A NOTE ON HILBERT-KUNZ MULTIPLICITY
MOHSEN ASGHARZADEH

arXiv:1603.04297v1 [] 14 Mar 2016

Abstract. In this note we first give a new bound on eHK (∼) the Hilbert-Kunz multiplicity of invariant rings, by the help of the Noether’s bound. Then, we simplify, extend
and present applications of the reciprocity formulae due to L. Smith. He proved the
formula over polynomial rings and his result is tight in the following sense: Over complete intersection rings with isolated singularity we show that the reciprocity formulae
”eHK (R/I) + eHK (R/J) = eHK (R/f )” is equivalent with p. dim(I) < ∞ when I is an
m-primary unmixed ideal linked to J along with a regular sequence f .

1. Introduction
Throughout this note rings are of prime characteristic p > 0. Let F n (−) be the PeskineSzpiro functor. The Hilbert-Kunz multiplicity over ∗-local (A, m) with an m-primary ideal
I is defined by
ℓ (F n (A/I))
.
n→∞
pn dim A
This introduced by Kunz. Monsky proved such a limit exists [7]. Hilbert-Kunz multiplicity
eHK (I, A) := lim

encodes the singularity in prime characteristic [14], et cetera. Some times computing
eHK (∼) is difficult and finding bounds may be a reasonable task. Also, eHK (∼) behaves
both similar to and different from those of Hilbert-Samuel multiplicity. In this note we
present a new sample for this behavior.
We use eHK (A) instead of eHK (m, A). We denote the usual Hilbert-Samuel multiplicity
by e(I). Suppose a finite group G acts linearly over a polynomial ring with n variables, denote the invariant ring by A and assume that the order of G is prime to p. First we observe
(n−1+|G|
)
n
eHK (A) ≤
, as an easy application of Noether’s bound from invariant theory. Over
|G|
2-dimensional quotient singularity, this has in the simple form eHK (k[X, Y ]G ) ≤
e(k[X, Y

]G )

|G|+1
2 .

So

≤ |G| + 1. We show the bound is sharp in some cases but not in general. We

do this in Section 3.
Section 4 motivated by the reciprocity formulae of Larry Smith. This result connects
the Hilbert-Kunz multiplicity of linked ideals I and J via the complete intersection ideal
a := (f ) over polynomial rings, i.e.,
eHK (I, R) + eHK (J, R) = eHK (a, R).
2010 Mathematics Subject Classification. 13D40; 13A35; 13A50.
Key words and phrases. Hilbert-Kunz theory; invariant theory; liaison theory; prime characteristic
methods.

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ASGHARZADEH

We first reprove Smith’s result by a short argument. Then we extend it from the polynomial rings to more general setting. We give corollaries of this result, see e.g. Corollary
4.2.
Suppose X, Y ⊂ P3 are curves directly linked by equations of degrees d and e. It may
be worth to recall that Peskine and Szpiro proved the degrees and genus of X and Y are
related by the following formulas:
i) deg X + deg Y = de
ii) g(X) − g(Y ) = (deg X − deg Y )(d + e − 4)/2.
Up to our knowledge the above result regards as a root of liaison theory. The following is
our main result:
Theorem 1.1. Let R be a local complete intersection with isolated singularity and of positive dimension. Let I be an unmixed primary ideal linked to J via the complete intersection
ideal a := (f ) of full length. The following are equivalent.
i) eHK (I, R) + eHK (J, R) = eHK (a, R).
ii) p. dim(I) < ∞.
To prove this result, and among other things, we use the corner powers of unmixed ideals
developed by Vraciu, and we apply the theory of generalized Hilbert-Kunz multiplicity that
introduced by Dao and Smirnov. In particular, the generalized Hilbert-Kunz multiplicity
has a new application to the classical Hilbert-Kunz multiplicity.
2. preliminary
All rings in this note are commutative and Noetherian. Let R be a ring of prime
characteristic p. The assignment a 7→ ap defines a ring homomorphism F : R → R. By
F n (R), we mean R as a group equipped with left and right scalar multiplication from R
given by
n

a.r ⋆ b = abp r, where a, b ∈ R and r ∈ F n (R).
Now we recall the Peskine-Szpiro functor. For a finitely generated R-module M , set
F n (M ) := M ⊗ F n (R).
n

