# A Volume = Multiplicity formula for $p$-families of ideals

Sudipta Das  1290 Frenger Mall, Las Cruces, New Mexico 88003-8001
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we work with certain families of ideals called $p$-families in
rings of prime characteristic. This family of ideals is present in the
theories of tight closure, Hilbert-Kunz multiplicity, and $F$-signature. For
each $p$-family of ideals, we attach a Euclidean object called $p$-body, which
is analogous to the Newton Okounkov body associated with a graded family of
ideals. Using the combinatorial properties of $p$-bodies and algebraic
properties of the Hilbert-Kunz multiplicity, we establish in this paper a
Volume = Multiplicity formula for $p$-families of $\mathfrak{m}_{R}$-primary
ideals in a Noetherian local ring $R$.

###### 2020 Mathematics Subject Classification:

Primary 13A18, 13D40, 13H15

##  1\. Introduction

Let $(R,\mathfrak{m},\mathbb{K})$ be a Noetherian local ring of dimension $d$,
with prime characteristic $p>0$, $I$ be an $\mathfrak{m}$-primary ideal and
$q=p^{e}$ for some $e\in\mathbb{N}$, and $I^{[q]}=\left(x^{q}\mid x\in
I\right)$, the $q$-th Frobenius power of $I$. We denote the
$\mathfrak{m}$-adic completion of $R$ by $\hat{R}$ and $\ell_{R}(-)$ denotes
the length as an $R$-module.

The Hilbert-Kunz theory deals with the question of how
$\ell_{R}\left(R/I^{[q]}\right)$ behaves as a function on $q$ and how
understanding this behavior leads us to have a better understanding of the
singularities of the ring. In his work, Kunz [9] introduced this study to
measure how close the ring $R$ is to be regular. Later P. Monsky [11] showed
that

$e_{HK}(I,R)=\lim_{e\to\infty}\dfrac{\ell_{R}(R/I^{[q]})}{q^{d}}$

exists for any $\mathfrak{m}$-primary ideal $I$, this positive real number is
called the Hilbert-Kunz Multiplicity of $I$. Contrary to the Hilbert-Samuel
multiplicity of an $\mathfrak{m}$-primary ideal $I$, which is given by
$e(I,R)=\lim_{k\to\infty}\dfrac{\ell_{R}(R/I^{k})}{k^{d}/d!}$, the Hilbert-
Kunz Multiplicity demonstrates much more complicated behavior [6]

In this paper, we work with certain families of ideals called $p$-families. A
$p$-family of ideals is a sequence of ideals
$I_{\bullet}=\\{I_{q}\\}_{q=1}^{\infty}$ with $I_{q}^{[p]}\subseteq I_{pq}$
for all $q$. Given a $p$-family of $\mathfrak{m}$-primary ideals, we define
the following limit as the volume of the $p$-family.

$\operatorname{Vol}_{R}(I_{\bullet})=\lim_{q\to\infty}\dfrac{\ell_{R}(R/I_{q})}{q^{d}}$

In [7], Jack Jeffries and Daniel J Hernández showed that
$\operatorname{Vol}_{R}(I_{\bullet})$ exists as a limit if and only if
$\operatorname{dim}N(\hat{R})<d$, where $N(\hat{R})$ denotes the nilradical of
$\hat{R}$. This provides an alternating proof of the existence of Hilbert Kunz
multiplicity and $F$-signature for a complete local domain ([7], Corollary
6.1,6.8).

This article aims to prove the following general Volume = Multiplicity formula
for $p$-families of $\mathfrak{m}$-primary ideals.

Theorem 5.3. Let $(R,\mathfrak{m},\mathbb{K})$ be a $d$-dimensional local ring
of characteristic $p>0$. If $\operatorname{dim}N(\hat{R})<d$, then for any
$p$-family of $\mathfrak{m}$-primary ideals
$I_{\bullet}=\\{I_{q}\\}_{q=1}^{\infty}$

$\operatorname{Vol}_{R}(I_{\bullet})=\lim_{q\to\infty}\frac{\ell_{R}(R/I_{q})}{q^{d}}=\lim_{q\to\infty}\frac{e_{HK}(I_{q},R)}{(q)^{d}}$

To prove our main result, we have followed the approach of Cutkosky’s
existence theorem ([2] [3]), which can be outlined as follows:

(1) Reduce to the case of a complete local domain.

(2) Using a suitable valuation on this complete local domain $R$, we attach to
every $p$-family of ideals a combinatorial structure, called $p$-system of
ideals (see Definition 3.11) and associate an Euclidean object called $p$-body
(see Definition 3.12).

(3) To compute the relevant $R$-module lengths, we prove an approximation
Theorem for $p$-systems of ideals. Since Lazarsfeld and Mustaţă in their paper
[10, Proposition 3.1] have proved similar approximation theorem and using it
recovered the Fujita Approximation Theorem [10, Theorem 3.3], we call it
Fujita Type Approximation Theorem for $p$-systems of ideals (see Theorem
3.18).

Although the Hilbert-Kunz multiplicity behaves much differently compared to
the Hilbert-Samuel multiplicity, a parallel analog of Theorem 5.3 for every
graded family $J_{\bullet}$ = $\\{J_{n}\\}_{n\in\mathbb{N}}$ (i.e., a sequence
of ideals $J_{\bullet}$ such that $J_{m}.J_{n}\subseteq J_{m+n}$) of
$\mathfrak{m}$-primary ideals and Hilbert-Samuel multiplicities has been
proven for valuation ideals associated to an Abhyankar valuation in a regular
local ring which is essentially of finite type over a field in [5] when $R$ is
a local domain which is essentially of finite type over an algebraically
closed field $\mathbb{K}$ with $R/\mathfrak{m}=\mathbb{K}$ in [10], it is
proven when $R$ is analytically unramified with perfect residue field in [3].
Finally in [4] Cutkosky settled this for any $d$-dimensional Noetherian local
ring $R$ with $\operatorname{dim}N(\hat{R})<d$.

##  2\. Preliminaries

This section reviews a few basic notions from semigroup theory, convex
geometry, and valuation theory.

Let $U$ and $V$ be arbitrary subsets of $\mathbb{R}^{d}$. If $U$ is Lebesgue
measurable, its measure $\operatorname{Vol}_{\mathbb{R}^{d}}(U)$ is called the
volume of $U.$ We define the Minkowski sum of two sets as $U+V:=\left\\{\
\boldsymbol{u}+\boldsymbol{v}\ |\
\boldsymbol{u}\in{U},\boldsymbol{v}\in{V}\right\\}.$

A convex cone is any subset of $\mathbb{R}^{d}$ that is closed under taking an
$\mathbb{R}$-linear combination of points with non-negative coefficients. Let
$Cone(U)\subseteq\mathbb{R}^{d}$ be the convex cone which is the closure of
the set of all linear combinations $\sum_{i}\lambda_{i}u_{i},$ with $u_{i}\in
U$ and $\lambda_{i}\in\mathbb{R}_{\geqslant 0}$. Note that a cone in
$\mathbb{R}^{d}$ has a non-empty interior if and only if the real vector space
it generates has dimension $d.$ We call such a cone full-dimensional.

A cone $C$ is pointed if it is closed and if there exists a vector
$\boldsymbol{a}\in{\mathbb{R}^{d}}$ such that
$(\boldsymbol{u},\boldsymbol{a})>0$, for all $\boldsymbol{u}\in
C\setminus\\{\boldsymbol{0}\\}$, where $(-,-)$ is the inner product in
$\mathbb{R}^{d}$.

