Tight closure of parameter ideals in local rings $F$-rational on the punctured spectrum

TL;DR

This paper investigates the behavior of tight closure of parameter ideals in certain local rings, showing independence of length from the choice of ideal under specific $F$-singularity conditions.

Contribution

It demonstrates that in $F$-injective local rings that are $F$-rational on the punctured spectrum, the length of the tight closure difference is independent of the parameter ideal chosen.

Findings

The length $ ext{length}_R(rak{q}^*/rak{q})$ is invariant under different parameter ideals.

The result applies to excellent equidimensional local rings of characteristic $p>0$.

Provides insight into the structure of tight closure in rings with specific $F$-singularities.

Abstract

Let $(R, \mathfrak{m}, k)$ be an excellent equidimensional local ring of characteristic $p>0$. The aim of this paper is to show that $\ell_R(\mathfrak{q}^*/\mathfrak{q})$ does not depend on the choice of parameter ideal $\mathfrak{q}$ provided $R$ is an $F$-injective local ring that is $F$-rational on the punctured spectrum.

Equations

$${\ell }_R(R/\mathfrak{q})-e(\mathfrak{q})\le \sum _{i=0}^{d-1}\left(\genfrac{}{}{0ex}{}{d-1}{i}\right){\ell }_R(H_{\mathfrak{m}}^i(R))$$

$${\ell }_R(M/\mathfrak{q}M)-e(\mathfrak{q},M)$$

$${\ell }_R(M/\mathfrak{q}M)-e(\mathfrak{q},M)\le \sum _{i=0}^{t-1}\left(\genfrac{}{}{0ex}{}{t-1}{i}\right){\ell }_R(H_{\mathfrak{m}}^i(M))$$

$${\ell }_R(M/\mathfrak{q}M)-e(\mathfrak{q},M)=\sum _{i=0}^{t-1}\left(\genfrac{}{}{0ex}{}{t-1}{i}\right){\ell }_R(H_{\mathfrak{m}}^i(M)).$$

$$(\underline{x})_{M}^{\lim}=\bigcup_{n>0}\big{(}(x_{1}^{n+1},\ldots,x_{t}^{n+1})M:_{M}(x_{1}\cdots x_{t})^{n}\big{)}$$

$$H^t(\underline{x};M)\cong M/(\underline{x})M\to H_{\mathfrak{m}}^t(M).$$

$${\ell }_R({\mathfrak{q}}_M^{\lim }/\mathfrak{q}M)\le \sum _{i=0}^{t-1}\left(\genfrac{}{}{0ex}{}{t}{i}\right){\ell }_R(H_{\mathfrak{m}}^i(M)),$$

$$R\to R/I\stackrel{\mathrm{id}\otimes F^e}{⟶}R/I\otimes F_{\ast }^e(R)$$

$$R\to R/I\to R/I\otimes F_{\ast }^e(R)\stackrel{\mathrm{id}\otimes F_{\ast }^e(c)}{⟶}R/I\otimes F_{\ast }^e(R)$$

$$M\to M/N\to M/N\otimes F_{\ast }^e(R)\stackrel{\mathrm{id}\otimes F_{\ast }^e(c)}{⟶}M/N\otimes F_{\ast }^e(R)$$

$$0\to R\to \underset{i=1}{\overset{t}{⨁}}{R}_{x_i}\to \cdots \to R_{x_1\dots x_d}\to 0.$$

$$H_{\mathfrak{m}}^d(R)\cong \underset{⟶}{\lim }R/(x_1^n,\dots ,x_d^n).$$

$$\underset{⟶}{\lim }(x_1^n,\dots ,x_d^n{)}^{\ast }/(x_1^n,\dots ,x_d^n)\cong 0_{H_{\mathfrak{m}}^d(R)}^{\ast }.$$

$$\underset{⟶}{\lim }(x_1^n,\dots ,x_d^n{)}^{\ast }/(x_1^n,\dots ,x_d^n{)}^{\lim }\cong 0_{H_{\mathfrak{m}}^d(R)}^{\ast }$$

$${\ell }_R({\mathfrak{q}}^{\ast }/\mathfrak{q})\le \sum _{i=0}^{d-1}\left(\genfrac{}{}{0ex}{}{d}{i}\right){\ell }_R(H_{\mathfrak{m}}^i(R))+{\ell }_R(0_{H_{\mathfrak{m}}^d(R)}^{\ast }).$$

$$F\text{-rational/}{\mathrm{Spec}}^{\circ }(R)\&F\text{-injective}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathfrak{q}}_{F_{\ast }^e(R)}^{\ast }/{\mathfrak{q}}_{F_{\ast }^e(R)}^{\lim }\to 0_{H_{\mathfrak{m}}^d(F_{\ast }^e(R))}^{\ast }$$

$$0\to R\to F_{\ast }^e(R)\to F_{\ast }^e(R)/R\to 0,$$

$$0\to H_{\mathfrak{m}}^i(R)\to H_{\mathfrak{m}}^i(S)\to H_{\mathfrak{m}}^i(S/R)\to 0$$

$${\ell }_R(H_{\mathfrak{m}}^i(S))={\ell }_R(H_{\mathfrak{m}}^i(R))+{\ell }_R(H_{\mathfrak{m}}^i(S/R))$$

$$0\to R/\mathfrak{q}\to S/\mathfrak{q}S\to (S/R)/\mathfrak{q}(S/R)\to 0.$$

$$\begin{array}{ccccccccc}0& \to & R/\mathfrak{q}& \to & S/\mathfrak{q}S& \to & (S/R)/\mathfrak{q}(S/R)& \to & 0\\ \beta \downarrow & & \alpha \downarrow & & \phi \downarrow & & & & \\ 0& \to & H_{\mathfrak{m}}^d(R)& \to & H_{\mathfrak{m}}^i(S)& \to & H_{\mathfrak{m}}^i(S/R)& \to & 0.\end{array}$$

$${\ell }_R(\mathrm{Ker}(\beta ))=\sum _{i=0}^{d-1}\left(\genfrac{}{}{0ex}{}{d}{i}\right){\ell }_R(H_{\mathfrak{m}}^i(R)).$$

$${\ell }_R(\mathrm{Ker}(\alpha ))=\sum _{i=0}^{d-1}\left(\genfrac{}{}{0ex}{}{d}{i}\right){\ell }_R(H_{\mathfrak{m}}^i(S)),$$

$${\ell }_R(\mathrm{Ker}(\phi ))=\sum _{i=0}^{d-1}\left(\genfrac{}{}{0ex}{}{d}{i}\right){\ell }_R(H_{\mathfrak{m}}^i(S/R)).$$