It is easy to see F n (R/a) = R/a[p ] . Let a be an ideal a with a generating set x :=
x1 , . . . , xr . The ith local cohomology of M with respect to a is defined by
Hai (M ) := lim ExtiR (R/an , M ).
−→
n

One may calculate it as the ith cohomology of Čech complex of a module M with respect
to x. The grade of a on M is defined by
gradeR (a, M ) := inf{i ∈ N0 |Hai (M ) 6= 0}.
We use depth(M ), when we deal with the maximal ideal of ∗-local rings. Our next auxiliary
tool is

A NOTE ON HILBERT-KUNZ MULTIPLICITY

3


M (n) := ℓ H 0 (F n (M ))
i) fgHK
m
ii) egHK (M ) := limn→∞

M
(n)
fgHK
,
pn dim R

if the limit exists and put ∞ otherwise. Following Dao and Smirnov [3], we call them
the generalized Hilbert-Kunz function and the generalized Hilbert-Kunz multiplicity. If
M and e
ℓ(M ) < ∞ we are in the situation of [7] and we use fHK
HK (M ), respectively.

If there is no doubt of confusion, we use eHK (R/a) instead of eHK (a, R). We need the
following two facts:
Fact 2.1. (See [2, Proposition 8.2.5]) Frobenius map commutes with the localization, i.e.,
RS ⊗R F n (M ) ≃ F n (RS ⊗R M ) for any multiplicative closed subset S ⊂ R.
Fact 2.2. Over regular rings the Frobenius map is flat, see [2, Corollary 8.2.8].
3. A bound on eHK (∼) of invariant rings
L
Throughout this section R := n≥0 Rn is a standard graded algebra of a field R0 := k
L
of prime characteristic p > 0 and m := n>0 Rn is the irrelevant ideal.
Notation 3.1. Let G be a finite group such that |G| is invertible in the field k, i.e.,
(p, |G|) = 1.
Discussion 3.2. Let A be a ring and ρ : G → Aut(A) be a representation of finite group G
by Aut(A) and |G| is invertible in A. Recall that a ⊂ A is said to be G-stable, if g(a) ∈ a
for all g ∈ G and a ∈ a. For any subset X ⊂ A by XA, we mean the ideal generated by
X.
Let us recall the following version of Noether’s bound due to D. Benson:
Lemma 3.3. (See [8, Lemma 2.3.1]) Adopt the above notation and let a be G-stable. Then
a|G| ⊆ (aG )A.
Suppose G acts on R by degree-preserving k-algebra homomorphisms. This means that
g(Rn ) ⊆ Rn for all g ∈ G and n ∈ N. Then
A := RG = {f ∈ R : g(f ) = f ∀g ∈ G}
L
is the ring of invariants. Denote i>0 Ai by n. Recall from [14] that:
Fact 3.4. eHK (A) =

dim(R/nR)
.
|G|

Proposition 3.5. Adopt the above notation. Then eHK (A) ≤

(n−1+|G|
)
n
|G|

.

Proof. As G acts on R by degree-preserving k-algebra homomorphisms, m is G-stable. In
view of Lemma 3.3, m|G| ⊆ (mG )R. Also, (mG )R ⊆ nR. Thus,
m|G| ⊆ (mG )R ⊆ nR.

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ASGHARZADEH

This induces a surjection
R/m|G| −→ R/nR −→ 0,
and so
dim(R/nR) ≤ dim(R/m|G| )
P|G|−1
= i=0 dim(Ri )

P|G|−1
= i=0 n+i−1
n−1

= n−1+|G|
.
n
Therefore, in view of Fact 3.4,
eHK (A) ≤

n−1+|G|
n



|G|

.


Corollary 3.6. Over 2-dimensional invariant rings, one has:
i) eHK (k[X, Y ]G ) ≤

|G|+1
2

ii) e(k[X, Y ]G ) ≤ |G| + 1.
Proof. Set n = 2 in Proposition 3.5. Part i) is now immediate. For part ii) we recall a
result of Monsky [7], i.e., over any ring A one observes
e(A)/(dim A)! ≤ eHK (A).
Combining this along with i) we get the claim.