If $C$ is a pointed cone and $\alpha$ is a non-negative real number, we
define:

$H=H_{\alpha}:=\left\\{\ \boldsymbol{u}\in{\mathbb{R}^{d}}\ |\
(\boldsymbol{u},\boldsymbol{a})<\alpha\ \right\\}$

a truncating half-space for $C.$

######  Remark 2.1.

Let $C$ be a pointed cone. If $\alpha>0,$ then a truncation of $C$ is defined
as $C\cap H_{\alpha}$. It is a non-empty, bounded subset of $C$.

######  Remark 2.2.

If $H$ is any truncating halfspace of a pointed cone $C$ then so is $qH$, for
all $q>0$.

We define a semigroup to be any subset of $\mathbb{Z}^{d}$ that contains
$\boldsymbol{0}$ and it is closed under addition; e.g. $\mathbb{N}$ is the
semigroup of non-negative integers. A semigroup $S$ is finitely generated if
there exist a finite subset $S_{0}\subseteq S$ such that every element of $S$
can be written as an $\mathbb{N}$-linear combination of elements of $S_{0}.$ A
subset $T\subseteq S$ of a semigroup $S,$ is an ideal of $S$ whenever $S+T$ is
contained in $T.$ A semigroup $S$ is called pointed if $\boldsymbol{a}\in S$
and $\boldsymbol{-a}\in S$ then $\boldsymbol{a}$ is $\boldsymbol{0}$.

######  Remark 2.3.

If $\operatorname{Cone}(S)$ is pointed, then so is $S$, but the converse is
generally false. However, it does hold if $S$ is assumed to be finitely
generated.

######  Remark 2.4.

For any $\mathbb{Z}$-linear embedding
$i:\mathbb{Z}^{d}\hookrightarrow\mathbb{R}$, there exists some unique
$\boldsymbol{a}\in\mathbb{R}^{d}$, whose coordinates are linearly independent
over $\mathbb{Q}$, such that
$i(\boldsymbol{u})=(\boldsymbol{a},\boldsymbol{u})$ for all
$u\in\mathbb{Z}^{d}$. We define
$\boldsymbol{u}\leqslant_{\boldsymbol{a}}\boldsymbol{v}$ whenever
$(\boldsymbol{a},\boldsymbol{u})\leqslant(\boldsymbol{a},\boldsymbol{v})$.

Let $\mathbb{F}^{\times}=\mathbb{F}\setminus\left\\{0\right\\}$. Fix,
$\boldsymbol{a}\in{\mathbb{R}^{d}},$ and fix the embedding
$\mathbb{Z}^{d}\hookrightarrow\mathbb{R}$ given by
$\boldsymbol{v}\to(\boldsymbol{a},\boldsymbol{v}),$ defined by
$\boldsymbol{a}$. An $\boldsymbol{a}$-valuation on a field $\mathbb{F}$ with
value group $\mathbb{Z}^{d}$ is a surjective group homomorphism:
$\vartheta:\mathbb{F}^{\times}\rightarrow\mathbb{Z}^{d}$ with the property
that

$\displaystyle\vartheta(xy)=\vartheta(x)+\vartheta(y),$
$\displaystyle\vartheta(x+y)\geqslant_{\boldsymbol{a}}\mathrm{min}\left\\{\vartheta(x),\vartheta(y)\right\\}.$

For any subset $N\subseteq\mathbb{F},$ with
$N^{\times}=N\setminus\left\\{0\right\\}$ we define
$\vartheta(N):=\vartheta(N^{\times}),$ and this is called the image of $N$
under $\vartheta.$
Given a point $\boldsymbol{u}\in{\mathbb{Z}^{d}},$ we define:

$\displaystyle\mathbb{F}_{\geqslant\boldsymbol{u}}:=\left\\{\
x\in{\mathbb{F}}\ |\ \vartheta(x)\geqslant_{\boldsymbol{a}}\boldsymbol{u}\
\right\\}\cup\left\\{0\right\\}\quad\text{and}\quad\mathbb{F}_{>\boldsymbol{u}}:=\left\\{\
x\in{\mathbb{F}}\ |\ \vartheta(x)>_{\boldsymbol{a}}\boldsymbol{u}\
\right\\}\cup\left\\{0\right\\}$

The local ring of $\vartheta$ is the subring
$(V_{\vartheta},m_{\vartheta},k_{\vartheta})$ of $\mathbb{F},$ such that
$V_{\vartheta}=\mathbb{F}_{\geqslant 0},$ and
$m_{\vartheta}=\mathbb{F}_{>\boldsymbol{0}}.$ A local domain $(D,m,k)$ is
dominated by $(V_{\vartheta},m_{\vartheta},k_{\vartheta}),$ if $(D,m,k)$ is a
local subring of $(V_{\vartheta},m_{\vartheta},k_{\vartheta})$, i.e.,
$\mathfrak{m}\subseteq\mathfrak{m}_{\vartheta}$.

######  Remark 2.5.

If $\boldsymbol{u}\in{\mathbb{Z}^{d}},$ then both
$\mathbb{F}_{\geqslant\boldsymbol{u}}$ and $\mathbb{F}_{>\boldsymbol{u}}$ are
modules over $V_{\vartheta}$ (as $V_{\vartheta}=\mathbb{F}_{\geqslant 0}$ and
$\vartheta(xy)=\vartheta(x)+\vartheta(y)$) and
$\mathbb{F}_{\geqslant\boldsymbol{u}}/\mathbb{F}_{>\boldsymbol{u}}$ is a
vector space over $k_{\vartheta}.$ Note that, if
$\boldsymbol{w}=\vartheta(f),$ for some non-zero $f\in{\mathbb{F}},$ then:

$\mathrm{dim}_{k_{\vartheta}}\left(\mathbb{F}_{\geq\boldsymbol{w}}/\mathbb{F}_{>\boldsymbol{w}}\right)=1.$

Indeed if $f$ is as above then for any $g$ such that
$\vartheta(g)=\boldsymbol{w},$ we have $g=\left(\dfrac{g}{f}\right)\cdot f$,
where $\dfrac{g}{f}\in k_{\vartheta}$ because
$\vartheta\left(\dfrac{g}{f}\right)=0$.

######  Remark 2.6.

Let $(D,m,k)$ be a local domain of dimension $d$ with fractional field
$\mathbb{F},$ and let
$\vartheta:\mathbb{F}^{\times}\twoheadrightarrow\mathbb{Z}^{d}$ be an
$\boldsymbol{a}$-valuation with value group $\mathbb{Z}^{d}.$ Note that
$S=\vartheta(D)$ is a subsemigroup of $\mathbb{Z}^{d},$ and the image of any
ideal $I$ in $D$ is a semigroup ideal of $S.$ Moreover as $\vartheta$ is
surjective, the Minkowski sum $S+(-S)$ is equal to $\mathbb{Z}^{d},$ as
$\vartheta$ is surjective. Therefore, the real vector space generated by
$\operatorname{Cone}(S)$ is $\mathbb{R}^{d}$, i.e., $\operatorname{Cone}(S)$
is full dimensional.

Observe that $D$ is dominated by $V_{\vartheta}$ if and only if for all
$\boldsymbol{x}\in{\vartheta(m)},$ we have
$\boldsymbol{x}>_{\boldsymbol{a}}\boldsymbol{0}$ and
$S=\vartheta(m)\cup\left\\{\boldsymbol{0}\right\\}$, i.e., any non-zero
$\boldsymbol{u}\in{S},$ satisfies $(\boldsymbol{a},\boldsymbol{u})>0.$

######  Definition 2.7.

Following Remark 2.6, if $D$ is dominated by $V_{\vartheta}$ and
$(\boldsymbol{a},\boldsymbol{u})>0$, for every
$\boldsymbol{u}\in{\operatorname{Cone}(S)\setminus\left\\{\boldsymbol{0}\right\\}}$,
then we say $D$ is strongly dominated by $\vartheta$.