$${\ell }_R(\mathrm{Ker}(\beta ))={\ell }_R(\mathrm{Ker}(\alpha ))+{\ell }_R(\mathrm{Ker}(\phi )).$$

$$\begin{array}{ccccccccc}0& \to & R/{\mathfrak{q}}^{\lim }& \stackrel{\nu }{\to }& S/{\mathfrak{q}}_S^{\lim }& \stackrel{\pi }{\to }& (S/R)/{\mathfrak{q}}_{(S/R)}^{\lim }& \to & 0\\ \overline{\beta }\downarrow & & \overline{\alpha }\downarrow & & \overline{\phi }\downarrow & & & & \\ 0& \to & H_{\mathfrak{m}}^d(R)& \stackrel{\mu }{\to }& H_{\mathfrak{m}}^i(S)& \stackrel{\theta }{\to }& H_{\mathfrak{m}}^i(S/R)& \to & 0,\end{array}$$

$$\overline{\beta }:{\mathfrak{q}}^{\ast }/{\mathfrak{q}}^{\lim }\to 0_{H_{\mathfrak{m}}^d(R)}^{\ast }$$

$$\overline{\phi }(\pi (\overline{z}))=\theta (\overline{\alpha }(\overline{z}))=\theta (\mu (y))=0.$$

$$A^{\Gamma }:=\bigcup _{e>0}{k}_e[[x_1,\dots ,x_d]].$$

Full text

2010 Mathematics Subject Classification: 13A35, 13D45, 13H10.

The 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 $(R,\mathfrak{m},k)$ be a local ring of dimension $d$ and $\mathfrak{q}$ a parameter ideal of $R$. The motivation of this paper comes from the theory of Buchsbaum rings. Recall that the length $\ell_{R}(R/\mathfrak{q})$ is always greater than or equal to the multiplicity $e(\mathfrak{q})$ for all parameter ideals $\mathfrak{q}$. Furthermore, $R$ is Cohen-Macaulay if and only if $\ell_{R}(R/\mathfrak{q})=e(\mathfrak{q})$ for some (and hence for all) $\mathfrak{q}$. The ring $R$ is called generalized Cohen-Macaulay if the difference $\ell_{R}(R/\mathfrak{q})-e(\mathfrak{q})$ is bound above for all $\mathfrak{q}$. More precisely, if $R$ is generalized Cohen-Macaulay then

[TABLE]

for all parameter ideals $\mathfrak{q}$, and the equality occurs for all parameter ideals $\mathfrak{q}$ contained in a large enough power of $\mathfrak{m}$. The ring $R$ is said to be Buchsbaum if $\ell_{R}(R/\mathfrak{q})-e(\mathfrak{q})$ does not depend on the choice of parameter ideal $\mathfrak{q}$.

Suppose $(R,\mathfrak{m},k)$ is an equidimensional excellent local ring of characteristic $p>0$. A classical result of Kunz [17] says that $R$ is regular if and only if the Frobenius endomorphism $F:R\to R,x\mapsto x^{p}$ is flat. Kunz’s theorem is the starting point to study the singularities of $R$ in terms of Frobenius homomorphism, say $F$-singularities. $F$-singularities appear in the theory of tight closure (cf. [13] for its introduction), which was systematically introduced by Hochster and Huneke around the mid 80’s [11]. The main objects of $F$-singularities are $F$-regularity, $F$-rationality, $F$-purity, and $F$-injectivity. Recall that $R$ is said to be $F$-rational if $\mathfrak{q}^{*}=\mathfrak{q}$ for all parameter ideals $\mathfrak{q}$, where $\mathfrak{q}^{*}$ is the tight closure of $\mathfrak{q}$. It should be noted that if $R$ is $F$-rational then it is Cohen-Macaulay and normal (here we assume that $R$ is excellent and equidimensional). In [9] Goto and Nakamura considered rings satisfying that $\ell_{R}(\mathfrak{q}^{*}/\mathfrak{q})$ is bounded above. They proved that $\ell_{R}(\mathfrak{q}^{*}/\mathfrak{q})$ is bounded above for every parameter ideal $\mathfrak{q}$ if and only if $R$ is $F$-rational on the punctured spectrum $\mathrm{Spec}^{\circ}(R)=\mathrm{Spec}(R)\setminus\{\mathfrak{m}\}$. It is worth to noting that these conditions imply the generalized Cohen-Macaulay property of $R$. In the present paper, we study the counterpart of Buchsbaum rings in the $F$-singularities realm.

Recall that the Frobenius endomorphism yields the (natural) Frobenius action on local cohomology $F:H^{i}_{\mathfrak{m}}(R)\to H^{i}_{\mathfrak{m}}(R)$ for all $i\geq 0$. We say $R$ is $F$-injective if the Frobenius action on all local cohomology modules $H^{i}_{\mathfrak{m}}(R)$ are injective (cf. [6]). Ma [21] showed that if an $F$-injective ring is generalized Cohen-Macaulay, then it is Buchsbaum. Therefore if $R$ is an $F$-injective local ring that is $F$-rational on the punctured spectrum, it is Buchsbaum. As the main result of this paper, we prove the following.

Main Theorem**.**

Let $(R,\mathfrak{m},k)$ be an equidimensional excellent local ring of characteristic $p>0$. Suppose $R$ is an $F$-injective local ring that is $F$-rational on the punctured spectrum. Then $\ell_{R}(\mathfrak{q}^{*}/\mathfrak{q})$ does not depend on the choice of parameter ideal $\mathfrak{q}$.

The paper is organized as follows. In the next section we collect results of generalized Cohen-Macaulay rings and of $F$-singularities used in this paper. Section 3 is devoted for a proof of the main theorem in the $F$-finite case. We prove the main theorem for any equidimensional excellent local ring in Section 4. In the last section we state several open questions for future research.

**Acknowledgement **.

The author is deeply grateful to Linquan Ma for his discussion about the $\Gamma$-construction. He is also grateful to Nguyen Cong Minh and Kei-ichi Watanabe for their useful comments on this work. The author is grateful to the referee for his/her careful reading and useful comments.

2. Preliminary

2.1. Buchsbaum and generalized Cohen-Macaulay modules

Let us recall the definition of Buchsbaum and generalized Cohen-Macaulay modules (cf. [26, 27]). Let $M$ be a finitely generated module over a local ring $(R,\mathfrak{m},k)$ and let $\mathfrak{q}$ be a parameter ideal of $M$. We denote by $e(\mathfrak{q},M)$ the multiplicity of $M$ with respect to $\mathfrak{q}$ (cf. [2] for details).

