arXiv:1308.5085v1 [] 23 Aug 2013

THE HILBERT-KUNZ FUNCTIONS OF TWO-DIMENSIONAL
RINGS OF TYPE ADE
DANIEL BRINKMANN
Abstract. We compute the Hilbert-Kunz functions of two-dimensional rings
of type ADE by using representations of their indecomposable, maximal CohenMacaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals.

Introduction
The central objects in this paper will be the two-dimensional rings of type ADE
(or ADE-rings for short), namely
• An := k[X, Y, Z]/(X n+1 − Y Z) with n ≥ 0,
• Dn := k[X, Y, Z]/(X 2 + Y n−1 + Y Z 2 ) with n ≥ 4,
• E6 := k[X, Y, Z]/(X 2 + Y 3 + Z 4 ),
• E7 := k[X, Y, Z]/(X 2 + Y 3 + Y Z 3 ) and
• E8 := k[X, Y, Z]/(X 2 + Y 3 + Z 5 ),
as well as their (X, Y, Z)-adic completions, where k denotes an algebraically closed
field. These rings were studied by several authors, e.g. Klein (in [Kle86]), du Val
(in [dV34a], [dV34b], [dV34c]), Brieskorn (in [Bri68]) or Artin (in [Art77], [Art66]),
where they appeared in various forms, e.g as quotient singularities or rational double
points. They also appear in string theory (cf. [Wit02] for a survey).
The goal is to compute (in positive characteristic) their Hilbert-Kunz functions



e
e
e
e 7→ dimk k[X, Y, Z]/ F, X p , Y p , Z p
,

where F denotes one of the defining polynomials above. Note that the (non-local)
rings of type ADE are homogeneous with respect to some positive grading.
Recall that the rings C[X, Y, Z]/(F ) of type ADE appear as rings of invariants of
C[x, y] by the actions of the finite subgroups of SL2 (C). The groups corresponding
to the singularities of type An , Dn , E6 , E7 resp. E8 are the cylic group with n + 1
elements, the binary dihedral group Dn−2 of order 4n − 8, the binary tetrahedral
group T of order 24, the binary octahedral group O of order 48 resp. the binary
icosahedral group I of order 120. If k is algebraically closed of characteristic p > 0
the groups above can be viewed as finite subgroups of SL2 (k), provided their order
is invertible in k. In these cases k[X, Y, Z]/(F ) is again the ring of invariants of
k[x, y] under the action of the corresponding group (cf. [LW12, Chapter 6, §2]).
Using this fact, Watanabe and Yoshida showed in [WY00] that the Hilbert-Kunz

Date: August 26, 2013.
2010 Mathematics Subject Classification. 13A35, 13A50, 13C14, 13D02, 13D40, 14J17, 14J60,
20G05 .
Key words and phrases. ADE singularity, Hilbert-Kunz function, vector bundle, Frobenius,
Hilbert-series, syzygy module, matrix factorization, maximal Cohen-Macaulay.
1

2

DANIEL BRINKMANN

1
, provided the order of the
multiplicity of rings of type ADE is given by 2 − |G|
corresponding group G is invertible modulo p. Moreover, by a result of Brenner
(cf. [Bre07]) the Hilbert-Kunz functions are of the form

e 7→ eHK ·p2e + γ(pe ),
where γ is an eventually periodic function. For example, the Hilbert-Kunz function
of a ring of type An is due to Kunz (cf. [Kun76, Example 4.3]) given by


1
r2
e 7→ 2 −
p2e − r +
,
n+1
n+1

where r is the smallest non-negative representative of the class of pe modulo n + 1.
We also want to mention that the Hilbert-Kunz functions of rings of type E6 and
E8 can be computed with the algorithm of Han and Monsky (cf. [HM93]), provided
the characteristic is known.
If R denotes an ADE-ring, the flat map
φ̄ : R → k[U, V, W ]/(φ(F )), X 7→ U deg(X) , Y 7→ V deg(Y ) , Z 7→ W deg(Z) ,

will allow us to switch between the non-standard graded situation we are interested
in and a standard-graded situation. Explictly, the corresponding Hilbert-Kunz
functions in these two cases are related in the following way (cf. [Bri13, Remark
1.22])
  e

e
e
dimk R/ X p , Y p , Z p
e
e
e 
dimk k[U, V, W ]/ φ(F ), U deg(X)·p , V deg(Y )·p , W deg(Z)·p
.
=
deg(X) · deg(Y ) · deg(Z)

This makes it possible to use strong tools on both sides. Namely, in the nonstandard graded situation, the ADE-rings are graded Cohen-Macaulay finite (cf.
[Knö87] and [BGS87]), meaning that there are up to isomorphism and to degree
shifts only finitely many indecomposable, graded, maximal Cohen-Macaulay modules. In the standard-graded case we can use the geometric interpretation of HilbertKunz functions due to Brenner and Trivedi (cf. [Bre06] resp. [Tri05]) and the theory
of vector bundles.
To be more precise, instead of computing the Hilbert-Kunz functions of the rings
R of type ADE (with respect to the ideal (X, Y, Z)), we might compute the
Hilbert-Kunz function of S := k[U, V, W ]/(φ(F )) with respect to the monomial
ideal I := (U deg(X) , V deg(Y ) , W deg(Z) ). For this purpose, we need to control the
Frobenius pull-backs of SyzProj(S) (I). The idea for this is the following. Since R is
e
e
e
graded Cohen-Macaulay finite and SyzR (X, Y, Z) as well as all SyzR (X p , Y p , Z p ),
e ≥ 1, are maximal Cohen-Macaulay there has to be an eventually periodicity
e
e
e
in the sense that the isomorphism class of SyzR (X p , Y p , Z p ), e ≫ 0, depends
only on the residue class of pe modulo a certain number. This periodicity of
e
e
e
the R-modules SyzR (X p , Y p , Z p ) translates via φ̄ to a periodicity of the Se
e
e
modules SyzS (U deg(X)p , V deg(Y )p , W deg(Z)p ), which makes it possible to compute
the Hilbert-Kunz function of S with respect to I.
In order to detect this periodicity, we have to find a full list of representatives of the
isomorphism classes of the indecomposable, maximal Cohen-Macaulay R-modules
as well as an (easy computable) invariant that allows us to distinguish between
these isomorphism classes.