##  3\. OK-valuation, OK-domain, $p$-systems and $p$-bodies

In this section, we review the constructions of OK-valuation, OK-domain,
$p$-systems, and $p$-bodies following the work of Cutkosky ([2],[3]) and
Jeffries and Hernández [7]. The main result of this section is Theorem 3.18.

######  Definition 3.1.

Let $(D,m,k)$ be a $d$-dimensional local domain with fractional field
$\mathbb{F}$. Fix the embedding $\mathbb{Z}^{d}\hookrightarrow\mathbb{R}$
defined by $\boldsymbol{a}\in\mathbb{R}^{d}.$
If an $\boldsymbol{a}$-valuation

$\vartheta:\mathbb{F}^{\times}\to\mathbb{Z}^{d}$

with value group $\mathbb{Z}^{d},$ and local ring
$(V_{\vartheta},m_{\vartheta},k_{\vartheta})$ satisfies the following
conditions:
i. $D$ is strongly dominated by $V_{\vartheta}$, (see Definition 2.7)
ii. the resulting extension of residue fields $k\hookrightarrow k_{\vartheta}$
is finite, and
iii. there exists a point $\boldsymbol{v}$ $\in{\mathbb{Z}^{d}}$ such that:

$D\cap\mathbb{F}_{\geqslant n\boldsymbol{v}}\subseteq m^{n},\ \forall
n\in{\mathbb{N}}$

then we say that $(V_{\vartheta},m_{\vartheta},k_{\vartheta})$ is OK-relative
to $D$.

######  Definition 3.2.

Let $D$ be a $d$-dimensional local domain with fraction field $\mathbb{F}$. If
there exists a valuation $\vartheta$ on $\mathbb{F}$ with value group
$\mathbb{Z}^{d}$, and valuation ring
$(V_{\vartheta},m_{\vartheta},k_{\vartheta})$ that is OK-relative to $D$, we
say that $D$ is an OK-domain.

######  Example 3.3.

Let $R=\mathbb{K}\llbracket x_{1},...,x_{d}\rrbracket$ is a power series ring
of dimension $d$, containing a field $\mathbb{K}$,
$\mathfrak{m}=(x_{1},...,x_{d})$ is the maximal ideal and $\mathbb{F}$ is the
quotient field of $R$. Take $a_{1},...,a_{d}$ be rationally independent real
numbers and without loss of generality assume $a_{i}>0$ for all $1\leqslant
i\leqslant d$. Define an ordering on $\mathbb{Z}^{d}$ by
$\boldsymbol{a}=(a_{1},...,a_{d})$.
Let $f\in R$; then we can write,

$f=\sum_{s_{i_{1},...,i_{d}}\in\mathbb{K},(i_{1},...,i_{d})\in\mathbb{N}^{d}}s_{i_{1},...,i_{d}}x_{1}^{i_{1}}...x_{d}^{i_{d}}.$

Since $\boldsymbol{a}$ has positive entries, this will imply that every subset
of $\mathbb{N}^{d}$ has a least element, i.e., there exist a well defined
least exponent vector $(i_{1}^{{}^{\prime}},...,i_{d}^{{}^{\prime}})$ among
its terms and we define,
$\vartheta(f)=(i_{1}^{{}^{\prime}},...,i_{d}^{{}^{\prime}})$.
We claim that the induced valuation
$\vartheta:\mathbb{F}^{\times}\to\mathbb{Z}^{d}$ is OK relative to $R$.
Let $S=\vartheta(R)$, therefore from the above definition it is clear that
$C=\operatorname{Cone}(S)$ is contained in the non-negative octant, then
$(\boldsymbol{u},\boldsymbol{a})>0$ for every nonzero $\boldsymbol{u}$ in $C$.
This shows that $R$ is strongly dominated by $\vartheta$, satisfying condition
i. Let $(V,\mathfrak{m}_{\vartheta},\mathbb{K}_{\vartheta})$ is the valuation
ring $\mathbb{F}_{\geqslant\boldsymbol{0}}$ with it’s maximal ideal
$\mathbb{F}_{>\boldsymbol{0}}$, then clearly
$\mathbb{K}_{\vartheta}=V/\mathfrak{m}_{\vartheta}\cong
R/\mathfrak{m}\cong\mathbb{K}$, which satisfies condition ii. Moreover, let
$a_{j}=\operatorname{max}\\{a_{1},...,a_{d}\\}$, then we claim that
$\vartheta(x_{j})=(0,..,1,0,..,0)=\boldsymbol{v}$ is the vector for condition
iii. Indeed let,

$\displaystyle f\in R\cap\mathbb{F}_{\geqslant
a\boldsymbol{v}}\implies\vartheta(f)\geqslant
a\boldsymbol{v}\implies(i_{1},...,i_{d})\geqslant(0,..,a,0,..,0)$
$\displaystyle\implies a_{1}i_{1}+...+a_{d}i_{d}\geqslant aa_{j}\implies
aa_{j}\leqslant a_{j}i_{1}+...+a_{j}i_{d}(\because
a_{j}=\operatorname{max}\\{a_{1},...,a_{d}\\})$ $\displaystyle\implies
a\leqslant i_{1}+...+i_{d}\implies R\cap\mathbb{F}_{\geqslant
a\boldsymbol{v}}\subseteq\mathfrak{m}^{a}$

This finishes the proof that every power series ring containing a field is an
OK-domain.

######  Example 3.4.

[7, Example 3.5]) This immediately implies a regular local ring $R$ containing
a field is an OK-domain.

######  Example 3.5.

[7, Corollary 3.7] An interesting class of OK-domain examples is excellent
local domains containing a field. In particular, a complete local domain
containing a field is an OK-domain.

For the rest of this section, we fix a $d$-dimensional local domain $D$ of
characteristic $p>0$ with residue field $k$ and fraction field $\mathbb{F},$ a
$\mathbb{Z}$-linear embedding of $\mathbb{Z}^{d}$ into $\mathbb{R}$ induced by
a vector $\boldsymbol{a}$ in $\mathbb{R}^{d}$, and a valuation
$\vartheta:\mathbb{F}^{\times}\twoheadrightarrow\mathbb{Z}^{d}$ that is OK
relative to $D$. We use $S$ to denote the semigroup $\vartheta(D)$ in
$\mathbb{Z}^{d}$, and $C$ to denote the closed cone in $\mathbb{R}^{d}$
generated by $S$. We follow the notation established in Definition 3.1.

######  Remark 3.6.

If $M$ is a $D$-submodule of $\mathbb{F},$ then the $D$-module structure on
$M$ induces a $k$-vector space structure on the quotient
$\dfrac{M\cap\mathbb{F}_{\geqslant\boldsymbol{u}}}{M\cap\mathbb{F}_{>\boldsymbol{u}}}$
and by definition this space is non-zero if and only if there exists an
element $x\in{M}$ with $\vartheta(x)=\boldsymbol{u}.$

######  Definition 3.7.

For a $D$-submodule $M$ of $\mathbb{F}$, we define:

$\vartheta^{(h)}(M)=\left\\{\boldsymbol{u}\in\mathbb{Z}^{d}\mid\mathrm{dim}_{k}\left(\dfrac{M\cap\mathbb{F}_{\geqslant\boldsymbol{u}}}{M\cap\mathbb{F}_{>\boldsymbol{u}}}\right)\geqslant
h\right\\},$

where $1\leqslant h\leqslant[k_{\vartheta}:k]$.