Definition 2.1**.**

Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak{m},k)$ such that $t=\dim M>0$. Then $M$ is called generalized Cohen-Macaulay, if the difference

[TABLE]

is bounded above, where $\mathfrak{q}$ ranges over the set of all parameter ideals of $M$.

Remark 2.2**.**

Let the notation be as in Definition 2.1.

1. (1)

It is well-known that $M$ is Cohen-Macaulay if and only if $H^{i}_{\mathfrak{m}}(M)=0$ for all $i<t$. Similarly, $M$ is generalized Cohen-Macaulay if and only if $H^{i}_{\mathfrak{m}}(M)$ is a finitely generated $R$-module for all $i<t$. 2. (2)

Suppose $R$ is equidimensional and is an image of a Cohen-Macaulay local ring. Then $M$ is generalized Cohen-Macaulay if and only if the non-Cohen-Macaulay locus of $M$ is isolated. 3. (3)

Let $M$ be a generalized Cohen-Macaulay $R$-module over $(R,\mathfrak{m},k)$ such that $t=\dim M>0$. Then

[TABLE]

for every parameter ideal $\mathfrak{q}$ of $M$.

Definition 2.3** (cf. [27]).**

Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak{m},k)$ such that $t=\dim M>0$. A parameter ideal $\mathfrak{q}$ of $M$ is called standard if

[TABLE]

An $\mathfrak{m}$-primary ideal $\mathfrak{a}$ is said to be standard if every parameter ideal contained in $\mathfrak{a}$ is standard. We say that $M$ is Buchsbaum if every parameter ideal of $M$ is standard, i.e. $\mathfrak{m}$ is a standard ideal of $M$.

Standard ideals admit a cohomological characterization as follows.

Remark 2.4**.**

1. (1)

The parameter ideal $\mathfrak{q}$ of $M$ is standard if and only if the canonical homomorphism $H^{i}(\mathfrak{q};M)\to H^{i}_{\mathfrak{m}}(M)$ is surjective for all $i<t$. Moreover $M$ is Buchsbaum if and only if the canonical homomorphism $H^{i}(\mathfrak{m};M)\to H^{i}_{\mathfrak{m}}(M)$ is surjective for all $i<t$, where the Koszul cohomology can be defined by any set of generators of $\mathfrak{m}$ ([27, Theorem 3.4]). 2. (2)

Let $M$ be a generalized Cohen-Macaulay module over $(R,\mathfrak{m},k)$ such that $d=\dim M>0$ and let $n\in\mathbb{N}$ be a positive integer such that $\mathfrak{m}^{n}\,H^{i}_{\mathfrak{m}}(M)=0$ for all $i<d$. Then every parameter element $x\in\mathfrak{m}^{2n}$ of $M$ admits the splitting property, i.e. $H^{i}_{\mathfrak{m}}(M/xM)\cong H^{i}_{\mathfrak{m}}(M)\oplus H^{i+1}_{\mathfrak{m}}(M)$ for all $i<d-1$. Furthermore, every parameter ideal contained in $\mathfrak{m}^{2n}$ is standard (cf. [4]).

We also need the notion of limit closure of parameter ideal in the sequel.

Definition 2.5**.**

Let $(R,\mathfrak{m},k)$ be a local ring, let $M$ be a finitely generated module with $t=\dim M$ and let $\underline{x}=x_{1},\ldots,x_{t}$ be a of system of parameters of $M$. The limit closure of $\underline{x}$ in $M$ is defined as a submodule of $M$:

[TABLE]

with the convention that $(\underline{x})_{M}^{\lim}=0$ when $t=0$. If $M=R$, then we simply write $(\underline{x})^{\lim}$.

From the definition, it is clear that $(\underline{x})M\subseteq(\underline{x})_{M}^{\lim}$.

Remark 2.6**.**

Let the notation be as in Definition 2.5.

1. (1)

The quotient $(\underline{x})_{M}^{\lim}/(\underline{x})M$ is the kernel of the canonical map

[TABLE]

This implies the following fact. Let $\mathfrak{q}=(x_{1},\ldots,x_{t})$ and put $\mathfrak{q}_{M}^{\lim}:=(\underline{x})^{\lim}_{M}$. Hence the notation $\mathfrak{q}_{M}^{\lim}$ is independent of the choice of $x_{1},\ldots,x_{t}$ which generate $\mathfrak{q}$. 2. (2)

It is known that $(\underline{x})M=(\underline{x})_{M}^{\lim}$ if and only if $\underline{x}$ forms an $M$-regular sequence. 3. (3)

It is shown that the Hochster’s monomial conjecture is equivalent to the claim that $\mathfrak{q}^{\lim}\neq R$ for every parameter ideal $\mathfrak{q}$ of $R$. 4. (4)

If $M$ is a generalized Cohen-Macaulay module of dimension $t>0$, then for every parameter ideal $\mathfrak{q}$ we have

[TABLE]

and the equality occurs if and only if $\mathfrak{q}$ is standard by Definition 2.3 and [3, Theorem 5.1] (see also [8, Proposition 3.6]).

2.2. Tight closure and $F$-singularities

In the rest of this paper, we always assume that $(R,\mathfrak{m},k)$ is an excellent equidimnesional local ring of characteristic $p>0$ and of dimension $d>0$. If we want to notationally distinguish the source and target of the $e$-th Frobenius endomorphism $F^{e}:R\overset{x\mapsto x^{p^{e}}}{\longrightarrow}R$, we will use $F_{*}^{e}(R)$ to denote the target. $F_{*}^{e}(R)$ is an $R$-bimodule, which is the same as $R$ as an abelian group and as a right $R$-module, that acquires its left $R$-module structure via the $e$-th Frobenius endomorphism $F^{e}$. By definition the $e$-th Frobenius endomorphism $F^{e}:R\to F_{*}^{e}(R)$ sending $x$ to $F_{*}^{e}(x^{p^{e}})=x\cdot F_{*}^{e}(1)$ is an $R$-homomorphism. We say $R$ is $F$-finite if $F_{*}(R)$ is a finite $R$-module. When $R$ is reduced, we will use $R^{1/p^{e}}$ to denote the ring whose elements are $p^{e}$-th roots of elements of $R$. Note that these notations (when $R$ is reduced) $F_{*}^{e}(R)$ and $R^{1/p^{e}}$ are used interchangeably in the literature.

Definition 2.7** ([11, 12, 13]).**

Let $R^{\circ}=R\setminus\cup_{\mathfrak{p}\in\mathrm{Min}R}\mathfrak{p}$. 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 $I$, $I^{*}=\{x\mid cx^{q}\in I^{[q]}\text{ for some }c\in R^{\circ}\text{ and for all }q\gg 0\}$.