THE HILBERT-KUNZ FUNCTIONS OF TWO-DIMENSIONAL RINGS OF TYPE ADE

3

The full list of representatives of the isomorphism classes of the indecomposable,
maximal Cohen-Macaulay R-modules should consist of first syzygy modules of
ideals. This is done with the help of Eisenbud’s theory of matrix factorizations,
which encodes maximal Cohen-Macaulay modules over hypersurface rings as a pair
of matrices. We will see how one might translate a matrix factorization into a first
syzygy module of an ideal. This will give the desired list of first syzygy modules
of ideals. For example in the case of a ring of type E8 , we obtain the following
theorem.
Theorem 1. Let R := k[X, Y, Z]/(X 2 + Y 3 + Z 5 ). The pairwise non-isomorphic
modules
M1 = SyzR (X, Y, Z),
M2 = SyzR (X, Y 2 , Y Z, Z 2 ),
M3 = SyzR (XY, XZ, Y 2 , Y Z 2 , Z 3 ),
M4 = SyzR (XY, XZ 2, Y 3 , Y 2 Z, Y Z 3 , Z 4 ),
M5 = SyzR (XY 2 , XY Z 2 , XZ 4 , Y 4 , Y 3 Z, Y 2 Z 3 , Z 5 ),
M6 = SyzR (X, Y 2 , Y Z, Z 3 ),
M7 = SyzR (XY, XZ, Y 2 , Y Z 2 , Z 4 ),
M8 = SyzR (X, Y, Z 2 )
give a complete list of representatives of the isomorphism classes of indecomposable,
non-free, maximal Cohen-Macaulay modules.
The second step is to find an invariant of modules which is easy to calculate and
detects the isomorphism class of the module. Computing the Hilbert-series of the
syzygy modules from our finite list, one recognizes that the Hilbert-series might be
a good candidate for this invariant. But we will have two further problems, namely
e
e
e
we need an algorithm to compute the Hilbert-series of SyzR (X p , Y p , Z p ), which
depends on the parameters p and e and the second problem is that in the cases
where R is of type D or E the Hilbert-series does not detect whether a given module
is indecomposable or not.
e
e
e
The problem that we need to know that the modules SyzR (X p , Y p , Z p ) are indecomposable is solved by using ringinclusions of the form ι : R → An , where R
is of type D or E. Translating everything to the corresponding punctured spectra
R and An , we can pull the OR -modules in question back to An using the map
induced by ι. Since every OAn -module is a direct sum of line bundles and since
we know how the Frobenius acts on line bundles, we will deduce from the commutativity of the pull-backs via the Frobenius and the map induced by ι, that all
e
e
e
SyzR (X p , Y p , Z p ) are indecomposable.
Last but not least we need to deal with the problem of computing the Hilbert-series
e
e
e
of SyzR (X p , Y p , Z p ). To do so, we will prove the following theorem.
Theorem 2. Let k be a field and
R := k[X, Y, Z]/(X d − F (Y, Z))
with deg(X) = α, deg(Y ) = β, deg(Z) = γ and F ∈ k[Y, Z] homogeneous of degree
dα (d, α, β, γ ∈ N≥1 ). Let a ∈ N≥1 and write a = d · q + r with 0 ≤ r ≤ d − 1

4

DANIEL BRINKMANN

and q ∈ N. For l ∈ N we use the abbreviation S(l) := SyzR (X l , Y b , Z c ). Then the
Hilbert-series of S(a) = SyzR (X a , Y b , Z c ) is given by

(tα·r − tα·d ) · HS(a−r) (t) + (1 − tα·r ) HS(a+d−r) (t)
.
1 − tα·d
By iteration this theorem reduces the task of computing the Hilbert-series of the
e
e
e
modules SyzR (X p , Y p , Z p ) to the computation of the Hilbert-series of the mode
e
e
pe −1
pe
pe
ules SyzR (X
, Y , Z ) and SyzR (X p +1 , Y p , Z p ) for only finitely many small
values of e, which can be done over the polynomial ring k[Y, Z]. Over P :=
k[Y, Z] these modules split as a direct sum P (m) ⊕ P (n) for some m, n ∈ Z by
Hilbert’s Syzygy Theorem. To compute the number |n − m| we will define a nonstandard graded analogue τ of Han’s δ-function and prove a connection between
τ and δ which allows us to compute τ effectively. To give an example, assume
R is of type E8 . Then SyzR (X 2a , Y b , Z c ) with non-negative a, b, c ∈ Z splits as
k[Y, Z](m) ⊕ k[Y, Z](n) and we have


b c
|m − n| = 30 · δ a, ,
.
3 5
At the end of this introduction, we want to give an outline of this article.
In Section 1 we give some basic notations and discuss what is known about the
Hilbert-Kunz functions of two-dimensional rings of type ADE.
In Section 2 the process how one gets syzygy modules out of matrix factorizations
will be explained.
e
e
e
In the third section we will prove that the modules SyzR (X p , Y p , Z p ) are indecomposable in almost all characteristics if R is of type D or E.
e
e
e
In Section 4 we explain how the Hilbert-series of the modules SyzR (X p , Y p , Z p )
can be computed.
Finally, in Section 5 we will see how everything works together to obtain the HilbertKunz functions of rings of type ADE.
We will use the ring of type E8 as a running example to illustrate the various steps
in the computation of the Hilbert-Kunz functions of two-dimensional ADE-rings.
HS(a) (t) =

Acknowledgements
Since the results in this paper are part of my PhD-thesis, I thank first of all my
advisor Holger Brenner for his patience and friendship during the last years.
I thank Almar Kaid and Axel Stäbler for many usefull discussions on Hilbert-Kunz
theory as well as on vector bundles and Alessio Caminata for the discussions on the
proof that two-dimensional ADE-rings are Cohen-Macaulay finite.
Moreover, I want to thank Igor Burban who brought the theory of matrix factorizations to my attention and Kevin Tucker who is responsible for the corollary about
the F -signature functions of the rings of type ADE.
1. Hilbert-Kunz functions via vector bundles
Definition 1.1. Let (R, m) be a local Noetherian ring of characteristic p > 0 and
dimension d. Let I = (f1 , . . . , fm ) be an m-primary ideal and let λR denote the
length function.
(1) The function HK(I, R, pe ) : N → N,

  e
pe
e 7−→ λR R/ f1p , . . . , fm

THE HILBERT-KUNZ FUNCTIONS OF TWO-DIMENSIONAL RINGS OF TYPE ADE

5

is called the Hilbert-Kunz function of R with respect to I.
(2) We call the limit
HK(I, R, pe )
,
lim
e→∞
ped
whose existence was proven by Monsky in [Mon83], the Hilbert-Kunz multiplicity of R with respect to I. We denote this limit by eHK (I, R) and
set eHK (R) := eHK (m, R) for short. We will call eHK (R) the Hilbert-Kunz
multiplicity of R.
e

e

e

p
).
Sometimes we will use the symbol (f1 , . . . , fm )[p ] to denote the ideal (f1p , . . . , fm

Remark 1.2. Note that for any finitely generated R-module M its length can be
dimk (M)
computed as λR (M ) = dim
if R is an algebra over some field k and dimk (R/m)
k (R/m)
is finite.
Next we give some basic facts on syzygy bundles, which are the vector bundles that
will appear in the application of vector bundles to Hilbert-Kunz theory.
Definition 1.3. Let k be a field and R an affine k-algebra. Let f1 , . . . , fn ∈ R and
X := Spec(R).
(1) We call the kernel of the map
f1 ,...,fn

n
OX
−→ OX

the sheaf of syzygies for f1 , . . . , fn and denote it by SyzX (f1 , . . . , fn ).
(2) Let IZ be the ideal sheaf of the closed subscheme Z := V (f1 , . . . , fn ) ⊆ X.
We call the short exact sequence
n
0 −→ SyzX (f1 , . . . , fn ) −→ OX

f1 ,...,fn

−→ IZ −→ 0

the presenting sequence of the sheaf of syzygies.

Definition 1.4. Let k be a field and R a standard-graded, affine k-algebra. Let
f1 , . . . , fn ∈ R be homogeneous elements with deg(fi ) = di . Let Y := Proj(R).
(1) We call the kernel of the map
n
M
i=1

f1 ,...,fn

OY (−di ) −→ OY

the sheaf of syzygies for f1 , . . . , fn and denote it by SyzY (f1 , . . . , fn ).
(2) Let IZ be the ideal sheaf of the closed subscheme Z := V+ (f1 , . . . , fn ) ⊆ Y .
We call the short exact sequence
n
M
f1 ,...,fn
OY (−di ) −→ IZ −→ 0
0 −→ SyzY (f1 , . . . , fn ) −→
i=1

the presenting sequence of the sheaf of syzygies.