######  Remark 3.8.

Let $g\in D^{\times},$ with $\boldsymbol{v}=\vartheta(g),$ and let
$\boldsymbol{u}\in\mathbb{Z}^{d}.$ Let
$I_{\bullet}=\\{I_{q}\\}_{q=1}^{\infty}$ be a sequence of ideals indexed by
$q=p^{e}$, then the map:

$\dfrac{I_{q}\cap\mathbb{F}_{\geqslant\boldsymbol{u}}}{I_{q}\cap\mathbb{F}_{>\boldsymbol{u}}}\rightarrow\dfrac{I_{q}\cap\mathbb{F}_{\geqslant\boldsymbol{u}+\boldsymbol{v}}}{I_{q}\cap\mathbb{F}_{>\boldsymbol{u}+\boldsymbol{v}}}\
:\ [x]\mapsto[gx]$

is a $k$-linear injection for all $\boldsymbol{u}\in{\mathbb{Z}^{d}}$ and for
all $I_{q}$ in that sequence. Therefore using Definition 3.7, we have
$\vartheta^{(h)}(I_{q})+S\subseteq\vartheta^{(h)}(I_{q})$ ,i.e.,
$\vartheta^{(h)}(I_{q})$ is an ideal of $S=\vartheta(D)$ for all $1\leqslant
h\leqslant[k_{\vartheta}:k]$.

######  Lemma 3.9.

[7, Lemma 3.11] For a $D$-submodule $M$ of $\mathbb{F}$ and
$\boldsymbol{v}\in{\mathbb{Z}^{d}},$ we have:

$\ell_{D}\left(M/(M\cap\mathbb{F}_{\geqslant\boldsymbol{v}})\right)=\sum_{h=1}^{[k_{\vartheta}:k]}\\#(\vartheta^{(h)}(M)\cap
H)$

where $H=\left\\{\boldsymbol{u}\in{\mathbb{R}^{d}}\
|(\boldsymbol{a},\boldsymbol{u})<(\boldsymbol{a},\boldsymbol{v})\right\\}.$

We now discuss the notions of $p$-system and $p$-bodies.

######  Definition 3.10.

A semigroup $S$ is called standard if $S-S=\mathbb{Z}^{d}$, and the full
dimensional cone generated by $S$ in $\mathbb{R}^{d}$ is pointed. Following
the discussion in Remark 2.6 and from Definition 3.1 the main example of a
standard semigroup in $\mathbb{Z}^{d}$ is $\vartheta(D)$ associated to an
$\boldsymbol{a}$-valuation $\vartheta$ that is OK relative to some
$d$-dimensional local domain $D$.

######  Definition 3.11.

A collection of subsets $T_{\bullet}=\\{T_{q}\\}_{q=1}^{\infty}$ of a
semigroup $S$ indexed by $q=p^{e}$ satisfying
i. $T_{q}$ is an ideal of S. ($T_{q}+S\subseteq S)$ and ii. $pT_{q}\subseteq
T_{pq}$ is called a $p$-system.

######  Definition 3.12.

The $p$-body associated to a given $p$-system of ideals $T_{\bullet}$ of a
semigroup in $\mathbb{Z}^{d}$ is defined by
$\Delta(S,T_{\bullet})=\bigcup_{q=1}^{\infty}\Delta_{q}(S,T_{\bullet})$ where
$\Delta_{q}(S,T_{\bullet})=\frac{1}{q}T_{q}+\operatorname{Cone}(S)$

######  Remark 3.13.

Although we have defined the notion of $p$-body for a $p$-system of ideals in
any semigroup, in our situation, we are mostly concerned with $p$-system of
ideals in the standard semigroup. Moreover, let $q_{1}=p^{e_{1}}$ and
$q_{2}=p^{e_{2}}$ and $e_{1}\leqslant e_{2}$, then
$p^{e_{2}-e_{1}}T_{p^{e_{1}}}\subseteq T_{p^{e_{2}}}$ (using Definition 3.12)
this implies
$\frac{T_{{p}^{e_{1}}}}{p^{e_{1}}}\subseteq\frac{T_{{p}^{e_{2}}}}{p^{e_{2}}}$.
Therefore the $p$-body associated to a given $p$-system of ideals is an
ascending union of sets.

######  Example 3.14.

[7, Example 4.5] Let $T$ be any subset of $S$, define $T_{q}=T+S$, for all
$q=p^{e}$, then:
i. Clearly $T_{q}+S\subseteq T_{q}$
ii. $pT_{q}=(pT+pS)\subseteq(pT+S)\subseteq(T+(p-1)T+S)\subseteq T+S$ which is
equal to $T_{pq}$.
Therefore $T_{q}$ is a $p$-system of ideals for all $q=p^{e}$. Moreover, we
have
$\Delta_{q}(S,T_{\bullet})=\left(\frac{1}{q}T+\operatorname{Cone}(S)\right)$
and the closure of $\Delta(S,T_{\bullet})$ equals $\operatorname{Cone}(S)$.

######  Example 3.15.

Let $D$ be an OK domain for an OK valuation $\vartheta$ and assume
$S=\vartheta(D)$ is the corresponding semigroup. Take an ideal $I$ in $D$ and
define $T_{q}=q\vartheta(I)+S$. Following the same argument as in Example
3.14, one can show that $\\{T_{q}\\}_{q=1}^{\infty}$ is a $p$-system of
ideals. Moreover
$\Delta(S,T_{\bullet})=\Delta_{q}(S,T_{\bullet})=\vartheta(I)+\operatorname{Cone}(S)$.

In the previous example, one can replace $\vartheta(I)$ with any arbitrary
subset $T$ of $S$, and the corresponding $p$-body would be
$T+\operatorname{Cone}(S)$. This shows that $p$ bodies need not be convex, and
in practical applications, mostly, they are not.

######  Remark 3.16.

Following Definition 3.12 one can see that every $p$-body $\Delta$ is the
union of countably many translates of $C=\operatorname{Cone}(S)$ therefore, it
is Lebesgue measurable. If $H$ is a truncating halfspace of $C$, then the
volume of ($\Delta\cap H$) is a well-defined real number.

The following Theorem describes
$\operatorname{Vol}_{\mathbb{R}^{d}}(\Delta\cap H)$ for any truncating
halfspace $H$ of $\operatorname{Cone}(S)$. This Theorem is one of the key
ingredients in the proof of our main Theorem.

######  Theorem 3.17.

[7, Theorem 4.10] For a standard semigroup $S$ in $\mathbb{Z}^{d}$, a
$p$-system $T_{\bullet}$ in $S$, and a truncating halfspace $H$ for
$\operatorname{Cone}(S)$, we have:

$\lim_{q\to\infty}\frac{\\#(T_{q}\cap
qH)}{q^{d}}=\operatorname{Vol}_{\mathbb{R}^{d}}(\Delta(S,T_{\bullet})\cap H)$

Lazarsfeld and Mustaţă in their paper [10, Proposition 3.1] have proved an
approximation theorem similar to the following Theorem and using this in a
global setup with $D$ a big divisor on an irreducible projective variety; they
have recovered the Fujita Approximation Theorem [10, Theorem 3.3]. Since our
approximation is similar to a different set of properties on the semigroup, we
call it Fujita Type Approximation Theorem for $p$ system of ideals.

######  Theorem 3.18.

Let S be a standard semigroup in $\mathbb{Z}^{d}$. If $T_{\bullet}$ is a
$p$-system of ideals in $S$, and let $H$ is any truncating halfspace for
$\operatorname{Cone}(S)$, then for any given $\epsilon>0$, there exists
$q_{0}$ such that if $q\geqslant q_{0}$ the following inequality hold.