Corollary 3.7. Suppose |G| = 2. One has
dim R+1
.
2
(dim R+1)!
.
2

i) eHK (RG ) ≤
ii) e(RG ) ≤

Let us show these bounds are sharp.
Example 3.8. Suppose |G| = 2 and n = 2. Denote the corresponding invariant ring by A.
We assume that A is not regular and such a thing happens. Then we get
3
e(A)
≤ eHK (A) ≤
!
(dim A)
2

(∗)

As the ring A is not regular, eHK (A) > 1. This along with (∗) yields:
i) eHK (A) = 3/2;
ii) e(A) = 3.
4. Smith’s reciprocity formulae
Recall that I is linked to J if there is a regular sequence f := f1 , . . . , fi in I ∩ J such
that I = (f : J) and J = (f : I). Smith proved a reciprocity formulae over polynomial
rings [12, Theorem 5.7]. First, we extend his result to Gorenstein rings by a new and short
argument.

A NOTE ON HILBERT-KUNZ MULTIPLICITY

5

Proposition 4.1. Let R be a d-dimensional Gorenstein ring, I and J primary to the
maximal ideal m linked via a regular sequence f := f1 , . . . , fd . Then
ℓ(R/I) + ℓ(R/J) = ℓ(R/f ).
Proof. Note that
1

ℓ(R/I) + ℓ(R/J) = ℓ(Hm0 (R/I)) + ℓ(R/J)
2

= ℓ(ExtdR (R/I, R)) + ℓ(R/J)
3

= ℓ(Ext0R (R/I, R/f )) + ℓ(R/J)
4

= ℓ(J/f ) + ℓ(R/J)
5

= ℓ(R/f ),
where:
1) Since R/I is of finite length.
2) This is the local duality.
3) Keep in mind that f is a regular sequence in the Jacobson radical of the ring. An
easy induction implies that
Ext0R (R/I, R/f ) ≃ ExtdR (R/I, R).
4) This is in the definition of linked ideals.
5) This follows by the following short exact sequence:
0 −→ J/f −→ R/f −→ R/J −→ 0.
The proof is now complete.



Corollary 4.2. Let (R, m) be a Gorenstein local ring of dimension zero with an ideal I
in which (0 : I) = I. Then ℓ(R) ∈ 2N.
Proof. As (0 : I) = I, we get that
2ℓ(R/I) = ℓ(Hm0 (R/I)) + ℓ(R/(0 : I))
= ℓ(HomR (R/I, R)) + ℓ(R/(0 : I))
= ℓ((0 : I)) + ℓ(R/(0 : I))
= ℓ(R).
So, ℓ(R) ∈ 2N.



Example 4.3. (i) Here we remark that the length of a zero-dimensional Gorenstein local
ring can be every thing, e.g., K[X]/(X n ). To see examples of high embedding dimension,
look at the zero-dimensional Gorenstein ring
R := K[X1 , . . . , Xn ]/(X12 − X22 , . . . , X12 − Xn2 , X1 X2 , X1 X3 , . . . , Xn−1 Xn ).
So, {1, X1 , . . . , Xn , X12 } is a base for R and {X1 , . . . , Xn } is a base for m/m2 .
(ii) Let (R, m, K) be a Gorenstein local ring containing a field with m2 = 0. By this
property, (0 : m) = m. In view of the above Corollary, ℓ(R) ∈ 2N. Let us check this by

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ASGHARZADEH

hand. As R is Gorenstein, its socle is principal. Therefore, m is principal. Denote its
generator by r. The assignment r 7→ X yields R ≃ K[X]/(X 2 ). In particular, ℓ(R) = 2
as claimed.
Remark 4.4. (See [6, Example 6.6]) The concept of numerically Roberts rings introduced
in [6]. Suppose R is a homomorphic image of a regular local ring and let d = dim R. The
following are numerically Roberts rings:
i) d < 2
ii) d = 2 and R is equi-dimensional.
iii) d = 3 and R is Gorenstien.
The following extends [12, Corollary 5.8] via a new argument.
Corollary 4.5. Let R be a d-dimensional Gorenstein ring, I and J be two ideals of finite
projective dimension linked through a regular sequence f := f1 , . . . , fd . Assume one of the
following holds:
i) Suppose in addition, R is standard graded over a field, I and J are homogenous.
ii) Suppose in addition, R is a local complete intersection ring.
iii) Suppose in addition, R is local and dim R < 4.
Then eHK (I, R) + eHK (J, R) = eHK (f , R).
Proof. i) Recall that over a graded ring of dimension d with a graded module M of finite
length and of finite projective dimension, a celebrated result of Peskine and Szpiro says:
ℓ(F n (M )) = pnd ℓ(M ).*
In particular, eHK (I, R) = ℓ(R/I). Now, Proposition 4.1 proves the desired claim.
ii) First we note that p. dim(R/f ) < ∞. Any complete intersection ring is a numerically
Roberts ring. In view of [6, Theorem 6.4], the Hilbert-Kunz multiplicity of an ideal primary
to the maximal ideal and of finite projective dimension is its colength. Now, Proposition
4.1 proves the desired claim.
iii) In view of the above Remark this follows in a similar vein as ii).



