# On the Hilbert-Samuel Coefficients of Frobenius Powers
of an ideal

Arindam Banerjee  Indian Institute of Technology Kharagpur, Kharagpur, India
<EMAIL_ADDRESS>,  Kriti Goel  University of Utah, Utah, USA 84112
<EMAIL_ADDRESS>and  J. K. Verma  Indian Institute of Technology
Bombay, Mumbai, India 400076<EMAIL_ADDRESS>
###### Abstract.

We provide suitable conditions under which the asymptotic limit of the
Hilbert-Samuel coefficients of the Frobenius powers of an
$\displaystyle{\mathfrak{m}}$-primary ideal exists in a Noetherian local ring
$\displaystyle(R,{\mathfrak{m}})$ with prime characteristic $\displaystyle
p>0.$ This, in turn, gives an expression of the Hilbert-Kunz multiplicity of
powers of the ideal. We also prove that for a face ring $\displaystyle R$ of a
simplicial complex and an ideal $\displaystyle J$ generated by pure powers of
the variables, the generalized Hilbert-Kunz function
$\displaystyle\ell(R/(J^{[q]})^{k})$ is a polynomial for all $\displaystyle
q,k$ and also give an expression of the generalized Hilbert-Kunz multiplicity
of powers of $\displaystyle J$ in terms of Hilbert-Samuel multiplicity of
$\displaystyle J.$ We conclude by giving a counter-example to a conjecture
proposed by I. Smirnov which connects the stability of an ideal with the
asymptotic limit of the first Hilbert coefficient of the Frobenius power of
the ideal.

The second author is supported by Fulbright-Nehru Postdoctoral Research
Fellowship.

Key words and phrases: Hilbert-Samuel coefficients, Hilbert-Kunz function,
Hilbert-Kunz multiplicity, Frobenius power, face ring of simplicial complex,
generalized Hilbert-Kunz function, generalized Hilbert-Kunz multiplicity.

2010 AMS Mathematics Subject Classification: 13A30, 13C14, 13C15, 13D40,
13F55.

##  1\. Introduction

This article is inspired by the work of I. Smirnov in [13], where the author
studies the Hilbert-Kunz multiplicity of powers of an ideal. Let
$\displaystyle(R,{\mathfrak{m}})$ be a $\displaystyle d$-dimensional
Noetherian local ring of prime characteristic $\displaystyle p$ and let
$\displaystyle I$ be an $\displaystyle{\mathfrak{m}}$-primary ideal. The
$\displaystyle q^{th}$-Frobenius power of $\displaystyle I$ is the ideal
$\displaystyle I^{[q]}=(x^{q}\mid x\in I)$ where $\displaystyle q=p^{e}$ for
$\displaystyle e\in{\mathbb{N}}.$ The function $\displaystyle
e\mapsto\ell_{R}(R/I^{[p^{e}]})$ is called the Hilbert-Kunz function of
$\displaystyle R$ with respect to $\displaystyle I$ and was first considered
by E. Kunz in [8]. In [10], P. Monsky showed that this function is of the form

$\displaystyle\ell_{R}(R/I^{[q]})=e_{HK}(I,R)q^{d}+O(q^{d-1}),$

where $\displaystyle e_{HK}(I,R)$ is a positive real number called the
Hilbert–Kunz multiplicity of $\displaystyle R$ with respect to $\displaystyle
I.$ We write $\displaystyle e_{HK}(R):=e_{HK}({\mathfrak{m}},R)$ and
$\displaystyle e_{HK}(I):=e_{HK}(I,R).$ The Hilbert-Kunz multiplicity of
powers of an ideal was first considered by D. Hanes in [4, Theorem 3.2]. He
proved that

$\displaystyle\ell\left(\frac{R}{(I^{[q]})^{k}}\right)=\left(\frac{e_{0}(I)}{d!}k^{d}+O(k^{d-1})\right)q^{d},$

where $\displaystyle e_{0}(I)$ denotes the Hilbert-Samuel multiplicity of
$\displaystyle R$ with respect to $\displaystyle I.$ Recall that the Hilbert-
Samuel function $\displaystyle H_{I}(n)$ of $\displaystyle R$ with respect to
$\displaystyle I$ is defined as $\displaystyle H_{I}(n)=\ell_{R}(R/I^{n}).$ It
is known that $\displaystyle H_{I}(n)$ is a polynomial function of
$\displaystyle n$ of degree $\displaystyle d$, for large $\displaystyle n.$ In
particular, there exists a polynomial $\displaystyle
P_{I}(x)\in{\mathbb{Q}}[x],$ called the Hilbert-Samuel polynomial, such that
$\displaystyle H_{I}(n)=P_{I}(n)$ for all large $\displaystyle n.$ Write

$\displaystyle
P_{I}(x)=e_{0}(I)\binom{x+d-1}{d}-e_{1}(I)\binom{x+d-2}{d-1}+\cdots+(-1)^{d}e_{d}(I),$

where $\displaystyle e_{i}(I)$ for $\displaystyle i=0,1,\ldots,d$ are
integers, called the Hilbert-Samuel coefficients of $\displaystyle I.$ The
leading coefficient $\displaystyle e_{0}(I)$ is called the multiplicity of
$\displaystyle I$ and $\displaystyle e_{1}(I)$ is called the Chern number of
$\displaystyle I.$

In [14], Trivedi proved that if $\displaystyle R$ is a standard graded ring of
dimension $\displaystyle d$ over a perfect field of characteristic
$\displaystyle p>0,$ and $\displaystyle I$ is a homogeneous ideal of finite
colength and generated in the same degree, then

$\displaystyle\displaystyle\lim\limits_{k\rightarrow\infty}\frac{e_{HK}(I^{k})-e_{0}(I^{k})/d!}{k^{d-1}}=\frac{e_{0}(I)}{2(d-2)!}-\lim\limits_{q\rightarrow\infty}\frac{e_{1}(I^{[q]})}{q^{d}(d-1)!}$

and the last limit exists. The result was proved in full generality by Smirnov
in [13, Proposition 2.5]. He proved that if $\displaystyle(R,{\mathfrak{m}})$
is a Noetherian local ring and $\displaystyle I$ is an
$\displaystyle{\mathfrak{m}}$-primary ideal, then

$\displaystyle
e_{HK}(I^{k})=e_{0}(I)\binom{k+d-1}{d}-\lim\limits_{q\rightarrow\infty}\frac{e_{1}(I^{[q]})}{q^{d}}\binom{k+d-2}{d-1}+o(k^{d-1}).$

The above equation motivates one to ask if other Hilbert coefficients may also
be appear in the expression.

One of the aims of this article is to provide some suitable conditions under
which the following questions, proposed by Smirnov in [13], can be answered in
the affirmative. Let $\displaystyle(R,{\mathfrak{m}})$ be a Noetherian local
ring of dimension $\displaystyle d$ with prime characteristic $\displaystyle
p>0.$ Let $\displaystyle I$ be an $\displaystyle{\mathfrak{m}}$-primary ideal.
Put $\displaystyle q=p^{e},$ for some $\displaystyle e\in{\mathbb{N}}.$

######  Question 1.1.

Does the limit
$\displaystyle\lim\limits_{q\rightarrow\infty}e_{i}(I^{[q]})/q^{d}$ exist for
all $\displaystyle i\geq 1$ $\displaystyle?$

######  Question 1.2.

Do we have for large $\displaystyle k,$ that

$\displaystyle
e_{HK}(I^{k})=\sum_{i=0}^{d}(-1)^{i}\binom{k+d-1-i}{d-i}\lim\limits_{q\rightarrow\infty}\frac{e_{i}(I^{[q]})}{q^{d}}\
?$

Recall that for an ideal $\displaystyle I$ in a ring $\displaystyle R,$ a
reduction of $\displaystyle I$ is an ideal $\displaystyle J\subseteq I$ such
that $\displaystyle JI^{n}=I^{n+1}$, for all large $\displaystyle n.$ A
minimal reduction of $\displaystyle I$ is a reduction of $\displaystyle I$
minimal with respect to inclusion. For a minimal reduction $\displaystyle J$
of $\displaystyle I$, we set $\displaystyle r_{J}(I)=\min\\{n\mid
I^{m+1}=JI^{m}\text{ for all }m\geq n\\}.$ The reduction number $\displaystyle
r(I)$ of $\displaystyle I$ is defined as

$\displaystyle r(I)=\min\\{r_{J}(I)\mid J\text{ is a minimal reduction of
}I\\}.$

An ideal with reduction number one is called a stable ideal.

It is easy to check that the questions 1.1 and 1.2 have affirmative answers
when $\displaystyle R$ is a Cohen-Macaulay local ring and $\displaystyle I$ is
a parameter ideal, or when $\displaystyle R$ is a Cohen-Macaulay local ring of
dimension $\displaystyle d\geq 2$ and $\displaystyle I$ is a stable ideal (see
[17, Proposition 1.7, Theorem 1.8]). A class of rings satisfying the latter
assumption includes rings with minimal multiplicity, taking $\displaystyle
I={\mathfrak{m}}.$ Recall that $\displaystyle R$ has minimal multiplicity if
$\displaystyle\text{emb}(R)-\dim R+1=e_{0}(R),$ where
$\displaystyle\text{emb}(R)$ denotes the embedding dimension of $\displaystyle
R.$

In section 2, we prove the following results which answer the above questions
in the affirmative under suitable assumptions.

######  Theorem 1.3.

Let $\displaystyle(R,{\mathfrak{m}})$ be a Buchsbaum ring of dimension
$\displaystyle d$ with prime characteristic $\displaystyle p$ and let
$\displaystyle I$ be an ideal generated by a system of parameters. Then for
all $\displaystyle i=1,\ldots,d,$
$\displaystyle\lim\limits_{q\rightarrow\infty}e_{i}(I^{[q]})/q^{d}=0$ and for
all $\displaystyle k\geq 1,$

$\displaystyle\displaystyle e_{HK}(I^{k})=e_{0}(I)\binom{k+d-1}{d}.$

######  Theorem 1.4.

Let $\displaystyle(R,{\mathfrak{m}})$ be a Cohen-Macaulay local ring of
dimension $\displaystyle d\geq 1$ and prime characteristic $\displaystyle p.$
Let $\displaystyle I$ be an $\displaystyle{\mathfrak{m}}$-primary ideal.
Suppose that $\displaystyle r=r(I)$ and
$\displaystyle\operatorname{depth}G(I^{[q]})\geq d-1$ for all large
$\displaystyle q.$ Then

  1. (1)

For all $\displaystyle i=1,2,\ldots,d$ the limit $\displaystyle
L_{i}(I):=\lim_{q\to\infty}e_{i}(I^{[q]})/q^{d}$ exists and

$\displaystyle
L_{i}(I)=\lim_{q\to\infty}\frac{e_{i}(I^{[q]})}{q^{d}}=\sum_{n=i}^{r}\binom{n-1}{i-1}\left[e_{0}(I)-\sum_{j=0}^{d}(-1)^{j}\binom{d}{j}e_{HK}(I^{n-j})\right].$

  2. (2)

For all $\displaystyle n\geq r-d+1,$

$\displaystyle\ell\left(\frac{R}{(I^{[q]})^{n}}\right)=\sum_{i=0}^{d}(-1)^{i}e_{i}(I^{[q]})\binom{n+d-1-i}{d-i}$

and hence for all $\displaystyle n\geq r-d+1,$ we have

$\displaystyle
e_{HK}(I^{n})=\sum_{i=0}^{d}(-1)^{i}L_{i}(I)\binom{n+d-1-i}{d-i}.$

The generalized Hilbert-Kunz function was introduced by Aldo Conca in [3]. Let
$\displaystyle(R,{\mathfrak{m}})$ be a $\displaystyle d$-dimensional
Noetherian local (resp. standard graded) ring with maximal (resp. maximal
homogeneous) ideal $\displaystyle{\mathfrak{m}}$ and $\displaystyle I$ be an
$\displaystyle{\mathfrak{m}}$-primary (resp. a graded
$\displaystyle{\mathfrak{m}}$-primary) ideal. Fix a set of generators of
$\displaystyle I$, say $\displaystyle I=(a_{1},a_{2},\ldots,a_{g}).$ We choose
these as homogeneous elements in case $\displaystyle R$ is a graded ring.
Define the $\displaystyle s^{th}$ Frobenius power of $\displaystyle I$ to be
the ideal $\displaystyle I^{[s]}=(a_{1}^{s},a_{2}^{s},\ldots,a_{g}^{s}).$ The
generalized Hilbert-Kunz function of $\displaystyle I$ is defined as
$\displaystyle HK_{I}(s)=\ell(R/I^{[s]}).$ The generalized Hilbert-Kunz
multiplicity is defined as
$\displaystyle\lim\limits_{s\rightarrow\infty}HK_{I}(s)/s^{d}$, whenever the
limit exists and is denoted by $\displaystyle e_{HK}(I)$.