$\lim_{e\to\infty}\frac{\\#((p^{e}T_{q}+S)\cap
p^{e}qH)}{p^{ed}q^{d}}\geqslant\operatorname{Vol}_{\mathbb{R}^{d}}(\Delta(S,T_{\bullet})\cap
H)-\epsilon$

###### Proof.

By Example 3.15 we notice that
$T_{\bullet}^{{}^{\prime}}=\\{p^{e}T_{q}+S\\}_{e=1}^{\infty}$ is a $p$-system
of ideals. Therefore using Theorem 3.17 and Remark 2.2 we have,

(3.1) $\lim_{e\to\infty}\frac{\\#((p^{e}T_{q}+S)\cap
p^{e}qH)}{p^{ed}}=\operatorname{Vol}_{\mathbb{R}^{d}}((T_{q}+\operatorname{Cone}(S))\cap
qH)$

Now note that
$\frac{\operatorname{Vol}_{\mathbb{R}^{d}}((T_{q}+\operatorname{Cone}(S)\cap
qH)}{q^{d}}=\operatorname{Vol}_{\mathbb{R}^{d}}((\frac{T_{q}}{q}+\operatorname{Cone}(S))\cap
H)$ and from Remark 3.13 and Remark 3.16, we know that these are an ascending
union of measurable sets. Therefore,

(3.2)
$\lim_{q\to\infty}\frac{\operatorname{Vol}_{\mathbb{R}^{d}}((T_{q}+\operatorname{Cone}(S))\cap
qH)}{q^{d}}=\operatorname{Vol}_{\mathbb{R}^{d}}(\Delta(S,T_{\bullet})\cap H)$

We get Equation 3.2 by using the sub-additivity and continuity from below
properties of the Lebesgue measure. Now combining Equation 3.1, Equation 3.2
for a chosen $\epsilon>0$, we can find $q_{0}$ such that for all $q\geqslant
q_{0}$
$\lim_{e\to\infty}\frac{\\#((p^{e}T_{q}+S)\cap
p^{e}qH)}{p^{ed}q^{d}}\geqslant\operatorname{Vol}_{\mathbb{R}^{d}}(\Delta(S,T_{\bullet})\cap
H)-\epsilon$ ∎

##  4\. p-families and a brief introduction to Hilbert Kunz Multiplicity

In this section, all rings will be commutative, Noetherian, and of prime
characteristic, $p>0$.

Let $I$ be an ideal of a ring $R$, generated by $\\{x_{1},...,x_{r}\\}$ then
$p^{e}$th Frobenius power of $I$ denoted by $I^{[p^{e}]}$ is generated by
$\\{x_{1}^{p^{e}},...,x_{r}^{p^{e}}\\}$.

######  Definition 4.1.

A sequence of ideals $I_{\bullet}=\\{I_{p^{e}}\\}_{e=0}^{\infty}$ is called a
$p$-family whenever $I_{q}^{[p]}\subseteq I_{pq}$, where $q=p^{e}$ for some
$e\geqslant 0$.

######  Remark 4.2.

We note that if $(R,\mathfrak{m})$ is local, then every term in this family is
$\mathfrak{m}$-primary if and only if $I_{1}$ is $\mathfrak{m}$ primary.

######  Example 4.3.

Let $I_{\bullet}=\\{J^{[p^{e}]}\\}_{e=0}^{\infty}$ for some fixed ideal $J$,
then $I_{p^{e}}^{[p]}=J^{[p^{e+1}]}=I_{p^{e+1}}$, therefore $I_{\bullet}$ is a
$p$-family.

######  Lemma 4.4.

Let $(R,\mathfrak{m})$ be a local ring and let
$I_{\bullet}=\\{I_{q}\\}_{q=1}^{\infty}$ be a $p$-family of
$\mathfrak{m}$-primary ideals. Then there exist a $c>0$ such that
$\mathfrak{m}^{cq}\subseteq I_{q}$ and $\mathfrak{m}^{cp^{e}q}\subseteq
I_{q}^{[p^{e}]}$ for all $q$ a power of $p$ and for all $e\geqslant 0$.

###### Proof.

We know from Remark 4.2 that there exist $c_{1}$ such that
$(\mathfrak{m}^{c_{1}})^{[q]}\subseteq I_{q}$ for all $q$, a power of $p$. Let
$\mathfrak{m}$ be generated by $b$ elements then using pigeon-hole principle,
$\mathfrak{m}^{bc_{1}p^{e}q}\subseteq(\mathfrak{m}^{c_{1}})^{[p^{e}q]}\subseteq((\mathfrak{m}^{c_{1}})^{[q]})^{[p^{e}]}\subseteq
I_{q}^{[p^{e}]}$, for all $q$ a power of $p$ and for all $e\geqslant 0$. So,
letting $c=bc_{1}$ we get our desired conclusion. ∎

We define the Frobenius map $F:R\to R$ by $F(r)=r^{p}$. This map turns $R$
into an $R$-module with a nonstandard action $r.x=r^{p}x$, and we call this
module $F_{*}R$. Inductively one can define $F_{*}^{e}R$. We define an
$R$-linear map $\phi_{e}:F_{*}^{e}R\to R$ which is a set map from $R$ to $R$
such that $\phi_{e}$ is additive and $\phi_{e}(x^{p^{e}}y)=x\phi(y)$.
The following is an interesting example of $p$-families.

######  Example 4.5.

Assume $R$ is reduced and $I$ is an $R$-ideal. Define for all $q=p^{e}$,
$J_{q}=\\{x\in R\mid\phi(x)\in I,\forall\phi\in Hom_{R}(F_{*}^{e}(R),R)\\}$.
Notice that $J_{q}$ is an ideal for each $q=p^{e}$. Since $R$ is reduced we
can identify $F_{*}^{e}(R)$ as $R^{\frac{1}{p^{e}}}$, the ring of $p^{e}$-th
roots of the elements in $R$. We claim that this is a $p$-family of ideals.
Indeed, let $\phi\in Hom_{R}(F_{*}^{e+1}R,R)$ and choose $r\in J_{q}$ then
$\psi(r^{\frac{1}{p^{e}}})\in I$ for all $\psi\in Hom_{R}(F_{*}^{e}R,R)$ . Now
$\phi\left((r^{p})^{\frac{1}{p^{e+1}}}\right)=\phi_{e}\left(r^{\frac{1}{p^{e}}}\right)\in
I$, as $r\in J_{q}$. This implies $J_{q}^{[p]}\subseteq J_{pq}$.

######  Remark 4.6.

If in the previous example we choose $I=\mathfrak{m}$, then we get a
$p$-family of $m$-primary ideals because $\mathfrak{m}^{[q]}\subseteq J_{q}$.
Use of this sequence of ideals was present in the work of Y. Yao [13] as well
as F. Enescu and I. Aberbach [1]. Later this sequence of ideals was used in
the work of K. Tucker [12] for proving that $F$-signature exists.

Let $(D,m,k)$ be a $d$-dimensional local OK-domain with OK-valuation
$\vartheta$. Choose any $p$ family $I_{\bullet}$ of ideals, then
i. $\vartheta(I_{q})$ is an ideal of the semigroup $\vartheta(D)=S$, and,
ii.
$p\vartheta(I_{q})\subseteq\vartheta(I_{q}^{[p]})\subseteq\vartheta(I_{pq})$.
Therefore, $\\{\vartheta(I_{q})\\}_{q=1}^{\infty}$ is a $p$-system of ideals
in $S$.

######  Remark 4.7.

Following Definition 3.7 and Remark 3.8, one can observe that although
$\vartheta^{(h)}(I_{q})$ is an ideal but we need the extra condition that
$\\{m_{1},...,m_{h}\\}$ are $k$-linear independent in $\vartheta^{(h)}(I_{q})$
implies $\\{m_{1}^{p},...,m_{h}^{p}\\}$ are $k$-linear independent in
$\vartheta^{(h)}(I_{pq})$ but it doesn’t always true unless we assume that $k$
is perfect. In ([7, Lemma 3.12]) they get around it by uniformly approximating
$\vartheta^{h}(I_{q})$ with $\vartheta(I_{q})$ and proving the following
result:

######  Proposition 4.8.

[7, Corollary 5.10] Let $D$ be a $d$-dimensional local OK-domain with OK-
valuation $\vartheta$. For a $p$-family of ideals $I_{\bullet}$ in $D$ we
have:

$\lim_{q\to\infty}\frac{(\vartheta^{(h)}(I_{q)}\cap
qH)}{q^{d}}=\operatorname{Vol}_{\mathbb{R}^{d}}(\Delta(S,\vartheta(I_{\bullet}))\cap
H)$

where $S=\vartheta(D)$, $C=\operatorname{Cone}(S)$ and $H$ is any truncating
halfspace of $C$.