Remark 2.8**.**

An element $x\in I^{F}$ if it is contained in the kernel of the composition

[TABLE]

for some $e\geq 0$. Similarly, an element $x\in I^{*}$ if it is contained in the kernel of the composition

[TABLE]

for some $c\in R^{\circ}$ and for all $e\gg 0$. In general, let $N$ be a submodule of an $R$-module $M$. The tight closure of $N$ in $M$, denoted by $N^{*}_{M}$, consists elements that are contained in the kernel of the composition

[TABLE]

for some $c\in R^{\circ}$ and for all $e\gg 0$.

Let $x_{1},\ldots,x_{d}$ be a system of parameters of $R$. Recall that local cohomology $H^{i}_{\mathfrak{m}}(R)$ may be computed as the homology of the Čech complex

[TABLE]

Then the Frobenius endomorphism $F:R\to R$ induces a natural Frobenius action $F:H^{i}_{\mathfrak{m}}(R)\to H^{i}_{\mathfrak{m}^{[p]}}(R)\cong H^{i}_{\mathfrak{m}}(R)$. There is a very useful way of describing the top local cohomology. It can be given as the direct limit of Koszul cohomologies

[TABLE]

Then for each $\overline{a}\in H^{d}_{\mathfrak{m}}(R)$, which is the canonical image of $a+(x_{1}^{n},\ldots,x_{d}^{n})$, we find that $F(\overline{a})$ is the canonical image of $a^{p}+(x_{1}^{pn},\ldots,x_{d}^{pn})$ (see [25]).

Remark 2.9**.**

Recall that we always assume $R$ is excellent and equidimensional.

1. (1)

An element $x\in 0^{*}_{H^{d}_{\mathfrak{m}}(R)}$ if there exists $c\in R^{\circ}$ such that $c\,F^{e}(x)=0$ for all $e\gg 0$. Let $x_{1},\ldots,x_{d}$ be a system of parameters of $R$. The direct system $\lim R/(x_{1}^{n},\ldots,x_{d}^{n})\cong H^{d}_{\mathfrak{m}}(R)$ induces the direct system of tight closures

[TABLE]

By [14, Remark 5.4] we have $(x_{1},\ldots,x_{d})^{\lim}\subseteq(x_{1},\ldots,x_{d})^{*}$ for all system of parameters $x_{1},\ldots,x_{d}$. By Remark 2.6 (1) we obtain the direct system

[TABLE]

with all maps in the direct system are injective. As a consequence, $\ell_{R}\,(\mathfrak{q}^{*}/\mathfrak{q}^{\lim})$ is bounded above if and only if $\ell_{R}(0^{*}_{H^{d}_{\mathfrak{m}}(R)})<\infty$. In this case $\max_{\mathfrak{q}}\{\ell_{R}\,(\mathfrak{q}^{*}/\mathfrak{q}^{\lim})\}=\ell_{R}(0^{*}_{H^{d}_{\mathfrak{m}}(R)})$, and $\ell_{R}\,(\mathfrak{q}^{*}/\mathfrak{q}^{\lim})=\ell_{R}(0^{*}_{H^{d}_{\mathfrak{m}}(R)})$ for all parameter ideals $\mathfrak{q}$ contained in a large enough power of maximal ideal. 2. (2)

If we consider the target of the Frobenius endomorphism $F:R\to R$ as an $R$-module $F_{*}(R)$ via the Frobebenius endomorphism, then the Frobenius action on $H^{i}_{\mathfrak{m}}(R)$ becomes an $R$-homomorphism $F:H^{i}_{\mathfrak{m}}(R)\to H^{i}_{\mathfrak{m}}(F_{*}(R))$ for all $i\geq 0$.

We now present $F$-singularities used in this paper.

Definition 2.10**.**

A local ring $(R,\mathfrak{m},k)$ is called $F$-rational if every parameter ideal is tight closed, i.e. $\mathfrak{q}^{*}=\mathfrak{q}$ for all $\mathfrak{q}$.

Remark 2.11**.**

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

The following is the main result of Goto and Nakamura [9, Theorem 1.1].

Theorem 2.12**.**

Let $(R,\mathfrak{m},k)$ be an equidimensional excellent local ring. Then the following are equivalent.

1. (1)

$\ell_{R}(\mathfrak{q}^{*}/\mathfrak{q})$* is bounded above;* 2. (2)

$R$* is $F$-rational on the punctured spectrum $\mathrm{Spec}^{\circ}(R)=\mathrm{Spec}(R)\setminus\{\mathfrak{m}\}$;* 3. (3)

$R$* is generalized Cohen-Macaulay and $\ell_{R}(0^{*}_{H^{d}_{\mathfrak{m}}(R)})<\infty$.*

In more detail, we have the following.

Proposition 2.13**.**

Let $(R,\mathfrak{m},k)$ be an equidimensional excellent local ring of characteristic $p>0$ and of dimension $d>0$ that is $F$-rational on the punctured spectrum. Then for every parameter ideal $\mathfrak{q}$ of $R$ we have

[TABLE]

Moreover, the equality occurs if and only if $\mathfrak{q}$ is a standard parameter ideal satisfying one of the following condition

1. (1)

The canonical map $\mathfrak{q}^{*}/\mathfrak{q}\to 0^{*}_{H^{d}_{\mathfrak{m}}(R)}$ is surjective; 2. (2)

The canonical map $\mathfrak{q}^{*}/\mathfrak{q}^{\lim}\to 0^{*}_{H^{d}_{\mathfrak{m}}(R)}$ is an isomorphism.

Furthermore, we have the equality for all parameter ideals $\mathfrak{q}$ contained in a large enough power of $\mathfrak{m}$.

Proof.

Note that $\mathfrak{q}\subseteq\mathfrak{q}^{\lim}\subseteq\mathfrak{q}^{*}$, so $\ell_{R}\,(\mathfrak{q}^{*}/\mathfrak{q})=\ell_{R}\,(\mathfrak{q}^{*}/\mathfrak{q}^{\lim})+\ell_{R}\,(\mathfrak{q}^{\lim}/\mathfrak{q})$. The assertion now follows from Remarks 2.6 (4) and 2.9 (1). ∎

Since every parameter ideal of a Buchsbaum ring is standard, we have

Proposition 2.14**.**

Let $(R,\mathfrak{m},k)$ be an equidimensional excellent local ring of characteristic $p>0$ and of dimension $d>0$. Suppose $R$ is a Buchsbaum ring that is $F$-rational on the punctured spectrum. Then the following are equivalent.

