Generalized Hilbert-Kunz function of the Rees algebra of the face ring of a simplicial complex

Abstract

Let R be the face ring of a simplicial complex of dimension d−1d-1 and R(n){\mathcal R}(\mathfrak{n}) be the Rees algebra of the maximal homogeneous ideal n\mathfrak{n} of R. We show that the generalized Hilbert-Kunz function HK(s)=ℓ(R(n)/(n,nt)[s])HK(s)=\ell({\mathcal R}(\mathfrak n)/(\mathfrak n, \mathfrak n t)^{[s]}) is given by a polynomial for all large s. We calculate it in many examples and also provide a Macaulay2 code for computing $HK(s).

Full text

GENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA OF
THE FACE RING OF A SIMPLICIAL COMPLEX
ARINDAM BANERJEE, KRITI GOEL, AND J. K. VERMA
Abstract. Let Rbe the face ring of a simplicial complex of dimension d−1 and R(n) be the
Rees algebra of the maximal homogeneous ideal nof R. We show that the generalized Hilbert-Kunz
function HK (s) = `(R(n)/(n,nt)[s]) is given by a polynomial for all large s. We calculate it in many
examples and also provide a Macaulay2 code for computing HK (s).
Dedicated to Roger Wiegand and Silvia Wiegand on the occasion of their 150th birthday
1. Introduction
The objective of this paper is to find the generalized Hilbert-Kunz function of the maximal homo-
geneous ideal of the Rees algebra of the maximal homogeneous ideal of the face ring of a simplicial
complex. The Hilbert-Kunz functions of the Rees algebra, associated graded ring and the extended
Rees algebra have been studied by K. Eto and K.-i. Yoshida in [3] and by K. Goel, M. Koley and
J. K. Verma in [5].
In order to recall one of the main results of Eto and Yoshida, we set up some notation first. Let
(R, m) be a Noetherian local ring of dimension dand of prime characteristic p. Let q=pewhere eis
a non-negative integer. The qth Frobenius power of an ideal Iis defined to be I[q]= (aq|a∈I).Let
Ibe an m-primary ideal. The Hilbert-Kunz function of Iis the function H KI(q) = `(R/I[q]).This
function, for I=m, was introduced by E. Kunz in [8] who used it to characterize regular local rings.
The Hilbert-Kunz multiplicity of an m-primary ideal Iis defined as eHK (I) = lim
q→∞ `(R/I[q])/qd.
It was introduced by P. Monsky in [10]. We refer the reader to an excellent survey paper of C.
Huneke [7] for further details.
Eto and Yoshida calculated the Hilbert-Kunz multiplicity of various blowup algebras of an ideal
under certain conditions. Put c(d) = (d/2) + d/(d+ 1)!.They proved the following.
Theorem 1.1. Let (R, m)be a Noetherian local ring of prime characteristic p > 0with d= dim R≥
1.Then for any m-primary ideal I, we have
eHK (R(I)) ≤c(d)·e(I).
2010 AMS Mathematics Subject Classification: Primary 13A30, 13D40, 13F55.
Key words and phrases: Generalized Hilbert-Kunz function, Generalized Hilbert-Kunz multiplicity, Stanley-Reisner
ring.
1
arXiv:2003.01438v1 [math.AC] 3 Mar 2020

2 ARINDAM BANERJEE, KRITI GOEL, AND J. K. VERMA
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).
It is natural to ask if there is a formula for the Hilbert-Kunz function and the Hilbert-Kunz
multiplicity of the maximal homogeneous ideal (m, It) of the Rees algebra R(I) = ⊕∞
n=0Intnwhere
Iis an m-primary ideal, in terms of invariants of the ideals mand I. In this paper we answer this
question for the Rees algebra of the maximal homogeneous ideal of the face ring of a simplicial
complex. In fact, we find its generalized Hilbert-Kunz function. The generalized Hilbert-Kunz
function was introduced by Aldo Conca in [2]. Let (R, m) be a d-dimensional Noetherian local
(resp. standard graded) ring with maximal (resp. maximal homogeneous) ideal mand Ibe an m-
primary (resp. a graded m-primary) ideal. Fix a set of generators of I, say I= (a1, a2, . . . , ag).We
choose these as homogeneous elements in case Ris a graded ring. Define the sth Frobenius power
of Ito be the ideal I[s]= (as
1, as
2, . . . , as
g).The generalized Hilbert-Kunz function of Iis defined
as HKI(s) = `(R/I[s]).The generalized Hilbert-Kunz multiplicity is defined as lim
s→∞ HKI(s)/sd,
whenever the limit exists.
We now describe the contents of the paper. Let ∆ be a simplicial complex of dimension d−1.
Let kbe any field, k[∆] denote the face ring of ∆ and nbe its maximal homogeneous ideal. Let
R(n) = ⊕∞
n=0nntnbe the Rees algebra of n.In section 2, we collect some preliminaries required for
estimation of the asymptotic reduction number in terms of the a-invariants of local cohomology
modules and Hilbert-Samuel polynomial of the maximal homogeneous ideal of the face ring of
a simplicial complex. Section 3 is devoted to the computation of the generalized Hilbert-Kunz
function HK(n,nt)(s), where (n,nt) is the maximal homogeneous ideal of the Rees algebra R(n).We
also estimate an upper bound on the postulation number of HK(n,nt)(s) in terms of a-invariants of
the local cohomology modules. This enables us to explicitly calculate the generalized Hilbert-Kunz
function for the Rees algebra in several examples such as the edge ideal of a complete bipartite graph,
the real projective plane and a few other examples of simplicial complexes. We have implemented
the formula for the Hilbert-Kunz function in an algorithm written in the language of Macaulay2.
2. Preliminaries
In this section we gather some results which we shall use in the later sections.
Let Rbe a ring and Ibe an Rideal. An ideal J⊆Iis called a reduction of Iif JIn=In+1 , for
all large n. A minimal reduction of Iis a reduction of Iminimal with respect to inclusion. For
a minimal reduction Jof I, we set rJ(I) = min{n|Im+1 =JImfor all m≥n}.The reduction
number of Iis defined as
r(I) = min{rJ(I)|Jis a minimal reduction of I}.
We shall use the following results to estimate the reduction number of powers of an ideal.

GENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 3
Theorem 2.1 ([9, 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,
where n(I)denotes the postulation number of I.
Theorem 2.2 ([4, Theorem 2.1]).Let (R, m)be a Noetherian local ring and let I⊆mbe an
R-ideal. Then rJ(In)is independent of Jand stable if nis large. In particular, for all n >
max{|ai(G(I))|:ai(G(I)) 6=−∞}, we get
rJ(In) = 


sif as(G(I)) ≥0,
s−1if as(G(I)) <0,
where Jis any minimal reduction of Inand sis the analytic spread of I.
Let Sbe a d-dimensional Cohen-Macaulay local ring and let Ibe a parameter ideal. Fix s∈N.
For a fixed set of generators of I, define functions
F(n) := HI(I[s], n) = `S I[s]
I[s]In!and H(n) := HI(S, n) = `SS
In=e(I)n+d−1
d
for all n. Note that if Sis 1-dimensional, then F(n) = H(n) for all n. In [5], the authors prove that
the function F(n) is a piecewise polynomial in n.
Theorem 2.3 ([5, Theorem 3.2]).Let Sbe a d-dimensional Cohen-Macaulay local ring and 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.
Let ∆ be a (d−1)-dimensional simplicial complex on nvertices and kbe a field. Let R=k[∆]
be the corresponding Stanley-Reisner ring and nbe its unique maximal homogeneous ideal. Let
h(∆) = (h0, h1, . . . , hd) and f(∆) = (f−1, f0, . . . , fd−1) denote the h-vector and f-vector of ∆
respectively. We use the following formula for the Hilbert-Samuel polynomial of nin the main
result.
Theorem 2.4 ([6, Theorem 6.2]).Set h(λ) = h0+h1λ+· · · +hdλd. Let h(i)(λ)denote the i-th
derivative of h(λ)with respect to λ. Then for all n≥1,
`R
nn=
d
X
i=0
(−1)ih(i)(1)
i!n+d−i−1
d−i.

4 ARINDAM BANERJEE, KRITI GOEL, AND J. K. VERMA
3. The generalized Hilbert-Kunz function of (n,nt)
Let S=k[x1, . . . , xr] be a polynomial ring in rvariables over a field kand let m= (x1, . . . , xr)
denote the maximal homogeneous ideal of S. Let Pj, for j= 1, . . . , α and α≥2, be distinct S-
ideals generated by subsets of {x1, . . . , xr}.Let I=∩α
j=1Pjand R=S/I . Let n=m/I denote the
maximal homogeneous ideal of R.
In this section, we find the generalized Hilbert-Kunz function of the maximal homogeneous ideal
(n,nt) of the Rees algebra R(n) of R. We begin by proving that for s, n ∈N,`S(S/I +m[s]mn) is
a piecewise polynomial in sand n. First we prove the following result which is a consequence of
Theorem 2.3.
Corollary 3.1. Let S=k[x1, . . . , xd]be a polynomial ring in dvariables over a field k. Let
m= (x1, . . . , xd)be its maximal homogeneous ideal. Let s, n ∈N.
(1) If d= 1, then `S
m[s]mn=s+n.
(2) If d= 2, then
`S
m[s]mn=




s2+n2+nif 1≤n≤s,
n+s+ 1
2if n≥s.
(3) If d≥3, then
`S
m[s]mn=





















sd+dn+d−1
dif 1≤n≤s,
sd+
d−1
X
i=1
(−1)i+1d
in−(i−1)s+d−1
dif s+ 1 ≤n≤(d−1)s−1,
n+s+d−1
dif n≥(d−1)s.
Proof. Let s, n ∈N.If d= 1, then S=k[x] and m= (x) implying that
`S
m[s]mn=`k[x]
(xs+n)=s+n.
Let d≥2.Since
`S
m[s]mn=`S
m[s]+` m[s]
m[s]mn!
and `(S/m[s]) = sd, the result follows from Theorem 2.3.
Let T=⊕n≥0Tnbe a Noetherian graded ring, where T0is an Artinian ring. Let M=⊕n≥0Mn
be a finitely generated graded T-module. Then `T0(Mn)<∞.The Hilbert series H(M, λ) of Mis
defined by H(M, λ) = X
n≥0
`T0(Mn)λn.

GENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 5
Theorem 3.2. Let Tbe a standard graded Artinian ring and let I1, . . . , Iα, for α≥2, be homoge-
neous T-ideals. Let I=∩α
i=1Ii.Then
HT
I, λ=
α
X
i=1
HT
Ii
, λ−
α
X
i,j=1
i<j
HT
Ii+Ij
, λ+· · · + (−1)α−1HT
Pα
i=1 Ii
, λ.
Proof. Apply induction on α. Let α= 2.Consider the following short exact sequence
0−→ T
I1∩I2
−→ T
I1MT
I2
−→ T
I1+I2
−→ 0.
Then
HT
I, λ=HT
I1∩I2
, λ=HT
I1
, λ+HT
I2
, λ−HT
I1+I2
, λ.
Let α > 2 and consider the short exact sequence
0−→ T
∩α
i=1Ii
−→ T
∩α−1
i=1 IiMT
Iα
−→ T
∩α−1
i=1 Ii+Iα
−→ 0.
Using induction hypothesis, it follows that
HT
∩α
i=1Ii
, λ=HT
∩α−1
i=1 Ii
, λ+HT
Iα
, λ−HT
∩α−1
i=1 Ii+Iα
, λ
=
α−1
X
i=1
HT
Ii
, λ−
α−1
X
i,j=1
i<j
HT
Ii+Ij
, λ+· · · + (−1)α−2H T
Pα−1
i=1 Ii
, λ!
+HT
Iα
, λ−
α−1
X
i=1
HT
Ii+Iα
, λ+
α−1
X
i,j=1
i<j
HT
Ii+Ij+Iα
, λ+· · ·
+ (−1)α−1H T
Pα−1
i=1 Ii+Iα
, λ!.
Rearranging the terms gives the required result. 
Corollary 3.3. Let S=k[x1, . . . , xr]be a polynomial ring in rvariables over a field kand let
m= (x1, . . . , xr)be the maximal homogeneous ideal of S. Let P1, . . . , Pα, for α≥2, be distinct
S-ideals generated by subsets of {x1, . . . , xr}.Let I=∩α
i=1Pi.Then for s, n ∈N,
`S
I+m[s]mn=
α
X
i=1
`S
Pi+m[s]mn−X
1≤i<j≤α
`S
Pi+Pj+m[s]mn+· · ·
+ (−1)α−1`S
Pα
i=1 Pi+m[s]mn.(3.1)
In particular, `S
I+m[s]mnis a piecewise polynomial in sand n.

6 ARINDAM BANERJEE, KRITI GOEL, AND J. K. VERMA
Proof. Since S/m[s]mnis a standard graded Artinian ring, using Theorem 3.2 it follows that
HS
I+m[s]mn, λ=
α
X
i=1
HS
Pi+m[s]mn, λ−
α
X
i,j=1
i<j
HS
Pi+Pj+m[s]mn, λ+· · ·
+ (−1)α−1HS
Pα
i=1 Pi+m[s]mn, λ.(3.2)
The modules involved on the right side of (3.2) are finite length S-modules. Put λ= 1 in (3.2) to
get (3.1). Observe that S/(Pi1+· · · +Pij), for i1, . . . , ij∈ {1, . . . , α}, is isomorphic to a polynomial
ring. Since image of min S/(Pi1+· · · +Pij) is a parameter ideal for all i1, . . . , ij∈ {1, . . . , α},
using Corollary 3.1 we obtain the required result. 
The following result is a generalization of A. Conca’s result ([2, Remark 2.2]).
Theorem 3.4. For s≥1, the generalized Hilbert-Kunz function of R=k[∆] is given by the
equation
`R
n[s]=
d
X
i=0
fi−1(s−1)i.
Proof. Observe that `(R/n[s]) = |V|, where
V={a= (a1, . . . , ar)∈Nr|0≤ai≤s−1 and Supp(a)∈∆}.
Therefore, for s≥1,
`R
n[s]=X
F∈∆
|{a∈V|Supp(a)∈F}| =X
F∈∆
(s−1)|F|=
d
X
i=0
fi−1(s−1)i.

We are now ready to prove the main result of the section. We first consider the general case.
3.1. The generalized Hilbert-Kunz function of (n,nt).
Theorem 3.5. Let S=k[x1, . . . , xr]be a polynomial ring in rvariables over a field kand let m
be the maximal homogeneous ideal of S. Let P1, . . . , Pα, for α≥2, be distinct S-ideals generated
by subsets of {x1, . . . , xr}.Let I=∩α
i=1Piand R=S/I . Suppose n=m/I denotes the maximal
homogeneous ideal of Rand dim(R) = d. Set δ= max{|ai(R)|:ai(R)6=−∞}.Then for s>δ,
`R(n)
(n,nt)[s]
is a polynomial in s.

GENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 7
Proof. Since Ris a standard graded ring, it follows that R'G(n).Let s > δ. Using Theorem 2.2,
it follows that
r(ns) = 


d−1 if ad(R)<0,
dif ad(R)=0.
In other words, r(ns) = d−j, where jis either 0 or 1 as per the above observation. As n[s]is
a minimal reduction of ns, we get, n[s]n(d−j)s=n(d−j+1)s.In other words, n[s]nn−s=nn, for all
n≥(d−j+ 1)s. This implies that
(n,nt)[s]= (n[s],n[s]ts) = s−1
M
n=0
n[s]nntn!+
M
n≥s
n[s]nn−stn