We will now discuss Hilbert-Kunz Multiplicity and some important properties.

For the rest of this section let $(R,\mathfrak{m},\mathbb{K})$ be a
$d$-dimensional local ring and $q=p^{e}$, for some $e\in\mathbb{N}$.

######  Definition 4.9.

Let $I$ be an $\mathfrak{m}$-primary ideal of $R$. We define Hilbert-Kunz
Multiplicity of $I$ by
$e_{HK}(I,R)=\lim_{q\to\infty}\dfrac{\ell\left(R/I^{[q]}\right)}{q^{d}}$

######  Remark 4.10.

Let $J$ be an $\mathfrak{m}$-primary ideal of $R$, then $\ell\left(R/J\right)$
is unaffected by completion. Therefore using the above definition, we have

$e_{HK}(J,R)=e_{HK}(J\hat{R},\hat{R})$

######  Lemma 4.11.

Let $I$ be an $\mathfrak{m}$-primary ideal of $R$ and $M$ be a finitely
generated $R$-module. Then there exists a constant $\alpha>0$ such that

$\ell_{R}\left(\frac{M}{I^{[p^{e}]}M}\right)\leqslant\alpha
p^{e.\operatorname{dim}M}$

###### Proof.

Since $I$ is $\mathfrak{m}$-primary there exist $c>0$ such that
$\mathfrak{m}^{c}\subseteq I$. Let $b$ is the minimal number of generators of
$I$, then $\mathfrak{m}^{p^{e}bc}\subseteq I^{p^{e}b}\subseteq I^{[p^{e}]}$
(using pigeon-hole principle). Therefore
$\ell_{R}\left(\frac{M}{I^{[p^{e}]}}\right)\leqslant\ell_{R}\left(\frac{M}{\mathfrak{m}^{p^{e}bc}}\right)$
and the later agrees with a polynomial in $p^{e}bc$ of degree
$\operatorname{dim}M$. If the leading coefficient of this polynomial is $a$
then choose $a_{0}>>a$ implies $\ell_{R}(\frac{M}{\mathfrak{m}^{p^{e}bc}})$ is
bounded above by ,
$a_{0}(b^{\operatorname{dim}M}c^{\operatorname{dim}M}p^{e\operatorname{dim}M})$
taking $\alpha=a_{0}b^{\operatorname{dim}M}c^{\operatorname{dim}M}$ will
finish the proof. ∎

######  Corollary 4.12.

Let $N$ be an ideal of $R$ and $A=R/N$, then for any $\mathfrak{m}$-primary
ideal $I$ of $R$ there exist $\beta>0$ such that

$0\leqslant\ell_{R}\left(R/I^{[p^{e}]}\right)-\ell_{A}\left(A/I^{[p^{e}]}\right)\leqslant\beta.{p^{e.\operatorname{dim}{N}}}$

###### Proof.

Take the following exact sequence

$0\to N/(N\cap I^{[p^{e}]})\to R/{I^{[p^{e}]}}\to A/{I^{[p^{e}]}A}\to 0$

Using the length formula, we have

(4.1)
$\ell_{R}\left(R/{I^{[p^{e}]}}\right)-\ell_{A}\left(A/{I^{[p^{e}]}}\right)=\ell_{R}\left(N/(N\cap
I^{[p^{e}]})\right)$

Note that
$\ell_{R}\left(A/{I^{[p^{e}]}}\right)=\ell_{A}\left(A/{I^{[p^{e}]}}\right)$
and $I^{[p^{e}]}N\subseteq I^{[p^{e}]}\cap N$. Now using Lemma 4.11, we have
from Equation 4.1,
$\ell_{R}\left(R/{I^{[p^{e}]}}\right)-\ell_{A}\left(A/{I^{[p^{e}]}}\right)\leqslant\ell_{R}\left(N/{I^{[p^{e}]}N}\right)\leqslant\beta
p^{e.\operatorname{dim}N}$ ∎

######  Definition 4.13.

Let $f,g:\mathbb{N}\to\mathbb{R}$ be real valued functions from the set of
non-negative integers. We say $f(n)=O(g(n))$ if there exists a positive
constant $C$ such that $|f(n)|\leqslant Cg(n)$ for all $n\gg 0$ and we say
$f(n)=o(g(n))$ if $\lim_{n\to\infty}\dfrac{f(n)}{g(n)}=0$

Now we will mention some results on Hilbert Kunz Multiplicity without proof;
for a detailed verification, we refer readers to [8].

######  Proposition 4.14.

Let

$0\to N\to M\to K\to 0$

be a short exact sequence of finitely generated $R$-modules. Then

(4.2)
$\ell_{R}\left(M/{I^{[q]}M}\right)=\ell_{R}\left(N/{I^{[q]}N}\right)+\ell_{R}\left(K/{I^{[q]}K}\right)+O(q^{d-1})$

Now dividing Equation 4.2 by $q^{d}$ and taking $q\to\infty$ we have

$e_{HK}(I,M)=e_{HK}(I,K)+e_{HK}(I,N)$

######  Theorem 4.15.

Let $I$ be an $\mathfrak{m}$-primary ideal of $R$ and $M$ is a finitely
generated $R$-module. Let $\Gamma$ be the set of minimal prime ideals $P$ of
$R$ such that $\operatorname{dim}\left(R/P\right)=\operatorname{dim}(R)$. Then

$e_{HK}(I,M)=\sum_{p\in\Gamma}e_{HK}\left(I,R/P\right)\ell(M_{P})$

The main idea to prove this result is to take a prime filtration of $M$, use
the fact that $e_{HK}(I,R/Q)=0$ if $\operatorname{dim}R/Q<\operatorname{dim}R$
and then use Proposition 4.14.

######  Remark 4.16.

In the Theorem 4.15, if we choose $M=R$ and further assume $R$ is reduced.
Then

$e_{HK}(I,R)=\sum_{p\in\Gamma}e_{HK}\left(I,R/P\right)$

because a reduced local ring with Krull dimension $0$ is a field, therefore
$\ell(R_{P})=1$

##  5\. Volume=Multiplicity Formula

In this section we prove our main result, a general Volume=Multiplicity
formula for $p$-families of $\mathfrak{m}$-primary ideals. We begin with the
following important lemma.

######  Lemma 5.1.

[7, Lemma 5.21] Let $(R,\mathfrak{m},\mathbb{K})$ be a $d$-dimensional reduced
local ring of positive characteristic, and $I_{\bullet}$ be a sequence of
ideals of $R$ indexed by the powers of $p$ such that
$\mathfrak{m}^{cq}\subseteq I_{q}$ for some positive integer $c$ and for all
$q=p^{e},e\in\mathbb{N}$. Let $P_{1},...,P_{n}$ be the minimal primes of $R$,
then there exists $\delta>0$ such that for all $q$,

$\left|\sum_{i=1}^{n}\ell_{R_{i}}\left(R_{i}/{I_{q}R_{i}}\right)-\ell_{R}\left(R/{I_{q}}\right)\right|\leqslant\delta.q^{d-1}$

where $R_{i}=R/P_{i}$, for all $i\in\\{1,...,n\\}$

######  Remark 5.2.

Suppose $(R,\mathfrak{m})$ be a $d$-dimensional reduced local ring of
chracteristic $p>0$ and $I$ is an $\mathfrak{m}$-primary ideal. Since $R$ is
reduced and $I_{\bullet}=\\{I^{[p^{e}]}\\}_{e=0}^{\infty}$ is a $p$-family of
$\mathfrak{m}$ primary ideals, using Lemma 5.1, Lemma 4.4 one can give an
alternative proof of Remark 4.16.