1. (1)

$\ell_{R}\,(\mathfrak{q}^{*}/\mathfrak{q})=\sum_{i=0}^{d-1}\binom{d}{i}\ell_{R}(H^{i}_{\mathfrak{m}}(R))+\ell_{R}(0^{*}_{H^{d}_{\mathfrak{m}}(R)})$* for all parameter ideals $\mathfrak{q}$;* 2. (2)

The canonical map $\mathfrak{q}^{*}/\mathfrak{q}\to 0^{*}_{H^{d}_{\mathfrak{m}}(R)}$ is surjective for all parameter ideals $\mathfrak{q}$; 3. (3)

The canonical map $\mathfrak{q}^{*}/\mathfrak{q}^{\lim}\to 0^{*}_{H^{d}_{\mathfrak{m}}(R)}$ is an isomorphism for all parameter ideals $\mathfrak{q}$.

The relation between Buchsbaum rings and $F$-singularities appears explicit in [21].

Definition 2.15**.**

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

Remark 2.16**.**

If $R$ is $F$-injective then it is reduced (cf. [23, Lemma 3.11]). Conversely, a reduced ring $R$ is $F$-injective if and only if the inclusion $R\hookrightarrow F_{*}^{e}(R)$ induces injective $R$-homomorphisms $H^{i}_{\mathfrak{m}}(R)\to H^{i}_{\mathfrak{m}}(F_{*}^{e}(R))$ for all $e,i\geq 0$.

Remark 2.17**.**

Recently, Ma [21, Corollary 3.5] showed that an $F$-injective generalized Cohen-Macaulay ring is Buchsbaum. Thus we have the following implications of singularities in this paper

[TABLE]

3. Proof of the main theorem in the $F$-finite case

We prove the main theorem for $F$-finite rings. Our method is inspired by the proof of Theorem 3.7 of [1, Arxiv: 1512.05374, Version 1]. Note that any $F$-finite ring is excellent by Kunz [18].

Theorem 3.1**.**

Let $(R,\mathfrak{m})$ be an equidimensional $F$-finite local ring of dimension $d>0$. Suppose $R$ is an $F$-injective local ring that is $F$-rational on the punctured spectrum. Then $\ell_{R}(\mathfrak{q}^{*}/\mathfrak{q})$ does not depend on the choice of parameter ideal $\mathfrak{q}$.

Proof.

By [21, Corollary 3.5] we have $R$ is Buchsbaum. By Proposition 2.14 we need only to show that the canonical map $\mathfrak{q}^{*}/\mathfrak{q}^{\lim}\to 0^{*}_{H^{d}_{\mathfrak{m}}(R)}$ is an isomorphism for any parameter ideal $\mathfrak{q}$. Let $e$ be a large enough positive integer such that canonical map $(\mathfrak{q}^{[p^{e}]})^{*}/(\mathfrak{q}^{[p^{e}]})^{\lim}\to 0^{*}_{H^{d}_{\mathfrak{m}}(R)}$ is an isomorphism (cf. Remark 2.9). Since the role of $\mathfrak{q}^{[p^{e}]}$ in $R$ is the same as that of $\mathfrak{q}$ in $F_{*}^{e}(R)$, so the canonical map

[TABLE]

is an isomorphism, where $\mathfrak{q}_{F_{*}^{e}(R)}^{*}$ is the tight closure of $\mathfrak{q}\,F_{*}^{e}(R)$ as a submodule of $F_{*}^{e}(R)$. Since $R$ is reduced, we have the short exact sequence of $R$-module

[TABLE]

where both $F_{*}^{e}(R)$ and $F_{*}^{e}(R)/R$ are Buchsbaum as $R$-modules (see the proofs of [21, Theorem 3.4] and [23, Theorem 4.17]). Set $S=F_{*}^{e}(R)$. By Remark 2.16 we have the short exact sequence of local cohomology

[TABLE]

for all $i\geq 0$. Thus

[TABLE]

for all $0\leq i\leq d-1$. By the proof of [23, Theorem 4.17] we have the following short exact sequence

[TABLE]

Therefore we have the following commutative diagram

[TABLE]

By Remark 2.9 (1), $\operatorname{Ker}(\beta)=\mathfrak{q}^{\lim}/\mathfrak{q}$. Since $R$ is Buchsbaum,

[TABLE]

Similarly, we have

[TABLE]

and

[TABLE]

Notice that $\ell_{R}\,(H^{i}_{\mathfrak{m}}(S))=\ell_{R}\,(H^{i}_{\mathfrak{m}}(R))+\ell_{R}\,(H^{i}_{\mathfrak{m}}(S/R))$ for all $0\leq i\leq d-1$, so

[TABLE]

Therefore the above commutative diagram induces the following commutative diagram

[TABLE]

where $\overline{\alpha},\overline{\beta}$ and $\overline{\varphi}$ are injective. As above we known that the restriction of $\overline{\alpha}$ on the tight closures $\mathfrak{q}_{S}^{*}/\mathfrak{q}^{\lim}_{S}\to 0^{*}_{H^{d}_{\mathfrak{m}}(S)}$ is an isomorphism. We claim the restriction map

[TABLE]

is an isomorphism. It is enough to show this map is surjective. Indeed, let $y$ be any element in $0^{*}_{H^{d}_{\mathfrak{m}}(R)}$. We have $\mu(y)\in 0^{*}_{H^{d}_{\mathfrak{m}}(S)}$. Thus there exists $z\in\mathfrak{q}^{*}_{S}$ such that $\overline{\alpha}(\overline{z})=\mu(y)$. We have

[TABLE]

Thus $\pi(\overline{z})=0$ since $\overline{\varphi}$ is injective. Therefore we have $x\in R$ such that $\nu(\overline{x})=\overline{z}\in\mathfrak{q}_{S}^{*}/\mathfrak{q}^{\lim}_{S}$. Note that the role of $\mathfrak{q}$ in $F_{*}(R)$ is the same that of $\mathfrak{q}^{[p^{e}]}$ in $R$. Now the condition of $x$ means $x^{p^{e}}\in(\mathfrak{q}^{[p^{e}]})^{*}$. Thus we have some $c\in R^{\circ}$ such that $cx^{p^{e+e^{\prime}}}\in\mathfrak{q}^{[p^{e+e^{\prime}}]}$ for all $e^{\prime}\gg 0$. Hence $x\in\mathfrak{q}^{*}$. Moreover $\mu(\overline{\beta}(\overline{x}))=\overline{\alpha}(\nu(x))=\mu(y)$. Thus $\overline{\beta}(\overline{x})=y$ since $\mu$ is injective, and so the canonical map $\mathfrak{q}^{*}/\mathfrak{q}^{\lim}\to 0^{*}_{H^{d}_{\mathfrak{m}}(R)}$ is surjective as desired. The proof is complete. ∎

We do not have the converse of the previous theorem by the following.