In sections $\displaystyle 3$ and $\displaystyle 4,$ we compute the
generalized Hilbert-Kunz function of powers of certain monomial ideals in face
rings of simplicial complexes. In particular, these results are characteristic
independent.

######  Theorem 1.5.

Let $\displaystyle K$ be a field and $\displaystyle
R=K[x_{1},\ldots,x_{r}]/I_{\Delta}$ be a $\displaystyle d$-dimensional face
ring of a simplicial complex $\displaystyle\Delta.$ Let $\displaystyle
J=(x_{1}^{v_{1}},\ldots,x_{r}^{v_{r}})R,$ where $\displaystyle v_{i}>0$ for
all $\displaystyle i.$ Then
$\displaystyle\lim\limits_{q\rightarrow\infty}e_{i}(J^{[q]})/q^{d}=0$ for all
$\displaystyle i=1,\ldots,d$ and for all $\displaystyle k\geq 1,$ the
generalized Hilbert-Kunz multiplicity

$\displaystyle e_{HK}(J^{k})=e_{0}(J)\binom{k+d-1}{d}.$

Smirnov, in the same paper, also proposed the following conjecture.

######  Conjecture 1.6.

Let $\displaystyle(R,{\mathfrak{m}})$ be a Cohen-Macaulay local ring. Then an
$\displaystyle{\mathfrak{m}}$-primary ideal $\displaystyle I$ is stable if and
only if
$\displaystyle\lim\limits_{q\rightarrow\infty}e_{1}(I^{[q]})/q^{d}=e_{0}(I)-e_{HK}(I).$

It was perhaps motivated by a result of Huneke and Ooishi, which characterized
the stability of an ideal using the Northcott’s inequality. In section 4, we
give a counter example to the conjecture. In fact, using Theorem 1.5 above, it
follows that for any ideal $\displaystyle J$ of a face ring $\displaystyle R$
of the form $\displaystyle(x_{1}^{v_{1}},\ldots,x_{r}^{v_{r}})R$ where
$\displaystyle v_{i}>0$ for all $\displaystyle i,$
$\displaystyle\lim\limits_{q\rightarrow\infty}e_{1}(J^{[q]})/q^{d}=e_{0}(J)-e_{HK}(J).$
This means that one side of the conjecture is always true for such ideals. In
particular, the conjecture is false whenever $\displaystyle J$ is not a stable
ideal. The following example illustrates one such case.

######  Example 1.7.

Let $\displaystyle\Delta$ be the simplicial complex

$\displaystyle x_{1}$$\displaystyle x_{2}$$\displaystyle x_{3}$$\displaystyle
x_{4}$

Let $\displaystyle K$ be a field. Then $\displaystyle
R=K[x_{1},x_{2},x_{3},x_{4}]/((x_{3},x_{4})\cap(x_{1},x_{3})\cap(x_{1},x_{4})\cap(x_{1},x_{2}))$
is the face ring of $\displaystyle\Delta.$ Observe that $\displaystyle R$ is a
$\displaystyle 2$-dimensional Cohen-Macaulay ring. Let
$\displaystyle{\mathfrak{n}}=(x_{1},x_{2},x_{3},x_{4})$ denote the maximal
ideal of $\displaystyle R.$ Using Theorem 1.5, it follows that $\displaystyle
4=e_{0}({\mathfrak{n}})=e_{HK}({\mathfrak{n}})$ and
$\displaystyle\lim\limits_{q\rightarrow\infty}e_{1}({\mathfrak{n}}^{[q]})/q^{2}=0.$
We prove that $\displaystyle{\mathfrak{n}}$ is not stable in section 4.

##  2\. Hilbert-Kunz multiplicity of powers of ideals

In this section we answer Smirnov’s questions, mentioned in the introduction,
in the affirmative in certain cases. We show that both the questions have
answers in the affirmative for parameter ideals in Buchsbaum local rings. We
then consider the questions for $\displaystyle{\mathfrak{m}}$-primary ideals
$\displaystyle I$ in Cohen-Macaulay local rings
$\displaystyle(R,{\mathfrak{m}})$ of dimension $\displaystyle d$ subject to
the condition $\displaystyle\operatorname{depth}G(I^{[q]})\geq d-1.$ This is a
strong condition. However, we show that it is satisfied in several examples
(in sections 3 and 4). We begin by considering the case of Buchsbaum local
rings.

######  Theorem 2.1.

Let $\displaystyle(R,{\mathfrak{m}})$ be a Buchsbaum ring of dimension
$\displaystyle d$ with prime characteristic $\displaystyle p$ and let
$\displaystyle I$ be a parameter ideal. Then for all $\displaystyle
i=1,\ldots,d,$
$\displaystyle\lim\limits_{q\rightarrow\infty}e_{i}(I^{[q]})/q^{d}=0$ and for
all $\displaystyle k\geq 1,$

$\displaystyle\displaystyle e_{HK}(I^{k})=e_{0}(I)\binom{k+d-1}{d}.$

###### Proof.

Since $\displaystyle I$ is a parameter ideal, then so is $\displaystyle
I^{[q]}$ for all $\displaystyle q.$ Therefore, using [15, Corollary 4.2], we
obtain that for all $\displaystyle i=1,\ldots,d,$

$\displaystyle\displaystyle
e_{i}(I^{[q]})=(-1)^{i}\sum_{j=0}^{d-i}\binom{d-i-1}{j-1}\ell(H^{i}_{{\mathfrak{m}}}(R)),$
(2.1)

where $\displaystyle\binom{d-i-1}{-1}=0$ if $\displaystyle i\neq d$ and
$\displaystyle\binom{-1}{-1}=1.$ Also, for all $\displaystyle k\geq 1,$

$\displaystyle\displaystyle\ell\left(\frac{R}{I^{[q]k}}\right)=e_{0}(I^{[q]})\binom{k+d-1}{d}+\sum_{i=1}^{d}e_{i}(I^{[q]})\binom{k+d-1-i}{d-i}.$
(2.2)

From (2.1), it follows that for all $\displaystyle i=1,\ldots,d,$
$\displaystyle e_{i}(I^{[q]})$ is independent of $\displaystyle q$ and hence
using (2.2), we get the required result. ∎

The next result answers Smirnov’s questions for
$\displaystyle{\mathfrak{m}}$-primary ideals $\displaystyle I$ in Cohen-
Macaulay local rings of prime characteristic $\displaystyle p$ and dimension
$\displaystyle d\geq 1$ and $\displaystyle\operatorname{depth}G(I^{[q]})\geq
d-1$ for all large $\displaystyle q.$ Recall that for a function
$\displaystyle f:{\mathbb{Z}}\to{\mathbb{Z}}$,
$\displaystyle\Delta(f)(n)=f(n)-f(n-1).$ Define
$\displaystyle\Delta^{t}(f)=\Delta(\Delta^{t-1}(f))$ for all $\displaystyle
t\geq 2.$ We shall use the following results due to S. Huckaba and T. Marley.

######  Theorem 2.2 ([5, 6]).

Let $\displaystyle(R,{\mathfrak{m}})$ be a Cohen-Macaulay local ring of
dimension $\displaystyle d\geq 1.$ Let $\displaystyle I$ be an
$\displaystyle{\mathfrak{m}}$-primary ideal with a minimal reduction
$\displaystyle J=(a_{1},a_{2},\ldots,a_{d}).$ Suppose that $\displaystyle
r=r(I)$ and $\displaystyle\operatorname{depth}G(I)\geq d-1.$ Then for all
$\displaystyle n\in{\mathbb{Z}},$

  1. (1)

$\displaystyle\Delta^{d}[P_{I}(n)-H_{I}(n)]=\ell(I^{n}/JI^{n-1})$.

  2. (2)

$\displaystyle\Delta^{d}P_{I}(n)=e_{0}(I)$ and
$\displaystyle\Delta^{d}(H_{I}(n))=\sum_{j=0}^{d}(-1)^{j}\binom{d}{j}\ell(R/I^{n-j}).$

  3. (3)

$\displaystyle e_{i}(I)=\sum_{n=i}^{r}\binom{n-1}{i-1}\ell(I^{n}/JI^{n-1})$
for $\displaystyle i=1,2,\ldots,d.$

######  Theorem 2.3.

Let $\displaystyle(R,{\mathfrak{m}})$ be a Cohen-Macaulay local ring of
dimension $\displaystyle d\geq 1$ and prime characteristic $\displaystyle p.$
Let $\displaystyle I$ be an $\displaystyle{\mathfrak{m}}$-primary ideal with a
minimal reduction $\displaystyle J=(a_{1},a_{2},\ldots,a_{d}).$ Suppose that
$\displaystyle r=r(I)$ and $\displaystyle\operatorname{depth}G(I^{[q]})\geq
d-1$ for all large $\displaystyle q.$ Then

  1. (1)

For all $\displaystyle i=1,2,\ldots,d$ the limit $\displaystyle
L_{i}(I):=\lim_{q\to\infty}e_{i}(I^{[q]})/q^{d}$ exists and

$\displaystyle
L_{i}(I)=\lim_{q\to\infty}\frac{e_{i}(I^{[q]})}{q^{d}}=\sum_{n=i}^{r}\binom{n-1}{i-1}\left[e_{0}(I)-\sum_{j=0}^{d}(-1)^{j}\binom{d}{j}e_{HK}(I^{n-j})\right].$

  2. (2)

For all $\displaystyle n\geq r-d+1,$

$\displaystyle\ell\left(\frac{R}{(I^{[q]})^{n}}\right)=\sum_{i=0}^{d}(-1)^{i}e_{i}(I^{[q]})\binom{n+d-1-i}{d-i}$

and hence for all $\displaystyle n\geq r-d+1,$ we have

$\displaystyle
e_{HK}(I^{n})=\sum_{i=0}^{d}(-1)^{i}L_{i}(I)\binom{n+d-1-i}{d-i}.$

###### Proof.

Since $\displaystyle JI^{r}=I^{r+1},$ it follows that $\displaystyle
J^{[q]}(I^{[q]})^{r}=(I^{[q]})^{r+1}.$ Hence $\displaystyle r(I^{[q]})\leq r$
for all $\displaystyle q.$ Since
$\displaystyle\operatorname{depth}G(I^{[q]})\geq d-1$ for all large
$\displaystyle q,$ $\displaystyle P_{I^{[q]}}(n)=H_{I^{[q]}}(n)$ for all
$\displaystyle n\geq r-d+1.$ For all $\displaystyle i=1,2,\ldots,d,$ by
Theorem 2.2(3),

$\displaystyle
e_{i}(I^{[q]})=\sum_{n=i}^{r}\binom{n-1}{i-1}\ell\left(\frac{(I^{[q]})^{n}}{J^{[q]}(I^{[q]})^{n-1}}\right).$

Let us find $\displaystyle\beta_{I}(n):=\ell(I^{n}/JI^{n-1}).$ By Theorem 2.2,
for all $\displaystyle n\in{\mathbb{Z}},$
$\displaystyle\Delta^{d}(P_{I}(n)-H_{I}(n))=\ell(I^{n}/JI^{n-1}).$ Therefore

$\displaystyle\beta_{I}(n)=e_{0}(I)-\Delta^{d}(H_{I}(n))=e_{0}(I)-\sum_{j=0}^{d}(-1)^{j}\binom{d}{j}\ell(R/I^{n-j}).$

Since $\displaystyle\operatorname{depth}G(I^{[q]})\geq d-1$ for large
$\displaystyle q,$ we have

$\displaystyle\beta_{I^{[q]}}(n)=q^{d}e_{0}(I)-\Delta^{d}(H_{I^{[q]}}(n))=q^{d}e_{0}(I)-\sum_{j=0}^{d}(-1)^{j}\binom{d}{j}\ell(R/(I^{[q]})^{n-j}).$

Divide by $\displaystyle q^{d}$ and then take the limit to get

$\displaystyle\lim_{q\to\infty}\frac{\beta_{I^{[q]}}(n)}{q^{d}}=e_{0}(I)-\sum_{j=0}^{d}(-1)^{j}\binom{d}{j}e_{HK}(I^{n-j}).$