The following is the Main Theorem of this paper.

######  Theorem 5.3.

Let $(R,\mathfrak{m},\mathbb{K})$ be a $d$-dimensional local ring of
characteristic $p>0$. If the $R$-module dimension of the nilradical of
$\hat{R}$ is less than $d$, then for any $p$-family of $\mathfrak{m}$-primary
ideals $I_{\bullet}=\\{I_{q}\\}_{q=1}^{\infty}$, we have

$\lim_{q\to\infty}\frac{\ell_{R}(R/I_{q})}{q^{d}}=\lim_{q\to\infty}\frac{e_{HK}(I_{q},R)}{(q)^{d}}$

###### Proof.

Step 1: The case of OK-Domain:

Assume $R$ is an OK-domain and let
$\vartheta:\mathbb{F}^{\times}\to\mathbb{Z}^{d}$ be the valuation that is OK-
relative to $R$ with respect to a $\mathbb{Z}$-linear embedding of
$\mathbb{Z}^{d}$ into $\mathbb{R}$ induced by a vector $\boldsymbol{a}$. Let
$(V,\mathfrak{m}_{\vartheta},\mathbb{K}_{\vartheta})$ be the associated
valuation ring, $S=\vartheta(R)$ be the corresponding semigroup in
$\mathbb{Z}^{d}$ and $C=\operatorname{Cone}(S)$.
Let $\boldsymbol{u}\in\mathbb{Z}^{d}$; then we have the following two exact
sequences:

(5.1)
$0\to\frac{I_{q}}{I_{q}\cap\mathbb{F}_{\geq\boldsymbol{u}}}\to\frac{R}{I_{q}\cap\mathbb{F}_{\geq\boldsymbol{u}}}\to\frac{R}{I_{q}}\to
0$ (5.2)
$0\to\frac{R\cap\mathbb{F}_{\geqslant\boldsymbol{u}}}{I_{q}\cap\mathbb{F}_{\geqslant\boldsymbol{u}}}\to\frac{R}{I_{q}\cap\mathbb{F}_{\geqslant\boldsymbol{u}}}\to\frac{R}{R\cap\mathbb{F}_{\geqslant\boldsymbol{u}}}\to
0$

From Equation 5.1, 5.2, we have

(5.3)
$\ell_{R}\left(R/{I_{q}}\right)=\ell_{R}\left(\frac{R\cap\mathbb{F}_{\geqslant\boldsymbol{u}}}{I_{q}\cap\mathbb{F}_{\geqslant\boldsymbol{u}}}\right)+\ell_{R}\left(\frac{R}{R\cap\mathbb{F}_{\geqslant\boldsymbol{u}}}\right)-\ell_{R}\left(\frac{I_{q}}{I_{q}\cap\mathbb{F}_{\geqslant\boldsymbol{u}}}\right)$

Let ${\boldsymbol{v}}\in S$ satisfy the last condition in Definition 3.1.
Choose the number $c>0$ from Lemma 4.4, define
$\boldsymbol{w}$=$c\boldsymbol{v}$, then $R\cap\mathbb{F}_{\geqslant
q\boldsymbol{w}}\subseteq\mathfrak{m}^{cq}\subseteq I_{q}$. Therefore

(5.4) $I_{q}\cap\mathbb{F}_{\geqslant
q\boldsymbol{w}}=R\cap\mathbb{F}_{\geqslant q\boldsymbol{w}}$

Let $H$ be the halfspace $\\{\boldsymbol{u}^{{}^{\prime}}\
|(\boldsymbol{a},\boldsymbol{u}^{{}^{\prime}})<(\boldsymbol{a},\boldsymbol{w})\\}$.
Then using Equation 5.3, 5.4 and Lemma 3.9 we have,

(5.5)
$\ell_{R}\left(R/{I_{q}}\right)=\sum_{h=1}^{[\mathbb{K}_{\vartheta}:\mathbb{K}]}\\#(\vartheta^{h}(R)\cap
qH)-\sum_{h=1}^{[\mathbb{K}_{\vartheta}:\mathbb{K}]}\\#(\vartheta^{h}(I_{q})\cap
qH)$

Fix $q^{{}^{\prime}}$ a power of $p$. I do the same calculation with respect
to the $p$-system of ideals
$J_{\bullet}=\\{I_{q^{{}^{\prime}}}^{[p^{e}]}\\}_{e=0}^{\infty}$ then using
Lemma 4.4 and Lemma 3.9, we have

(5.6)
$\ell_{R}\left(R/{I_{q^{{}^{\prime}}}^{[p^{e}]}}\right)=\sum_{h=1}^{[\mathbb{K}_{\vartheta}:\mathbb{K}]}\\#(\vartheta^{h}(R)\cap
p^{e}q^{{}^{\prime}}H)-\sum_{h=1}^{[\mathbb{K}_{\vartheta}:\mathbb{K}]}\\#(\vartheta^{h}(I_{q^{{}^{\prime}}}^{[p^{e}]})\cap
p^{e}q^{{}^{\prime}}H)$

Note that we can consider $J_{q}^{{}^{\prime}}={R}$ for all $q=p^{e}$. It is
clearly a $p$-family of ideals. Therefore using Proposition 4.8 and Equation
5.5, we have

(5.7)
$\lim_{q\to\infty}\frac{\ell_{R}(R/I_{q})}{q^{d}}=[\mathbb{K}_{\vartheta}:\mathbb{K}].(\operatorname{Vol}_{\mathbb{R}^{d}}(\Delta(S,\vartheta(J_{\bullet}^{{}^{\prime}})\cap
H)-\operatorname{Vol}_{\mathbb{R}^{d}}(\Delta(S,\vartheta(I_{\bullet})\cap
H))$

Similarly applying Proposition 4.8 and Equation 5.6 for the $p$-system of
ideals $J_{\bullet}=\\{I_{q^{{}^{\prime}}}^{[p^{e}]}\\}_{e=0}^{\infty}$, we
have

(5.8)
$\lim_{e\to\infty}\frac{\ell_{R}\left(R/I_{q^{{}^{\prime}}}^{[p^{e}]}\right)}{p^{ed}(q^{{}^{\prime}})^{d}}=[\mathbb{K}_{\vartheta}:\mathbb{K}].(\operatorname{Vol}_{\mathbb{R}^{d}}(\Delta(S,\vartheta(J_{\bullet}^{{}^{\prime}})\cap
H)-\operatorname{Vol}_{\mathbb{R}^{d}}(\Delta(S,\vartheta(J_{\bullet})\cap
H))$

Using Definition 3.12, we have

(5.9)
$\Delta(S,\vartheta(J_{\bullet}))=\bigcup_{e=0}^{\infty}\frac{1}{p^{e}}\vartheta\left(I_{q^{{}^{\prime}}}^{[p^{e}]}\right)+\operatorname{Cone}(S).$

Now
$\vartheta\left(I_{q^{{}^{\prime}}}^{[p^{e}]}\right)\subseteq\vartheta(I_{q^{{}^{\prime}}p^{e}})$,
as $\\{I_{q}\\}$ is a $p$-system of ideals and
$\frac{1}{p^{e}}\vartheta\left(I_{q^{{}^{\prime}}}^{[p^{e}]}\right)\subseteq\frac{1}{p^{e}}\vartheta(I_{q^{{}^{\prime}}p^{e}})\subseteq\frac{1}{p^{e}q^{{}^{\prime}}}\vartheta(I_{q^{{}^{\prime}}p^{e}})$.
Therefore
$\bigcup_{e=0}^{\infty}\frac{1}{p^{e}}\vartheta\left(I_{q^{{}^{\prime}}}^{[p^{e}]}\right)+\operatorname{Cone}(S)\subseteq\bigcup_{e=0}^{\infty}\frac{1}{p^{e}q^{{}^{\prime}}}\vartheta(I_{q^{{}^{\prime}}p^{e}})+\operatorname{Cone}(S)\subseteq\bigcup_{q=1}^{\infty}\frac{1}{q}\vartheta(I_{q})+\operatorname{Cone}(S)=\Delta(S,\vartheta(I_{\bullet})\cap\mathbb{H})$.
From this, we conclude