Note that the sheaf SyzX (f1 , . . . , fn ) in the affine case as well as SyzY (f1 , . . . , fn )
in the projective case are nothing but the sheafification of the (graded) R-module
SyzR (f1 , . . . , fn ). Therefore we may define in the graded case for any m ∈ Z the
sheaf of syzygies of total degree m of f1 , . . . , fn , denoted by SyzY (f1 , . . . , fn )(m),
as the sheafification of the R-module SyzR (f1 , . . . , fn )(m). Note that the equality
SyzY (f1 , . . . , fn )(m) = SyzY (f1 , . . . , fn ) ⊗ OY (m)

6

DANIEL BRINKMANN

holds, since R is standard-graded. Moreover, if R has dimension at least two and
satisfies S2 the sheaf SyzY (f1 , . . . , fn )(m) encodes the homogeneous component of
degree m of the R-module SyzR (f1 , . . . , fn ). We get this homogeneous component
back by taking global sections (cf. [Har87, Exercise III.3.5])
Γ (Y, Syz (f1 , . . . , fn )(m)) ∼
= Syz (f1 , . . . , fn )m .
Y

R

Recall some basic properties of syzygy modules. The proofs might be found in
[Bri13, Chapter 1.2.3].
Proposition 1.5. Let R be a standard-graded, affine k-domain and pick some
homogeneous elements f, f1 , . . . , fn ∈ R with f 6= 0.
(1) The following isomorphism holds
Syz (f f1 , . . . , f fn ) ∼
= Syz (f1 , . . . , fn )(− deg(f )).
R

R

(2) If f, fn is an R-regular sequence, we have
Syz (f f1 , . . . , f fn−1 , fn ) ∼
= Syz (f1 , . . . , fn )(− deg(f )).
R

R

(3) If deg(f fi ) = deg(fn ) for some i ∈ {1, . . . , n − 1}, we have
Syz (f1 , . . . , fn ) ∼
= Syz (f1 , . . . , fn−1 , fn + f fi ).
R

R

Proposition 1.6. Let k be an algebraically closed field and R a standard-graded
k-algebra of dimension at least two and Y := Proj(R). Let f1 , . . . , fn ∈ R be
homogeneous elements with deg(fi ) = di . Assume that at least one of the fi is a
non-zero divisor. Then the following statements hold.
(1) The rank of SyzY (f1 , . . . , fn ) is n − 1.
(2) If the ideal generated by the fi is R+ -primary, then the morphism
n
M
f1 ,...,fn
OY (−di ) −→ OY
i=1

is surjective and SyzY (f1 , . . . , fn ) is locally free.
(3) The sheaf SyzY (f1 , . . . , fn ) is locally free on U := D+ (f1 , . . . , fn ) ⊆ Y .
(4) If Y is an irreducible curve and the ideal (f1 , . . . , fn ) is R+ -primary, then
!
n
X
di .
det(Syz (f1 , . . . , fn )(m)) ∼
= OY (n − 1)m −
Y

i=1

In particular, we have

deg(SyzY (f1 , . . . , fn )(m)) =

(n − 1)m −

n
X
i=1

di

!

· deg(Y ).

(5) Let φ : R → S be a morphism of degree d of normal, standard-graded kdomains of dimension two. Let g : Y := Proj(S) → X be the corresponding
morphism of smooth projective curves. Then the following holds
g ∗ (Syz (f1 , . . . , fn )(m)) ∼
= Syz (φ(f1 ), . . . , φ(fn ))(dm).
X

Y

(6) Assume that char(k) = p > 0, that Y is an integral curve and that the ideal
(f1 , . . . , fn ) is R+ -primary. For the iterated pull-backs under the Frobenius,
we have
q
Fe∗ (Syz (f1 , . . . , fn )(m)) ∼
= Syz (f , . . . , f q )(mq),
X

where q = pe and e ∈ N.

X

1

n

THE HILBERT-KUNZ FUNCTIONS OF TWO-DIMENSIONAL RINGS OF TYPE ADE

7

The next lemma shows how syzygy bundles arise in Hilbert-Kunz theory. For a
proof see [Bre06] or [Tri05].
Lemma 1.7. Let k be an algebraically closed field of positive characteristic p,
let R be a standard-graded k-domain of dimension at least two satisfying S2 and
let Y = Proj(R). Let I = (f1 , . . . , fn ) be a homogeneous R+ -primary ideal with
deg(fi ) = di . Then for every q = pe ≥ 1 and m ∈ Z the following formula holds
n

  e
X
p
0
pe
h0 (X, OX (m − pe · di ))
=h (X, OX (m)) −
dimk R/ f1 , . . . , fn
m

0

+h





i=1
e

e

X, SyzX f1p , . . . , fnp




(m) .

If R = k[X, Y, Z]/(F ) is a standard-graded, two-dimensional, normal domain and
I is generated by powers of X, Y and Z, we will use Lemma 1.7 to provide explicit
formulas for HK(I, R, pe ) in the two cases where SyzProj(R) (I) admits a Frobenius
periodicity up to twist resp. where the higher Frobenius pull-backs of SyzProj(R) (I)
split as a direct sum of twisted structure sheafs.
Definition 1.8. Let X be a scheme over an algebraically closed field k of characteristic p > 0. Let S be a vector bundle over X. Assume there are s < t ∈ N such that
the Frobenius pull-backs Fe∗ (S) of S are pairwise non-isomorphic for 0 ≤ e ≤ t − 1
and Ft∗ (S) ∼
= Fs∗ (S). We say that S admits a (s, t)-Frobenius periodicity. Finally,
the bundle S admits a Frobenius periodicity if there are s < t ∈ N such that S
admits a (s, t)-Frobenius periodicity.
Remark 1.9. If X is a projective scheme only bundles of degree zero might admit
a Frobenius periodicity in the strong sence above. For bundles of non-zero degree
we will use the following weaker definition of Frobenius periodicity. We say that S
admits a (s, t)-Frobenius periodicity if
′

Fe∗ (S) ≇ Fe ∗ (S)(m)

and Ft∗ (S) ∼
= Fs∗ (S)(n)

hold for all 0 ≤ e′ < e ≤ t − 1, all m ∈ Z and some n ∈ Z.
Lemma 1.10. Let R := k[X, Y, Z]/(F ) be a normal standard-graded domain over
an algebraically closed field k with char(k) = p > 0. Fix the positive natural numbers
α, β, γ as well as the ideal I = (X α , Y β , Z γ ). Assume that for some e ∈ N one has
 e
(1.1)
SyzR I [p ] ∼
= SyzR (X a , Y b , Z c )(−n)
for some n ∈ Z and some positive integers a, b, c such that at least one of the
inequalities a < αpe , b < βpe , c < γpe holds. With D := dimk (R/(X a , Y b , Z c )) the
value at e of the Hilbert-Kunz function of R with respect to I is given by

d · Q(α, β, γ) · p2e − d · Q(a, b, c)
+ D,
4
where Q denotes the quadratic form
HK(I, R, pe ) =

(a, b, c) 7−→ 2(ab + ac + bc) − a2 − b2 − c2 .
Proof. Computing the degrees of the C := Proj(R)-bundles corresponding to the
R-modules in (1.1), we find
n=

q(α + β + γ) − a − b − c
.
2

8

DANIEL BRINKMANN

By assumption we have the following diagram
0

// SyzR (I [q] )

// R(−α · q) ⊕ R(−β · q) ⊕ R(−γ · q)

// I [q]

// 0

// R(−n − a) ⊕ R(−n − b) ⊕ R(−n − c)

// J

// 0,

∼
=

0



// SyzR (J)

with J := (X a , Y b , Z c ). Sheafifying the second row, twisting by OC (m) and taking
global dimensions yields
h0 (C, SyzC (X a , Y b , Z c )(m − n))

(1.2) = h0 (C, OC (m − n − a)) + h0 (C, OC (m − n − b)) + h0 (C, OC (m − n − c))
− h0 (C, OC (m − n)) + dimk (R/(X a , Y b , Z c ))m−n .