Therefore we have

$\displaystyle\displaystyle
L_{i}(I)=\lim_{q\to\infty}\frac{e_{i}(I^{[q]})}{q^{d}}=\sum_{n=i}^{r}\binom{n-1}{i-1}\left[e_{0}(I)-\sum_{j=0}^{d}(-1)^{j}\binom{d}{j}e_{HK}(I^{n-j})\right].$
(2.3)

Since $\displaystyle r(I^{[q]})\leq r,$ for all $\displaystyle n\geq r-d+1,$
$\displaystyle H_{I^{[q]}}(n)=P_{I^{[q]}}(n).$ Hence for all $\displaystyle
n\geq r-d+1,$

$\displaystyle\ell(R/(I^{[q]})^{n})=\sum_{i=0}^{d}(-1)^{i}e_{i}(I^{[q]})\binom{n+d-1-i}{d-i}.$

Divide by $\displaystyle q^{d}$ and take the limit to see that for all
$\displaystyle n\geq r-d+1,$

$\displaystyle
e_{HK}(I^{n})=\sum_{i=0}^{d}(-1)^{i}L_{i}(I)\binom{n+d-1-i}{d-i}.$

∎

######  Corollary 2.4.

Let $\displaystyle(R,{\mathfrak{m}})$ be a one-dimensional Cohen-Macaulay
local ring of prime characteristic $\displaystyle p.$ Let $\displaystyle
J=(a)$ be a minimal reduction of an $\displaystyle{\mathfrak{m}}$-primary
ideal $\displaystyle I$ with $\displaystyle JI^{r}=I^{r+1}.$ Then
$\displaystyle L_{1}(I)=0.$

###### Proof.

Put $\displaystyle i=1$ in the equation 2.3 to get

$\displaystyle\displaystyle
L_{1}(I)=\lim_{q\to\infty}\frac{e_{1}(I^{[q]})}{q^{d}}=$
$\displaystyle\displaystyle\
\sum_{n=1}^{r}\left[e_{0}(I)-\sum_{j=0}^{1}(-1)^{j}e_{HK}(I^{n-j})\right]$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\
re_{0}(I)-\sum_{n=1}^{r}\left[e_{HK}(I^{n})-e_{HK}(I^{n-1})\right]$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\
re_{0}(I)-e_{HK}(I^{r})$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\ 0.$

∎

######  Corollary 2.5.

Let $\displaystyle(R,{\mathfrak{m}})$ be a $\displaystyle 2$-dimensional
Cohen-Macaulay local ring of characteristic $\displaystyle p$. Let
$\displaystyle J=(a,b)$ be a minimal reduction of an
$\displaystyle{\mathfrak{m}}$-primary ideal $\displaystyle I$ with
$\displaystyle r_{J}(I)=r.$ Let
$\displaystyle\operatorname{depth}G(I^{[q]})\geq 1$ for all large
$\displaystyle q.$ Then

$\displaystyle\displaystyle L_{1}(I)=$ $\displaystyle\displaystyle\
re_{0}(I)-e_{HK}(I^{r})+e_{HK}(I^{r-1})$ $\displaystyle\displaystyle
L_{2}(I)=$ $\displaystyle\displaystyle\
\binom{r}{2}e_{0}(I)-(r-1)e_{HK}(I^{r})+re_{HK}(I^{r-1})$
$\displaystyle\displaystyle e_{HK}(I^{n})=$ $\displaystyle\displaystyle\
e_{0}(I)\binom{n+1}{2}-L_{1}(I)n+L_{2}(I)\mbox{ for all }n\geq r-1.$

###### Proof.

Put $\displaystyle d=2$ and $\displaystyle i=1$ in 2.3 to get

$\displaystyle\displaystyle L_{1}(I)=$
$\displaystyle\displaystyle\sum_{n=1}^{r}\left[e_{0}(I)-\sum_{j=0}^{2}(-1)^{j}\binom{2}{j}e_{HK}(I^{n-j})\right]$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\
re_{0}(I)-\sum_{n=1}^{r}\left[e_{HK}(I^{n})-2e_{HK}(I^{n-1})+e_{HK}(I^{n-2})\right]$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\
re_{0}(I)-e_{HK}(I^{r})+e_{HK}(I^{r-1}).$

Now put $\displaystyle d=2$ and $\displaystyle i=2$ in 2.3 to get

$\displaystyle\displaystyle L_{2}(I)=$ $\displaystyle\displaystyle\
\sum_{n=2}^{r}(n-1)\left[e_{0}(I)-\sum_{j=0}^{2}(-1)^{j}\binom{2}{j}e_{HK}(I^{n-j})\right]$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\
\binom{r}{2}e_{0}(I)-\sum_{n=2}^{r}(n-1)\left[e_{HK}(I^{n})-2e_{HK}(I^{n-1})+e_{HK}(I^{n-2})\right]$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\
\binom{r}{2}e_{0}(I)-(r-1)e_{HK}(I^{r})+re_{HK}(I^{r-1}).$

From [9, Theorem 2.15], it also follows that the postulation number of
$\displaystyle I^{[q]},$ $\displaystyle n(I^{[q]})=r(I^{[q]})-2\leq r-2.$
Thus, for all $\displaystyle n\geq r-1,$

$\displaystyle\displaystyle\ell\left(\frac{R}{I^{n[q]}}\right)=e_{0}(I^{[q]})\binom{n+1}{2}-e_{1}(I^{[q]})n+e_{2}(I^{[q]}).$

Divide by $\displaystyle q^{2}$ and take the limit to see that for all
$\displaystyle n\geq r-1$

$\displaystyle e_{HK}(I^{n})=e_{0}(I)\binom{n+1}{2}-L_{1}(I)n+L_{2}(I).$

∎

######  Example 2.6.

Let $\displaystyle R=K[[x,y,z]]/(x^{3}-y^{3}z)$, where $\displaystyle K$ is a
field of characteristic $\displaystyle 3.$ Let
$\displaystyle{\mathfrak{m}}=(x,y,z)$ denote the maximal ideal of
$\displaystyle R.$ Then $\displaystyle I=(y,z)$ is a minimal reduction of
$\displaystyle{\mathfrak{m}}$ and the reduction number of
$\displaystyle{\mathfrak{m}},$ $\displaystyle r({\mathfrak{m}})=2.$ Set
$\displaystyle q=3^{e},$ $\displaystyle e\in{\mathbb{N}}.$ Then $\displaystyle
r({\mathfrak{m}}^{[q]})\leq 2.$ We show that $\displaystyle
G({\mathfrak{m}}^{[q]})$ is Cohen-Macaulay for all large $\displaystyle q.$
Using Valabrega-Valla’s result ([16]), it is sufficient to show that for all
$\displaystyle n\geq 1,$

$\displaystyle(y^{q},z^{q})\cap({\mathfrak{m}}^{[q]})^{n}=(y^{q},z^{q})({\mathfrak{m}}^{[q]})^{n-1}.$

Since $\displaystyle r({\mathfrak{m}}^{[q]})\leq 2,$ it is enough to show that
$\displaystyle(y^{q},z^{q})\cap({\mathfrak{m}}^{[q]})^{2}=(y^{q},z^{q}){\mathfrak{m}}^{[q]}.$
Consider

$\displaystyle\displaystyle(y^{q},z^{q})\cap({\mathfrak{m}}^{[q]})^{2}=(y^{q},z^{q})\cap(x^{2q},x^{q}y^{q},x^{q}z^{q},y^{2q},y^{q}z^{q},z^{2q}).$

Let

$\displaystyle
ay^{q}+bz^{q}=cx^{2q}+dx^{q}y^{q}+ex^{q}z^{q}+fy^{2q}+gy^{q}z^{q}+hz^{2q},$

for some $\displaystyle a,b,c,d,e,f,g,h\in R.$ Then

$\displaystyle
y^{q}\big{(}a-dx^{q}-fy^{q}-gz^{q}\big{)}=z^{q}\big{(}-b+ex^{q}+hz^{q}\big{)}+cx^{2q}.$

Since $\displaystyle x^{3}=y^{3}z,$ it follows that

$\displaystyle
x^{2q}=(x^{3})^{2.3^{e-1}}=(y^{3}z)^{2.3^{e-1}}=y^{2q}z^{2q/3}.$

Therefore,

$\displaystyle a-dx^{q}-fy^{q}-gz^{q}\in(z^{q},y^{2q}):y^{q}=(z^{q},y^{q})$

and hence, $\displaystyle a\in{\mathfrak{m}}^{[q]}.$ Similarly, $\displaystyle
b\in{\mathfrak{m}}^{[q]}.$ This proves that $\displaystyle
G({\mathfrak{m}}^{[q]})$ is Cohen-Macaulay for all large $\displaystyle q.$

Note that $\displaystyle e_{0}({\mathfrak{m}})=3$ and $\displaystyle
e_{HK}({\mathfrak{m}})=3$ (see [3, Theorem 3.1]). In order to compute
$\displaystyle e_{HK}({\mathfrak{m}}^{2})$, we compute
$\displaystyle\ell(K[x,y,z]/(x^{3}-y^{3}z,x^{2q},y^{2q},z^{2q},x^{q}y^{q},x^{q}z^{q},y^{q}z^{q})).$
It is easy to check that
$\displaystyle\\{x^{3}-y^{3}z,y^{2q},y^{q}z^{q},z^{2q}\\}$ is a Gröbner basis
of the ideal $\displaystyle(x^{3}-y^{3}z)+{\mathfrak{m}}^{2[q]}$ (This
involves assigning different weights to the variables, $\displaystyle\deg
x=2,$ $\displaystyle\deg y=1,$ $\displaystyle\deg z=3$, in order to make the
polynomial $\displaystyle x^{3}-y^{3}z$ homogeneous). Hence, the Hilbert-Kunz
function

$\displaystyle\ell\left(\frac{K[[x,y,z]]}{(x^{3}-y^{3}z)+{\mathfrak{m}}^{2[q]}}\right)=\ell\left(\frac{K[x,y,z]}{(x^{3},y^{2q},y^{q}z^{q},z^{2q})}\right)=9q^{2}.$

Therefore, $\displaystyle e_{HK}({\mathfrak{m}}^{2})=9.$ Using Corollary 2.5,
we obtain that for all $\displaystyle k\geq 1,$

$\displaystyle\displaystyle
e_{HK}({\mathfrak{m}}^{k})=e_{0}({\mathfrak{m}})\left[\binom{k+1}{2}-2k+1\right]-(k-2)\
e_{HK}({\mathfrak{m}})+(k-1)\ e_{HK}({\mathfrak{m}}^{2}).$

##  3\. Face rings of simplicial complexes

Let $\displaystyle\Delta$ be a $\displaystyle(d-1)$-dimensional simplicial
complex on $\displaystyle r$ vertices. Let $\displaystyle K$ be a field and
$\displaystyle R=K[\Delta]$ be the face ring of simplicial complex
$\displaystyle\Delta.$ Then $\displaystyle R\simeq S/I$ is a $\displaystyle
d$-dimensional ring, where $\displaystyle S=k[x_{1},\ldots,x_{r}]$ and
$\displaystyle I=I_{\Delta}$ is the Stanley-Reisner ideal. The ideal
$\displaystyle I$ can be viewed as intersection of face ideals:

$\displaystyle I=\bigcap_{\begin{subarray}{c}F\in\Delta\\\ F\text{ is a
facet}\end{subarray}}{\mathfrak{p}}_{F},$

where $\displaystyle{\mathfrak{p}}_{F}=(x_{i}\mid i\notin F).$ Let
$\displaystyle{\mathfrak{n}}={\mathfrak{m}}/I$ denote the maximal homogeneous
ideal of $\displaystyle R$, where $\displaystyle{\mathfrak{m}}$ denotes the
maximal homogeneous ideal of $\displaystyle S.$ Set $\displaystyle
J=(x_{i}^{v_{i}}\mid v_{i}>0\text{ for all }1\leq i\leq r).$ We compute the
generalized Hilbert-Kunz function and Hilbert-Kunz multiplicity of powers of
$\displaystyle J$ and in particular, of $\displaystyle{\mathfrak{n}}.$ We
prove a few lemmas first. Note that there is no restriction on the
characteristic of the field $\displaystyle K$ in this section.