= s−1
M
n=0
n[s]nntn!+

(d−j+1)s−1
M
n=s
n[s]nn−stn
+
M
n≥(d−j+1)s
nntn
.
Therefore, for s>δ,
`R(n)
(n,nt)[s]=
s−1
X
n=0
`nn
n[s]nn+
(d−j+1)s−1
X
n=s
`nn
n[s]nn−s
=
s−1
X
n=0
`R
n[s]nn+
(d−j+1)s−1
X
n=s
`R
n[s]nn−s−
(d−j+1)s−1
X
n=0
`R
nn
=
s−1
X
n=1
`S
I+m[s]mn+
(d−j+1)s−1
X
n=s+1
`S
I+m[s]mn−s−
(d−j+1)s−1
X
n=1
`R
nn+ 2 `R
n[s]
= 2
s−1
X
n=1
`S
I+m[s]mn+
(d−j)s−1
X
n=s
`S
I+m[s]mn−
(d−j+1)s−1
X
n=1
`R
nn+ 2 `R
n[s].
The result now follows from Corollary 3.3, Theorem 2.4 and Theorem 3.4.
3.2. The generalized Hilbert-Kunz function of (n,nt)for Cohen-Macaulay k[∆].
Theorem 3.6. Let S=k[x1, . . . , xr]be a polynomial ring in rvariables over a field kand let m
be the maximal homogeneous ideal of S. Let P1, . . . , Pα, for α≥2, be distinct S-ideals generated
by subsets of {x1, . . . , xr}.Let I=∩α
i=1Piand R=S/I . Suppose n=m/I denotes the maximal
homogeneous ideal of Rand dim(R) = d. Suppose that Ris Cohen-Macaulay. Then
`R(n)
(n,nt)[s]
is given by a polynomial for s≥1.
Proof. Since Ris a standard graded ring, it follows that R'G(n).Let h(∆) = (h0, . . . , hd) denote
the h-vector of R. Note that −d<n(n)≤0.If n(n) = −d, then h0= 1 and hi= 0 for all i6= 0,

8 ARINDAM BANERJEE, KRITI GOEL, AND J. K. VERMA
implying that 0 = h1=r−d. It follows that Iis a height zero ideal, which is not true. Hence,
−d<n(n)≤0.
Suppose n(n) = 0.Using Theorem 2.1, it follows that r(Is) = d, for all s≥1.Using the same
arguments as in the proof of Theorem 3.5, it follows that for s≥1,
`R(n)
(n,nt)[s]= 2
s−1
X
n=1
`S
I+m[s]mn+
ds−1
X
n=s
`S
I+m[s]mn−
(d+1)s−1
X
n=1
`R
nn+ 2 `R
n[s].
The result now follows from Corollary 3.3, Theorem 2.4 and Theorem 3.4.
Now suppose that n(n)<0.If s < |n(n)|,write |n(n)|=k1s+k2, where k2∈ {0,1, . . . , s −1}.
Using Theorem 2.1, it follows that
r(ns) = 








d−k1if s < |n(n)|, k2= 0,
d−k1−1 if s < |n(n)|, k26= 0,
d−1 if s≥ |n(n)|.
In other words, r(ns) = d−j, where j∈ {1, k1, k1+ 1}as per the above observation. Using the
same arguments as in the proof of Theorem 3.5, we are done. 
4. Examples
In this section, we illustrate the above results using some examples.
Example 4.1. Let ∆ be the simplicial complex
x1x2x3x4
Then R=k[x1, x2, x3, x4]/((x1, x2)∩(x3, x4)) is the face ring of ∆.Observe that Ris a 2-dimensional
ring with f-vector f(∆) = (1,4,2) and h-vector h(∆) = (1,2,−1).Set S=k[x1, x2, x3, x4], P1=
(x1, x2), P2= (x3, x4).Since depth(R) = 1, it follows that a0(R) = −∞.In order to find a1(R) and
a2(R), we consider the following short exact sequence.
0−→ S
P1∩P2
−→ S
P1MS
P2
−→ S
P1+P2
−→ 0
Using the corresponding long exact sequence of local cohomology modules, it follows that
H1
n(R)'H0
m(S/(P1+P2)) and H2
n(R)'H2
(x3,x4)(k[x3, x4]) ⊕H2
(x1,x2)(k[x1, x2]).
This implies that a1(R) = 0 and a2(R) = −2.Hence, δ= max{|ai(R)|:ai(R)6=−∞} = 2.Since
a2(R)<0, using Theorem 3.5 it follows that for all s > 2,
`R(n)
(n,nt)[s]= 2
s−1
X
n=1
`S
I+m[s]mn−
2s−1
X
n=1
`R
nn+ 2 `R
n[s].

GENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 9
From Corollary 3.3, Theorem 2.4 and Theorem 3.4, we obtain
`R(n)
(n,nt)[s]= 2
s−1
X
n=1 `S
P1+m[s]mn+`S
P2+m[s]mn−`S
P1+P2+m[s]mn
−
2s−1
X
i=1 "2
X
i=0
(−1)ih(i)(1)
i!n+ 1 −i
2−i#+ 2
2
X
i=0
fi−1(s−1)i.
Substituting the values and using Corollary 3.1, we get
`R(n)
(n,nt)[s]= 2
s−1
X
n=1 2(s2+n2+n)−1−
2s−1
X
n=1 2n+ 1
2−1+ 21 + 4(s−1) + 2(s−1)2.
Simplifying the above expression, we obtain that for all s > 2,
`R(n)
(n,nt)[s]=8
3s3−2
3s−1
= 16s+ 2
3−16s+ 1
2+ 2s−1.
Example 4.2. Let ∆ be the simplicial complex
x1x2
x3
x4
Then R=k[x1, x2, x3, x4]/((x1)∩(x3, x4)) is the face ring of ∆.Observe that Ris a 3-dimensional
ring with f-vector f(∆) = (1,4,4,1) and h-vector h(∆) = (1,1,−1,0).Set S=k[x1, x2, x3, x4],
P1= (x3, x4), P2= (x1).Since depth(R) = 2, it follows that a0(R) = a1(R) = −∞.In order to
find a2(R) and a3(R), we consider the following short exact sequence.
0−→ S
P1∩P2
−→ S
P1MS
P2
−→ S
P1+P2
−→ 0
Using the corresponding long exact sequence of local cohomology modules, we get
H3
n(R)'H3
(x2,x3,x4)(k[x2, x3, x4]) and 0 →H1
(x2)(k[x2]) →H2
n(R)→H2
(x1,x2)(k[x1, x2]) →0.
This implies that a2(R) = −1 and a3(R) = −3.Hence, δ= max{|ai(R)|:ai(R)6=−∞} = 3.Since
a3(R)<0, using Theorem 3.5 it follows that for all s > 3,
`R(n)
(n,nt)[s]= 2
s−1
X
n=1
`S
I+m[s]mn+
2s−1
X
n=s
`S
I+m[s]mn−
3s−1
X
n=1
`R
nn+ 2 `R
n[s].

10 ARINDAM BANERJEE, KRITI GOEL, AND J. K. VERMA
From Corollary 3.3, Theorem 2.4 and Theorem 3.4, we obtain
`R(n)
(n,nt)[s]= 2
s−1
X
n=1 `S
P1+m[s]mn+`S
P2+m[s]mn−`S
P1+P2+m[s]mn
+
2s−1
X
n=s`S
P1+m[s]mn+`S
P2+m[s]mn−`S
P1+P2+m[s]mn
−
3s−1
X
i=1 "3
X
i=0
(−1)ih(i)(1)
i!n+d−i−1
d−i#+ 2
3
X
i=0
fi−1(s−1)i.
Substituting the values and using Corollary 3.1, we get
`R(n)
(n,nt)[s]= 2
s−1
X
n=1 (s2+n2+n) + s3+ 3n+ 2
3−(s+n)
+
2s−1
X
n=sn+s+ 1
2+s3+ 3n+ 2
3−3n−s+ 2
3−(s+n)
−
3s−1
X
n=1 n+ 2
3+n+ 1
2−n+ 21 + 4(s−1) + 4(s−1)2+ (s−1)3.
Simplifying the above expression, we obtain that for all s > 3,
`R(n)
(n,nt)[s]=13
8s4+13
12s3−9
8s2−7
12s
= 39s+ 3
4−52s+ 2
3+ 14s+ 1
2.
Example 4.3. Let ∆ be the simplicial complex
x1x2
x3
x4
Then R=k[x1, x2, x3, x4]/((x3, x4)∩(x1, x3)∩(x1, x4)∩(x1, x2)) is the face ring of ∆.Observe that R
is a 2-dimensional Cohen-Macaulay ring with f-vector f(∆) = (1,4,4) and h-vector h(∆) = (1,2,1).
This implies that n(n) = 0.Using Theorem 3.6, it follows that for s≥1,
`R(n)
(n,nt)[s]= 2
s−1
X
n=1
`S
I+m[s]mn+
2s−1
X
n=s
`S
I+m[s]mn−
3s−1
X
n=1
`R
nn+ 2 `R
n[s].
Substituting, we get
`R(n)
(n,nt)[s]= 2
s−1
X
n=1 4(s2+n2+n)−4(s+n)+1+
2s−1
X
n=s4n+s+ 1
2−4(s+n)+1
−
3s−1
X
n=1 4n+ 1
2−4n+ 1+ 21 + 4(s−1) + 4(s−1)2.

GENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 11
Simplifying the above expression, we obtain that for all s≥1,
`R(n)
(n,nt)[s]=16
3s3−4s2−4
3s+ 1
= 32s+ 2
3−40s+ 1
2+ 8s+ 1.
Example 4.4. Let ∆ be a 1-dimensional simplicial complex on rvertices, for some r≥3 :
x1x2x3xr−1xr
For i= 1, . . . , r −1, set Pi={x1, . . . , xr}\{xi, xi+1}.Then R=k[x1, . . . , xr]/∩r−1
i=1 Piis the
face ring of ∆.It is a two-dimensional Cohen-Macaulay ring with f-vector f(∆) = (1, r, r −1) and
h-vector h(∆) = (1, r −2,0).Since the a-invariant a2(R) = −1, using Theorem 3.6, it follows that
for s≥1,
`R(n)
(n,nt)[s]
= 2
s−1
X
n=1
`S
I+m[s]mn−
2s−1
X
n=1
`R
nn+ 2 `R
n[s]
= 2
s−1
X
n=1




r−1
X
i=1
`S
Pi+m[s]mn−
r−1
X
i,j=1
i<j
`S
Pi+Pj+m[s]mn+· · · + (−1)r−2` S
Pr−1
i=1 Pi+m[s]mn!