Recall that a module M is called generalized Cohen-Macaulay if Hmi (M ) is finitely
generated for all i < dim M . Recall from [3]:
Definition 4.6. Let M be generalized Cohen-Macaulay and i < dim M . The i-th generalized Hilbert-Kunz multiplicity of M is
eigHK (M )


ℓ Hmi (F n (M ))
:= lim
,
n→∞
pn dim R

if it exists and put ∞ otherwise.
*There are 3-dimensional Cohen-Macaulay rings of characteristic p such that ℓ(F n (M )) 6= p3n ℓ(M ), for

a suitable module of finite length, by [11].

A NOTE ON HILBERT-KUNZ MULTIPLICITY

7

Remark 4.7. Let R be a regular ring and M a module of finite length. Then F n (M ) is of
finite length. Indeed, we do induction by ℓ := ℓ(M ). If ℓ = 1, then M = R/m and the
claim is clear in this case. Now suppose, inductively, that ℓ > 1 and the result has been
proved for modules of length less than ℓ. Look at the exact sequence
0 −→ N −→ M −→ R/m −→ 0
where ℓ(N ) = ℓ − 1. By inductive hypothesis F n (N ) is of finite length. By Fact 2.2, there
is an exact sequence
0 −→ F n (N ) −→ F n (M ) −→ F n (R/m) −→ 0.
The claim is now clear.
Proposition 4.8. Let R be a regular ring of dimension d, I and J be two ideals linked
via a regular sequence f := f1 , . . . , fg for some g < d. Suppose in addition that R/I is
generalized Cohen-Macaulay of dimension m. Then for all 1 ≤ i ≤ m − 1 one has
eigHK (R/I) = em−i
gHK (R/J).
n

n

n

Proof. Set f p := f1p , . . . , fgp . As R is regular, the Frobenius map is flat. Hence
n

n

n

n

n

1) I [p ] = (f p : J [p ] ),
2) J [p

n]

= (f p : I [p ] ) and

n

3) f p is a regular sequence.
Thus I [p

n]

and J [p

n]

n

are linked through f p . As Supp(F n (M )) = Supp(M )), one has

dim(F n (M )) = dim(M ). In particular, dim R/I [p

n]

= m. Also, exact functors commute

with the local cohomology modules. By the above Remark, Fact 2.2 and in view of
n

F n (Hmi (R/I)) ≃ Hmi (F n (R/I)) ≃ Hmi (R/I [p ] ),
n

we observe that Hmi (R/I [p ] ) is finitely generated for all i < m. Let 1 ≤ i ≤ m − 1. In
view of [13, Corollary 3.3],
n

n

Hmi (R/I [p ] ) ≃ Hom(Hmm−i (R/J [p ] ), E).
So,
n

n

ℓ(Hmi (R/I [p ] )) = ℓ(Hmm−i (R/J [p ] )).
This yields eigHK (R/I) = em−i
gHK (R/J) as claimed.



We cite the next result for the convenience of the reader.
Lemma 4.9. (Avramov-Miller, Dutta) Let R be a local complete intersection and M a
finitely generated R-module. Then the vanishing of ToriR (M, F n (R)) for one value each of
i > 0 and n > 0 implies that M has finite projective dimension.