Doing the same with the first row and substituting the term

 

h0 C, SyzC I [q] (m)
with the right hand side of (1.2), we get

dimk (R/(X αq , Y βq , Z γq ))
" x
x
x
X
X
X
= lim
h0 (OC (m − βq))
h0 (OC (m − αq)) −
h0 (OC (m)) −
x→∞

−
+

m=0
x
X

m=0

= lim

x→∞

−
+

h0 (OC (m − γq)) −

h (OC (m − n − b)) +

"

x
X

m=0
x−γq
X
0

h0 (OC (m)) −

h (OC (m)) +

m=0
x−n−c
X

(1.3) = lim

x→∞

m=0

0

m=0

−

m=0
x
X

m=0

x
X

"

h0 (OC (m)) −
x
X

m=x−n+1

x−βq
X

m=x−n−b+1

m=0

m=0

m=0
x−n
X

By Riemann-Roch we have

h0 (OC (m − n − a))

#

h (OC (m − n − c)) + D

h0 (OC (m)) −

x−βq
X

h0 (OC (m)

m=0

x−n−b
X

h0 (OC (m))

m=0

#

h0 (OC (m)) + D

m=0

h (OC (m) −

m=0

0

h0 (OC (m)) +

h0 (OC (m)) −

0

h0 (OC (m − n)) +

x
X

x−αq
X

x−n−a
X

m=0

x
X

x−αq
X

h0 (OC (m))

m=x−n−a+1
x−γq
X

m=x−n−c+1

0

#

h (OC (m)) + D.

h0 (OC (m)) = m · deg(C) + 1 − g + h1 (OC (m)),
where g is the genus of C. Substituting this in Equation (1.3), the summands
involving the genus cancel and the sums over h1 (OC (m)) vanish for x ≫ 0 due to

THE HILBERT-KUNZ FUNCTIONS OF TWO-DIMENSIONAL RINGS OF TYPE ADE

9

Serre. The surviving summands can be expressed as
d · Q(α, β, γ) · q 2 − d · Q(a, b, c)
+ D.
4

Similarly, one can show the following.
Lemma 1.11. With the notations from Lemma 1.10, assume that there is an e ∈ N
such that
 e
(1.4)
SyzR I [p ] = R(−n) ⊕ R(−l) for some n, l ∈ Z.

Then the value at e of the Hilbert-Kunz function of R with respect to I is given by

d · Q(α, β, γ) · p2e + d · |n − l|2
.
4
e
Note that the assumption (1.4) means that the quotient R/I [p ] has finite projective
e′
dimension and this carries over to all quotients R/I [p ] for e′ ≥ e.
HK(I, R, pe ) =

2. Matrix factorizations and representations as first syzygy
modules of ideals
Let S := k[X, Y, Z] and let R := S/(F ) be a ring of type ADE. The goal is
to construct isomorphisms between maximal Cohen-Macaulay R-modules and first
syzygy modules of homogeneous ideals of R. To be more precise, given a full
set M1 , . . . , Mn of representatives for the isomorphism classes of indecomposable,
maximal Cohen-Macaulay R-modules, we want to find for each i non-zero elements
f1 , . . . , fm ∈ R such that
Mi ∼
= SyzR (f1 , . . . , fm ).
This has to be possible, since first syzygy modules of ideals are second syzygy
modules of quotient rings.
We will denote the punctured spectrum of R by U . Note that all maximal CohenMacaulay R-modules are locally free on U since Spec(R) has an isolated singularity
at the origin.
Now we describe a way to find a representation of a maximal Cohen-Macaulay
module, given by a matrix factorization, as a syzygy bundle on U .
At first we recall the definition of a matrix factorization and some general facts.
Our reference for matrix factorizations is Chapter 7 of the textbook [Yos90]. For
the interaction of the graded and the local situation see [Yos90, Chapter 15].
Definition 2.1. Let S be a polynomial ring. Let φ, ψ ∈ Matn (S) with φψ =
ψφ = F · En , for a non-zero element F ∈ (S+ )2 . Then the pair (φ, ψ) is a matrix
factorization for F of size n. It is called reduced if all entries are non-units.
Remark 2.2. Let (φ, ψ) be a matrix factorization for some F of size n.
(1) The module coker(φ) is a maximal Cohen-Macaulay R := S/(F )-module,
which is non-trivial if and only if (φ, ψ) is reduced.
(2) If F is irreducible, the determinant of φ is u · F i , where u ∈ S is a unit and
0 ≤ i ≤ n. Moreover, the rank of the R-module coker(φ) equals i.
(3) The pair (φT , ψ T ) is again a matrix factorization for F of size n and
∼ coker(φ)∨ .
coker(φT ) =

10

DANIEL BRINKMANN

(4) It is possible to define direct sums as well as morphisms of matrix factorizations. Then the set of matrix factorizations for F becomes an additive
category and Eisenbud has shown in [Eis80] that the equivalence classes of
reduced, indecomposable matrix factorizations are in one to one correspondence with the isomorphism classes of non-free, indecomposable, maximal
Cohen-Macaulay R-modules.
Construction 2.3. For a (reduced) matrix factorization (φS , ψS ) of size n, we
have a two periodic free resolution of M := coker(φS ) as an R-module:
ψ

φ

ψ

φ

. . . −→ Rn −→ Rn −→ Rn −→ Rn −→ M −→ 0,
where φ and ψ arise from φS and ψS by taking their entries modulo F . Note that
we can identify M with coker(φ).
Therefore we get
M∼
= Rn / im(φ) ∼
= Rn / ker(ψ) ∼
= im(ψ).
For any subset J of {1, . . . , n} we denote by ψ J the matrix obtained from ψ by
keeping all columns whose index belongs to J (and deleting all other columns).
If coker(φ) ∼
= im(ψ) has rank m as an R-module, the sheaf im(ψ)∼ |U gives a locally
free OU -module of rank m. Therefore a representing first syzygy module of an ideal
has to be a syzygy module of an (m + 1)-generated ideal.
Suppose that we can choose a J ⊆ {1, . . . , n} of cardinality m+1 such that im(ψ)∼ |U
1
and F := im(ψ J )∼ |U are isomorphic as sheaves of OU -modules
, where the isomor
J
phism is the map induced by the natural inclusion im ψ ⊆ im(ψ). A necessary
condition for this is that the matrix ψ J has full rank - namely m - in every nonzero point. Otherwise there would be at least one point u ∈ U such that the stalk
(im(ψ J )∼ )u has rank at most m − 1.
Since im(ψ)∼ |U comes from a maximal Cohen-Macaulay R-module, it is locally free
as OU -module, giving that
ψJ

m+1
OU
−→ F −→ 0

is a surjection of locally free OU -modules of ranks m + 1 and m. Therefore the
kernel L has to be locally free of rank one. This gives an isomorphism
L ⊗ det(F ) ∼
= OU

of the determinants. We distinguish whether the determinant of F is trivial or not.
Case 1. Assume det(F ) ∼
= OU . Then the line bundle L has to be trivial. Hence
(2.1)

η

ψJ

m+1
−→ F −→ 0
0 −→ OU −→ OU

is a short exact sequence and η has to be the multiplication by a column vector
(g1 , . . . , gm+1 )T with gi ∈ R. But then Sequence (2.1) is exactly the dual of the
presenting sequence of SyzU (g1 , . . . , gm+1 ), hence there is an isomorphism
SyzU (g1 , . . . , gm+1 ) ∼
= F ∨.
Moreover, the ideal generated by the gi has to be R+ -primary, since the determinant
of the syzygy bundle is trivial.
1Note that we have no argument why this should be possible, but it works in all examples.