−
2s−1
X
n=1
2
X
i=0
(−1)ih(i)(1)
i!n+ 1 −i
2−i+ 2
2
X
i=0
fi−1(s−1)i.
Observe that in this case, using Corollary 3.1, it follows that `(S/(Pi+m[s]mn)) = s2+n2+n, for
all 1 ≤n≤s−1 and for all i= 1, . . . , r −1.For 1 ≤i<j≤r−1, if {xi, xi+1 }∩{xj, xj+1} 6=∅,
then S/(Pi+Pj)'k[x] and there are r−2 such instances. Otherwise, S/(Pi+Pj)'k. It is also
easy to observe that S/(Pi1+· · · +Piu)'k, for all u≥3 and i1, . . . , iu∈ {1, . . . , r −1}.Therefore,
`R(n)
(n,nt)[s]
= 2
s−1
X
n=1 (r−1)(s2+n2+n)−h(r−2)(s+n) + r−1
2−(r−2)i+r−1
3+· · · + (−1)r−2
−
2s−1
X
n=1 (r−1)n+ 1
2−(r−2)n+ 2 1 + r(s−1) + (r−1)(s−1)2.

12 ARINDAM BANERJEE, KRITI GOEL, AND J. K. VERMA
Since,
r−1
X
i=2
(−1)ir−1
i=r−2,simplifying the above expression we get
`R(n)
(n,nt)[s]=4
3(r−1)s3−(r−2)s2−(r−1)
3s
= 8(r−1)s+ 2
3−2(5r−6)s+ 1
2+ (2r−3)s.
We need some terminologies for the next example.
Definition 4.5. Let Gbe a finite simple graph with vertices V=V(G) = {x1, . . . , xn}and the
edges E=E(G). The edge ideal of I(G)of Gis defined to be the ideal in K[x1, . . . , xn]generated
by the square free quadratic monomials representing the edges of G, i.e.,
I(G) = hxixj|xixj∈Ei.
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 Gwith the set of all minimal vertex covers C, the edge ideal I(G) has the primary
decomposition:
I(G) = \
{xi1,...,xiu}∈C
(xi1, . . . , xiu).
For example, when Gis a five cycle, the primary decomposition of the edge ideal
I(G)=(x1x2, x2x3, x3x4, x4x5, x5x1)
= (x1, x2, x4)∩(x1, x3, x5)∩(x1, x3, x4)∩(x2, x3, x5)∩(x2, x4, x5).
Example 4.6 (Complete Bipartite Graphs). A complete bipartite graph Kα,β 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.
x1
x2
xα
y1
y2
yβ
Figure 1. Kα,β
Let S=k[x1, . . . , xα, y1, . . . , yβ],where 3 ≤α≤β. The edge ideal of Kα,β is the ideal I=xiyj|
1≤i≤α, 1≤j≤β.Observe that R=S/I is a β-dimensional ring. Let P1= (x1, . . . , xα),
P2= (y1, . . . , yβ).Then I=P1∩P2. Note that Iis the Stanley-Reisner ideal of the union of an
α-simplex and a β-simplex.

GENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 13
In order to find the a-invariants we consider the following short exact sequence.
0−→ S
P1∩P2
−→ S
P1MS
P2
−→ S
P1+P2
−→ 0
Using the corresponding long exact sequence of local cohomology modules, it follows that
H1
n(R)'H0
mS
P1+P2,Hα
n(R)'Hα
(x1,...xα)(k[x1, . . . , xα]) and Hβ
n(R)'Hβ
(y1,...,yβ)(k[y1, . . . , yβ]).
Therefore, a1(R) = 0, aα(R) = −αand aβ(R) = −β. Hence, δ= max{|ai(R)|:ai(R)6=−∞} =β.
Since aβ(R)<0, using Theorem 3.5 it follows that for all s > β,
`R(n)
(n,nt)[s]= 2
s−1
X
n=1
`S
I+m[s]mn+
(β−1)s−1
X
n=s
`S
I+m[s]mn−
βs−1
X
n=1
`R
nn+ 2 `R
n[s].
From Corollary 3.3, Theorem 2.4 and Theorem 3.4, we obtain
`R(n)
(n,nt)[s]= 2
s−1
X
n=1 `S
P1+m[s]mn+`S
P2+m[s]mn−`S
P1+P2+m[s]mn
+
(β−1)s−1
X
n=s`S
P1+m[s]mn+`S
P2+m[s]mn−`S
P1+P2+m[s]mn
−
βs−1
X
n=1 "β
X
i=0
(−1)ih(i)(1)
i!n+β−i−1
β−i#+ 2
β
X
i=0
fi−1(s−1)i.
As the f-vector is f(∆) = 1, α +β, α
2+β
2,...,α
α+β
α,β
α+ 1,...,β
β and the
h-vector can be computed using [1, Lemma 5.1.8], substituting the values and using Corollary 3.1,
it follows that for all s > β,
`R(n)
(n,nt)[s]= 2
s−1
X
n=1 sβ+βn+β−1
β+sα+αn+α−1
α−1
+
(β−1)s−1
X
n=ssβ+
β−1
X
i=1
(−1)i+1β
in−(i−1)s+β−1
β−1
+
(α−1)s−1
X
n=ssα+
α−1
X
i=1
(−1)i+1α
in−(i−1)s+α−1
α+
(β−1)s−1
X
n=(α−1)sn+s+α−1
α
−
βs−1
X
n=1 β
X
i=0
(−1)ih(i)(1)
i!n+β−i−1
β−i+ 21 +
β
X
i=1 β
i(s−1)i+
α
X
i=1 α
i(s−1)i.
In particular, when α= 3 and β= 4,we obtain that for all s > 4,
`R(n)
(n,nt)[s]=61
30s5+19
24s4−1
12s3−7
24s2−9
20s−1.

