Hilbert-Kunz function and Hilbert-Kunz multiplicity of some ideals of the Rees algebra

Abstract

We prove that the Hilbert-Kunz function of the ideal (I,It) of the Rees algebra R(I)\mathcal{R}(I), where I is an m\mathfrak{m}-primary ideal of a 1-dimensional local ring (R,m)(R,\mathfrak{m}), is a quasi-polynomial in e, for large e. For s∈Ns \in \mathbb{N}, we calculate the Hilbert-Samuel function of the R-module I[s]I^{[s]} and obtain an explicit description of the generalized Hilbert-Kunz function of the ideal (I,It)R(I)(I,It)\mathcal{R}(I) when I is a parameter ideal in a Cohen-Macaulay local ring of dimension d≥2d \geq 2, proving that the generalized Hilbert-Kunz function is a piecewise polynomial in this case.

Full text

arXiv:1911.03889v1 [math.AC] 10 Nov 2019
HILBERT-KUNZ FUNCTION AND HILBERT-KUNZ MULTIPLICITY
OF SOME IDEALS OF THE REES ALGEBRA
KRITI GOEL, MITRA KOLEY, AND J. K. VERMA
Abstract. We prove that the Hilbert-Kunz function of the ideal (I , I t) of the Rees algebra R(I),
where Iis an m-primary ideal of a 1-dimensional local ring (R, m), is a quasi-polynomial in e, for
large e. For s∈N, we calculate the Hilbert-Samuel function of the R-module I[s]and obtain an
explicit description of the generalized Hilbert-Kunz function of the ideal (I , It)R(I) when Iis a
parameter ideal in a Cohen-Macaulay local ring of dimension d≥2, proving that the generalized
Hilbert-Kunz function is a piecewise polynomial in this case.
1. Introduction
Let (R, m) be a d-dimensional Noetherian local ring of positive prime characteristic pand let Ibe
an m-primary ideal. The qth-Frobenius power of Iis the ideal I[q]= (xq|x∈I) where q=pefor
e∈N.The function e7→ ℓR(R/I[pe]) is called the Hilbert-Kunz function of the ideal Iand was first
considered by E. Kunz in [7]. In [9], P. Monsky showed that this function is of the form
ℓR(R/I[q]) = eH K (I, R)qd+O(qd−1)
where eHK (I , R) is a positive real number called the Hilbert–Kunz multiplicity of Iin R. Besides
the mysterious leading coefficient, the behavior of the Hilbert–Kunz function is also unpredictable.
Monsky proved that in the case of 1-dimensional rings, the Hilbert-Kunz function ℓR(R/I[pe]) =
eHK (I , R)q+δe, where δeis a periodic function of e, for large e. Precisely, take R=k[[X, Y ]]/(X5−
Y5), where kis a field of prime characteristic p≡ ±2(mod 5).Let mbe the maximal ideal of R.
Monsky [9] showed that for large e,ℓR(R/m[q]) = 5q+αm(e), where αm(e) = −4 when eis even
and αm(e) = −6 when eis odd. In [5], the authors determined the Hilbert-Kunz function of the
maximal ideal of the ring R=Z/p[[x1,...,xs]]/(xd1
1+···+xds
s), where diare positive integers and
s=d+ 1 ≥3.They proved that if ℓR(R/m[q]) = eH K (R)qd+δe, then eHK (R) is rational and δe
is an eventually periodic function of ewhenever s= 3 or p= 2.But when s= 4 and p= 5 with
d1=··· =ds= 4, then ℓR(R/m[5e]) = (168/61) ·53e−(107/61) ·3e.In [1], H. Brenner proved the
following result
The first author is supported by a UGC fellowship, Govt. of India.
Key words and phrases: Hilbert-Kunz function, Hilbert-Kunz multiplicity, Rees algebra, generalized Hilbert-Kunz
function.
2010 AMS Mathematics Subject Classification: 13A30, 13A35, 13D40.
1

2 KRITI GOEL, MITRA KOLEY, AND J. K. VERMA
Theorem 1.1. Let kdenote the algebraic closure of a finite field of characteristic p. Let Rdenote
a normal two-dimensional standard-graded k-domain and let Idenote a homogeneous R+-primary
ideal. Then the Hilbert–Kunz function of Ihas the form ℓR(R/I[q]) = eH K (I, R)q2+δe, where
eHK (I , R)is a rational number and δeis an eventually periodic function.
Several other authors including D. Brinkmann, V. Trivedi and P. Teixeira have worked on similar
problems.
We recall a few definitions. Let (R, m) be a Noetherian local ring of dimension dand Ibe an
m-primary ideal of R. The Hilbert-Samuel function HI(n)of Iis defined as HI(n) = ℓR(R/In).
It is known that HI(n) is a polynomial function of nof degree d, for large n. In particular, there
exists a polynomial PI(x)∈Q[x] such that HI(n) = PI(n) for all large n. Write
PI(x) = e0(I)x+d−1
d−e1(I)x+d−2
d−1+···+ (−1)ded(I),
where ei(I) for i= 0,1,...,d are integers, called the Hilbert coefficients of I. The leading coefficient
e0(I) = e(I) is called the multiplicity of Iand e1(I) is called the Chern number of I. The postulation
number of Iis defined as
n(I) = max{n|HI(n)6=PI(n)}.
The notion of reduction of an ideal was introduced by D. G. Northcott and D. Rees. Let J⊆I
be ideals of R. If J In=In+1 for some n, then Jis called a reduction of I. The reduction number
rJ(I) of Iwith respect to Jis the smallest nsuch that J I n=In+1 .The reduction number of Iis
defined as
r(I) = min{rJ(I)|Jis a minimal reduction of I}.
Moreover, if R/mis infinite, then minimal reduction of Iexists.
In this paper, we calculate the Hilbert-Kunz function of certain ideals of the Rees algebra. Let
R(I) = ⊕n≥0Intndenote the Rees algebra of I. The Hilbert-Kunz multiplicity of various blowup
algebras was estimated by K. Eto and K.-i. Yoshida in their paper [3]. Put c(d) = (d/2)+d/(d+1)!.
They proved the following.
Theorem 1.2. Let (R, m)be a Noetherian local ring of characteristic p > 0with d= dim R≥1.
Then for any m-primary ideal I, we have
eHK (R(I)) ≤c(d)·e(I).
Moreover, equality holds if and only if eHK (R) = e(I).When this is the case, eHK (R) = e(R)and
eHK (I) = e(I).
Since a precise formula for the multiplicity of the maximal graded ideal of the Rees algebra is known
(see [12]), it is natural to ask if there is a formula for the Hilbert-Kunz multiplicity of (m, It), when
Iis an m-primary ideal, in terms of invariants of the ideals mand I. We aim to answer this question
in some cases. In section 2, we begin by proving the following result.

HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 3
Theorem 1.3. Let (R, m)be a 1-dimensional Noetherian local ring with prime characteristic p > 0.
Let I, J be m-primary ideals. Then eH K ((J, It)R(I)) = e(J).
Next, we prove that the Hilbert-Kunz function of the ideal (I , It) of the Rees algebra R(I), where
Iis an m-primary ideal of a 1-dimensional ring, is a quasi-polynomial in e, for large e. Recall that
a quasi-polynomial of degree dis a function f:Z→Cof the form
f(n) = cd(n)nd+cd−1(n)nd−1+···+c0(n),
where each ci(n) is a periodic function and cd(n) is not identically zero. Equivalently, fis a
quasi-polynomial if there exists an integer N > 0 (namely, a common period of c0, c1,...,cd) and
polynomials f0, f1,...,fN−1such that f(n) = fi(n) if n≡i(mod N) (see [10]).
Theorem 1.4. Let Rbe a 1-dimensional Noetherian local ring with prime characteristic p > 0.
Let Ibe an m-primary ideal. Let rbe the reduction number of Iand ρbe the postulation number
of I. Put I= (I, It)R(I).Let q=pe, where e∈Nis large.
(1) If ρ+ 1 ≤r, then
ℓRR(I)
I[q]=e(I)q2−e(I)r
2+r·e1(I) +
r−1
X
n=0
ℓRR
In+ 2
r−1
X
n=0
αI(In, e).
(2) If r < ρ + 1, then
ℓRR(I)
I[q]=e(I)q2−e(I)r(r−1) −ρ(ρ+ 1)
2+ (2r−ρ−1)e1(I) + β+ 2
r−1
X
n=0
αI(In, e),
where β=
r−1
X
n=0
ℓRR
In−
ρ
X
n=r
ℓRR
Inis a constant and αI(In, e) = ℓR(In/I[q]In)−e(I)qis a
periodic function in e, for large e. In other words, ℓR(R(I)/I[q])is a quasi-polynomial for large e.
In the above setup, if Iis a parameter ideal and Jis an m-primary ideal of a Cohen-Macaulay local
ring R, then we prove that for large e,
ℓRR(I)
(J, It)[q]=q2e(J) + q·αJ(e),
where αJ(e) = ℓR(R/J[q])−e(J)qis a periodic function in e. As a consequence of the above
results, we calculate the Hilbert-Kunz function of the ideals (m,mt)R(m) and (m, I t)R(I), when
R=k[[X, Y ]]/(X5−Y5), where kis a field of prime characteristic p≡ ±2(mod 5), mis the maximal
ideal of R,I= (x) is a parameter ideal and xis the image of Xin R.
Let Rbe a d-dimensional Noetherian ring. Let Ibe an ideal of finite co-length. Aldo Conca
introduced the concept of generalized Hilbert-Kunz function in [2]. For s∈N, let I[s]= (as
1,...,as
n)
where {a1, a2,...,an}is a fixed set of generators of I. Then the generalized Hilbert-Kunz function
is defined as
HKR,I (s) = ℓRR
I[s].

4 KRITI GOEL, MITRA KOLEY, AND J. K. VERMA
The generalized Hilbert-Kunz multiplicity is defined as lim
s→∞ HKR,I (s)/sdwhenever the limit exists.
If char(R) = p > 0,then the generalized Hilbert-Kunz function (multiplicity) coincides with the
Hilbert-Kunz function (multiplicity) and is independent of the choice of the generators of I.
In section 3, we find the generalized Hilbert-Kunz function of the ideal I= (I, I t) in R(I), when
Iis generated by a regular sequence in a d-dimensional Cohen-Macaulay local ring. Our approach
requires knowledge of the Hilbert-Samuel function of the R-module I[s].We obtain an explicit
description of the function F(n) = ℓR(I[s]/I[s]In) for a fixed s∈Nand then use some properties
of Stirling numbers of the second kind to prove the following result.
Theorem 1.5. Let Rbe a d-dimensional Cohen-Macaulay local ring and d≥2.Let Ibe a parameter
ideal of Rand I= (I, It)R(I).Let s∈N.
(1) Let s < d. Write d=k1s+k2where k2∈ {0,1,...,s−1}.If k2= 0, then
ℓRR(I)
I[s]= (d−k1+1)sd+1e(I)+d·e(I)s+d−1
d+ 1 −
d−1
X
i=0 (−1)id
ie(I)(d−i−k1+ 1)s+d−1
d+ 1 .
If k26= 0,then
ℓRR(I)
I[s]= (d−k1)sd+1e(I)+d·e(I)s+d−1
d+ 1 −
d−1
X
i=0 (−1)id
ie(I)(d−i−k1)s+d−1
d+ 1 .
(2) Let s≥d. Then
ℓRR(I)
I[s]=dsd+1e(I) + d·e(I)s+d−1
d+ 1 −
d−1
X
i=0 (−1)id
ie(I)(d−i)s+d−1
d+ 1 .
In other words, for slarge,
ℓRR(I)
I[s]=c(d)e(I)sd+1 +e(I)d−2
2 1
(d−1)! −1sd+e(I)d(d−1)(3d−10)
24(d−1)! sd−1+··· ,
implying that the generalized Hilbert-Kunz multiplicity eH K ((I, I t)R(I)) = c(d)·e(I).
As a consequence, we obtain the following result.
Corollary 1.6. Let (R, m)be a d-dimensional regular local ring with d≥2.Then for s≥d,
ℓRR(m)
(m,mt)[s]=c(d)e(m)sd+1+e(m)d−2
2 1
(d−1)! −1sd+e(m)d(d−1)(3d−10)
24(d−1)! sd−1+··· .
Acknowledgements: We thank K.-i. Watanabe and Anurag Singh for several discussions and
their lectures at IIT Bombay on Hilbert-Kunz multiplicity and positive characteristic methods.

HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 5
2. The Hilbert-Kunz function in dimension 1
Let (R, m) be a 1-dimensional Noetherian local ring with prime characteristic p > 0. In this section,
we calculate the Hilbert-Kunz function and Hilbert-Kunz multiplicity of certain ideals of the Rees
ring R(I), where Iis an m-primary ideal of R.
Let I, J be m-primary ideals of R. We begin by calculating eHK ((J, It)R(I)), the Hilbert-Kunz
multiplicity of the ideal (J, It) in R(I).Recall that if a Noetherian ring Rhas prime characteristic
p, then x∈Ris said to be in the tight closure I∗of an ideal Iif there exists c∈Ro:= {a∈R|a /∈
pfor any minimal prime p⊂R}such that cxq∈I[q]for all large q=pe.In [6, Theorem 8.17(a)],
M. Hochster and C. Huneke proved that if J⊆Iare m-primary ideals of Rsuch that I∗=J∗,
then eHK (I) = eH K (J).
For an ideal Iin a ring R, the integral closure of I, denoted by I, is the ideal which consists of
elements x∈Rsuch that xsatisfies an equation of the form xn+a1xn−1+··· +an= 0 for some
ai∈Ii, 1 ≤i≤n. The following generalization of the Brian¸con-Skoda theorem was given by
Hochster and Huneke.
Theorem 2.1 ([6, Theorem 5.4]).Let Rbe a Noetherian ring of prime characteristic pand let I
be an ideal of positive height generated by nelements. Then for every m∈N,In+m⊆(Im+1)∗.
It now follows that the tight closure and integral closure of principal ideals coincide.
Theorem 2.2. Let (R, m)be a 1-dimensional Noetherian local ring with prime characteristic p > 0.
Let I, J be m-primary ideals. Then eH K ((J, It)R(I)) = e(J).
Proof. We may assume that Rhas an infinite residue field. Let (x), (y) be minimal reductions of
Jand Irespectively in R. We claim that eHK ((J, I t)R(I)) = eHK ((x, yt)R(I)).Note that it is
sufficient to show that ((J, It)R(I))∗= ((x, yt)R(I))∗.Using generalized Brian¸con-Skoda theorem,
it follows that ((x)R(I))∗=(x)R(I) and ((yt)R(I))∗= (yt)R(I).As (x)R(I) and (yt)R(I) are
reductions of JR(I) and (It)R(I) respectively, we have
((x)R(I))∗=(x)R(I) = JR(I) and ((yt)R(I))∗= (yt)R(I) = (I t)R(I).
Since JR(I) = ((x)R(I))∗⊆(JR(I))∗, we get ((x)R(I))∗= (JR(I))∗.Similarly, ((yt)R(I))∗=
((It)R(I))∗.Consider
((x, yt)R(I))∗=((x)R(I))∗+ ((yt)R(I))∗∗=(JR(I))∗+ ((It)R(I))∗∗= ((J, It)R(I))∗.
This proves the claim. Therefore, eH K ((J, It)R(I)) = eH K ((x, yt)R(I)).As x, yt form a system of
parameters in R(I)(m,It), it follows that
eHK ((J, I t)R(I)) = eHK ((x, yt)R(I)) = e((x, yt)R(I)) = e((J, It)R(I)) = e(J),
where the last equality follows from [12, Theorem 3.1]. 

6 KRITI GOEL, MITRA KOLEY, AND J. K. VERMA
Let Ibe an m-primary ideal of R. We prove that ℓR(R(I)/(I , It)[pe]) is a quasi-polynomial in e, for
large e.
Theorem 2.3. Let (R, m)be a 1-dimensional Noetherian local ring with prime characteristic p > 0.
Let Ibe an m-primary ideal. Let rbe the reduction number of Iand ρbe the postulation number
of I. Put I= (I, It)R(I).Let q=pe, where e∈Nis large.
(1) If ρ+ 1 ≤r, then
ℓRR(I)
I[q]=e(I)q2−e(I)r
2+r·e1(I) +
r−1
X
n=0
ℓRR
In+ 2
r−1
X
n=0
αI(In, e).
(2) If r < ρ + 1, then
ℓRR(I)
I[q]=e(I)q2−e(I)r(r−1) −ρ(ρ+ 1)
2+ (2r−ρ−1)e1(I) + β+ 2
r−1
X
n=0
αI(In, e),
where β=
r−1
X
n=0
ℓRR
In−
ρ
X
n=r
ℓRR
Inis a constant and αI(In, e) = ℓR(In/I[q]In)−e(I)qis a
periodic function in e, for large e. In other words, ℓR(R(I)/I[q])is a quasi-polynomial for large e.
Proof. Fix q=pelarge. Observe that
I[q]= (I[q], I[q]tq) = q−1
M
n=0
I[q]Intn!+
M
n≥q
I[q]In−qtn
.
We may assume that Rhas an infinite residue field. Let (x) be a minimal reduction of Iand let r
be the reduction number of I . Then xkIl=Ik+lfor all k≥1 and l≥r. Write I=x+J, for some
ideal J⊆I. Then I[q]In−q= (xq+J[q])In−q=In, for all n≥q+r. Therefore,
I[q]= q−1
M
n=0
I[q]Intn!+ q+r−1
M
n=q
I[q]In−qtn!+
M
n≥q+r
Intn

and hence
ℓRR(I)
I[q]=
r−1
X
n=0
ℓRIn
I[q]In+
q−1
X
n=r
ℓRIn
I[q]In+
q+r−1
X
n=q
ℓRIn
I[q]In−q
= 2
r−1
X
n=0
ℓRIn
I[q]In+
q−1
X
n=r
ℓRR
I[q]In+
r−1
X
n=0
ℓRR
In−
q+r−1
X
n=r
ℓRR
In.
For n≥r,I[q]In= (xq+J[q])In=In+q.Using [9, Theorem 3.11], we can write ℓR(In/I[q]In) =
e(I, In)q+αI(In, e), where αI(In, e) is a periodic function of e. Using the associativity formula for
e(I), one may check that e(I , In) = e(I , R).Thus,
ℓRR(I)
I[q]= 2
r−1
X
n=0 e(I)q+αI(In, e)+
q−1
X
n=r
ℓRR
In+q+
r−1
X
n=0
ℓRR
In−
q+r−1
X
n=r
ℓRR
In.

HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 7
Case 1: Let ρ+ 1 ≤r. Then
ℓRR(I)
I[q]= 2rq ·e(I) + 2
r−1
X
n=0
αI(In, e) +
2q−1
X
n=r+qe(I)n−e1(I)+
r−1
X
n=0
ℓRR
In−
q+r−1
X
n=re(I)n−e1(I)
= 2rq ·e(I) + 2
r−1
X
n=0
αI(In, e) + e(I)2q
2−2r+q
2+r
2+r·e1(I) +
r−1
X
n=0
ℓRR
In
=e(I)q2−e(I)r
2+r·e1(I) +
r−1
X
n=0
ℓRR
In+ 2
r−1
X
n=0
αI(In, e).
Case 2: Let r < ρ + 1.Then
ℓRR(I)
I[q]= 2rq ·e(I) + 2
r−1
X
n=0
αI(In, e) +
2q−1
X
n=r+qe(I)n−e1(I)+
r−1
X
n=0
ℓRR
In−
ρ
X
n=r
ℓRR
In
−
q+r−1
X
n=ρ+1 e(I)n−e1(I)
= 2rq ·e(I) + 2
r−1
X
n=0
αI(In, e) + e(I)2q
2−2r+q
2+ρ+ 1
2+ (2r−ρ−1)e1(I) + β
=e(I)q2−e(I)r(r−1) −ρ(ρ+ 1)
2+ (2r−ρ−1)e1(I) + β+ 2
r−1
X
n=0
αI(In, e),
where β=
r−1
X
n=0
ℓRR
In−
ρ
X
n=r
ℓRR
Inis a constant. Let Nbe a common period of αI(In, e), for
0≤n≤r−1.Then it follows that ℓR(R(I)/I[q])−e(I)q2is a periodic function of e. In other
words, ℓR(R(I)/I[q]) is a quasi-polynomial in e, for large e. 
In particular, if Ris a 1-dimensional Cohen-Macaulay local ring we get the following result.
Corollary 2.4. Let (R, m)be a 1-dimensional Cohen-Macaulay local ring with prime characteristic
p > 0.Let Ibe an m-primary ideal. Let rbe the reduction number of I. Put I= (I , It)R(I).Then
for q=pe, where e∈Nis large,
ℓRR(I)
I[q]=e(I)q2−e(I)r
2+r·e1(I) +
r−1
X
n=0
ℓRR
In+ 2
r−1
X
n=0
αI(In, e),
where αI(In, e) = ℓR(In/I[q]In)−e(I)qis a periodic function in e, for large e. In other words,
ℓR(R(I)/I[q])is a quasi-polynomial for large e.
Proof. Since Ris Cohen-Macaulay, using [8, Theorem 2.15] it follows that the postulation number
of I,ρ=r−1.Substitute the same in Theorem 2.3 to conclude. 
Next, we calculate the Hilbert-Kunz function of the ideal J= (J, I t)R(I) when Iis a parameter
ideal and Jis an m-primary ideal of a Cohen-Macaulay ring R.

8 KRITI GOEL, MITRA KOLEY, AND J. K. VERMA
Theorem 2.5. Let (R, m)be a 1-dimensional Cohen-Macaulay local ring with prime characteristic
p > 0.Let I= (a)be a parameter ideal and Jbe an m-primary ideal. Put J= (J, I t)R(I).Then
for q=pe, where e∈Nis large,
ℓRR(I)
J[q]=q2e(J) + q·αJ(e),
where αJ(e) = ℓR(R/J [q])−e(J)qis a periodic function in e.
Proof. Observe that
J[q]= (J[q], I[q]tq) = q−1
M
n=0
J[q](an)tn!+
M
n≥q
(an)tn

which implies that
ℓRR(I)
J[q]=
q−1
X
n=0
ℓR(an)
J[q](an)=q·ℓRR
J[q]=e(J)q2+q·αJ(e).

We illustrate the above results in the following example.
Example 2.6. Let R=k[[X, Y ]]/(X5−Y5), where kis a field of prime characteristic p≡
±2(mod 5).Let mbe the maximal ideal of R. Put q=pefor some e∈N.Monsky [9] showed
that for large e,ℓR(R/m[q]) = 5q+αm(e), where αm(e) = −4 when eis even and αm(e) = −6 when
eis odd.
(1) We first calculate ℓR(R(m)/(m,mt)[q]).It is easy to check that e(m) = 5 and e1(m) = 10.Let
I= (x), where xdenotes the image of Xin R. Then Iis a minimal reduction of mand r(m) = 4.
Using Corollary 2.4, it follows that
ℓRR(m)
(m,mt)[q]= 5q2+ 10 +
3
X
n=0
ℓRR
mn+ 2
3
X
n=0
αm(mn, e) = 5q2+ 20 + 2
3
X
n=0
αm(mn, e).(2.1)
For n= 1,2,3,writing
ℓRmn
m[q]mn=ℓR m[q]
m[q]mn!−ℓRR
mn+ℓRR
m[q]=
n−1
X
i=0
µ(m[q]mi)−ℓRR
mn+ℓRR
m[q],
we obtain
αm(m, e) = 


−3 if eis even
−5 if eis odd,
αm(m2, e) = 


−2 if eis even
−3 if eis odd,αm(m3, e) = −1.
Substituting in (2.1), we get ℓR(R(m)/(m,mt)[q]) = 5q2+βe, where βe= 0 when eis even and
βe=−10 when eis odd.

HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 9
(2) We now calculate ℓR(R(I)/(m, It)[q]).Using Theorem 2.5, we get
ℓRR(I)
(m, It)[q]=e(m)q2+q·αm(e) = 5q2+


−4qif eis even
−6qif eis odd.
One can also verify this using the following arguments. Observe that R(I)≃k[[X, Y ]][Z]/(X5−
Y5).In order to find ℓ(k[X, Y , Z]/(X5−Y5, X q, Y q, Zq)), we find the Gr¨obner basis of the ideal
Mq= (X5−Y5, Xq, Y q, Z q) in k[X, Y, Z ].Let ‘>’ be any monomial ordering on k[X, Y, Z] with
X > Y > Z. Since qis large, the S-polynomials are:
S(X5−Y5, Xq) = Xq−5Y5, S(X5−Y5, X q−5Y5) = Xq−10Y10,...
Therefore, Gr¨obner basis of Mqis G={X5−Y5, Xq, Y q, Zq, X q−5iY5i|q−5i > 0}.Observe that
ℓRR(I)
(m, It)[q]=ℓk[X, Y, Z]
in>(Mq)=ℓk[X, Y, Z]
(X5, Y q, Z q, Xq−5iY5i|q−5i < 5) .
We now explore the condition q−5isuch that q−5i < 5.Since qis of the form (5k+ 2)eor (5k+3)e
for some k, e ∈N, it is sufficient to find the values (5k+ 2)eand (5k+ 3)emodulo 5. First, consider
the case (5k+ 2)emodulo 5. Using the binomial theorem, we only need to find 2e(mod 5).
If eis even, then 2e≡1,4(mod 5).This implies that Xq−5iY5i=XY q−1or Xq−5iY5i=X4Yq−4.
In either of the case, we get ℓR(R(I)/(m, It)[q]) = 5q2−4q.
If eis odd, then 2e≡2,3(mod 5).This implies that Xq−5iY5i=X2Yq−2or Xq−5iY5i=X3Yq−3.
In either of the case, ℓR(R(I)/(m, It)[q]) = 5q2−6q. We get the same conclusion in the case (5k+ 3)e
modulo 5.
3. The Hilbert-Kunz function in dimension ≥2.
Let Rbe a Cohen-Macaulay Stanley-Reisner ring of a simplicial complex over an infinite field
with prime characteristic p > 0.Let mbe the maximal homogeneous ideal of Rand Ibe an ideal
generated by a linear system of parameters. It is proved in [4, Theorem 6.1] that m=I∗.Therefore,
using [3, Corollary 4.5] we get eH K ((m,mt)R(m)) = eHK ((I, I t)R(I)).Thus in order to calculate
eHK (R(m)),it is sufficient to calculate eHK ((I , It)R(I)).We start with a more general setup.
In this section, we find the generalized Hilbert-Kunz function of the ideal (I, I t)R(I), where Iis
a parameter ideal in a Cohen-Macaulay local ring. It turns out that in this case, the generalized
Hilbert-Kunz function is eventually a polynomial. Let G(I) = ⊕n≥0In/In+1 be the associated
graded ring of I.
We shall use the following result to find the reduction number of powers of an m-primary ideal.
Theorem 3.1 ([8, Corollary 2.21]).Let (R, m)be a d-dimensional Cohen-Macaulay local ring with
infinite residue field and Ibe an m-primary ideal such that grade(G(I)+)≥d−1.Then for k≥1,
r(Ik) = ⌊n(I)
k⌋+d.

10 KRITI GOEL, MITRA KOLEY, AND J. K. VERMA
Let Rbe a Cohen-Macaulay local ring and Ibe a parameter ideal. Fix s∈N.For a fixed set of
generators of I, define functions
F(n) := HI(I[s], n) = ℓR I[s]
I[s]In!and H(n) := HI(R, n) = ℓRR
In=e(I)n+d−1
d
for all n. Note that if Ris 1-dimensional, then F(n) = H(n) for all n.
Theorem 3.2. Let Rbe a d-dimensional Cohen-Macaulay local ring and let Ibe a parameter ideal.
Let d≥2.For a fixed s∈N,
F(n) = 












d·H(n)if 1≤n≤s,
d−1
X
i=1
(−1)i+1d
iH(n−(i−1)s)if s+ 1 ≤n≤(d−1)s−1,
H(n+s)−sde(I)if n≥(d−1)s.
Proof. Case 1: Let 1 ≤n≤s. Let I= (x1,...,xd).Consider the following epimorphism
R
In⊕dφ
−→ I[s]
I[s]In→0
(a1+In,...,ad+In)7→ (a1xs
1+···+adxs
d) + I[s]In.
Let (a1+In,...,ad+In)∈ker(φ).Then a1xs
1+···+adxs
d∈I[s]Inimplies that
a1xs
1+···+adxs
d=b1xs
1+···+bdxs
d,
where b1,...,bd∈In.Since xs
1,...,xs
dis an R-regular sequence, it follows that (ai−bi)∈I[s]⊆In
for all i= 1,...,d. Thus a1,...,ad∈In.Therefore, ker(φ) = 0 and I[s]/I[s]In≃(R/I n)⊕d.Hence
F(n) = d·H(n), for all 1 ≤n≤s.
Case 2: Let s+ 1 ≤n≤(d−1)s−1.Let x∗
idenote the image of xiin I/I2, for all i= 1,...,d.
As G(I) is Cohen-Macaulay, (x∗)[s]= (xs
1)∗,...,(xs
d)∗∈Is/Is+1 is a G(I)-regular sequence. Hence
the following exact sequence is obtained from the Koszul complex of G(I) with respect to (x∗)[s].
0→G(I)(−(d−1)s)→G(I)(−(d−2)s)(d
1)→ · · · → G(I)(d
1)→G(I)(s)→H0((x∗)[s];G(I)(s)) →0.
As we have an exact sequence of graded G(I)-modules, taking the nth -graded component of each
of these modules gives us the following exact sequence
0→In−(d−1)s
In−(d−1)s+1 → In−(d−2)s
In−(d−2)s+1 !(d
1)
→ · · · → In
In+1 (d
1)
→In+s
In+s+1 →In+s
I[s]In+In+s+1 →0.
For s≤n≤(d−1)s−1,it follows that
ℓRIn+s
I[s]In+In+s+1 =ℓRIn+s
In+s+1 +
d−1
X
i=1
(−1)id
iℓR In−(i−1)s
In−(i−1)s+1 !

HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 11
which implies
ℓRR
I[s]In+In+s+1 =ℓRR
In+s+1 +
d−1
X
i=1
(−1)id
iℓR In−(i−1)s
In−(i−1)s+1 !.(3.1)
We can also write
ℓRR
I[s]In+In+s+1 =ℓRR
I[s]+ℓR I[s]
I[s]In!−ℓR I[s]In+In+s+1
I[s]In!
=ℓRR
I[s]+ℓR I[s]
I[s]In!−ℓRIn+s+1
I[s]In+1 (3.2)
since using Valabrega-Valla criterion ([11, Corollary 2.7]),
I[s]In+In+s+1
I[s]In≃In+s+1
(I[s]∩In+s)∩In+s+1 ≃In+s+1
I[s]∩In+s+1 ≃In+s+1
I[s]In+1 .
Combining equations (3.1) and (3.2), we get
ℓR I[s]
I[s]In!−ℓR I[s]
I[s]In+1 !=
d−1
X
i=1
(−1)id
iℓR In−(i−1)s
In−(i−1)s+1 !
and hence for all s≤n≤(d−1)s−1,
F(n+ 1) −F(n) =
d−1
X
i=1
(−1)i+1d
iH(n−(i−1)s+ 1) −H(n−(i−1)s).
Adding the above equality from sto t−1, for any tsuch that s+ 1 ≤t≤(d−1)s−1, we get
F(t)−F(s) =
d−1
X
i=1
(−1)i+1d
i[H(t−(i−1)s)−H((2 −i)s)]
=d[H(t)−H(s)] +
d−1
X
i=2
(−1)i+1d
iH(t−(i−1)s).
Since F(s) = d·H(s),we get the result for F(n), s+ 1 ≤n≤(d−1)s−1.
Case 3: Let n≥s(d−1).As Ris a Cohen-Macaulay ring and Iis a parameter ideal, it follows
that the associated graded ring G(I) is a polynomial ring and its a-invariant ad(G(I)) = −d < 0.
Using Theorem 3.1, it follows that r(Is) = d−1 if s≥d. Let s < d. Write d=k1s+k2, where
k2∈ {0,1,...,s−1}.Then using Theorem 3.1 it follows that
r(Is) = ⌊−k1−k2
s⌋+d=


d−k1if k2= 0,
d−k1−1 if k26= 0.
In all these cases, it follows that I[s]In=In+sfor all n≥(d−1)s. Thus for all n≥(d−1)s,
F(n) = ℓR I[s]
In+s!=ℓRR
In+s−ℓRR
I[s]=H(n+s)−sde(I).


12 KRITI GOEL, MITRA KOLEY, AND J. K. VERMA
Observe that F(n) is the Hilbert-Samuel function of Ifor the R-module I[s].Let n(I) denote
the postulation number of Iin Rand PI(n) denote the Hilbert-Samuel polynomial of Iin R.
Using Theorem 3.2, it follows that F(n) is a polynomial, given by P(n+s)−sde(I),for n >
max{n(I) + s+ 1,(d−1)s}.
We recall some properties of Stirling numbers of the second kind.
Remark 3.3 ([10, Chapter 1, Sec. 1.4]).The Stirling number of the second kind, denoted by
S(n, k), admits the following characterizations:
(1) S(n, k) is equal to the number of partitions of the set [n] = {1,...,n}into kblocks.
(2) S(n, k) = 1
k!
k
X
i=0
(−1)k−ik
iin.
Using (1), it is easy to check that
(a) d+ 1
2=S(d+ 1, d) = 1
d!
d
X
i=0
(−1)d−id
iid+1,
(b) S(d, d) = 1,and
(c) S(d−1, d) = 0.
Lemma 3.4. For d≥2,put β1= (d−2)/2and β2= (d−1)(3d−10)/24. Then for any α∈N,
αq +d−1
d+ 1 =αd+1
(d+ 1)!qd+1 +αdβ1
d!qd+αd−1β2
(d−1)!qd−1+··· ,
Proof.
αq +d−1
d+ 1 =(αq +d−1)(αq +d−2) ···(αq −1)
(d+ 1)!
=αd+1
(d+ 1)! q−(1 −d)
αq−(2 −d)
α···q−1
α=αd+1
(d+ 1)! ·f(q).
If we write f(q) = qd+1 −σ1qd+σ2qd−1+···+ (−1)d+1 σd+1, then
σ1=
d+1
X
n=1 (n−d)
α=1
α(d+ 2)(d+ 1)
2−d(d+ 1)=−(d−2)(d+ 1)
2α=−(d+ 1)β1
α.
Let w2=
d+1
X
n=1 n−d
α2
.Then using Newton’s identities, w2=σ2
1−2σ2.Consider
w2=
d+1
X
n=1 n−d
α2
=1
α2
d+1
X
n=1 n2−2nd +d2
=1
α2(d+ 1)(d+ 2)(2d+ 3)
6−d(d+ 2)(d+ 1) + d2(d+ 1)
=1
α2(d2+ 3d+ 2)(2d+ 3)
6−d(d2+ 3d+ 2) + d3+d2
=1
α22d3+ 9d2+ 13d+ 6
6−2d2−2d=2d3−3d2+d+ 6
6α2.

HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 13
Then
σ2=σ2
1−w2
2=1
2(d−2)2(d+ 1)2
4α2−2d3−3d2+d+ 6
6α2
=1
2d4−2d3−3d2+ 4d+ 4
4α2−2d3−3d2+d+ 6
6α2
=3d4−10d3−3d2+ 10d
24α2=d(d+ 1)(d−1)(3d−10)
24α2=d(d+ 1)β2
α2.
Therefore,
αq +d−1
d+ 1 =αd+1
(d+ 1)!qd+1 +αdβ1
d!qd+αd−1β2
(d−1)!qd−1+··· ,
where β1= (d−2)/2 and β2= (d−1)(3d−10)/24. 
Theorem 3.5. Let Rbe a d-dimensional Cohen-Macaulay local ring and d≥2.Let Ibe a parameter
ideal of Rand I= (I, It)R(I).Let s∈N.
(1) Let s < d. Write d=k1s+k2where k2∈ {0,1,...,s−1}.If k2= 0, then
ℓRR(I)
I[s]= (d−k1+1)sd+1e(I)+d·e(I)s+d−1
d+ 1 −
d−1
X
i=0 (−1)id
ie(I)(d−i−k1+ 1)s+d−1
d+ 1 .
If k26= 0,then
ℓRR(I)
I[s]= (d−k1)sd+1e(I)+d·e(I)s+d−1
d+ 1 −
d−1
X
i=0 (−1)id
ie(I)(d−i−k1)s+d−1
d+ 1 .
(2) Let s≥d. Then
ℓRR(I)
I[s]=dsd+1e(I) + d·e(I)s+d−1
d+ 1 −
d−1
X
i=0 (−1)id
ie(I)(d−i)s+d−1
d+ 1 .
In other words, for slarge,
ℓRR(I)
I[s]=c(d)e(I)sd+1 +e(I)d−2
2 1
(d−1)! −1sd+e(I)d(d−1)(3d−10)
24(d−1)! sd−1+··· ,
implying that the generalized Hilbert-Kunz multiplicity eH K ((I, I t)R(I)) = c(d)·e(I).
Proof. We first find ℓR(R(I)/I[s]).Consider
I[s]= (I[s], I[s]ts) = s−1
M
n=0
I[s]Intn!+
M
n≥s
I[s]In−stn
.
Since G(I) is Cohen-Macaulay and the a-invariant ad(G(I)) = −d < 0, using Theorem 3.1 it follows
that r(Is) = d−1 if s≥d. Let s < d. Write d=k1s+k2, where k2∈ {0,1,...,s−1}.Then using
Theorem 3.1 it follows that
r(Is) = ⌊−k1−k2
s⌋+d=


d−k1if k2= 0,
d−k1−1 if k26= 0.

14 KRITI GOEL, MITRA KOLEY, AND J. K. VERMA
Case 1: Let s < d. Write d=k1s+k2, where k2∈ {0,1,...,s−1}.Then as observed above, r(Is) =
d−k1−j, j ∈ {0,1}.As I[s]is a minimal reduction of Is, we get, I[s]I(d−k1−j)s=I(d−k1−j+1)s.In
other words, I[s]In−s=In, for all n≥(d−k1−j+ 1)s. Therefore,
I[s]= s−1
M
n=0
I[s]Intn!+

(d−k1−j+1)s−1
M
n=s
I[s]In−stn
+
M
n≥(d−k1−j+1)s
Intn
.
Consider
ℓRR(I)
I[s]=
s−1
X
n=0
ℓRIn
I[s]In+
(d−k1−j+1)s−1
X
n=s
ℓRIn
I[s]In−s
=
s−1
X
n=0
ℓRR
I[s]In+
(d−k1−j+1)s−1
X
n=s
ℓRR
I[s]In−s−
(d−k1−j+1)s−1
X
n=0
ℓRR
In
= (d−k1−j+ 1)s·ℓRR
I[s]+ 2
s−1
X
n=0
ℓR I[s]
I[s]In!+
(d−k1−j)s−1
X
n=s
ℓR I[s]
I[s]In!
−
(d−k1−j+1)s−1
X
n=1
e(I)n+d−1
d.
As s < d, we get k1≥1.Thus, d−k1−j≤d−1.Using Theorem 3.2, we get
ℓRR(I)
I[s]
= (d−k1−j+ 1)sd+1e(I) + 2d·e(I)
s−1
X
n=1 n+d−1
d+d·e(I)
(d−k1−j)s−1
X
n=sn+d−1
d
+
(d−k1−j)s−1
X
n=s"d−1
X
i=2
(−1)i+1d
ie(I)n−(i−1)s+d−1
d#−
(d−k1−j+1)s−1
X
n=1
e(I)n+d−1
d
= (d−k1−j+ 1)sd+1e(I) + 2d·e(I)s+d−1
d+ 1 +d·e(I)(d−k1−j)s+d−1
d+ 1 −s+d−1
d+ 1 
−
d−1
X
i=2 (−1)id
ie(I)(d−j−k1−i+ 1)s+d−1
d+ 1 −e(I)(d−k1−j+ 1)s+d−1
d+ 1 
= (d−k1−j+ 1)sd+1e(I) + d·e(I)s+d−1
d+ 1 −
d−1
X
i=0 (−1)id
ie(I)(d−j−k1−i+ 1)s+d−1
d+ 1 .
Case 2: Let s≥d. Then r(Is) = d−1.As I[s]is a minimal reduction of Is, we get, I[s]I(d−1)s=Ids.
In other words, I[s]In−s=In, for all n≥ds. Therefore,
I[s]= s−1
M
n=0
I[s]Intn!+ ds−1
M
n=s
I[s]In−stn!+
M
n≥ds
Intn
.

HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 15
Consider
ℓRR(I)
I[s]=
s−1
X
n=0
ℓRIn
I[s]In+
ds−1
X
n=s
ℓRIn
I[s]In−s
=
s−1
X
n=0
ℓRR
I[s]In+
ds−1
X
n=s
ℓRR
I[s]In−s−
ds−1
X
n=0
ℓRR
In
=ds ·ℓRR
I[s]+ 2
s−1
X
n=0
ℓR I[s]
I[s]In!+
(d−1)s−1
X
n=s
ℓR I[s]
I[s]In!−
ds−1
X
n=1
ℓRR
In
=dsd+1e(I) + 2
s−1
X
n=1
ℓR I[s]
I[s]In!+
(d−1)s−1
X
n=s
ℓR I[s]
I[s]In!−
ds−1
X
n=1
e(I)n+d−1
d.
Using Theorem 3.2, we get
ℓRR(I)
I[s]=dsd+1e(I) + 2d·e(I)
s−1
X
n=1 n+d−1
d+d·e(I)
(d−1)s−1
X
n=sn+d−1
d
+
(d−1)s−1
X
n=s"d−1
X
i=2
(−1)i+1d
ie(I)n−(i−1)s+d−1
d#−
ds−1
X
n=1
e(I)n+d−1
d
=dsd+1e(I) + 2d·e(I)s+d−1
d+ 1 +d·e(I)(d−1)s+d−1
d+ 1 −s+d−1
d+ 1 
−
d−1
X
i=2 (−1)id
ie(I)(d−i)s+d−1
d+ 1 −e(I)ds +d−1
d+ 1 
=dsd+1e(I) + d·e(I)s+d−1
d+ 1 −
d−1
X
i=0 (−1)id
ie(I)(d−i)s+d−1
d+ 1 .(3.3)
This implies that the Hilbert-Kunz function is a polynomial for s≥d. The coefficient of sd+1 in
the above expression is
e(I)"d+d
(d+ 1)! −
d−1
X
i=0
(−1)id
i(d−i)d+1
(d+ 1)! #=d·e(I)"1 + 1
(d+ 1)! −
d−1
X
i=0
(−1)id
i(d−i)d+1
d(d+ 1)! #
=d·e(I)1 + 1
(d+ 1)! −1
d(d+ 1)S(d+ 1, d).
Using Remark 3.3, we get eHK ((I , It),R(I)) = e(I)·c(d).We now use Lemma 3.4 to find coefficients
of sdand sd−1in the expression (3.3). Coefficient of sdin expression (3.3) is
e(I)"dβ1
d!−
d
X
i=0
(−1)id
i(d−i)dβ1
d!#=e(I)d−2
2 1
(d−1)! −S(d, d)
=e(I)d−2
2 1
(d−1)! −1

16 KRITI GOEL, MITRA KOLEY, AND J. K. VERMA
and coefficient of sd−1is
e(I)"dβ2
(d−1)! −
d
X
i=0
(−1)id
i(d−i)d−1β2
(d−1)! #=e(I)(d−1)(3d−10)
24  d
(d−1)! −dS(d−1, d)
=e(I)d(d−1)(3d−10)
24(d−1)! .

Corollary 3.6. Let (R, m)be a d-dimensional regular local ring with d≥2.Then for s≥d,
ℓRR(m)
(m,mt)[s]=c(d)e(m)sd+1+e(m)d−2
2 1
(d−1)! −1sd+e(m)d(d−1)(3d−10)
24(d−1)! sd−1+··· .
Proof. Use I=min Theorem 3.5.
Corollary 3.7. Let (R, m)be a Cohen-Macaulay local ring of dimension dwith positive prime
characteristic p > 0.If there exists a parameter ideal Iof Rsuch that I∗=m,then
eHK (R(m)) = c(d)e(m).
Proof. If there exists a parameter ideal Iof Rsuch that I∗=m,then using the same arguments
as in the proof of [3, Corollary 4.5], it follows that ((I, I t)R(m))∗= ((m,mt)R(m))∗and hence
eHK ((m,mt)R(m)) = eH K ((I, It)R(I)).Therefore,
eHK (R(m)) = eH K ((m,mt)R(m)) = eHK ((I , It)R(I)) = c(d)e(I) = c(d)e(m).
The latter equality holds as I∗=mimplies that I=m.
Corollary 3.8. Let Rbe a d-dimensional Cohen-Macaulay Stanley-Reisner ring of a simplicial
complex over an infinite field with prime characteristic p > 0.Let mbe a maximal homogeneous
ideal of R. Then eHK (R(m)) = c(d)·fd−1, where fd−1is the number of facets in the simplicial
complex.
Proof. Let Ibe an ideal generated by a linear system of parameters. It is proved in [4, Theorem
6.1] that m=I∗.Therefore, using Corollary 3.7 it follows that eHK (R(m)) = c(d)·e(m).Since
e(m) = fd−1, we are done. 
Example 3.9. Let k[[X1, X2, X3]] be a power series ring in 3variables over a field k. Let I=
(Xn1
1, Xn2
2, Xn3
3), where n1, n2, n3∈N.Then
ℓRR(I)
(I, It)[2] =e(I)"32 + 3 −
2
X
i=0 (−1)i3
i6−2i
4#= 23 ·e(I) = 23 ·n1n2n3
and for s≥3,
ℓRR(I)
I[s]=n1n2n3·"3s4+ 3s+ 2
4−
2
X
i=0 (−1)i3
i(3 −i)s+ 2
4#.

HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 17
In particular,
ℓRR(m)
(m,mt)[s]="3s4+ 3s+ 2
4−
2
X
i=0 (−1)i3
i(3 −i)s+ 2
4#
=13
8s4−1
4s3−1
8s2−1
4s,
for s≥3.
Example 3.10. Let R=k[[X, Y, Z]]/(XY −Zn)for some positive integer n≥2and let Ibe a
parameter ideal of R. Then for s≥2,
ℓRR(I)
I[s]=e(I)"2s3+ 2s+ 1
3−
1
X
i=0 (−1)i2
i(2 −i)s+ 1
3#
=e(I)4
3s3−1
3s.
References
[1] Holger Brenner. The Hilbert-Kunz function in graded dimension two. Comm. Algebra,
35(10):3199–3213, 2007. 1
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[5] C. Han and P. Monsky. Some surprising Hilbert-Kunz functions. Math. Z., 214(1):119–135,
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[8] Thomas John Marley. Hilbert functions of ideals in Cohen-Macaulay rings. ProQuest LLC,
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[9] P. Monsky. The Hilbert-Kunz function. Math. Ann., 263(1):43–49, 1983. 1,6,8
[10] Richard P. Stanley. Enumerative combinatorics. Vol. 1, volume 49 of Cambridge Studies in
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18 KRITI GOEL, MITRA KOLEY, AND J. K. VERMA
[12] J. K. Verma. Rees algebras and mixed multiplicities. Proc. Amer. Math. Soc., 104(4):1036–
1044, 1988. 2,5
Indian Institute of Technology Bombay, Mumbai, India 400076
E-mail address:kritigoel.maths@gmail.com
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, India
400005
E-mail address:mitrak@math.tifr.res.in
Indian Institute of Technology Bombay, Mumbai, India 400076
E-mail address:jkv@math.iitb.ac.in