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ASGHARZADEH

Let (R, m) be a characteristic p Gorenstein ring. If I ⊳ R is an unmixed ideal of height
g, let a ⊂ I be a Gorenstein ideal of finite projective dimension and of height g (e.g.,
a parameter ideal), and let J := (a : I). Now, by the help of [15, Proposition 1], the
following is well-define:
Notation 4.10. Following [15], define I <q> := (a[q] : J [q] ) for all q = pn .
Lemma 4.11. Let (R, m) be a ring of prime characteristic p, let I be an unmixed ideal
and q = pn . The following holds:
i) If R is Gorenstein, I [q] ⊆ I <q> ∀q and the equality holds when p. dim(I) < ∞.
ii) Suppose R is complete intersection. Then I [q] = I <q> ∀q ⇔ p. dim(I) < ∞.
iii) Suppose R is complete intersection. Then I [q] = I <q> ∃q ⇔ p. dim(I) < ∞.
Proof. i): This is in [15, Proposition 2].
ii): First note that complete intersection rings are Gorenstein. The desired claim is in
[15, Proposition 2] when the ring is hypersurface. Here, we sketch that proof. Let N be
the cokernel of the natural inclusion R/I ֒→ ⊕n R/a defined via the matrix (f1 , . . . , fn ).
Vraciu’s main trick is as follows:
n
I <q> /I [q] = TorR
1 (N, F (R))

(∗),

see the proof of [15, Proposition 2]. Now, suppose I <q> = I [q] . Apply this throughout
n
the complete intersection property to conclude that TorR
1 (N, F (R)) = 0. So, Lemma 4.9

implies that p. dimR (N ) < ∞. By definition of N , we observe p. dimR (R/I) < ∞.
iii) This is similar to ii).



In what follows we will use the following two results on the concept of generalized
Hilbert-Kunz functions.
Lemma 4.12. (See [3, Corollary 4.7]) Let R be a local complete intersection with isolated
singularity. The following are equivalent:
i) egHK (M ) = 0,
M (n) = 0 for all n,
ii) fgHK

iii) p. dimR M < dim R.
Lemma 4.13. (See [3, Lemma 3.10]) Let R be a formally equi-dimensional local ring of

n
0
positive depth, then Hm0 TorR
1 (M, F (R)) = Hm (Syz(M )).
Corollary 4.14. Suppose in addition to Lemma 4.13, R has isolated singularity. Then
n
0
TorR
1 (M, F (R)) = Hm (Syz(M )).
n
Proof. It is enough to show TorR
1 (M, F (R)) is of finite length. Equivalently,
n
TorR
1 (M, F (R))p = 0

∀p 6= m.

A NOTE ON HILBERT-KUNZ MULTIPLICITY

9

Let p be a non-maximal prime ideal. Due to Fact 2.2 and the isolated singularity,
R

Tor1 p (∼, F n (Rp )) = 0.
By Fact 2.1,
R

p
n
n
TorR
1 (M, F (R))p ≃ Tor1 (Mp , F (Rp )) = 0,

and this completes the argument.



Now we are ready to prove the following:
Theorem 4.15. Let R be a local complete intersection with isolated singularity and of
positive dimension. Let I be an unmixed ideal linked to J throughout a regular sequence f
of full length. The following are equivalent:
i) eHK (I, R) + eHK (J, R) = eHK (f , R),
ii) p. dim(I) < ∞.
Proof. i) ⇒ ii): First we remark that any Cohen-Macaulay ring is formally equidimensional. Suppose on the contrary that p. dim(I) = ∞. Let d := dim R. Let N be the
cokernel of the natural inclusion R/I ֒→ ⊕d R/a defined via the matrix (f1 , . . . , fd ). Then
by Lemma 4.11,
n
0 6= I <q> /I [q] = TorR
1 (N, F (R)).

On the other hand by Corollary 4.14

Syz(N )
n
ℓ TorR
1 (N, F (R)) = fgHK (n).
We note that p. dim(N ) = ∞ and the same thing holds for its syzygies. By Lemma 4.12,
ℓ(I <q> /I [q] )
6= 0.
n→∞
qd
lim

In the light of [15, Proposition 3] we observe
ℓ(R/I <q> ) + ℓ(R/J [q] ) = ℓ(R/f [q] ).
Putting this along with i) we get that
ℓ(I <q> /I [q] )
= 0,
n→∞
qd
lim

a contradiction.
ii) ⇒ i): This is in Corollary 4.5.



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ASGHARZADEH

References
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E-mail address: mohsenasgharzadeh@gmail.com