14 ARINDAM BANERJEE, KRITI GOEL, AND J. K. VERMA
Sometimes, certain invariants of the Stanley-Reisner ring may depend on the characteristic of the
ring. Triangulation of the real projective plane is one such example where the Cohen-Macaulay
property of the ring is characteristic dependent. We prove that in this example, the Hilbert-Kunz
function is characteristic independent.
Example 4.7 (Triangulation of real projective plane). Let ∆ be the triangulation of the real
projective plane.
Let kbe a field and Rbe the corresponding Stanley-Reisner ring of ∆.It is known that Ris Cohen-
Macaulay if and only if chark6= 2.The f-vector of Ris f(∆) = (1,6,15,10) and h-vector of Ris
h(∆) = (1,3,6,0).Let char k6= 2.Then Ris Cohen-Macaulay and n(n) = −1.Using Macaulay2
code, we obtain that for s≥1,
`R(n)
(n,nt)[s]= 390s+ 3
4−720s+ 2
3+ 372s+ 1
2−41s. (4.1)
We save the code in a file named as HKPolySC.m2 and make the following session in Macaulay2.
i1 : S = QQ[a..f];
i2 : I = ideal"abe, ade, acd, bcd, bdf, abf, acf, cef, bce, def";
i3 : loadPackage"Depth"
i4 : loadPackage"SimplicialComplexes"
i5 : loadPackage"SimplicialDecomposability"
i6 : load"HKPolySC.m2"
i7 : HKPolySC(I)
The postulation number is: -1
Enter a number bigger than or equal to the absolute value of the postulation number: 2
The value of the Hilbert-Kunz function at the point 2 is: 104
Do you wish to enter one more point? (true/false): true
Enter a number bigger than or equal to the absolute value of the postulation number: 3
The value of the Hilbert-Kunz function at the point 3 is: 759
Do you wish to enter one more point? (true/false): true
Enter a number bigger than or equal to the absolute value of the postulation number: 4
The value of the Hilbert-Kunz function at the point 4 is: 2806

GENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 15
Do you wish to enter one more point? (true/false): true
Enter a number bigger than or equal to the absolute value of the postulation number: 5
The value of the Hilbert-Kunz function at the point 5 is: 7475
Do you wish to enter one more point? (true/false): true
Enter a number bigger than or equal to the absolute value of the postulation number: 6
The value of the Hilbert-Kunz function at the point 6 is: 16386
Do you wish to enter one more point? (true/false): false
One may check that if char k= 2, then the a-invariant of Ris negative and depth(R)=2.Using
Theorem 3.5 it follows that `(R(n)/(n,nt)[s]) has the same formula as in (4.1) for s > δ, where
δ= max{|a2(R)|,|a3(R)|}.This proves that the Hilbert-Kunz function is characteristic independent
in this example.
5. Macaulay2 code for Cohen-Macaulay Stanley-Reisner rings
In this section we present a Macaulay2 code which uses the idea of Theorem 3.6 to calculate the
value of the generalized Hilbert-Kunz function at a point. The code requires Macaulay2 packages
SimplicialComplexes, SimplicialDecomposability and Depth. The code accepts the Stanley-Reisner
ideal as an input. It then calculates the postulation number, after ensuring that the corresponding
ring is Cohen-Macaulay, and prompts the user to enter a point according to the postulation number
calculated. The value of the generalized Hilbert-Kunz function at the point is produced as an output
and the user is given a choice to enter more points.
HKPolySC = (SCIdeal) -> (
polyRing := ring SCIdeal;
Step 1: Check if the Stanley-Reisner ring is Cohen-Macaulay
if isCM(polyRing/SCIdeal) == false then error "Stanley-Reisner ring is not Cohen-Macaulay";
dimSC := dim (polyRing/SCIdeal);
SComplex := simplicialComplex monomialIdeal SCIdeal;
fvect := fVector(SComplex);
hvect := hVector(SComplex);
Step 2: Calculate the derivatives of the polynomial corresponding to the h-vector at 1
Diffh = (i) -> (
TT := QQ[tt];

16 ARINDAM BANERJEE, KRITI GOEL, AND J. K. VERMA
hPoly := sum(0..dimSC, j -> (hvect#j)*(tt^j));
for j from 1 to i do (
hPoly = diff(tt, hPoly)
);
sub(sub(hPoly, TT/(tt-1)), QQ)
);
Find the list of minimal primes
MinPrimeList := primaryDecomposition SCIdeal;
numPrime := #MinPrimeList;
SubsetList := subsets toList (0..(numPrime-1));
The function CombinationList outputs the list containing subsets of {0,...,(numPrime −1)}of
cardinality j
CombinationList = (j) -> (
jCombi = {};
for i from 0 to 2^(numPrime)-1 do (
if #(SubsetList#i) == j then jCombi = append(jCombi,SubsetList#i)
);
jCombi
);
Ring required for the output polynomial
OutputRing = QQ[s];
Redefining the binomial function
binom = (aa, bb) -> (
if aa > 0 then return binomial(aa,bb)
else if (aa == 0 and bb == 0) then return 1
else return 0
);
Step 3: Calculate and print the postulation number

GENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 17
PostNum := -position(toList apply(0..dimSC, i-> dimSC-i), i-> hvect#i !=0);
<<"The postulation number is: "<< PostNum <<endl;
Step 4: Obtain the point from the user as an input and calculate the Hilbert-Kunz polynomial at
that point
pointer := true;
while pointer == true do(
point = read "Enter a number bigger than or equal to the absolute value of the
postulation number: ";
point = value point;
Step 5: The function FunctionF calculates length as in Corollary 3.1.
FunctionF = (QtI, n) -> (
dimQt = dim (polyRing/QtI);
use OutputRing;
if dimQt == 0 then return 1
else if dimQt == 1 then return point + n
else if dimQt == 2 then (
if n <= point then return point^2 + n^2 + n
else return (n + point + 1)*(n + point)/2
)
else (
if n <= point then return point^dimQt + dimQt*binom(n+dimQt-1,dimQt)
else if (point+1 <= n and n <= (dimQt-1)*point-1) then
return point^dimQt + sum(1..(dimQt-1), i ->
((-1)^(i+1))*binom(dimQt,i)*binom(n-(i-1)*point+dimQt-1,dimQt))
else return binom(n+point+dimQt-1,dimQt)
)
);
Step 6: The function AltSumLength calculates length as in Corollary 3.3
AltSumLength = (n) -> (
polySum = 0;
for i from 1 to numPrime do (
CL := toList CombinationList(i);
midSum = 0;

18 ARINDAM BANERJEE, KRITI GOEL, AND J. K. VERMA
for j from 0 to #CL-1 do (
midIdeal = sum(0..(i-1), k -> MinPrimeList#(CL#j#k));
midSum = midSum + FunctionF(midIdeal, n);
);
polySum = polySum + (-1)^(i+1)*midSum;
);
polySum
);
Step 7: Calculate the Hilbert-Kunz polynomial at the point
use OutputRing;
if PostNum == 0 then
polyPoint = 2*sum(1..(point-1), n -> AltSumLength(n))
+ sum(point..(dimSC*point-1), n -> AltSumLength(n))
- sum(1..((dimSC+1)*point-1), n ->
sum(0..dimSC,i->(-1)^i*Diffh(i)*(1/i!)*binom(n+dimSC-i-1,dimSC-i)))
+ 2*sum(0..dimSC, n -> (fvect#(n-1))*(point-1)^n)
else (
polyPoint1 = 2*sum(1..(point-1), n -> AltSumLength(n));
if (dimSC == 2) then (polyPoint2 = 0;)
else (polyPoint2 = sum(point..((dimSC-1)*point-1), n -> AltSumLength(n)););
polyPoint3 = sum(1..(dimSC*point-1), n ->
sum(0..dimSC, i -> (-1)^i*Diffh(i)*(1/i!)*binom(n+dimSC-i-1,dimSC-i)));
polyPoint4 = 2*sum(0..dimSC, n -> (fvect#(n-1))*((point-1)^n));
polyPoint = polyPoint1 + polyPoint2 - polyPoint3 + polyPoint4;
);
<<"The value of the Hilbert-Kunz polynomial at the point " << point << " is: "
<< polyPoint << endl;
pointer = read "Do you wish to enter one more point? (true/false): ";
pointer = value pointer;
)
)
If the a-invariant of the ring is known, the above code can also be used for the non Cohen-Macaulay
case with minor modifications.

GENERALIZED HILBERT-KUNZ FUNCTION OF THE REES ALGEBRA 19
References
1. Winfried Bruns and J¨urgen Herzog. Cohen-Macaulay rings, volume 39 of Cambridge Studies
in Advanced Mathematics. Cambridge University Press, Cambridge, 1993.
2. Aldo Conca. Hilbert-Kunz function of monomial ideals and binomial hypersurfaces.
Manuscripta Math., 90(3):287–300, 1996.
3. Kazufumi Eto and Ken-ichi Yoshida. Notes on Hilbert-Kunz multiplicity of Rees algebras.
Comm. Algebra, 31(12):5943–5976, 2003.
4. Le Tuan Hoa. Reduction numbers and Rees algebras of powers of an ideal. Proc. Amer. Math.
Soc., 119(2):415–422, 1993.
5. Kriti Goel, Mitra Koley, and J. K. Verma. Hilbert-Kunz function and Hilbert-Kunz multiplicity
of some ideals of the Rees algebra. arXiv preprint arXiv:1911.03889.
6. Kriti Goel, Vivek Mukundan and J. K. Verma. Tight closure of powers of ideals and tight
Hilbert polynomials. Mathematical Proceedings of the Cambridge Philosophical Society, 1-21.
7. Craig Huneke. Hilbert-Kunz multiplicity and the F-signature. Commutative algebra, 485–525,
Springer, New York, 2013.
8. Ernst Kunz. On Noetherian rings of characteristic p.Amer. J. Math., 98(4):999–1013, 1976.
9. Thomas John Marley. Hilbert functions of ideals in Cohen-Macaulay rings. ProQuest LLC,
Ann Arbor, MI, 1989. Thesis (Ph.D.)–Purdue University.
10. Paul Monsky. The Hilbert-Kunz function. Math. Ann., 263(1):43–49, 1983.
Ramakrishna Mission Vivekananda Educational and Research Institute, Belur, India
E-mail address:123.arindam@gmail.com
Indian Institute of Technology Bombay, Mumbai, India 400076
E-mail address:kritigoel.maths@gmail.com
Indian Institute of Technology Bombay, Mumbai, India 400076
E-mail address:jkv@math.iitb.ac.in