THE HILBERT-KUNZ FUNCTIONS OF TWO-DIMENSIONAL RINGS OF TYPE ADE

11

Case 2. Now assume that L is a non-trivial line bundle, hence det(F ) ∼
= L∨ .
m+1
Dualizing the sequence 0 → L → OU
→ F → 0 yields
(2.2)

(ψ J )T

m+1
0 −→ F ∨ −→ OU
→ L∨ −→ 0.

Since L∨ is a line bundle on U there exists an embedding L∨ ֒→ OU . With this
embedding, we can extend Sequence (2.2) to
(2.3)

(ψ J )T

µ

m+1
0 −→ F ∨ −→ OU
−→ OU −→ T −→ 0,

where T is a non-zero torsion sheaf and the map µ is the multiplication by a row
vector (g1 , . . . , gm+1 ). In this case, Sequence (2.3) is the presenting sequence of
the syzygy bundle SyzU (g1 , . . . , gm+1 ), where the ideal (g1 , . . . , gm+1 ) is not R+ primary, since T is non-zero.
In both cases the column vector (g1 , . . . , gm+1 )T generates the kernel of ψ J . In order

to compute the gi , we only need to find an element (f1 , . . . , fm+1 )T ∈ ker ψ J .
Then (f1 , . . . , fm+1 )T is just a multiple of (g1 , . . . , gm+1 )T and equality (up to
multiplication by a unit) holds if we can prove that the fi are coprime. For example
this condition is automatically fulfilled if the ideal generated by the fi is R+ primary. Anyway, the syzygy modules of the two ideals are isomorphic.
All in all, our task is to find a vector (f1 , . . . , fm+1 )T in the kernel of ψ J . This
gives an isomorphism
g∨ ∼
M
= SyzU (f1 , . . . , fm+1 ).
= F∨ ∼

This isomorphism extends to a global isomorphism of R-modules, since all involved
modules are reflexive (cf. [BD08, Lemma 3.6])
We will illustrate this construction by an explicit computation.
Example 2.4. Let F = X 2 + Y 3 + Z 5 be the equation of an E8 -singularity. The
pair of matrices

 

X −Y −Z
0
X
Y
Z
0

 2
−Y 2
0
−Z 
X
0
Z 

 , Y 4 X

(φ, ψ) := 
4
−Z
0
X
Y 
0
X −Y  Z
0
Z 4 −Y 2 X
0
−Z 4 Y 2 X

is a reduced (indecomposable) matrix factorization for F of size 4 and the corresponding module coker(φ) has rank two. The following computations show that
over U the first column of ψ lies in the span generated by the other three columns
of ψ



  
−Y
−Z
X
 Z 4  0  Y 2 
Y2 
X
+
 = ,
·
·
X  0  X  X  Z 4 
Z4
−Y 2
0

  


X
−Y
0
 Z 4 −Z  Y 2 
X 
X
+
 = .
− ·
·
Y  0 
Y  Y  Z 4 
0
Z4
X

Now we have to find three non-zero elements f1 , f2 and f3 such that (f1 , f2 , f3 )T
belongs to the kernel of ψ {2,3,4} .

12

DANIEL BRINKMANN

The first row (−Y, −Z, 0) of ψ {2,3,4} gives the relation −Y ·f1 −Z ·f2 = 0 from which
we conclude (since Y and Z are coprime in R) that f1 = Z · g and f2 = −Y · g
for some non-zero g ∈ R. From the second row (X, 0, −Z) we get the relation
Z · X · g − Z · f3 = 0. This gives f3 = X · g. Looking at the relations given by
the rows three and four of ψ {2,3,4} , we see that g can be chosen as 1. We get the
isomorphism
{2,3,4} ∼ ∨ ∼
∼
im(ψ)∼ |∨
) |U = SyzU (Z, −Y, X),
U = im(ψ

extending to the global isomorphism
im(ψ)∨ ∼
= SyzR (Z, −Y, X).
It is easy to see that SyzR (Z, −Y, X) is self-dual since coker(φ)∨ ∼
= coker(φT ) and
(α, α) : (φ, ψ) −→ (φT , ψ T ) with


0 0 0 −1
0 0 1 0 

α := 
0 −1 0 0 
1 0 0 0
is an equivalence of matrix factorizations. Hence,

coker(φ) ∼
= im(ψ) ∼
= SyzR (Z, −Y, X).
In fact, one can delete any column from ψ, giving the isomorphisms
im(ψ) ∼
= SyzR (Z, −Y, X)
∼
= Syz (Z, X, Y 2 )
R

∼
= SyzR (Y, X, −Z 4 )
∼ Syz (X, −Y 2 , −Z 4 ).
=
R

Note that if coker(φ) has rank one, the isomorphism coker(φ) ∼
= im(ψ) is already
enough to obtain an isomorphism of the form coker(φ) ∼
= SyzR (f1 , f2 ), since for a
suitable choice of f1 , f2 the columns of ψ are directly the generators SyzR (f1 , f2 ).
A complete list of non-equivalent matrix factorizations representing all isomorphism
classes of indecomposable, maximal Cohen-Macaulay R-modules can be found in
[KST07] resp. [LW12, Chapter 9, §4]. Using these lists, one obtains the following
theorems whose proofs are analogue to the previous examples and can be found
explicitly in [Bri13, Chapter 3]. The statements of the form det(SyzR (I)∼ |U ) ∼
=
SyzR (J)∼ |U follow by showing I ∨∨ = J. We should mention that the enumeration
of the modules in the subsequent theorems corresponds to the enumeration of the
matrix factorizations as given in the above references. It corresponds also to the
enumeration of the vertices in the corresponding Dynkin diagrams.
Theorem 2.5. Let R = k[X, Y, Z]/(X n+1 − Y Z) with n ≥ 0. The pairwise nonisomorphic modules Mm := SyzR (X m , Z) for m = 1, . . . , n give a complete list
of representatives of the isomorphism classes of indecomposable, non-free, maximal
∨ ∼
Cohen-Macaulay modules. Moreover, we have Mm
= (X m , Y ).
= Mn+1−m and Mm ∼

THE HILBERT-KUNZ FUNCTIONS OF TWO-DIMENSIONAL RINGS OF TYPE ADE

13

Theorem 2.6. Let R := k[X, Y, Z]/(X 2 + Y n−1 + Y Z 2 ) and n ≥ 4. The pairwise
non-isomorphic modules
M1 = SyzR (X, Y ),
Mm = SyzR (X, Y m/2 , Z)
Mm = SyzR (X, Y

