the Frobenius test exponent for parameter ideals of $R$, denoted by $Fte(R)$, is the least integer (if exists) $e$ satisfying that $(\mathfrak{q}^{F})^{[p^{e}]}=\mathfrak{q}^{[p^{e}]}$ for every parameter ideal $\mathfrak{q}$, where $\mathfrak{q}^{F}$ is the Frobenius closure of $\mathfrak{q}$. It is asked by Katzman and Sharp that whether $Fte(R)<\infty$ for every (equidimensional) local ring (cf. [8]). If $R$ is Cohen-Macaulay then $Fte(R)=HSL(R)$. Moreover the question of Katzman and Sharp has affirmative answers when $R$ is either generalized Cohen-Macaulay by [5] or $F$-nilpotent by [14] (see the next section for the details). The main result of the present paper is as follows.

Theorem 1.1**.**

Let $(R,\mathfrak{m})$ be a local ring of dimension $d$ with the embedding dimension $v$. Then

1. (1)

If $R$ is $F$-nilpotent then

[TABLE]

where $Q=p^{Fte(R)}$. 2. (2)

Suppose $Fte(R)<\infty$. Then

[TABLE]

where $Q=p^{Fte(R)}$.

We will prove the above theorem in the last section. In the next section we collect some useful materials.

2. Preliminaries

$F$-singularities. We firstly give the definition of the tight closure and the Frobenius closure of ideals.

Definition 2.1** ([3, 4]).**

Let $R$ have characteristic $p$. We denote by $R^{\circ}$ the set of elements of $R$ that are not contained in any minimal prime ideal.Then for any ideal $I$ of $R$ we define

1. (1)

The Frobenius closure of $I$, $I^{F}=\{x\mid x^{Q}\in I^{[Q]}\text{ for some }Q=p^{e}\}$, where $I^{[Q]}=(x^{Q}\mid x\in I)$. 2. (2)

The tight closure of $I$, $I^{*}=\{x\mid cx^{Q}\in I^{[Q]}\text{ for some }c\in R^{\circ}\text{ and for all }Q=p^{e}\gg 0\}$.

We next recall some classes of $F$-singularities mentioned in this paper.

Definition 2.2**.**

A local ring $(R,\mathfrak{m})$ is called $F$-rational if it is a homomorphic image of a Cohen-Macaulay local ring and every parameter ideal is tight closed, i.e. $\mathfrak{q}^{*}=\mathfrak{q}$ for all $\mathfrak{q}$.

Definition 2.3**.**

A local ring $(R,\mathfrak{m})$ is called $F$-pure if the Frobenius endomorphism $F:R\to R,x\mapsto x^{p}$ is a pure homomorphism. If $R$ is $F$-pure, then it is proved that every ideal $I$ of $R$ is Frobenius closed, i.e. $I^{F}=I$ for all $I$.

The Frobenius endomorphism of $R$ induces the natural Frobenius action on local cohomology $F:H^{i}_{\mathfrak{m}}(R)\to H^{i}_{\mathfrak{m}}(R)$ for all $i\geq 0$. By a similar way, we can define the Frobenius closure and tight closure of zero submodule of local cohomology, and denote by $0^{F}_{H^{i}_{\mathfrak{m}}(R)}$ and $0^{*}_{H^{i}_{\mathfrak{m}}(R)}$ respectively.

Definition 2.4**.**

1. (1)

A local ring $(R,\mathfrak{m})$ is called $F$-injective if the Frobenius action on $H^{i}_{\mathfrak{m}}(R)$ is injective, i.e. $0^{F}_{H^{i}_{\mathfrak{m}}(R)}=0$, for all $i\geq 0$. 2. (2)

A local ring $(R,\mathfrak{m})$ is called $F$-nilpotent if the Frobenius actions on all lower local cohomologies $H^{i}_{\mathfrak{m}}(R)$, $i\leq d-1$, and $0^{*}_{H^{d}_{\mathfrak{m}}(R)}$ are nilpotent, i.e. $0^{F}_{H^{i}_{\mathfrak{m}}(R)}=H^{i}_{\mathfrak{m}}(R)$ for all $i\leq d-1$ and $0^{F}_{H^{d}_{\mathfrak{m}}(R)}=0^{*}_{H^{d}_{\mathfrak{m}}(R)}$.