Example 3.2**.**

Let $R=k[X,Y,Z]/(X^{3}+Y^{3}+Z^{3})$, where $k$ is a perfect field of characteristic $0<p\neq 3$. The ring $R$ is a Gorenstein ring with isolated singular. The test ideal of $R$ is the maximal ideal (cf. [14, Example 4.8]). We have $\ell_{R}\,(\mathfrak{q}^{*}/\mathfrak{q})=1$ for all parameter ideals $\mathfrak{q}$. However $R$ is $F$-injective if and only if $p\equiv 1\,\mathrm{mod}\,3$.

4. Proof of the main theorem for excellent local rings

We will prove the main theorem mentioned in the introduction. The key ingredient is using the $\Gamma$-construction of Hochster and Huneke [12] to reduce the problem to the case of $F$-finite rings of the previous section. We briefly recall the construction. Let $(R,\mathfrak{m},k)$ be a complete local ring with coefficient field $k$ of characteristic $p>0$ and of dimension $d$. Let us fix a $p$-basis $\Lambda$ of $k\subset R$ and let $\Gamma\subset\Lambda$ be a cofinite subset (we refer the reader to [22] for the definition of a $p$-basis). We denote by $k_{e}$ (or $k_{\Gamma,e}$ to signify the dependence on the choice of $\Gamma$) the purely inseparable extension field $k[\Gamma^{1/p^{e}}]$ of $k$, which is obtained by adjoining $p^{e}$-th roots of all elements in $\Gamma$. Next, fix a system of parameters $x_{1},\ldots,x_{d}$ of $R$. Then the natural map $A:=k[[x_{1},\ldots,x_{d}]]\to R$ is module-finite. Let us put

[TABLE]

Then the natural map $A\to A^{\Gamma}$ is faithfully flat and purely inseparable and $\mathfrak{m}_{A}A^{\Gamma}$ is the unique maximal ideal of $A^{\Gamma}$. Now we set $R^{\Gamma}:=A^{\Gamma}\otimes_{A}R$. Then $R\to R^{\Gamma}$ is faithfully flat and purely inseparable. The maximal ideal of $R$ expands to the maximal ideal of $R^{\Gamma}$. The crucial fact about $R^{\Gamma}$ is that it is an $F$-finite local ring (see [12, (6.6) Lemma]). Moreover, we can preserve some good properties of $R$ if we choose a sufficiently small cofinite subset $\Gamma$. For example, if $R$ is reduced, so is $R^{\Gamma}$ for any sufficiently small choice of $\Gamma$ cofinite in $\Lambda$. For each prime ideal $\mathfrak{p}$ of $R$, $\sqrt{\mathfrak{p}R^{\Gamma}}$ is a prime ideal of $R^{\Gamma}$. Furthermore if we choose $\Gamma$ small enough, $\mathfrak{p}R^{\Gamma}$ is also prime. We need several lemmas.

Lemma 4.1** ([5], Lemma 2.9).**

Let $(R,\mathfrak{m},k)$ be a complete local ring that is $F$-injective. Then for any sufficiently small choice of $\Gamma$ cofinite in $\Lambda$, $R^{\Gamma}$ is $F$-injective.

The next result can be proven by the same method used in [20, Proposition 5.6], so we omit the detail proof. Note that we use [28, Lemma 2.3, Theorem 3.5] to replace the roles of [20, Proposition 5.4, Lemma 5.5] in the proof of [20, Proposition 5.6].

Lemma 4.2**.**

Let $(R,\mathfrak{m},k)$ be a complete local ring that is $F$-rational on the punctured spectrum. Then for any sufficiently small choice of $\Gamma$ cofinite in $\Lambda$, $R^{\Gamma}$ is $F$-rational on the punctured spectrum.

Let $M$ be an $R$-module with a Frobenius action $F$. A submodule $N$ of $M$ is called F-compatible if $F(N)\subseteq N$. An $R$-module $M$ with a Frobenius action is said to be simple if it has no nontrivial $F$-compatible submodules. If $R$ is a complete local domain, $0^{*}_{H^{d}_{\mathfrak{m}}(R)}$ is the unique maximal proper $F$-compatible submodule of $H^{d}_{\mathfrak{m}}(R)$ by Smith [25]. Hence $H^{d}_{\mathfrak{m}}(R)/0^{*}_{H^{d}_{\mathfrak{m}}(R)}$ is simple. In general, for reduced equidimensional excellent local rings Smith [25, Proposition 2.5] showed that $0^{*}_{H^{d}_{\mathfrak{m}}(R)}$ is the unique maximal proper $F$-compatible submodule of $H^{d}_{\mathfrak{m}}(R)$ which is annihilated by an element $c\in R^{\circ}$.

Lemma 4.3**.**

Let $(R,\mathfrak{m},k)$ be an equidimensional complete local ring. Then for any sufficiently small choice of $\Gamma$ cofinite in $\Lambda$, $0^{*}_{H^{d}_{\mathfrak{m}}(R^{\Gamma})}=0^{*}_{H^{d}_{\mathfrak{m}}(R)}\otimes_{R}R^{\Gamma}$.

Proof.

It is obvious that we can assume $R$ is reduced. Set $(0)=\mathfrak{p}_{1}\cap\cdots\cap\mathfrak{p}_{n}$ the primary decomposition of $(0)$. We prove by induction on $n$ the following claim.

Claim. $H^{d}_{\mathfrak{m}}(R)/0^{*}_{H^{d}_{\mathfrak{m}}(R)}\cong H^{d}_{\mathfrak{m}}(R_{1})/0^{*}_{H^{d}_{\mathfrak{m}}(R_{1})}\oplus\cdots\oplus H^{d}_{\mathfrak{m}}(R_{n})/0^{*}_{H^{d}_{\mathfrak{m}}(R_{n})}$ as $R$-modules with Frobenius actions, where $R_{i}=R/\mathfrak{p}_{i}$ for all $1\leq i\leq n$.