(5.10)
$\operatorname{Vol}_{\mathbb{R}^{d}}(\Delta(S,\vartheta(J_{\bullet})\cap
H))\leqslant\operatorname{Vol}_{\mathbb{R}^{d}}(\Delta(S,\vartheta(I_{\bullet})\cap
H))$

Moreover
$p^{e}\vartheta(I_{q^{{}^{\prime}}})+S\subseteq\vartheta(I_{q^{{}^{\prime}}}^{[p^{e}]})$.
Indeed for $\boldsymbol{s}_{1}\in\vartheta(I_{q^{{}^{\prime}}})$ and
$\boldsymbol{s}_{2}\in S$, we have $\vartheta(x)=\boldsymbol{s}_{1}$ for some
$x\in I_{q^{{}^{\prime}}}$ and $\vartheta(y)=\boldsymbol{s}_{2}$ for some
$y\in R$. Therefore
$p^{e}\boldsymbol{s}_{1}+\boldsymbol{s}_{2}=\vartheta(x^{[p^{e}]}\times y)=$
and $x^{[p^{e}]}\times y\in I_{q^{{}^{\prime}}}^{[p^{e}]}$.

Using the observation above and Theorem 3.18, we have for any given
$\epsilon>0$, there exist $q_{0}$ such that for all $q^{{}^{\prime}}\geqslant
q_{0}$,

(5.11)
$\operatorname{Vol}_{\mathbb{R}^{d}}(\Delta(S,\vartheta(I_{\bullet})\cap
H)-\epsilon\leqslant\lim_{e\to\infty}\frac{\\#(p^{e}\vartheta(I_{q^{{}^{\prime}}})+S)\cap
p^{e}q^{{}^{\prime}}H)}{p^{ed}(q^{{}^{\prime}})^{d}}\leqslant\lim_{e\to\infty}\frac{\\#(\vartheta(I_{q^{{}^{\prime}}}^{[p^{e}]}\cap
p^{e}q^{{}^{\prime}}H)}{p^{ed}(q^{{}^{\prime}})^{d}}$

Now using Equation 5.10 and 5.11 in Equation 5.7 and 5.8 we have for all
$q^{{}^{\prime}}\geqslant q_{0}$,

$\lim_{q\to\infty}\frac{\ell_{R}(R/I_{q})}{q^{d}}\leqslant\lim_{e\to\infty}\frac{\ell_{R}(R/I_{q^{{}^{\prime}}}^{[p^{e}]})}{p^{ed}(q^{{}^{\prime}})^{d}}=\dfrac{e_{HK}(I_{q^{{}^{\prime}}},R)}{(q^{{}^{\prime}})^{d}}\leqslant\lim_{q\to\infty}\frac{\ell_{R}(R/I_{q})}{q^{d}}+\epsilon$

Therefore letting $q^{{}^{\prime}}\to\infty$, we obtain the result in OK-
domain case.
Step 2: Reduction to the OK-domain case:

Since $\ell(R/J)$ is unaffected by completion for any $\mathfrak{m}$-primary
ideal $J$, we may assume that $R$ is complete.
Using [7, Corollary 5.18] and Corollary 4.12 by choosing $R=\hat{R}$ and
$N=N(\hat{R})$ with the condition that $\operatorname{dim}N<d$ we can
therefore assume that $R$ is complete and reduced.
Now suppose that the minimal primes of the complete reduced ring $R$ are
$\\{P_{1},...,P_{n}\\}$. Let $R_{i}=R/P_{i}$ be complete local domain for all
$1\leqslant i\leqslant n$. By Remark 4.16, we have

(5.12)
$\dfrac{e_{HK}(I_{q},R)}{(q)^{d}}=\sum_{i=1}^{n}\dfrac{e_{HK}(I_{q}R_{i},R_{i})}{(q)^{d}}$

We also have from Lemma 5.1

(5.13)
$\lim_{q\to\infty}\dfrac{\ell_{R}({R/I_{q}})}{q^{d}}=\sum_{i=1}^{n}\lim_{q\to\infty}\dfrac{\ell_{R_{i}}(R_{i}/I_{q}R_{i})}{q^{d}}$

Since each $R_{i}$ is a complete local domain and therefore an OK-domain by
Example 3.5, using Equation 5.12 and 5.13 we have

$\displaystyle\lim_{q\to\infty}\dfrac{\ell_{R}({R/I_{q}})}{q^{d}}$
$\displaystyle=\sum_{i=1}^{n}\lim_{q\to\infty}\dfrac{\ell_{R_{i}}(R_{i}/I_{q}R_{i})}{q^{d}}$
$\displaystyle=\sum_{i=1}^{n}\lim_{q\to\infty}\dfrac{e_{HK}(I_{q}R_{i},R_{i})}{(q)^{d}}$
$\displaystyle=\lim_{q\to\infty}\dfrac{e_{HK}(I_{q},R)}{(q)^{d}}$

∎

##  6\. Acknowledgement

I would like to thank my advisor Dr. Jonathan Montaño, for constant
encouragement and many helpful suggestions; specifically, our Starbucks
discussions made this a journey to remember. The author was partially
supported by NSF Grant DMS $\\#2001645$.

## References

  * [1] Ian M Aberbach and Florian Enescu.  The structure of f-pure rings.  Mathematische Zeitschrift, 250(4):791–806, 2005.
  * [2] Steven Cutkosky.  Multiplicities associated to graded families of ideals.  Algebra & Number Theory, 7(9):2059–2083, 2013.
  * [3] Steven Dale Cutkosky.  Asymptotic multiplicities of graded families of ideals and linear series.  Advances in Mathematics, 264:55–113, 2014.
  * [4] Steven Dale Cutkosky.  A general volume= multiplicity formula.  Acta Mathematica Vietnamica, 40(1):139–147, 2015.
  * [5] Lawrence Ein, Robert Lazarsfeld, and Karen E Smith.  Uniform approximation of abhyankar valuation ideals in smooth function fields.  American Journal of Mathematics, 125(2):409–440, 2003.
  * [6] Chungsim Han and Paul Monsky.  Some surprising hilbert-kunz functions.  Mathematische Zeitschrift, 214(1):119–135, 1993.
  * [7] Daniel J Hernández and Jack Jeffries.  Local okounkov bodies and limits in prime characteristic.  Mathematische Annalen, 372(1):139–178, 2018.
  * [8] Craig Huneke.  Hilbert–kunz multiplicity and the f-signature.  In Commutative algebra, pages 485–525. Springer, 2013.
  * [9] Ernst Kunz.  On noetherian rings of characteristic p.  American Journal of Mathematics, 98(4):999–1013, 1976.
  * [10] Robert Lazarsfeld and Mircea Mustață.  Convex bodies associated to linear series.  In Annales scientifiques de l’École normale supérieure, volume 42, pages 783–835, 2009.
  * [11] Paul Monsky.  The hilbert-kunz function.  Mathematische Annalen, 263(1):43–49, 1983.
  * [12] Kevin Tucker.  F-signature exists.  Inventiones mathematicae, 190(3):743–765, 2012.
  * [13] Yongwei Yao.  Observations on the f-signature of local rings of characteristic p.  Journal of Algebra, 299(1):198–218, 2006.