######  Lemma 3.1.

Let $\displaystyle S=K[x_{1},\ldots,x_{r}]$ be a polynomial ring in
$\displaystyle r$ variables over a field $\displaystyle K$ and let
$\displaystyle J=(x_{1}^{v_{1}},\ldots,x_{r}^{v_{r}})$ be an ideal of
$\displaystyle S$ such that $\displaystyle v_{i}>0$ for all $\displaystyle i.$
Let $\displaystyle{\mathfrak{p}}_{1},\ldots,{\mathfrak{p}}_{\alpha}$, for
$\displaystyle\alpha\geq 2$, be distinct $\displaystyle S$-ideals generated by
subsets of $\displaystyle\\{x_{1},\ldots,x_{r}\\}.$ Let $\displaystyle
I=\cap_{i=1}^{\alpha}\ {\mathfrak{p}}_{i}.$ Then for $\displaystyle
q,n\in{\mathbb{N}}$,

$\displaystyle\displaystyle\ell\left(\frac{S}{I+(J^{[q]})^{n}}\right)$
$\displaystyle\displaystyle=\sum_{i=1}^{\alpha}\ell\left(\frac{S}{{\mathfrak{p}}_{i}+(J^{[q]})^{n}}\right)-\sum_{1\leq
i<j\leq\alpha}\ell\left(\frac{S}{{\mathfrak{p}}_{i}+{\mathfrak{p}}_{j}+(J^{[q]})^{n}}\right)+\cdots$
$\displaystyle\displaystyle\hskip
199.16928pt+(-1)^{\alpha-1}\ell\left(\frac{S}{\sum_{i=1}^{\alpha}{\mathfrak{p}}_{i}+(J^{[q]})^{n}}\right).$
(3.1)

###### Proof.

The result follows using the same arguments as in the proof of [1, Corollary
3.3]. ∎

######  Lemma 3.2.

Let $\displaystyle S=K[x_{1},\ldots,x_{r}]$ be a polynomial ring in
$\displaystyle r$ variables over a field $\displaystyle K$ and let
$\displaystyle J=(x_{1}^{v_{1}},\ldots,x_{r}^{v_{r}})$ where $\displaystyle
v_{i}>0$ for all $\displaystyle i.$ Then for all $\displaystyle
q,k\in{\mathbb{N}},$

$\displaystyle\ell\left(\frac{S}{(J^{[q]})^{k}}\right)=q^{r}\binom{k+r-1}{r}\prod_{i=1}^{r}v_{i}.$

###### Proof.

Since $\displaystyle S$ is a polynomial ring, $\displaystyle J$ and hence
$\displaystyle J^{[q]}$ is an ideal generated by a homogeneous system of
parameters in $\displaystyle S.$ Using the Cohen-Macaulay property of
$\displaystyle S$, it follows that for all $\displaystyle q,k\in{\mathbb{N}},$

$\displaystyle\ell\left(\frac{S}{(J^{[q]})^{k}}\right)=q^{r}\binom{k+r-1}{r}\prod_{i=1}^{r}v_{i}.$

∎

######  Theorem 3.3.

Let $\displaystyle R$ be a $\displaystyle d$-dimensional face ring of a
simplicial complex. Let $\displaystyle
J=(x_{1}^{v_{1}},\ldots,x_{r}^{v_{r}})R$ where $\displaystyle v_{i}>0$ for all
$\displaystyle i.$ Then the generalized Hilbert-Kunz function
$\displaystyle\ell(R/(J^{[q]})^{k})$ is a polynomial for all $\displaystyle
q,k\in{\mathbb{N}}.$

###### Proof.

Write $\displaystyle R=S/I$, where $\displaystyle S=K[x_{1},\ldots,x_{r}]$ and
$\displaystyle I=\cap_{i=1}^{\alpha}{\mathfrak{p}}_{i}.$ Then for all
$\displaystyle q,k\in{\mathbb{N}},$

$\displaystyle\displaystyle\ell\left(\frac{R}{(J^{[q]})^{k}}\right)$
$\displaystyle\displaystyle=\ell\left(\frac{S}{I+(J^{[q]})^{k}}\right)$
$\displaystyle\displaystyle=\sum_{i=1}^{\alpha}\ell\left(\frac{S}{{\mathfrak{p}}_{i}+(J^{[q]})^{k}}\right)-\sum_{1\leq
i<j\leq\alpha}\ell\left(\frac{S}{{\mathfrak{p}}_{i}+{\mathfrak{p}}_{j}+(J^{[q]})^{k}}\right)+\cdots+(-1)^{\alpha-1}\ell\left(\frac{S}{\sum_{i=1}^{\alpha}{\mathfrak{p}}_{i}+(J^{[q]})^{k}}\right),$
(3.2)

using Lemma 3.1. The result follows using Lemma 3.2. ∎

From Theorem 3.3, it follows that for all $\displaystyle q\geq 1,$ the
Hilbert-Samuel function of the ideal $\displaystyle J^{[q]}$ coincides its
Hilbert-Samuel polynomial for all $\displaystyle k\geq 1.$ In particular, we
can write, for all $\displaystyle q,k\geq 1,$

$\displaystyle\displaystyle\ell\left(\frac{R}{(J^{[q]})^{k}}\right)=\sum_{i=0}^{d}(-1)^{i}e_{i}(J^{[q]})\binom{k+d-1-i}{d-i}.$
(3.3)

######  Theorem 3.4.

Let $\displaystyle R$ be a $\displaystyle d$-dimensional face ring of a
simplicial complex and let $\displaystyle
J=(x_{1}^{v_{1}},\ldots,x_{r}^{v_{r}})$ be an ideal of $\displaystyle R$ such
that $\displaystyle v_{i}>0$ for all $\displaystyle i.$ Then
$\displaystyle\lim\limits_{q\rightarrow\infty}e_{i}(J^{[q]})/q^{d}=0$ for all
$\displaystyle i=1,\ldots,d$ and for all $\displaystyle k\geq 1,$ the
generalized Hilbert-Kunz multiplicity

$\displaystyle e_{HK}(J^{k})=e_{0}(J)\binom{k+d-1}{d}.$

###### Proof.

(i) Comparing the equations (3.3) and (3) and using Lemma 3.2, it follows that
for all $\displaystyle i=1,\ldots,d,$ the coefficients $\displaystyle
e_{i}(J^{[q]})$ are of the form $\displaystyle\beta_{i}q^{d-i}$, for some
$\displaystyle\beta_{i}\in{\mathbb{Z}}.$ In particular, for all $\displaystyle
i=1,\ldots,d$,

$\displaystyle\lim\limits_{q\rightarrow\infty}\frac{e_{i}(J^{[q]})}{q^{d}}=\lim\limits_{q\rightarrow\infty}\frac{\beta_{i}q^{d-i}}{q^{d}}=0.$

(ii) In (3.3), divide by $\displaystyle q^{d}$ and take limit $\displaystyle
q\rightarrow\infty.$ Using part (i) it follows that for all $\displaystyle
k\geq 1,$

$\displaystyle e_{HK}(J^{k})=\sum_{i=0}^{d}(-1)^{i}\binom{k+d-1-i}{d-i}\
\lim\limits_{q\rightarrow\infty}\frac{e_{i}(J^{[q]})}{q^{d}}=e_{0}(J)\binom{k+d-1}{d}.$

∎

This answers the questions ([13, Question 1.2, Question 1.3]) posed by I.
Smirnov for ideals, primary for the maximal ideal, generated by pure powers of
the variables and in particular, for the maximal homogeneous ideal of face
rings of simplicial complexes.

Let $\displaystyle J=(x_{1}^{v_{1}},\ldots,x_{r}^{v_{r}})$ be an ideal of
$\displaystyle R=K[\Delta]$ such that $\displaystyle v_{i}>0$ for all
$\displaystyle i$ and let $\displaystyle{\mathfrak{n}}$ denote the maximal
homogeneous ideal of $\displaystyle R.$ From Lemma 3.2 and Theorem 3.3 it
follows that for all $\displaystyle q,k\geq 1,$

$\displaystyle\ell\left(\frac{R}{(J^{[q]})^{k}}\right)=\ell\left(\frac{R}{({\mathfrak{n}}^{[q]})^{k}}\right)\prod_{i=1}^{r}v_{i}.$

Therefore, we only compute the generalized Hilbert-Kunz function of
$\displaystyle{\mathfrak{n}}^{k}$ in the following examples. We use the
following result to check the stability of an ideal.

######  Theorem 3.5 ([7, Theorem 2.1], [11, Theorem 3.3]).

For any $\displaystyle{\mathfrak{m}}$-primary ideal $\displaystyle I$ of a
Cohen-Macaulay local ring $\displaystyle A$, $\displaystyle I$ is a stable
ideal if and only if $\displaystyle e_{1}(I)=e_{0}(I)-\ell(A/I).$

######  Example 3.6.

Let $\displaystyle\Delta$ be the $\displaystyle 1$-dimensional simplicial
complex on $\displaystyle r$ vertices, for some $\displaystyle r\geq 3:$

$\displaystyle x_{1}$$\displaystyle x_{2}$$\displaystyle x_{3}$$\displaystyle
x_{r-1}$$\displaystyle x_{r}$

For $\displaystyle i=1,\ldots,r-1$, set
$\displaystyle{\mathfrak{p}}_{i}=\big{(}\\{x_{1},\ldots,x_{r}\\}\setminus\\{x_{i},x_{i+1}\\}\big{)}.$
Let $\displaystyle I=\cap_{i=1}^{r-1}\ {\mathfrak{p}}_{i}.$ Then
$\displaystyle R=K[x_{1},\ldots,x_{r}]/I$ is the face ring of
$\displaystyle\Delta.$ As $\displaystyle\Delta$ is a shellable simplicial
complex, it follows that $\displaystyle R$ is a two-dimensional Cohen-Macaulay
ring with $\displaystyle f$-vector $\displaystyle f(\Delta)=(1,r,r-1)$ (see
[2, Definition 5.1.11, Theorem 5.1.13]). Let $\displaystyle{\mathfrak{n}}$ be
the maximal homogeneous ideal of $\displaystyle R$. From Theorem 3.3 it
follows that for all $\displaystyle q,k\geq 1$,

$\displaystyle\displaystyle\ell\left(\frac{R}{{\mathfrak{n}}^{[q]k}}\right)=$
$\displaystyle\displaystyle\
\ell\left(\frac{S}{I+{\mathfrak{m}}^{[q]k}}\right)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\
\sum_{i=1}^{r-1}\ell\left(\frac{S}{{\mathfrak{p}}_{i}+{\mathfrak{m}}^{[q]k}}\right)-\sum_{1\leq
i<j\leq
r-1}\ell\left(\frac{S}{{\mathfrak{p}}_{i}+{\mathfrak{p}}_{j}+{\mathfrak{m}}^{[q]k}}\right)+\cdots+(-1)^{r-2}\
\ell\left(\frac{S}{\sum_{i=1}^{r-1}{\mathfrak{p}}_{i}+{\mathfrak{m}}^{[q]k}}\right).$

For $\displaystyle 1\leq i<j\leq r-1$, if
$\displaystyle\\{x_{i},x_{i+1}\\}\cap\\{x_{j},x_{j+1}\\}\neq\emptyset$, then
$\displaystyle S/({\mathfrak{p}}_{i}+{\mathfrak{p}}_{j})\simeq K{[x]}$, i.e.
it is a polynomial ring over $\displaystyle K$ in one variable and there are
$\displaystyle r-2$ such instances. Otherwise, $\displaystyle
S/({\mathfrak{p}}_{i}+{\mathfrak{p}}_{j})\simeq K.$ Therefore, using Lemma
3.2, we get

$\displaystyle\sum_{1\leq i<j\leq
r-1}\ell\left(\frac{S}{{\mathfrak{p}}_{i}+{\mathfrak{p}}_{j}+{\mathfrak{m}}^{[q]k}}\right)=(r-2)kq-\left[\binom{r-1}{2}-(r-2)\right].$

It is also easy to observe that $\displaystyle
S/({\mathfrak{p}}_{i_{1}}+\cdots+{\mathfrak{p}}_{i_{u}})\simeq K,$ for all
$\displaystyle u\geq 3$ and $\displaystyle
i_{1},\ldots,i_{u}\in\\{1,\ldots,r-1\\}.$ Hence, for $\displaystyle q,k\geq
1,$

$\displaystyle\displaystyle\ell\left(\frac{R}{{\mathfrak{n}}^{[q]k}}\right)=$
$\displaystyle\displaystyle\
(r-1)\binom{k+1}{2}q^{2}-(r-2)kq-\left[\binom{r-1}{2}-(r-2)\right]+\binom{r-1}{3}+\cdots+(-1)^{r-2}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\
(r-1)\binom{k+1}{2}q^{2}-(r-2)kq,$ (3.4)

where the last equality follows as for any $\displaystyle n\in{\mathbb{N}},$
$\displaystyle\sum_{i=2}^{n}(-1)^{i}\binom{n}{i}=n-1.$ This implies that

$\displaystyle e_{0}({\mathfrak{n}}^{[q]})=(r-1)q^{2},\
e_{1}({\mathfrak{n}}^{[q]})=(r-2)q,\ e_{2}({\mathfrak{n}}^{[q]})=0\text{ and
}e_{HK}({\mathfrak{n}}^{k})=(r-1)\binom{k+1}{2}.$

Since $\displaystyle
e_{1}({\mathfrak{n}}^{[q]})=e_{0}({\mathfrak{n}}^{[q]})-\ell(R/{\mathfrak{n}}^{[q]})$,
from Theorem 3.5 it follows that $\displaystyle r({\mathfrak{n}}^{[q]})=1$ and
hence $\displaystyle G({\mathfrak{n}}^{[q]})$ is Cohen-Macaulay for all
$\displaystyle q.$

######  Example 3.7.

Let $\displaystyle\Delta$ be the the simplicial complex related to an
$\displaystyle n$-cycle and $\displaystyle S=K[x_{1},\ldots,x_{n}]$. Let
$\displaystyle I=I_{\Delta},R=S/I$. Let $\displaystyle{\mathfrak{n}}$ be the
maximal homogeneous ideal of $\displaystyle R$ and
$\displaystyle{\mathfrak{m}}=(x_{1},\ldots,x_{n})S$.