Indeed, we have nothing to do when $n=1$. Suppose $n>1$ and set $R^{\prime}=R/(\mathfrak{p}_{1}\cap\cdots\cap\mathfrak{p}_{n-1})$. We have the following short exact sequence

[TABLE]

where $R^{\prime\prime}=R/(\mathfrak{p}_{1}\cap\cdots\cap\mathfrak{p}_{n-1}+\mathfrak{p}_{n})$. This short exact sequence induces the following exact sequence of local cohomology with homomorphisms are compatible with Frobenius actions

[TABLE]

Note that $\dim R^{\prime\prime}<d$, so $\mathrm{Im}(\delta)$ is an $F$-compatible submodule of $H^{d}_{\mathfrak{m}}(R)$ of dimension less than $d$, and so that $\mathrm{Im}(\delta)\subseteq 0^{*}_{H^{d}_{\mathfrak{m}}(R)}$. Therefore we obtain the following short exact sequence by restricting on tight closures

[TABLE]

Thus we have the following commutative diagram

[TABLE]

Hence $H^{d}_{\mathfrak{m}}(R)/0^{*}_{H^{d}_{\mathfrak{m}}(R)}\cong H^{d}_{\mathfrak{m}}(R^{\prime})/0^{*}_{H^{d}_{\mathfrak{m}}(R^{\prime})}\oplus H^{d}_{\mathfrak{m}}(R_{n})/0^{*}_{H^{d}_{\mathfrak{m}}(R_{n})}$. The Claim now follows from the inductive hypothesis.

We continue to prove the lemma. Notice that for all $i\leq n$, $H^{d}_{\mathfrak{m}}(R_{i})/0^{*}_{H^{d}_{\mathfrak{m}}(R_{i})}$ is simple by Smith’s result. For any sufficiently small choice of $\Gamma$ cofinite in $\Lambda$ we have $R^{\Gamma}$ is reduced (so $\mathfrak{p}_{i}R^{\Gamma}$ is prime for all $i\leq n$). We can even assume that $(H^{d}_{\mathfrak{m}}(R_{i})/0^{*}_{H^{d}_{\mathfrak{m}}(R_{i})})\otimes_{R}R^{\Gamma}$ is simple as an $R^{\Gamma}$-module with a Frobenius action for all $i\leq n$ by [20, Lemma 5.8]. Thus

[TABLE]

is a direct sum of $n$ simple $R^{\Gamma}$-modules with Frobenius actions. On the other hand $0^{*}_{H^{d}_{\mathfrak{m}}(R)}\otimes_{R}R^{\Gamma}\subseteq 0^{*}_{H^{d}_{\mathfrak{m}}(R^{\Gamma})}$ and $H^{d}_{\mathfrak{m}}(R^{\Gamma})/0^{*}_{H^{d}_{\mathfrak{m}}(R^{\Gamma})}$ is also a direct sum of $n$ simple $R^{\Gamma}$-modules with Frobenius actions by the Claim. Therefore $0^{*}_{H^{d}_{\mathfrak{m}}(R^{\Gamma})}=0^{*}_{H^{d}_{\mathfrak{m}}(R)}\otimes_{R}R^{\Gamma}$. The proof is complete. ∎

Lemma 4.4**.**

Let $(R,\mathfrak{m},k)$ be an equidimensional complete local ring that is $F$-rational on the punctured spectrum. Then for any sufficiently small choice of $\Gamma$ cofinite in $\Lambda$, $(\mathfrak{q}R^{\Gamma})^{*}=\mathfrak{q}^{*}R^{\Gamma}$ for all parameter ideals $\mathfrak{q}$ of $R$.

Proof.

By Lemmas 4.2 and 4.3, $R^{\Gamma}$ is $F$-rational on the punctured spectrum and $0^{*}_{H^{d}_{\mathfrak{m}}(R^{\Gamma})}=0^{*}_{H^{d}_{\mathfrak{m}}(R)}\otimes_{R}R^{\Gamma}$ for any sufficiently small choice of $\Gamma$ cofinite in $\Lambda$. Since $R\to R^{\Gamma}$ is faithfully flat and $\mathfrak{m}R^{\Gamma}$ is the maximal ideal of $R^{\Gamma}$, we have $\ell_{R}(H^{i}_{\mathfrak{m}}(R))=\ell_{R^{\Gamma}}(H^{i}_{\mathfrak{m}}(R^{\Gamma}))$ for all $i\leq d-1$, and

[TABLE]

Let $\mathfrak{q}=(x_{1},\ldots,x_{d})$ be any parameter ideal of $R$. By Proposition 2.13 there exists a positive integer $k$ such that

[TABLE]

where $\mathfrak{q}_{k}=(x_{1}^{k},\ldots,x_{d}^{k})$ is a parameter ideal of $R$. Since $\mathfrak{q}_{k}^{*}R^{\Gamma}\subseteq(\mathfrak{q}_{k}R^{\Gamma})^{*}$ and $R\to R^{\Gamma}$ is faithfully flat, we have $\ell_{R^{\Gamma}}((\mathfrak{q}_{k}R^{\Gamma})^{*}/\mathfrak{q}_{k}R^{\Gamma})\geq\ell_{R}(\mathfrak{q}_{k}^{*}/\mathfrak{q}_{k})$. Therefore

[TABLE]

By Proposition 2.13 again we have

[TABLE]

Hence $\ell_{R^{\Gamma}}((\mathfrak{q}_{k}R^{\Gamma})^{*}/\mathfrak{q}_{k}R^{\Gamma})=\ell_{R}(\mathfrak{q}_{k}^{*}/\mathfrak{q}_{k})$, and so $(\mathfrak{q}_{k}R^{\Gamma})^{*}=\mathfrak{q}_{k}^{*}R^{\Gamma}$.

We next show that $(\mathfrak{q}R^{\Gamma})^{*}=\mathfrak{q}^{*}R^{\Gamma}$. Set $x=x_{1}\ldots x_{d}$. By [7, Proposition 3.3] we have $\mathfrak{q}^{*}=\mathfrak{q}_{k}^{*}:_{R}x^{k-1}$ and $(\mathfrak{q}R^{\Gamma})^{*}=(\mathfrak{q}_{k}R^{\Gamma})^{*}:_{R^{\Gamma}}x^{k-1}$. Moreover $(\mathfrak{q}_{k}R^{\Gamma})^{*}=\mathfrak{q}_{k}^{*}R^{\Gamma}$ and $R\to R^{\Gamma}$ is a flat extension, so we obtain $(\mathfrak{q}R^{\Gamma})^{*}=\mathfrak{q}^{*}R^{\Gamma}$. This completes the proof. ∎

We now prove the main result of this paper.

Proof of the main theorem.