Remark 2.5**.**

1. (1)

It is well known that an equidimensional local ring $R$ is $F$-rational if and only if it is Cohen-Macaulay and $0^{*}_{H^{d}_{\mathfrak{m}}(R)}=0$. 2. (2)

An excellent equidimensional local ring is $F$-rational if and only if it is both $F$-injective and $F$-nilpotent. 3. (3)

Suppose every parameter ideal of $R$ is Frobenius closed. Then $R$ is $F$-injective (cf. [13, Main Theorem A]). In particular, an $F$-pure ring is $F$-injective. 4. (4)

An excellent equidimensional local ring $R$ is $F$-nilpotent if and only if $\mathfrak{q}^{*}=\mathfrak{q}^{F}$ for every parameter ideal $\mathfrak{q}$ (cf. [12, Theorem A]).

$F$-invariants. We will bound the multiplicity of a local ring of prime characteristic in terms of the Frobenius test exponent for parameter ideals of $R$. Let $I$ be an ideal of $R$. The Frobenius test exponent of $I$, denoted by $Fte(I)$, is the smallest number $e$ satisfying that $(I^{F})^{[p^{e}]}=I^{[p^{e}]}$. By the Noetherianess of $R$, $Fte(I)$ exists (and depends on $I$). In general, there is no upper bound for the Frobenius test exponents of all ideals in a local ring by the example of Brenner [1]. In contrast, Katzman and Sharp [8] showed the existence of a uniform bound of Frobenius test exponents if we restrict to the class of parameter ideals in a Cohen-Macaulay local ring. For any local ring $(R,\mathfrak{m})$ of prime characteristic $p$ we define the Frobenius test exponent for parameter ideals, denoted by $Fte(R)$, is the smallest integer $e$ such that $(\mathfrak{q}^{F})^{[p^{e}]}=\mathfrak{q}^{[p^{e}]}$ for every parameter ideal $\mathfrak{q}$ of $R$, and $Fte(R)=\infty$ if we have no such integer. Katzman and Sharp raised the following question.

Question 1**.**

Is $Fte(R)$ a finite number for any (equidimensional) local ring?

The Frobenius test exponent for parameter ideals is closely related to an invariant defined by the Frobenius actions on the local cohomology modules $H^{i}_{\mathfrak{m}}(R)$, namely the Hartshorne-Speiser-Lyubeznik number of $H^{i}_{\mathfrak{m}}(R)$. The Hartshorne-Speiser-Lyubeznik number of $H^{i}_{\mathfrak{m}}(R)$ is a nilpotency index of Frobenius action on $H^{i}_{\mathfrak{m}}(R)$ and it is defined as follows

[TABLE]

By [2, Proposition 1.11] and [10, Proposition 4.4] $HSL(H^{i}_{\mathfrak{m}}(R))$ is well defined (see also [15]). The Hartshorne-Speiser-Lyubeznik number of $R$ is $HSL(R)=\max\{HSL(H^{i}_{\mathfrak{m}}(R))\mid i=0,\ldots,d\}$.

Remark 2.6**.**

1. (1)

If $R$ is Cohen-Macaulay then $Fte(R)=HSL(R)$ by Katzman and Sharp [8]. In general, the authors of this paper proved in [7] that $Fte(R)\geq HSL(R)$. Moreover, Shimomoto and the second author [13, Main Theorem B] constructed a local ring satisfying that $HSL(R)=0$, i.e. $R$ is $F$-injective, but $Fte(R)>0$. 2. (2)

Huneke, Katzman, Sharp and Yao [5] gave an affirmative answer for Question 1 for generalized Cohen-Macaulay rings. 3. (3)

Recently, the second author [14] provided a simple proof for the theorem of Huneke, Katzman, Sharp and Yao. By the same method he also proved that $Fte(R)<\infty$ if $R$ is $F$-nilpotent. Very recently, Maddox [11] extended this result for generalized $F$-nilpotent rings.

3. Proof of the main result

This section is devoted to prove the main result of this paper. Without loss of generality we will assume that $R$ is complete with an infinite residue field. We need the following key lemma.

Lemma 3.1**.**

Let $(R,\mathfrak{m})$ be a local ring of dimension $d$, and $\mathfrak{q}$ a parameter ideal.

1. (1)

If $R$ is $F$-nilpotent then $\overline{\mathfrak{q}^{d}}\subseteq\mathfrak{q}^{F}$, where $\overline{I}$ is the integral closure of ideal $I$. 2. (2)

In general we have $\overline{\mathfrak{q}^{d+1}}\subseteq\mathfrak{q}^{F}$.

Proof.