$\displaystyle x_{1}$$\displaystyle x_{2}$$\displaystyle x_{3}$$\displaystyle
x_{4}$$\displaystyle x_{n}$

For $\displaystyle i=1,\ldots,n-1$, set
$\displaystyle{\mathfrak{p}}_{i}=\big{(}\\{x_{1},\ldots,x_{n}\\}\setminus\\{x_{i},x_{i+1}\\}\big{)}$
and $\displaystyle{\mathfrak{p}}_{n}=(x_{2},\ldots,x_{n-1}).$ Then
$\displaystyle I=\cap_{i=1}^{n}\ {\mathfrak{p}}_{i}.$ From Theorem 3.3 it
follows that for all $\displaystyle q,k\geq 1,$

$\displaystyle\displaystyle\ell\left(\frac{R}{{\mathfrak{n}}^{[q]k}}\right)=\sum_{i=1}^{n}\ell\left(\frac{S}{{\mathfrak{p}}_{i}+{\mathfrak{m}}^{[q]k}}\right)-\sum_{1\leq
i<j\leq
n}\ell\left(\frac{S}{{\mathfrak{p}}_{i}+{\mathfrak{p}}_{j}+{\mathfrak{m}}^{[q]k}}\right)+\cdots+(-1)^{n-1}\
\ell\left(\frac{S}{\sum_{i=1}^{n}{\mathfrak{p}}_{i}+{\mathfrak{m}}^{[q]k}}\right).$

Using the same arguments as in the above example, it follows that

$\displaystyle\displaystyle\ell\left(\frac{R}{{\mathfrak{n}}^{[q]k}}\right)=$
$\displaystyle\displaystyle\
n\binom{k+1}{2}q^{2}-nkq-\left[\binom{n}{2}-n\right]+\binom{n}{3}+\cdots+(-1)^{n-1}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\
n\binom{k+1}{2}q^{2}-nkq+1.$ (3.5)

This implies that for all $\displaystyle k\geq 1,$

$\displaystyle e_{0}({\mathfrak{n}}^{[q]})=nq^{2},\
e_{1}({\mathfrak{n}}^{[q]})=nq,\ e_{2}({\mathfrak{n}}^{[q]})=1\text{ and
}e_{HK}({\mathfrak{n}}^{k})=n\binom{k+1}{2}.$

We now prove that
$\displaystyle\operatorname{depth}G({\mathfrak{n}}^{[q]})\geq 1$ for all
$\displaystyle q.$ We use the following result by M.E. Rossi and G. Valla.

######  Theorem 3.8 ([12, Corollary 1.7]).

Let $\displaystyle I$ be an $\displaystyle{\mathfrak{m}}$-primary ideal of a
Cohen-Macaulay local ring $\displaystyle(R,{\mathfrak{m}})$ of dimension
$\displaystyle d$. If $\displaystyle e_{0}(I)=\ell(I/I^{2})+(1-d)\ell(R/I)+1,$
then $\displaystyle\operatorname{depth}G(I)\geq d-1.$

From equation (3.7), we get that for all $\displaystyle k\geq 1,$

$\displaystyle\ell\left(\frac{({\mathfrak{n}}^{[q]})^{k}}{{({\mathfrak{n}}^{[q]})^{k+1}}}\right)=nq^{2}{{k+2}\choose
2}-nq(k+1)+1-nq^{2}{{k+1}\choose 2}+nqk-1=nq^{2}(k+1)-nq.$

Hence

$\displaystyle\ell({\mathfrak{n}}^{[q]}/({\mathfrak{n}}^{[q]})^{2})+(1-d)\ell(R/{\mathfrak{n}}^{[q]})+1=2nq^{2}-nq-(nq^{2}-nq+1)+1=nq^{2}=e_{0}({\mathfrak{n}}^{[q]}).$

Using Theorem 3.8, we get depth $\displaystyle G({\mathfrak{n}}^{[q]})\geq 1$
for all $\displaystyle q.$

Note that Theorem 3.3 and Theorem 3.4 also hold for edge ideals of graphs.

######  Definition 3.9.

Let $\displaystyle G$ be a finite simple graph with vertices $\displaystyle
V=V(G)=\\{x_{1},\ldots,x_{n}\\}$ and the edges $\displaystyle E=E(G)$. The
edge ideal $\displaystyle I(G)$ of $\displaystyle G$ is defined to be the
ideal in $\displaystyle K[x_{1},\ldots,x_{n}]$ generated by the square free
quadratic monomials representing the edges of $\displaystyle G$, i.e.,

$\displaystyle I(G)=\langle x_{i}x_{j}\mid x_{i}x_{j}\in E\rangle.$

A vertex cover of a graph is a set of vertices such that every edge has at
least one vertex belonging to that set. A minimal vertex cover is a vertex
cover such that none of its subsets is a vertex cover. For any graph
$\displaystyle G$ with the set of all minimal vertex covers $\displaystyle C$,
the edge ideal $\displaystyle I(G)$ has the primary decomposition:

$\displaystyle I(G)=\bigcap_{\\{x_{i_{1}},\ldots,x_{i_{u}}\\}\in
C}(x_{i_{1}},\ldots,x_{i_{u}}).$

For example, when $\displaystyle G$ is a five cycle, the primary decomposition
of the edge ideal $\displaystyle I(G)$ is

$\displaystyle
I(G)=(x_{1}x_{2},x_{2}x_{3},x_{3}x_{4},x_{4}x_{5},x_{5}x_{1})=(x_{1},x_{2},x_{4})\cap(x_{1},x_{3},x_{5})\cap(x_{1},x_{3},x_{4})\cap(x_{2},x_{3},x_{5})\cap(x_{2},x_{4},x_{5}).$

######  Example 3.10.

Let $\displaystyle\Delta$ be the simplicial complex

$\displaystyle x_{1}$$\displaystyle x_{2}$$\displaystyle x_{3}$$\displaystyle
x_{4}$

Then $\displaystyle R=K[x_{1},x_{2},x_{3},x_{4}]/((x_{1})\cap(x_{3},x_{4}))$
is the face ring of $\displaystyle\Delta.$ Observe that $\displaystyle R$ is
not a Cohen-Macaulay ring. Let $\displaystyle I=(x_{1})\cap(x_{3},x_{4})$.
Here the defining ideal is the edge ideal of a line graph with three vertices.
Set $\displaystyle S=K[x_{1},x_{2},x_{3},x_{4}],$
$\displaystyle{\mathfrak{p}}_{1}=(x_{1}),$ and
$\displaystyle{\mathfrak{p}}_{2}=(x_{3},x_{4}).$ Therefore, for $\displaystyle
q,k\geq 1,$

$\displaystyle\displaystyle\ell\left(\frac{R}{{\mathfrak{n}}^{[q]k}}\right)=\ell\left(\frac{S}{I+{\mathfrak{m}}^{[q]k}}\right)=$
$\displaystyle\displaystyle\
\ell\left(\frac{S}{{\mathfrak{p}}_{1}+{\mathfrak{m}}^{[q]k}}\right)+\ell\left(\frac{S}{{\mathfrak{p}}_{2}+{\mathfrak{m}}^{[q]k}}\right)-\ell\left(\frac{S}{{\mathfrak{p}}_{1}+{\mathfrak{p}}_{2}+{\mathfrak{m}}^{[q]k}}\right)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\
\binom{k+2}{3}q^{3}+\binom{k+1}{2}q^{2}-kq.$

This implies that $\displaystyle e_{0}({\mathfrak{n}}^{[q]})=q^{3},$
$\displaystyle e_{1}({\mathfrak{n}}^{[q]})=-q^{2},$ $\displaystyle
e_{2}({\mathfrak{n}}^{[q]})=-q,$ $\displaystyle e_{3}({\mathfrak{n}}^{[q]})=0$
and $\displaystyle e_{HK}({\mathfrak{n}}^{k})=\binom{k+2}{3}.$

######  Example 3.11 (Complete Bipartite Graphs).

A complete bipartite graph $\displaystyle K_{\alpha,\beta}$ is a graph whose
set of vertices is decomposed into two disjoint sets such that no two vertices
within the same set are adjacent and that every pair of vertices in the two
sets are adjacent.

Let $\displaystyle S=K[x_{1},\ldots,x_{\alpha},y_{1},\ldots,y_{\beta}],$ where
$\displaystyle 1\leq\alpha\leq\beta.$ Then the edge ideal $\displaystyle
I=I(K_{\alpha,\beta})=\big{(}x_{i}y_{j}\mid 1\leq i\leq\alpha,1\leq
j\leq\beta\big{)}$ and $\displaystyle R=S/I$ is a
$\displaystyle\beta$-dimensional ring. Let
$\displaystyle{\mathfrak{p}}_{1}=(x_{1},\ldots,x_{\alpha})$,
$\displaystyle{\mathfrak{p}}_{2}=(y_{1},\ldots,y_{\beta}).$ Thus
$\displaystyle I={\mathfrak{p}}_{1}\cap{\mathfrak{p}}_{2}$. Note that
$\displaystyle I$ is the Stanley-Reisner ideal of the union of an
$\displaystyle\alpha$-simplex and a $\displaystyle\beta$-simplex. We compute
the generalized Hilbert-Kunz function of $\displaystyle{\mathfrak{n}}^{k}$ for
different cases of $\displaystyle\alpha$ and $\displaystyle\beta.$ Recall that
for all $\displaystyle q,k\geq 1,$

$\displaystyle\ell\left(\frac{R}{{\mathfrak{n}}^{[q]k}}\right)=\ell\left(\frac{S}{I+{\mathfrak{m}}^{[q]k}}\right)=\
\ell\left(\frac{S}{{\mathfrak{p}}_{1}+{\mathfrak{m}}^{[q]k}}\right)+\ell\left(\frac{S}{{\mathfrak{p}}_{2}+{\mathfrak{m}}^{[q]k}}\right)-\ell\left(\frac{S}{{\mathfrak{p}}_{1}+{\mathfrak{p}}_{2}+{\mathfrak{m}}^{[q]k}}\right).$

Case 1: Let $\displaystyle\beta=\alpha.$ Then for $\displaystyle q,k\geq 1,$

$\displaystyle\displaystyle\ell\left(\frac{R}{{\mathfrak{n}}^{[q]k}}\right)=2\binom{k+\beta-1}{\beta}q^{\beta}-1.$
(3.6)

This implies that $\displaystyle e_{0}({\mathfrak{n}}^{[q]})=2q^{\beta},$
$\displaystyle e_{\beta}({\mathfrak{n}}^{[q]})=(-1)^{\beta+1}$ and
$\displaystyle e_{HK}({\mathfrak{n}}^{k})=2\binom{k+\beta-1}{\beta}.$

Case 2: Let $\displaystyle\beta\geq\alpha+1.$ Then for $\displaystyle q,k\geq
1,$

$\displaystyle\displaystyle\ell\left(\frac{R}{{\mathfrak{n}}^{[q]k}}\right)=\binom{k+\beta-1}{\beta}q^{\beta}+\binom{k+\alpha-1}{\alpha}q^{\alpha}-1.$

This implies that $\displaystyle e_{0}({\mathfrak{n}}^{[q]})=q^{\beta},$
$\displaystyle
e_{\beta-\alpha}({\mathfrak{n}}^{[q]})=(-1)^{\beta-\alpha}q^{\alpha}$,
$\displaystyle e_{\beta}({\mathfrak{n}}^{[q]})=(-1)^{\beta+1}$ and
$\displaystyle e_{HK}({\mathfrak{n}}^{k})=\binom{k+\beta-1}{\beta}.$

##  4\. Counter-example to Smirnov’s conjecture

Smirnov proposed that following conjecture in [13].