(m+1)/2

if m ∈ {2, . . . , n − 2} is even,
, Y Z)

if m ∈ {2, . . . , n − 2} is odd,

Mn−1 = SyzR (X, Z − iY (n−2)/2 ) and

Mn = SyzR (X, Z + iY (n−2)/2 ) if n is even, or

Mn−1 = SyzR (Z, X + iY (n−1)/2 ) and
Mn = SyzR (Z, X − iY (n−1)/2 ) if n is odd

give a complete list of representatives of the isomorphism classes of indecomposable,
non-free, maximal Cohen-Macaulay modules. Moreover, these modules are self-dual
∼
∨
∼
|U and Mn∼ |U
with the exception Mn−1
= Mn if n is odd. The determinants of Mn−1
∼
are M1 |U . For the rank one modules we have the isomorphisms M1 ∼
= (X, Y ),
Mn−1 ∼
=
= (X, Z + iY (n−2)/2 ) if n is even and Mn−1 ∼
= (X, Z − iY (n−2)/2 ) and Mn ∼
(Z, X − iY (n−1)/2 ) and Mn ∼
= (Z, X + iY (n−1)/2 ) if n is odd.
Theorem 2.7. Let R := k[X, Y, Z]/(X 2 + Y 3 + Z 4 ). The pairwise non-isomorphic
modules
M1 = SyzR (X, Y, Z),
M2 = SyzR (X, Y 2 , Y Z, Z 2 ),
M3 = SyzR (iX + Z 2 , Y 2 , Y Z),
M4 = SyzR (−iX + Z 2 , Y 2 , Y Z),
M5 = SyzR (−iX + Z 2 , Y ),
M6 = SyzR (iX + Z 2 , Y )
give a complete list of representatives of the isomorphism classes of indecomposable,
non-free, maximal Cohen-Macaulay modules. Moreover, the modules M1 and M2
are self-dual, M3∨ ∼
= M4 and M5∨ ∼
= M6 . The Determinants of M3∼ |U resp. M4∼ |U
∼
∼
are M5 |U resp. M6 |U . For the rank one modules we have the isomorphisms
M5 ∼
= (−iX + Z 2 , Y ).
= (iX + Z 2 , Y ) and M6 ∼
Theorem 2.8. Let R := k[X, Y, Z]/(X 2 +Y 3 +Y Z 3 ). The pairwise non-isomorphic
modules
M1 = SyzR (X, Y, Z),
M2 = SyzR (X, Y 2 , Y Z, Z 2 ),
M3 = SyzR (XY, XZ, Y 2 , Y Z 2 , Z 3 ),
M4 = SyzR (X, Y 2 , Y Z),
M5 = SyzR (XY, XZ, Y 2 , Y Z 2 ),
M6 = SyzR (X, Y, Z 2 ),
M7 = SyzR (X, Y )
give a complete list of representatives of the isomorphism classes of indecomposable,
non-free, maximal Cohen-Macaulay modules. Moreover, all Mj are self-dual and

14

DANIEL BRINKMANN

det(Mj∼ |U ) ∼
= M7∼ |U for j ∈ {4, 5, 7}. The rank one module M7 is isomorphic to
the ideal (X, Y ).
Theorem 2.9. Let R := k[X, Y, Z]/(X 2 + Y 3 + Z 5 ). The pairwise non-isomorphic
modules
M1 = SyzR (X, Y, Z),
M2 = SyzR (X, Y 2 , Y Z, Z 2 ),
M3 = SyzR (XY, XZ, Y 2 , Y Z 2 , Z 3 ),
M4 = SyzR (XY, XZ 2, Y 3 , Y 2 Z, Y Z 3 , Z 4 ),
M5 = SyzR (XY 2 , XY Z 2 , XZ 4 , Y 4 , Y 3 Z, Y 2 Z 3 , Z 5 ),
M6 = SyzR (X, Y 2 , Y Z, Z 3 ),
M7 = SyzR (XY, XZ, Y 2 , Y Z 2 , Z 4 ),
M8 = SyzR (X, Y, Z 2 )
give a complete list of representatives of the isomorphism classes of indecomposable,
non-free, maximal Cohen-Macaulay modules. Moreover, all Mj are self-dual.
Note that most of the maximal Cohen-Macaulay can be realized as first syzygy
modules of monomial ideals. Moreover, for the modules of rank at least two these
ideals tend to be R+ -primary.
3. Relations between the maximal Cohen-Macaulay modules over the
ADE-rings
Considering the rings of type ADE as rings of invariants under finite subgroups of
SL2 (k), we get ring inclusions among them by the actions of normal subgroups as
shown in Table 3.1 below.
We use the generators of the finite subgroups of SL2 (C), given in [BD08]. Note that
these representations are well-defined in positive characteristic, if the group order
is invertible (modulo the characteristic).
Let us very briefly recall some facts from invariant theory. If G is a group, acting on a ring R by ring automorphisms and H is a normal subgroup of G, the
ring of invariants RG is a subring of RH , the quotient G/H acts on RH and
G/H


one has RG = RH
. Let X := Spec RG and Y = Spec RH (or the
punctured resp. projective spectra). The inclusion RG ֒→ RH induces a morphism ι : Y → X. Given an OX -module F , its pull-back ι∗ (F ) is an OY module. If R = k[x, y] and G ( SL2 (k) is finite, we want to understand what
happens to the sheaves associated to the indecomposable, non-free, maximal CohenMacaulay modules under these pull-backs. For this purpose let i : R ֒→ S with
R := k[x, y]G = k[U, V, W ]/F1 (U, V, W ) and S := k[x, y]H = k[X, Y, Z]/F2 (X, Y, Z)
be an inclusion between two (different) rings of type ADE, where H is a normal
subgroup of G. This inclusion is given by sending U , V , W to homogeneous polynomials in the variables X, Y , Z and induces a morphism
ι : S := Spec(S) \ {(X, Y, Z)} −→ R := Spec(R) \ {(U, V, W )}.

THE HILBERT-KUNZ FUNCTIONS OF TWO-DIMENSIONAL RINGS OF TYPE ADE

15


/ D2n
Z/(4n)
O
✉: O
t|4n ✉✉
✉
✉✉
, ✉✉✉
Z/(t) r
O ■■
■■ t|2n
■■
■■
■$ 

?

/ D? n
2|t
Z/(2n)
: O
✉✉
✉✉
✉
✉✉
 ? , ✉✉
2|n
Z/(2)l  r
✻✻■■
✻✻ ■■■
✻✻ ■■■
✻✻ ■$  ?

/ D2  
/ T
✻✻ Z/(4) 
_
✻✻
✻✻
✻✻


I
O
Table 3.1. Inclusions of the finite subgroups (up to conjugation)
of SL2 (C). All direct inclusions are inclusions of normal subgroups.

Let M := SyzR (f1 , . . . , fm+1 ) be one of the indecomposable, non-free, maximal
Cohen-Macaulay R-modules from Theorems 2.5-2.9. We obtain the isomorphism
ι∗ (M ∼ |R ) ∼
= SyzS (i(f1 ), . . . , i(fm+1 ))∼ |S .