(1) By the Briançon-Skoda type theorem [3, Theorem 5.6] we have $\overline{\mathfrak{q}^{d}}\subseteq\mathfrak{q}^{*}$. The assertion now follows from Remark 2.5(4).

(2) The assertion follows from [9, Theorem 2.2]111In fact Katzman and Zhang [9, Theorem 2.2] needed to assume that every $c\in R^{\circ}$ is a non-zero divisor, i.e. $R$ has no embedded primes. However, we can easily remove this condition by passing to the quotient ring $R/\mathfrak{a}$, where $\mathfrak{a}$ is the intersection of primary ideals corresponding to minimal primes in a primary decomposition of the zero ideal.. ∎

We prove the main result of this paper.

Theorem 3.2**.**

Let $(R,\mathfrak{m})$ be a local ring of dimension $d$ with the embedding dimension $v$. Then

1. (1)

If $R$ is $F$-nilpotent then

[TABLE]

where $Q=p^{Fte(R)}$. 2. (2)

Suppose $Fte(R)<\infty$ (e.g. $R$ is generalized Cohen-Macaulay or generalized $F$-nilpotent). Then

[TABLE]

where $Q=p^{Fte(R)}$.

Proof.

Because the proofs of two assertions are almost the same, we will only prove (1). Since $R$ is $F$-nilpotent we have $Fte(R)<\infty$ by Remark 2.6(3). Let $\mathfrak{q}=(x_{1},\ldots,x_{d})$ be a minimal reduction of $\mathfrak{m}$. By Lemma 3.1(1) we have $\mathfrak{m}^{d}\subseteq\overline{\mathfrak{m}^{d}}=\overline{\mathfrak{q}^{d}}\subseteq\mathfrak{q}^{F}$. On the other hand we have $(\mathfrak{q}^{F})^{[Q]}=\mathfrak{q}^{[Q]}$ by the definition of $Fte(R)$. Thus $(\mathfrak{m}^{d})^{[Q]}\subseteq\mathfrak{q}^{[Q]}$. Extend $x_{1},\ldots,x_{d}$ to a minimal set of generators $x_{1},\ldots,x_{d},y_{1},\ldots,y_{v-d}$ of $\mathfrak{m}$. Now $R/\mathfrak{q}^{[Q]}$ is spanned by monomials

[TABLE]

where $0\leq\alpha_{1},\ldots,\alpha_{d},\gamma_{1},\ldots,\gamma_{v-d}<Q$ and $0\leq\beta_{1}+\cdots+\beta_{v-d}<d$. The number of such monomials is $Q^{v}\binom{v-1}{d-1}$ so $\ell_{R}(R/\mathfrak{q}^{[Q]})\leq Q^{v}\binom{v-1}{d-1}$.

Since $\mathfrak{q}$ is a parameter ideal we have $e(\mathfrak{q}^{[Q]})=Q^{d}e(\mathfrak{q})=Q^{d}e(R)$ and $e(\mathfrak{q}^{[Q]})\leq\ell_{R}(R/\mathfrak{q}^{[Q]})$. Hence

[TABLE]

The proof is complete. ∎

Finally, we present an example to prove that we can not remove $Q=p^{e}$ in the previous theorem.

Example 3.3**.**

Let $R=\mathbb{F}_{p}[[X^{4},X^{3}Y,XY^{3},Y^{4}]]$. It is easy to see that $\dim R=2$ and $e(R)=4$. Moreover, we can check $H^{1}_{\mathfrak{m}}(R)\cong\mathbb{F}_{p}$ and the Frobenius action on $H^{1}_{\mathfrak{m}}(R)$ is nilpotent. Thus $HSL(H^{1}_{\mathfrak{m}}(R))=1$. Let $S=\mathbb{F}_{p}[[X^{4},X^{3}Y,X^{2}Y^{2},XY^{3},Y^{4}]]$, the integral closure of $R$. We have that $S$ is $F$-regular and $H^{2}_{\mathfrak{m}}(R)\cong H^{2}_{\mathfrak{m}}(S)$. Therefore $0^{*}_{H^{2}_{\mathfrak{m}}(R)}=0$ and $R$ is $F$-nilpotent. We have $Fte(R)\leq 2$ by the main theorem of [14]222We believe that $Fte(R)=1$.. We can not omit $Q^{v-d}$ in Theorem 3.2 (1) since

[TABLE]

**Acknowledgement **.

The authors are grateful to the referee for careful reading and valuable comments.