######  Conjecture 4.1.

Let $\displaystyle(R,{\mathfrak{m}})$ be a Cohen-Macaulay local ring. Then an
$\displaystyle{\mathfrak{m}}$-primary ideal $\displaystyle I$ is stable if and
only if
$\displaystyle\lim\limits_{q\rightarrow\infty}e_{1}(I^{[q]})/q^{d}=e_{0}(I)-e_{HK}(I).$

From section 3, we know that if $\displaystyle R$ is a $\displaystyle
d$-dimensional face ring of a simplicial complex and $\displaystyle
J=(x_{1}^{v_{1}},\ldots,x_{r}^{v_{r}})$ is an $\displaystyle R$-ideal such
that $\displaystyle v_{i}>0$ for all $\displaystyle i$, then
$\displaystyle\lim\limits_{q\to\infty}e_{1}(J^{[q]})/q^{d}=0$ and
$\displaystyle e_{HK}(J)=e_{0}(J).$ This means that
$\displaystyle\lim\limits_{q\rightarrow\infty}e_{1}(J^{[q]})/q^{d}=e_{0}(J)-e_{HK}(J)$
is always true for such ideals. In particular, the conjecture is false
whenever $\displaystyle J$ is not a stable ideal. The following example
illustrates one such case.

######  Example 4.2.

Let $\displaystyle\Delta$ be the simplicial complex

$\displaystyle x_{1}$$\displaystyle x_{2}$$\displaystyle x_{3}$$\displaystyle
x_{4}$

Then $\displaystyle
R=K[x_{1},x_{2},x_{3},x_{4}]/((x_{3},x_{4})\cap(x_{1},x_{3})\cap(x_{1},x_{4})\cap(x_{1},x_{2}))$
is the face ring of $\displaystyle\Delta.$ Observe that $\displaystyle R$ is a
$\displaystyle 2$-dimensional Cohen-Macaulay ring with $\displaystyle
f$-vector $\displaystyle f(\Delta)=(1,4,4).$ Let
$\displaystyle{\mathfrak{n}}=(x_{1},x_{2},x_{3},x_{4})$ denote the maximal
ideal of $\displaystyle R.$ Using Theorem 3.4, it follows that $\displaystyle
4=e_{0}({\mathfrak{n}})=e_{HK}({\mathfrak{n}})$ and
$\displaystyle\lim\limits_{q\rightarrow\infty}e_{1}({\mathfrak{n}}^{[q]})/q^{2}=0.$
We prove that $\displaystyle{\mathfrak{n}}$ is not stable. Consider the
Hilbert series of $\displaystyle R$,

$\displaystyle
H(R,z)=1+\frac{4z}{1-z}+\frac{4z^{2}}{(1-z)^{2}}=\frac{1+2z+z^{2}}{(1-z)^{2}}.$

As $\displaystyle R$ is Cohen-Macaulay and degree of the numerator above is 2,
we get $\displaystyle r({\mathfrak{n}}^{[q]})=2.$

We also observe that in this case,
$\displaystyle\operatorname{depth}G({\mathfrak{n}}^{[q]})\geq 1.$ Set
$\displaystyle{\mathfrak{p}}_{1}=(x_{3},x_{4}),$
$\displaystyle{\mathfrak{p}}_{2}=(x_{1},x_{3}),$
$\displaystyle{\mathfrak{p}}_{3}=(x_{1},x_{4})$ and
$\displaystyle{\mathfrak{p}}_{4}=(x_{1},x_{2}).$ Then $\displaystyle
I=\cap_{i=1}^{4}\ {\mathfrak{p}}_{i}$ and $\displaystyle R=S/I,$ where
$\displaystyle S=K[x_{1},\ldots,x_{4}].$ From Theorem 3.3 it follows that for
all $\displaystyle q,k\geq 1,$

$\displaystyle\displaystyle\ell\left(\frac{R}{{\mathfrak{n}}^{[q]k}}\right)=$
$\displaystyle\displaystyle\
\sum_{i=1}^{4}\ell\left(\frac{S}{{\mathfrak{p}}_{i}+{\mathfrak{m}}^{[q]k}}\right)-\sum_{1\leq
i<j\leq
4}\ell\left(\frac{S}{{\mathfrak{p}}_{i}+{\mathfrak{p}}_{j}+{\mathfrak{m}}^{[q]k}}\right)+\sum_{1\leq
i<j<u\leq
4}\ell\left(\frac{S}{{\mathfrak{p}}_{i}+{\mathfrak{p}}_{j}+{\mathfrak{p}}_{u}+{\mathfrak{m}}^{[q]k}}\right)$
$\displaystyle\displaystyle\hskip
236.15796pt-\ell\left(\frac{S}{\sum_{i=1}^{4}{\mathfrak{p}}_{i}+{\mathfrak{m}}^{[q]k}}\right)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\
4\binom{k+1}{2}q^{2}-(5kq+1)+(kq+3)-1$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\ 4\binom{k+1}{2}q^{2}-4kq+1.$ (4.1)

This implies that

$\displaystyle e_{0}({\mathfrak{n}}^{[q]})=4q^{2},\
e_{1}({\mathfrak{n}}^{[q]})=4q,\ e_{2}({\mathfrak{n}}^{[q]})=1\text{ and
}e_{HK}({\mathfrak{n}}^{k})=4\binom{k+1}{2}.$

From equation (4.2), we get that for all $\displaystyle k\geq 1,$

$\displaystyle\ell\left(\frac{({\mathfrak{n}}^{[q]})^{k}}{{({\mathfrak{n}}^{[q]})^{k+1}}}\right)=4q^{2}{{k+2}\choose
2}-4q(k+1)+1-4q^{2}{{k+1}\choose 2}+4qk-1=4q^{2}(k+1)-4q.$

Hence

$\displaystyle\ell({\mathfrak{n}}^{[q]}/({\mathfrak{n}}^{[q]})^{2})+(1-d)\ell(R/{\mathfrak{n}}^{[q]})+1=8q^{2}-4q-(4q^{2}-4q+1)+1=4q^{2}=e_{0}({\mathfrak{n}}^{[q]}).$

Using Theorem 3.8, we get depth $\displaystyle G({\mathfrak{n}}^{[q]})\geq 1$
for all $\displaystyle q.$

######  Remark 4.3.

In the above example, $\displaystyle G({\mathfrak{n}})\simeq R$ is Cohen-
Macaulay and $\displaystyle\operatorname{depth}G({\mathfrak{n}}^{[q]})\geq
d-1$ for all $\displaystyle q.$ This indicates that the conjecture is false
even if $\displaystyle\operatorname{depth}G({\mathfrak{n}}^{[q]})\geq d-1.$

The above observation indicates that Conjecture 4.1 does not help in
characterizing the stability of ideals in rings, as in the setup of section 3.
We end the section by characterizing the stability of ideals in certain
classes of Cohen-Macaulay simplicial complexes and graphs. We use Theorem 3.5
for the same.

######  Examples 4.4.

(1) Let $\displaystyle R$ be the face ring of the simplicial complex on
$\displaystyle r$ vertices, $\displaystyle r\geq 3,$ as in Example 3.6.

$\displaystyle x_{1}$$\displaystyle x_{2}$$\displaystyle x_{3}$$\displaystyle
x_{r-1}$$\displaystyle x_{r}$

By putting $\displaystyle q=1$ in (3.6), we get for all $\displaystyle k\geq
1,$

$\displaystyle\ell(R/{\mathfrak{n}}^{k})=(r-1)\binom{k+1}{2}-(r-2)k.$

Hence,

$\displaystyle
e_{0}({\mathfrak{n}})-\ell(R/{\mathfrak{n}})=(r-1)-\big{(}r-1-(r-2)\big{)}=r-2=e_{1}({\mathfrak{n}}),$

implying that $\displaystyle{\mathfrak{n}}$ is a stable ideal. Let
$\displaystyle J=(x_{1}^{v_{1}},\ldots,x_{r}^{v_{r}})$ be an ideal of
$\displaystyle R$ such that $\displaystyle v_{i}>0$ for all $\displaystyle i.$
Since
$\displaystyle\ell(R/(J^{[q]})^{k})=\ell(R/({\mathfrak{n}}^{[q]})^{k})\prod_{i=1}^{r}v_{i},$
it follows that $\displaystyle J$ is a stable ideal.

(2) Let $\displaystyle R$ be the face ring of the simplicial complex related
to an $\displaystyle n$-cycle, as in Example 3.7. By putting $\displaystyle
q=1$ in (3.7), we get for all $\displaystyle k\geq 1,$

$\displaystyle\ell(R/{\mathfrak{n}}^{k})=n\binom{k+1}{2}-nk+1.$

Hence,

$\displaystyle e_{0}({\mathfrak{n}})-\ell(R/{\mathfrak{n}})=n-1\neq
e_{1}({\mathfrak{n}}),$

implying that $\displaystyle{\mathfrak{n}}$ is not a stable ideal. Similarly,
$\displaystyle J$ is not a stable ideal, where $\displaystyle
J=(x_{1}^{v_{1}},\ldots,x_{r}^{v_{r}})$ such that $\displaystyle v_{i}>0$ for
all $\displaystyle i.$

######  Theorem 4.5.

Consider a complete graph, $\displaystyle K_{r},$ on $\displaystyle r$
vertices, where $\displaystyle r\geq 1.$ Let $\displaystyle
S=K[x_{1},\ldots,x_{r}]$ and $\displaystyle I=I(K_{r})$ denote the edge ideal
of $\displaystyle K_{r}.$ Let $\displaystyle R=S/I$ and $\displaystyle
J=(x_{1}^{v_{1}},\ldots,x_{r}^{v_{r}})$ be an ideal of $\displaystyle R$ such
that $\displaystyle v_{i}>0$ for all $\displaystyle i.$ Then $\displaystyle J$
is a stable ideal.

###### Proof.

We begin by observing that any minimal vertex cover of $\displaystyle K_{r}$
contains exactly $\displaystyle r-1$ vertices. Hence, $\displaystyle
I=\cap_{i=1}^{r}(x_{1},\ldots,\hat{x_{i}},\ldots,x_{r}).$ Set
$\displaystyle{\mathfrak{p}}_{i}=(x_{1},\ldots,\hat{x_{i}},\ldots,x_{r}),$ for
all $\displaystyle i=1,\ldots,r.$ Let $\displaystyle{\mathfrak{n}}$ denote the
maximal homogeneous ideal of $\displaystyle R.$ Note that
$\displaystyle{\mathfrak{p}}_{i}+{\mathfrak{p}}_{j}={\mathfrak{n}}$ for all
$\displaystyle i\neq j.$ Since
$\displaystyle\ell(R/(J^{[q]})^{k})=\ell(R/({\mathfrak{n}}^{[q]})^{k})\prod_{i=1}^{r}v_{i},$
we compute $\displaystyle\ell(R/({\mathfrak{n}}^{[q]})^{k}).$ From Theorem
3.3, it follows that for all $\displaystyle q,k\geq 1,$

$\displaystyle\displaystyle\ell\left(\frac{R}{{\mathfrak{n}}^{[q]k}}\right)=$
$\displaystyle\displaystyle\
\sum_{i=1}^{r}\ell\left(\frac{S}{{\mathfrak{p}}_{i}+{\mathfrak{m}}^{[q]k}}\right)-\sum_{1\leq
i<j\leq
r}\ell\left(\frac{S}{{\mathfrak{p}}_{i}+{\mathfrak{p}}_{j}+{\mathfrak{m}}^{[q]k}}\right)+\cdots+(-1)^{r-1}\
\ell\left(\frac{S}{\sum_{i=1}^{r}{\mathfrak{p}}_{i}+{\mathfrak{m}}^{[q]k}}\right)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\
rkq-\binom{r}{2}+\binom{r}{3}+\cdots+(-1)^{r-1}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\ rkq-(r-1).$

This implies that
$\displaystyle\ell(R/(J^{[q]})^{k})=rkq\prod_{i=1}^{r}v_{i}-(r-1)\prod_{i=1}^{r}v_{i}.$
In particular,
$\displaystyle\ell(R/J^{k})=rk\prod_{i=1}^{r}v_{i}-(r-1)\prod_{i=1}^{r}v_{i}.$
Hence,

$\displaystyle
e_{0}(J)-e_{1}(J)=r\prod_{i=1}^{r}v_{i}-(r-1)\prod_{i=1}^{r}v_{i}=\ell(R/J),$

implying that $\displaystyle J$ is a stable ideal. ∎

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# Rationality of holomorphic vertex operator algebras

Masahiko Miyamoto 111Partially supported by the Grants-in-Aids for Scientific
Research, No.21K03195 and 18K18708, The Ministry of Education, Science and
Culture in Japan and Academia Sinica in Taiwan Institute of Mathematics,
University of Tsukuba

###### Abstract

We prove that if $V$ is a unitary simple holomorphic vertex operator algebra
of CFT-type, then $V$ is rational, that is, all ${\mathbb{N}}$-gradable
$V$-modules are direct sums of copies of $V$.