Suppose $R$ is an $F$-injective local ring that is $F$-rational on the punctured spectrum. We will show that $\ell_{R}(\mathfrak{q}^{*}/\mathfrak{q})$ does not depend on the choice of parameter ideal $\mathfrak{q}$. It is not hard to see that the $\mathfrak{m}$-adic complement $\widehat{R}$ is also an $F$-injective local ring that is $F$-rational on the punctured spectrum. Moreover $\ell_{R}(\mathfrak{q}^{*}/\mathfrak{q})=\ell_{\widehat{R}}((\mathfrak{q}\widehat{R})^{*}/\mathfrak{q}\widehat{R})$ (cf. [13, Exercise 4.1]). Therefore we can assume henceforth that $R$ is complete. Let $\mathfrak{q}$ be any parameter ideal of $R$. By Proposition 2.13 we need only to show that

[TABLE]

Taking a sufficiently small choice $\Gamma$ cofinite in $\Lambda$ satisfying Lemmas 4.1, 4.2, 4.3, and 4.4. We have that $R^{\Gamma}$ is an $F$-injective local ring that is $F$-rational on the punctured spectrum and $\mathfrak{q}^{*}R^{\Gamma}=(\mathfrak{q}R^{\Gamma})^{*}$ for all parameter ideals $\mathfrak{q}$. Since $R^{\Gamma}$ is $F$-finite,

[TABLE]

by Theorem 3.1. However $\ell_{R^{\Gamma}}((\mathfrak{q}R^{\Gamma})^{*}/\mathfrak{q}R^{\Gamma})=\ell_{R}(\mathfrak{q}^{*}/\mathfrak{q})$, so

[TABLE]

The proof is complete. ∎

5. Open Questions

Recall that the parameter test ideal of $R$ is $\tau^{R}=\cap_{\mathfrak{q}}(\mathfrak{q}:_{R}\mathfrak{q}^{*})$, where $\mathfrak{q}$ runs over all parameter ideals. Notice that $\tau^{R}$ is an $\mathfrak{m}$-primary ideal if and only if $R$ is $F$-rational on the punctured spectrum (so $R$ is generalized Cohen-Macaulay). Moreover any parameter ideal contained in $\tau^{R}$ is standard by [14, Remark 5.11]. Based on Proposition 2.13 we have the following natural question111I thank Professor Kei-ichi Watanabe for this question..

Question 1**.**

Let $(R,\mathfrak{m},k)$ be an equidimensional excellent local ring that is $F$-rational on the punctured spectrum. Is it true that for every parameter ideal $\mathfrak{q}$ is contained in $\tau^{R}$ we have

[TABLE]

Let $A$ be an Artinian $R$-module with a Frobenius action $F:A\to A$. Then we define the Frobenius closure $0^{F}_{A}$ of the zero submodule of $A$ is the submodule of $A$ consisting all element $z$ such that $F^{e}(z)=0$ for some $e\geq 0$. $0^{F}_{A}$ is the nilpotent part of $A$ by the Frobenius action. By [10, Proposition 1.11] and [19, Proposition 4.4] there exists a non-negative integer $e$ such that $0^{F}_{A}=\operatorname{Ker}(A\overset{F^{e}}{\longrightarrow}A)$ for all $i\geq 0$ (see also [24]). The smallest of such integers is called the Harshorne-Speiser-Lyubeznik number of $A$ and denoted by $HSL(A)$. We define the Harshorne-Speiser-Lyubeznik number of a local ring $(R,\mathfrak{m})$ as follows 222The Harshorne-Speiser-Lyubeznik number of $R$ is often defined in terms only the top local cohomology module $H^{d}_{\mathfrak{m}}(R)$. However, we think that it should be defined by all local cohomology modules. Moreover, our’s definition is more suitable with $F$-singularities. Indeed, it is clear that $R$ is $F$-injective if and only if $HSL(R)=0$.

[TABLE]

The $HSL(R)$ is closely related with the Frobenius test exponent of parameter ideals (see [15, 16]). Recall that the Frobenius test exponent of $R$, here we denote by $Fte(R)$, is the smallest non-negative integer $e$ such that $(\mathfrak{q}^{F})^{[p^{e}]}=\mathfrak{q}^{[p^{e}]}$ for all parameter ideals $\mathfrak{q}$, and $Fte(R)=\infty$ if we have no such $e$ 333If $R$ is Cohen-Macaulay, then Katman and Sharp [16] showed that $Fte(R)$ is just $HSL(R)$. In this paper, they posed the question that whether $Fte(R)$ is an integer for any local ring $R$. This question holds true for generalized Cohen-Macaulay rings, but it is still open in general. Recently, the author proved that $Fte(R)\geq HLS(R)$ for any local ring $R$.. The relation between $0^{F}_{H^{i}_{\mathfrak{m}}(R)}$ and $\mathfrak{q}^{F}$ appears explicit in [23]. In more details, if $\mathfrak{q}^{F}=\mathfrak{q}$ for all parameter ideals $q$ then $0^{F}_{H^{i}_{\mathfrak{m}}(R)}=0$ for all $i\geq 0$, i.e. $R$ is $F$-injective. Although the converse is not true for non-equidimensinal local rings, we believe the following question has an affirmative answer.

Question 2**.**

Let $(R,\mathfrak{m},k)$ be an equidimensional excellent local ring of characteristic $p>0$. Then is it true that $\mathfrak{q}^{F}=\mathfrak{q}$ for all parameter ideals $\mathfrak{q}$, i.e. $Fte(R)=0$, if and only if $0^{F}_{H^{i}_{\mathfrak{m}}(R)}=0$ for all $i\geq 0$, i.e. $HSL(R)=0$.

Notice that Ma [21] gave a positive answer for the above question for generalized Cohen-Macaulay rings. Suppose $R$ is an excellent equidimension local ring that is $F$-injective on the punctured spectrum. Then we can check that $\ell_{R}\,(0^{F}_{H^{i}_{\mathfrak{m}}(R)})<\infty$ for all $i\geq 0$. Inspired by the main result of this paper, we ask the following question.

Question 3**.**

Let $(R,\mathfrak{m},k)$ be an excellent generalized Cohen-Macaulay local ring that is $F$-injective on the punctured spectrum. Is it true that

[TABLE]

for all parameter ideals $\mathfrak{q}$.

If $R$ is a generalized Cohen-Macaulay local ring (of characteristic $p>0$), then some power of $\mathfrak{m}$ is a standard ideal of $R$ (cf. Remark 2.4 (2)). Hence there exists a non-negative integer $e$ such that every parameter ideal of $\mathfrak{m}^{[p^{e}]}$ is standard. This condition is equivalent to the condition that $F_{*}^{e}(R)$ is a Buchsbaum $R$-module provided $R$ is $F$-finite. I thank Nguyen Cong Minh for the following question.

Question 4**.**

Let $(R,\mathfrak{m},k)$ be a generalized Cohen-Macaulay local ring of characteristic $p>0$. Does there exist an integer $e$ (that is bounded above by the Frobenius invariants of $R$ such as $HSL(R)$ and $Fte(R)$) such that $\mathfrak{m}^{[p^{e}]}$ is a standard ideal of $R$.