Now, the OS -module SyzS (i(f1 ), . . . , i(fm+1 ))∼ |S has to be the sheaf associated
to a (not necessarily indecomposable) maximal Cohen-Macaulay S-module N and
we want to have an explicit description of N . To get this description, we do the
following manipulations on P := SyzS (i(f1 ), . . . , i(fm+1 )) which change P only by
an isomorphism (cf. Proposition 1.5).
Step 1: If the i(fj ) have a common factor A, replace all i(fj ) by i(fj )/A.
Step 2: If B is a common factor of i(f1 ), . . . , i(fˇj ), . . . , i(fm+1 ) and the ideal generated by B and the i(fj ) is (X, Y, Z)-primary, replace all i(fk ), k 6= j, by
i(fk )/B.
In both steps, we allow to replace i(fj ) by i(fj ) + g · i(fl ) for g ∈ S and j 6= l or by
a constant multiple λ · i(fj ) with λ ∈ k × .
Example 3.1. Since the group Z/(2n) is a normal divisor of the group Dn , we
obtain a ring inclusion
φ : k[X, Y, Z]/(X 2 + Y n+1 + Y Z 2 ) −→ k[U, V, W ]/(U 2n − V W ),
√
given by X 7→ 2i · U (V − W ), Y 7→ n+1 −1 · U 2 , Z 7→ 12 · (V + W ). Lets name
the rings above by their type of singularity, hence φ : Dn+2 → A2n−1 . Then φ
induces a map ψ between the punctured spectra of these two rings, called Dn+2
and A2n−1 . Now consider the module M := SyzDn+2 (X, Y, Z), which is the module
M2 from Theorem 2.6. We denote the restriction to Dn+2 of the sheafification of
SyzDn+2 (X, Y, Z) by SyzDn+2 (X, Y, Z) (and similarly for sheaves over A2n−1 coming

16

DANIEL BRINKMANN

from syzygy modules over A2n−1 . Then (provided p 6= 2)
√
n+1
−1 · U 2 , 12 · (V + W ))
ψ ∗ (SyzDn+2 (X, Y, Z)) ∼
= SyzA2n−1 ( 2i · U (V − W ),
∼
(U (V − W ), U 2 , V + W )
= Syz
A2n−1

∼
= SyzA2n−1 (U V, U 2 , V + W )
∼
(V, U, V + W )
= Syz

(3.1)

A2n−1

∼
= SyzA2n−1 (V, U, W )
∼
(V, U ) ⊕ Syz
= Syz
A2n−1

A2n−1 (W, U ),

where the isomorphism (3.1) is due to “Step 2”, since (U, V + W ) is (U, V, W )primary.
Computing φ∗ (SyzR (f1 , . . . , fm )∼ |R ) for all R of type ADE, for all possible φ as
shown in Table 3.1 and for all SyzR (f1 , . . . , fm ) from Theorems 2.5-2.9 as in the
previous example, we obtain in surprisingly many cases a first syzygy module which
is listed explicitly in the Theorems 2.5-2.9.
If this is not the case, we compute generators for P = SyzS (i(f1 ), . . . , i(fm+1 )) and
look for relations among these generators. These relations will detect the module
P ∨ . To compute generators for P , we will work over the factorial domain k[x, y]
(the ring onP
which the group acts) as follows. Let (a1 , . . . , am+1 ) ∈ P . Then treat
the relation aj ·i(fj ) = 0 over k[x, y], meaning that we write the polynomials i(fj )
in the variables x, y. Then oneP
cancels the common multiple of the i(fj ), written
in x, y. The result is a relation
aj · gj (x, y) = 0, where the gj are coprime. From
this relation one can compute the aj (as elements in S!), using the fact that k[x, y]
is factorial. We give an example.
Example 3.2. From the inclusion Z/(3) → Z/(9), we obtain a ring inclusion
φ : A8 = k[X, Y, Z]/(X 9 − Y Z) → A2 = k[U, V, W ]/(U 3 − V W ),

given by X 7→ U , Y 7→ V 3 , Z 7→ W 3 . Using the same notation philosophy as before,
the map φ induces a morphism ψ : A2 → A8 between the punctured spectra. The
module SyzA8 (X, Z) appears in Theorem 2.5 and
ψ ∗ (SyzA8 (X, Z)) ∼
= SyzA2 (U, W 3 ).
In this case we cannot use our reduction rules, since the ideal (U, W ) is not
(U, V, W )-primary. Therefore, we use the process described before this example.
The group Z/(3) acts on k[x, y] by x 7→ ǫx and y 7→ ǫ2 y, where ǫ is a primitive 3rd
root of unity. The ring of invariants is A2 with U = xy, V = x3 , W = y 3 . Now
pick an element (s1 , s2 ) ∈ SyzA2 (U, W 3 ), hence
(3.2)

s1 · U + s2 · W 3 = 0.

On k[x, y] this relation becomes
s1 · xy + s2 · y 9 = 0
and is equivalent to
(3.3)

s1 · x + s2 · y 8 = 0.

Over k[x, y] the solutions of (3.3) have the form
(3.4)

(s1 , s2 ) = c · (−y 8 , x)

THE HILBERT-KUNZ FUNCTIONS OF TWO-DIMENSIONAL RINGS OF TYPE ADE

17

for some c ∈ k[x, y]. The solutions of (3.2) are of the form (3.4) with c such that cy 8
as well as cx are polynomials in x3 , y 3 and xy. In this case all monomials appearing
in c have to be multiples of either y or x2 . Hence, the set of solutions of (3.2) is
generated as A2 -module by τ1 := y(−y 8 , x) = (−W 3 , U ) and τ2 := x2 (−y 8 , x) =
(−U 2 W 2 , V ). These generators satisfy the global relation
V · τ1 − U · τ2 = 0.

This shows

SyzA2 (U, W 3 ) ∼
= SyzA2 (U, W ),
= SyzA2 (U, V )∨ ∼
where the second isomorphism is due to Theorem 2.5.
Using this approach and Macaulay2, one can compute for each ring R of type ADE
and for each (representative of one of the isomorphism classes of the) indecomposable, maximal Cohen-Macaulay R-module its pull-backs along all morphisms from
Table 3.1. These computations are explicitly done in [Bri13, Chapter 5]. At this
point we only state the results that are necessary for the later argumentation.
Dn+2
M1
M2m
M2m−1
Mn+1 , n even
Mn+2 , n even
Mn+1 , n odd
Mn+2 , n odd

A2n−1
OA2n−1
M2m−1 ⊕ M2(n−m)+1
M2m−2 ⊕ M2(n−m)+2
Mn
Mn
Mn
Mn

A1
OA1
M21
2
OA
1
2
OA
1
2
OA
1
M1
M1

An−1
OAn−1
Mr ⊕ Mn−r , r := 2m − 1
Mr ⊕ Mn−r , r := 2m − 2
OAn−1
OAn−1
OAn−1
OAn−1

Table 3.2. The table shows the pull-back behavior of the Mi on Dn+2 .

E6
A1

M1
M21

M2
3
OA
1

M3
M21

M4
M21

M5
OA1

M6
OA1

Table 3.3. The table shows the pull-back behavior of the Mi on E6 .

E7
A1

M1
M21

M2
3
OA
1

M3
M41

M4
2
OA
1

M5
3
OA
1

M6
M21

M7
OA1

Table 3.4. The table shows the pull-back behavior of the Mi on E7 .

E8
A1

M1
M21

M2
3
OA
1

M3
M41

M4
5
OA
1

M5
M61

M6
3
OA
1

M7
4
OA
1

M8
M21

Table 3.5. The table shows the pull-back behavior of the Mi on E8 .