##  1 Introduction

A concept of vertex operator algebras (shortly VOA) was introduced by
Borcherds [1] to solve the moonshine conjecture in finite group theory. It is
now considered to be a rigorous algebraic definition of a chiral algebra in
conformal field theory. In the last two decades, there has been quite good
progress on the classification of holomorphic VOAs $V$ of CFT-type and with
central charge $24$ (for example, see [2] and [6]). In these works, to use
modular invariance properties and to get the fact that weight one space is a
reductive Lie algebra, most papers were assuming strongly regular conditions
on $V$, that is, positivity (or unitarity) of an invariant bilinear form,
rationality (i.e., semisimplicity for ${\mathbb{N}}$-gradable modules), and
$C_{2}$-cofiniteness on $V$ (or finiteness of simple (weak) modules). For the
definition of unitarity, see §2. Compared with the classification of
unimodular positive definite even lattices [7], it is natural to assume
positivity over ${\mathbb{R}}$ (or unitarity over ${\mathbb{C}}$) for an
invariant bilinear form on $V$, but is it necessary to assume the rationality
and $C_{2}$-cofiniteness? For some holomorphic VOAs of CFT-type with some
stronger assumptions, their rationalities are already known, see [4]. In this
paper, we will show that rationality comes from only the positivity (or
unitary) of an invariant bilinear form of $V$ for a holomorphic VOA $V$ of
CFT-type without any additional assumptions. We note that every holomorphic
VOA has a symmetric invariant bilinear form $\langle\,,\,\rangle$ by [5].

###### Theorem 1

If $V=\coprod_{n=0}^{\infty}V_{n}$ is a unitary simple holomorphic VOA of CFT-
type, then $V$ is rational, that is, all ${\mathbb{N}}$-gradable $V$-modules
are direct sums of copies of $V$, where holomorphic means that $V$ itself is
the only one simple ${\mathbb{N}}$-gradable $V$-module.

The author expresses great thanks to T. Abe, C. H. Lam, and H. Yamauchi for
suggesting many improvements to this note and also checking the proof. He
thanks the staff of Academia Sinica in Taiwan for their hospitality. The
author also expresses thanks to the referee for his/her advice.

##  2 Two propositions and preliminary results

Throughout this paper, $(V,Y,{\bf 1},\omega)$ denotes a simple holomorphic VOA
over ${\mathbb{C}}$. In this paper, a $V$-module $M$ is called
${\mathbb{N}}$-gradable if there is an ${\mathbb{N}}$-grading structure
$M=\oplus_{m=0}^{\infty}M_{(m)}$ such that $v_{m}(M_{(n)})\subseteq M_{(n+{\rm
wt}(v)-m-1)}$ for all $v\in V_{{\rm wt}(v)}$. We call $M_{(0)}$ the top level
of $M=\oplus_{m=0}^{\infty}M_{(m)}$. We also use ${\rm wt}(u)$ to denote an
eigenvalue of $L(0)$ on an eigenvector $u\in M$ and $M_{t}$ denotes the
eigenspace of $L(0)$ with an eigenvalue $t$ for a $V$-module $M$. For a simple
${\mathbb{N}}$-gradable module $M$, it is easy to see that there is
$-t\in{\mathbb{C}}$ such that $M_{(n)}=M_{n-t}$ for all $n$. Let $A(V)$ be the
Zhu algebra of a VOA $V$ introduced by Zhu [9]. Then for an
${\mathbb{N}}$-gradable $V$-module $M=\oplus_{m=0}^{\infty}M_{(m)}$, $M_{(0)}$
is an $A(V)$-module. One of the crucial results of Zhu’s theory we will use is
that every simple $A(V)$-module is a top level of an ${\mathbb{N}}$-gradable
simple $V$-module. Furthermore, for any $A(V)$-module $Z$ and a $V$-module $M$
generated from $Z$, we can define a canonical ${\mathbb{N}}$-grading
$M=\coprod_{m=0}^{\infty}M_{(m)}$ on $M$ such that the top level $M_{(0)}$ of
$M$ is $Z$. In particular, since $A(V)$ is a an $A(V)$-module, if $V$ is
rational, then $A(V)$ is a direct sum of simple $A(V)$-modules. Namely, $A(V)$
is semisimple.

If $V$ is simple and holomorphic, then $V=\coprod_{n=-t}^{\infty}V_{n}$ (with
$\dim V_{-t}\not=0$) is the only simple ${\mathbb{N}}$-gradable module and
$A(V)/J(A(V))={\rm End}(V_{-t})\cong{\rm Mat}_{s,s}({\mathbb{C}})$, where
$s=\dim V_{-t}$ and $J(A(V))$ denotes the Jacobson radical of $A(V)$. We note
that if $A(V)$ is Artinian, then most results in this section are well-known,
but we have to treat the other cases, for example, the case where $\dim
J(A(V))=\infty$. We will find that $J(A(V))=0$ implies the rationality of $V$
(see Lemma 2). Therefore, we study $J(A(V))$ for a while. Since
$A(V)/J(A(V))\cong{\rm End}(V_{-t})$, $J(A(V))$ is the unique maximal both-
side ideal and is the intersection of all maximal left ideals and coincides
with the intersection of all maximal right ideals. In particular, for
$\alpha\in J(A(V))$, $1-\alpha$ is a unit element of $A(V)$. As an
application, we have the following elementary lemma:

###### Lemma 1

For an ${\mathbb{N}}$-graded $V$-module $M=\coprod_{m=0}^{\infty}M_{(m)}$ and
any $0\not=a\in M_{(0)}$, we have $J(A(V))a\not=A(V)a$.

[Proof] Suppose false, then there is $\alpha\in J(A(V))$ such that $\alpha
a=a$, that is, $(1-\alpha)a=0$, which contradicts the fact that $1-\alpha$ is
a unit element. $\square$

We recall the notion of unitary VOAs from [3].

######  Definition 1 ([3])

Let $(V,Y,{\bf 1},\omega)$ be a VOA over ${\mathbb{C}}$ and let $\phi:V\to V$
be an anti-linear involution of $V$ $($i.e, $\phi(\lambda
v)=\bar{\lambda}\phi(u)$, $\phi({\bf 1})={\bf 1}$, $\phi(\omega)=\omega$,
$\phi(u_{n}v)=\phi(u)_{n}\phi(v)$ for any $u,v\in V$, $n\in{\mathbb{Z}}$, and
$\phi$ has order $2$ $)$. Then $(V,\phi)$ is said to be unitary if there
exists a positive-definite Hermitian form $(\cdot,\cdot)_{V}:V\times
V\to{\mathbb{C}}$, which is ${\mathbb{C}}$-linear on the first vector and
anti-${\mathbb{C}}$-linear on the second vector, such that the following
invariant property holds of any $a,u,v\in V$:

$(Y(e^{zL(1)}(-z^{-2})^{L(0)}a,z^{-1})u,v)_{V}=(u,Y(\phi(a),z)v)_{V},$ $None$

where $L(n)$ is defined by $Y(\omega,z)=\sum_{n\in{\mathbb{Z}}}L(n)z^{-n-2}$.

In our case, since $V$ is a holomorphic VOA, $V$ has a non-singular invariant
bilinear form $\langle\cdot,\cdot\rangle$ satisfying (2.1) by [5]. Hence if we
put $(u,v)_{V}=\langle u,\phi(v)\rangle$ for $v,u\in V$, then
$(\cdot,\cdot)_{V}$ becomes a desired Hermitian form. Therefore, $V$ is
unitary if and only if there is an anti-linear involution $\phi$ of $V$ such
that $\langle v,\phi(v)\rangle>0$ for all $0\not=v\in V$. We fix an anti-
linear involution $\phi$ of $V$.

######  Definition 2 ([4])

Let $M$ be a (weak) $V$-module. We call an element $\rho\in M$ a vacuum-like
element if $L(-1)\rho=0$.

If $\rho$ is a vacuum-like element, we can prove $v_{n}\rho=0$ for $n\geq 0$
by induction. Because, by the definition of a (weak) $V$-module, there is $m$
such that $v_{s}\rho=0$ for $s\geq m$. Then $0=L(-1)v_{m}\rho=-mv_{m-1}\rho$
and so we have $v_{n}\rho=0$ for $n\geq 0$. In this case,
$V\\!\cdot\\!\rho=\\{v_{-1}\rho\mid v\in V\\}$ becomes a $V$-module which is
isomorphic to $V$, see Lemma 9.6 in [4]. Here and there, we use a notation
$V\\!\cdot\\!S$ to denote a $V$-submodule generated from a subset $S$ of a
$V$-module.

In this section, we will show the following propositions in a general setting,
which will play important roles in the proof of Theorem 1.

###### Proposition 1

Assume that a VOA $V$ of CFT-type has a non-split extension

$0\to V\cong W\to\tilde{V}\xrightarrow{\psi}V\to 0$

of $V$-modules with a $V$-homomorphism $\psi$ and $W\cong V$. Set
$\tilde{V}_{[n]}=\\{\tilde{v}\in\tilde{V}\mid\exists m\in{\mathbb{N}}\mbox{
such that }(L(0)-n)^{m}\tilde{v}=0\\}$ the genelarized eigenspace of $L(0)$
with eigenvalue $n$. Let $\tilde{{\bf 1}}\in\tilde{V}_{[0]}$ be an inverse
image of the vacuum element ${\bf 1}\in V$ by $\psi$ and set
$u=L(-1)\tilde{{\bf 1}}$. Then $u\not=0$ and by viewing $u\in V_{1}$, $u(0)$
acts on $V_{k}$ as a nilpotent operator for each $k$.

[Proof] We note $u\in W_{1}$. If $u=0$, then $\tilde{{\bf 1}}$ is a vacuum-
like element and $\tilde{V}$ decomposes into a direct sum of two $V$-modules
$V\cong W$ and $U=\\{v_{-1}\tilde{{\bf 1}}\mid v\in V\\}$, which contradicts
the assumption. So, we have $0\not=u\in W\cong V$ and we view $u$ as an
element of $V$. Set $P=\\{v\in V\mid\exists m\in{\mathbb{N}}\mbox{ s.t.
}u(0)^{m}v=0\\}$. Since
$u(0)^{m}(v_{k}w)=\sum\binom{m}{j}(u(0)^{m-j}v)_{k}(u(0)^{j}w)$, $P$ is a
subVA of $V$. Furthermore, since $\omega(2)u\in V_{0}={\mathbb{C}}{\bf 1}$ and
${\bf 1}(-3)=0$, we have

$\begin{array}[]{rl}u(0)\omega=&u(0)\omega(-1){\bf 1}=-[\omega(-1),u(0)]{\bf
1}\cr=&-\\{(\omega(0)u)(-1)-(\omega(1)u)(-2)+(\omega(2)u)(-3)+\\!\cdots\\}{\bf
1}\cr=&\\{u(-2)-u(-2)+0\\}{\bf 1}=0.\end{array}$

Hence, $P$ is a full subVOA of $V$. Assume $P\not=V$ and let $\rho$ be a
lowest weight homogeneous element outside of $P$. Then $P\not\ni
u(0)\rho=-\rho(0)u-\sum_{k=1}^{\infty}\frac{(-L(-1))^{k}}{k!}\rho(k)u$ by the
assumption of $\rho$. For $k\geq 1$, since ${\rm wt}(\rho(k)u)<{\rm wt}(\rho)$
and $\omega\in P$, we have $\rho(k)u\in P$ and
$\sum_{k=1}^{\infty}\frac{(-L(-1))^{k}}{k!}\rho(k)u\in P$. Therefore, we have
$\rho(0)u\not\in P$. Viewing it in $V=W\subseteq\tilde{V}$,
$\rho(0)u=\rho(0)\omega(0)\tilde{{\bf 1}}=\omega(0)\rho(0)\tilde{{\bf 1}}$.
However, since $\psi(\rho(0)\tilde{{\bf 1}})=0$ and $\rho(0)\tilde{{\bf 1}}\in
V_{{\rm wt}(\rho)-1}$, we have $\rho(0)\tilde{{\bf 1}}\in P$. Since $P$ is a
full subVOA, we have $\rho(0)u=L(-1)\rho(0)\tilde{{\bf 1}}\in P$, which is a
contradiction. Therefore, $P=V$ and $u(0)$ acts on $V_{k}$ nilpotently for
every $k\in{\mathbb{Z}}$. $\square$

Since $V$ is of CFT-type, $v(0)\tilde{{\bf 1}}\in W_{0}=V_{0}$ for all $v\in
V_{1}$. Hence we have

$v(0)u=v(0)L(-1)\tilde{{\bf 1}}\in L(-1){\mathbb{C}}{\bf 1}=0$

for all $v\in V_{1}$. In particular, $u\in Z(V_{1})$, where $Z(V_{1})$ denotes
the center of Lie algebra $V_{1}$. Therefore, $u,\phi(u)\in Z(V_{1})$.

Since $\langle u,\phi(u)\rangle>0$, we have $\langle
u+\phi(u),u+\phi(u)\rangle>0$ or $\langle u-\phi(u),u-\phi(u)\rangle>0$. We
can choose $w\in\\{{\mathbb{C}}(u\pm\phi(u))\\}$ such that $\langle
w,w\rangle=1$. Since $[u(0),\phi(u)(0)]=0$, $w(0)$ also acts on $V_{k}$
nilpotently.

Let ${\rm VA}(w)$ be a vertex subalgebra generated by $w$. Then
$\omega^{1}=\frac{w(-1)w}{2}$ becomes a conformal element of ${\rm VA}(w)$. By
the relation $[w(m),w(n)]=\delta_{m+n,0}$, we have ${\rm VA}(w)\cong M(1)$ as
VOAs, where $M(1)$ denotes a VOA of free-boson type with central charge $1$.

###### Proposition 2

Under the assumptions in Proposition 1, if $V$ is a unitary VOA, then $V={\rm
VA}(w)\otimes{\rm Com}({\rm VA}(w),V)$, where ${\rm Com}({\rm
VA}(w),V)=\\{v\in V\mid\alpha(n)v=0\mbox{ for all }\alpha\in{\rm VA}(w)\mbox{
and }n\geq 0\\}$. In particular, $V$ is not holomorphic.

[Proof] Set $Q=\\{v\in V\mid w(0)v=0\\}$. As we explained in the proof of
Proposition 1, $Q$ is a subVOA and $\phi$-invariant, since $\phi(w)\in\pm w$.
Since $V$ has a positive definite form $(,)_{V}$, $Q^{\perp}$ is also
$w(0)$-invariant . Since $w(0)$ acts on $(Q^{\perp})_{k}$ nilpotently for
$k\in{\mathbb{Z}}$, if $Q^{\perp}\not=0$, then $Q^{\perp}$ contains a nonzero
element $v$ such that $w(0)v=0$. However, since $(,)_{V}$ is positive definite
and $0\not=v\in Q\cap Q^{\perp}$, we have a contradiction. Hence, we have
$V=Q$, that is, $w(0)$ acts on $V$ as zero.

Since $w(0)$ acts on $V$ as zero and $V$ has an invariant bilinear form
$\langle,\rangle$, $V$ is a direct sum of simple $M(1)$-modules $J$ isomorphic
to $M(1)$, that is, there is $b\in V$ with $\omega^{1}(1)b=0$ such that
$J=M(1)b$. Since $b\in{\rm Com}({\rm VA}(w),V)$, we have $V\cong{\rm
VA}(w)\otimes{\rm Com}({\rm VA}(w),V)$. We next show that
$\omega-\omega^{1}\in{\rm Com}({\rm VA}(w),V)$, where
$\omega^{1}=\frac{w(-1)w}{2}$. Then $w(0)\omega^{1}=w(2)\omega^{1}=0$ and
$w(1)\omega^{1}=w$. On the other hand, $w(1)\omega=\omega_{1}w=w$. Since
$\omega(2)w\in{\mathbb{C}}{\bf 1}$, there is $r\in{\mathbb{C}}$ such that
$\omega(2)w=r{\bf 1}$. Then $r=\langle\omega(2)w,{\bf 1}\rangle=\langle
w,\omega(0){\bf 1}\rangle=0$ and so $r=0$, that is, $\omega(2)w=0$. Therefore,
$w(m)(\omega-\omega^{1})=0$ for all $m\geq 0$ and $\omega-\omega^{1}\in{\rm
Com}({\rm VA}(w),V)$. Since $M(1)$ is not holomorphic, neither is $V$.
$\square$

##  3 Proof of Theorem 1

We first prove the following result.

###### Proposition 3

If $V$ is a simple holomorphic irrational VOA of CFT-type, then there are an
indecomposable $V$-module $\tilde{V}$ and its submodule $W$ such that
$\tilde{V}/W\cong V\cong W$ as $V$-modules.

[Proof] We start the proof of Proposition 3 by the following case:

###### Lemma 2

Let $V=\coprod_{m=-t}^{\infty}V_{m}$ be a simple holomorphic VOA of CFT-type
and assume that $A(V)$ is semisimple, then all ${\mathbb{N}}$-gradable
$V$-modules $M=\coprod_{n=0}^{\infty}M_{(n)}$ are semisimple. In particular,
$V$ is rational.

[Proof] We note that since $V$ is a holomorphic VOA of CFT-type and $A(V)$ is
semisimple, $A(V)={\rm End}(V_{0})\cong{\mathbb{C}}$ and $\bar{\omega}=0$,
where $\bar{\omega}$ is the image of $\omega$ in $A(V)$. Let
$M=\coprod_{n=0}^{\infty}M_{(n)}$ be an ${\mathbb{N}}$-gradable $V$-module.
Using the notation in [4], $\Omega(M)$ denotes the set of singular elements of
a $V$-module $M$, that is, $\Omega(M)=\\{w\in M\mid v_{m}w=0\mbox{ for all
}m\geq{\rm wt}(v)\mbox{ and }v\in V\\}$. Since $A(V)$ acts on $\Omega(M)$,
$L(0)$ acts on $\Omega(M)$ as zero. Furthermore, since $0$ is the lowest
eigenvalue of $L(0)$ on a $V$-module, $\Omega(M)$ coincides with the
generalized eigen subspace of $L(0)$ in $M$ with eigenvalue $0$.

Since $A(V)\cong{\mathbb{C}}$, $\Omega(M)$ is a direct sum of copies of a
simple $A(V)$-module $V_{0}\cong{\mathbb{C}}$ and furthermore a submodule
$V\\!\cdot\\!\Omega(M)$ of $M$ generated from $\Omega(M)$ is a direct sum of
copies of $V$-module $V$. We will show $M\subseteq V\\!\cdot\\!\Omega(M)$.
Clearly, $M_{(0)}\subseteq\Omega(M)$ and so $V\\!\cdot\\!M_{(0)}\subseteq
V\\!\cdot\\!\Omega(M)$. If $M_{(1)}\not\subseteq V\\!\cdot\\!M_{(0)}$, then
$M_{(1)}/(V\\!\cdot\\!M_{(0)}\cap M_{(1)})$ is a top level of
$V\\!\cdot\\!M_{(1)}/(V\\!\cdot\\!M_{(0)}\cap V\\!\cdot\\!M_{(1)})$. Therefore
$A(V)$ acts on $M_{(1)}/(V\\!\cdot\\!M_{(0)}\cap M_{(1)})$. We hence have
$(L(0))M_{(1)}\subseteq V\\!\cdot\\!M_{(0)}\cap M_{(1)}$. Since the
eigenvalues of $L(0)$ is $1$ on $V\\!\cdot\\!M_{(0)}\cap M_{(1)}$ and $0$ on
$M_{(1)}/(V\\!\cdot\\!M_{(0)}\cap M_{(1)})$, we have $M_{(1)}=(\Omega(M)\cap
M_{(1)})\oplus(V\\!\cdot\\!M_{(0)}\cap M_{(1)})$. In particular,
$M_{(0)}\oplus M_{(1)}\subseteq V\\!\cdot\\!\Omega(M)$ and the eigenvalues of
$L(0)$ on $M_{(1)}$ are $\\{0,1\\}$ at most. Therefore the eigenvalues of
$L(0)$ on $M_{(2)}\cap V\\!\cdot\\!(M_{(0)}+M_{(1)})$ are $\\{1,2\\}$ at most
and $0$ on $M_{(2)}/(M_{(2)}\cap V\\!\cdot(M_{(0)}+M_{(1)}))$. Iterating these
steps, we have $M_{(m)}\subseteq
V\\!\cdot\\!\Omega(M)+V\\!\cdot\\!\sum_{j=0}^{m-1}M_{(j)}$ for all $m$ and the
eigenvalues of $L(0)$ on $M_{(m)}$ are $\\{0,1,...,m\\}$ at most. We hence
have $M\subseteq V\\!\cdot\\!\Omega(M)$. Since we have already shown that
$V\\!\cdot\\!\Omega(M)$ is a direct sum of simple modules, $M$ is also a
direct sum of simple modules. $\square$

So if $V$ is a simple holomorphic irrational VOA of CFT-type, then $A(V)$ is
not semisimple, that is, $J(A(V))\not=0$.

###### Lemma 3

Let $R$ be an associative algebra with an identity $1$ and $J(R)$ a Jacobson
radical, that is, the intersection of all maximal left ideals and assume
$R/J(R)\cong{\rm Mat}_{s,s}({\mathbb{C}})$. If $T\subseteq J(R)$ is a left
ideal of $R$ such that $J(R)/T$ has a complement in $R/T$, then $T=J(R)$.

[Proof] If $R/T=M/T\oplus J(R)/T$, then $R=M+J(R)$ and so there are $m\in M$
and $p\in J(R)$ such that $1=m+p$. In this case, $m=1-p\in M$ is a unit of $R$
and so $M=R$ and $T=J(R)$. $\square$

Viewing $A(V)$ as an $A(V)$-module, set
$M=V\\!\cdot\\!A(V)=\coprod_{m=0}^{\infty}M_{(m)}$ with $M_{(0)}=A(V)$. Then
we have a non-split exact sequence

$0\to V\\!\cdot\\!J(A(V))\to M\xrightarrow{\psi}V\\!\cdot\\!(A(V)/J(A(V)))\to
0.$

We note $A(V)/J(A(V)\cong{\mathbb{C}}$ and $V\\!\cdot\\!(A(V)/J(A(V)))\cong V$
as $V$-modules. Hence $\psi(M_{(k)})=V_{k}$ for $k\in{\mathbb{Z}}$. Let
$\bar{{\bf 1}}$ be the vacuum element of $V\\!\cdot\\!(A(V)/J(A(V)))$ and let
$\tilde{{\bf 1}}$ be an inverse image of $\bar{{\bf 1}}$ in $M_{(0)}$ and set
$u=L(-1)\tilde{{\bf 1}}$. Let $W$ be a submodule of $V\\!\cdot\\!J(A(V))$
containing $u$. Then, the image in $M/W$ of $\tilde{{\bf 1}}$ is a vacuum-like
element and so $V\\!\cdot\\!(A(V)/J(A(V)))$ becomes a direct summand of $M/W$.
In particular, $A(V)/J(A(V))$ is a direct summand of $M_{(0)}/W_{(0)}$. By the
above lemma, we have $W=V\\!\cdot\\!J(A(V))$. In particular,
$V\\!\cdot\\!u=V\\!\cdot\\!J(A(V))$. Let ${\cal F}$ be the set of
$V$-submodules of $V\\!\cdot\\!J(A(V))$ that does not contain $u$. By Zorn’s
lemma, ${\cal F}$ has a maximal element, say, $Q$. Then
$V\\!\cdot\\!J(A(V))/Q=V\\!\cdot\\!\bar{u}$ and $V\\!\cdot\\!\bar{u}$ is a
simple module, where $\bar{u}$ is the image of $u$ in $M/Q$. Since $M/Q$ is
indecomposable module and $M/V\\!\cdot\\!J(A(V))\cong V$, we have a desired
indecomposable $V$-module.

This completes the proof of Proposition 3. $\square$

We now start the proof of Theorem 1. If $V$ is a simple holomorphic irrational
VOA of CFT-type, Then, $V$ satisfies the assumptions in Propositions 1 and 2
by Proposition 3. Therefore, $V$ is not holomorphic, which contradicts the
assumption.

This completes the proof of Theorem 1.

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