18

DANIEL BRINKMANN

Using these direct computations, we can show that the Frobenius pull-backs of the
syzygy modules of the maximal ideals stay indecomposable in almost all characteristics if the singularity is of type D or E. First recall the following lemma whose
proof can be found in [BK05, Lemma 1.2.6].
Lemma 3.3. If X is separable and k is algebraically closed, we have
F∗ (L) ∼
= Lp
for every invertible sheaf L on X.
Lemma 3.4. Let U be the punctured spectrum of a two-dimensional ring R of type
D or E and let F : U → U be the (restricted) Frobenius morphism. Assume that
the order of the group corresponding to the singularity is invertible in the ground
field. Then Fe∗ (SyzU (m)) is indecomposable for all e ∈ N.
Proof. Recall from Section 2 that SyzU (m) is isomorphic to M2 if R is of type
D and M1 if R is of type E. Assume first that R is of type D2n+2 or E. If
Fe∗ (SyzU (m)) splits for some e ≥ 1, its pull-back to A1 (along the morphism ψ,
induced by the inclusion into A1 ) gets trivial, since all line bundles pull back to
the structure sheaf by Tables 3.2-3.5. But the pull-back of SyzU (m) along ψ is
non-trivial (in fact, it is always M21 by the direct computations) and its Frobenius
pull-back stays non-trivial in odd characteristics. This gives a contradiction, since
the two types of pull-backs commute.
Assume that R is of type Dn+2 with n odd. Since Fe∗ (SyzU (m)) might split as
Mn+1 ⊕ Mn+2 and both pull back to M1 on A1 , the argument above gives no
contradiction. In this case, we have to compare the pull-backs to An−1 . Then all
line bundles from Dn+2 pull back to the structure sheaf of An−1 , but ψ ∗ (M2 ) ∼
=
M1 ⊕ Mn−1 . The Frobenius pull-backs on the right hand side do not get trivial,
since n is invertible modulo p.
Moreover, if R is of type D, the Frobenius pull-backs of M2 are again of the form
M2m , since the M2m−1 get trivial after pulling them back to A1 .

By the previous lemma all Frobenius pull-backs of SyzU (m) are again indecomposable of rank two. Since in all cases det(Fe∗ (SyzU (m))) ∼
= OU , we obtain directly
Fe∗ (SyzU (m)) ∼
= SyzU (m) if R is of type E6 , since in this case there is only one
isomorphism class of indecomposable rank two modules, whose representation is a
syzygy module of an m-primary ideal. In the E7 case Fe∗ (SyzU (m)) has to be isomorphic to either M1 or M6 and in the E8 case Fe∗ (SyzU (m)) has to be isomorphic
to either M1 or M8 . In both cases this depends on the characteristic as we will
see in the next section.
4. Detecting isomorphism classes by the Hilbert-series
Now we need an invariant of the isomorphism classes of the indecomposable, maximal Cohen-Macaulay R-modules. It turns out that the Hilbert-series does quite a
good job in this case.
For example if R is of type E8 , there are two isomorphism classes of indecomposable,
maximal Cohen-Macaulay R-modules of rank two, namely
M1 = SyzR (X, Y, Z)

and

M8 = SyzR (X, Y, Z 2 ).

THE HILBERT-KUNZ FUNCTIONS OF TWO-DIMENSIONAL RINGS OF TYPE ADE

19

Their Hilbert-series are given by
HM1 (t) =

t16 + t21 + t25 + t30
(1 − t10 )(1 − t6 )

and

HM8 (t) =

t22 + t25 + t27 + t30
.
(1 − t10 )(1 − t6 )

e
e
e
Since by Lemma 3.4 for all primes p ≥ 7, the modules M e := SyzR X p , Y p , Z p
are indecomposable, maximal Cohen-Macaulay modules, their Hilbert-series have
to be, up to a factor tl , the Hilbert-series of either M1 or M8 . To compute the
Hilbert-series of M e in dependence on the two parameters p and e, we will make
use of the following statements.

Lemma 4.1. Let k be a field and
R := k[X, Y, Z]/(X d − F (Y, Z))
with deg(X) = α, deg(Y ) = β, deg(Z) = γ and F ∈ k[Y, Z] homogeneous of degree
dα (d, α, β, γ ∈ N≥1 ). Let a ∈ N≥1 and write a = d · q + r with 0 ≤ r ≤ d − 1 and
q ∈ N. For all s ∈ Z we have a short exact sequence
0 −→ SyzR (X a+d−2r , Y b , Z c )(s − α · r)
ψ

−→ SyzR (X a−r , Y b , Z c )(s − α · r) ⊕ SyzR (X a+d−r , Y b , Z c )(s)
φ

−→ SyzR (X a , Y b , Z c )(s) −→ 0,
where the maps are defined via
ψ(h1 , h2 , h3 ) :=((X d−r · h1 , h2 , h3 ), (−h1 , −X r · h2 , −X r · h3 ))

φ((f1 , f2 , f3 ), (g1 , g2 , g3 )) :=(f1 + X d−r · g1 , X r · f2 + g2 , X r · f3 + g3 ).

Proof. The injectivity of ψ is clear and the exactness at the middle spot is straightforward. The proof that φ is surjective is essentially the same as in [Bre05, Lemma
1.1]. See also [Bri13, Chapter 4] for a detailed proof and a slight generalization. 
Remark 4.2. Note that the two first syzygy modules SyzR (X a−r , Y b , Z c ) and
SyzR (X a+d−r , Y b , Z c ) appearing in the middle spot are already defined over k[Y, Z],
hence they split by Hilberts Syzygy Theorem as a direct sum of two (degree shifted)
copies of R.
Theorem 4.3. The notations are the same as in Lemma 4.1. For l ∈ N we
use the abbreviation S(l) := SyzR (X l , Y b , Z c ). Then the Hilbert-series of S(a) =
SyzR (X a , Y b , Z c ) is given by
HS(a) (t) =

(tα·r − tα·d ) · HS(a−r) (t) + (1 − tα·r ) HS(a+d−r) (t)
.
1 − tα·d

Proof. Let a′ := a + d − 2r = dq + d − r and r′ := d − r. Since a′ + d − 2r′ = a,
Theorem 4.1 yields
HS(a) (t) = tα·r HS(a−r) (t) + HS(a+d−r) (t) − tα·r HS(a′ ) (t) and
′

′

HS(a′ ) (t) = tα·r HS(a−r) (t) + HS(a+d−r) (t) − tα·r HS(a) (t).

Substituting HS(a′ ) (t) in the first formula and solving for HS(a) (t) gives the result.


20

DANIEL BRINKMANN

Turning back to the question of computing the Hilbert-series of
 e

e
e
M e = SyzR X p , Y p , Z p

if R is type E8 , we obtain via Theorem 4.3

(t15 − t30 ) · HSyzR (X pe −1 ,Y pe ,Z pe ) (t) + (1 − t15 ) HSyzR (X pe +1 ,Y pe ,Z pe ) (t)
.
1 − t30
This shows that it is enough to compute the Hilbert-series of the k[Y, Z]-modules


 e

pe −1
e
e
e
e
Syzk[Y,Z] X p −1 , Y p , Z p ∼
= Syzk[Y,Z] (−Y 3 − Z 5 ) 2 , Y p , Z p ,

 e


pe +1
e
e
e
e
Syzk[Y,Z] X p +1 , Y p , Z p ∼
= Syzk[Y,Z] (−Y 3 − Z 5 ) 2 , Y p , Z p .
HM e (t) =

By Hilbert’s Syzygy Theorem, modules of the form


pe ∓1
e
e
(4.1)
Syzk[Y,Z] (Y 3 + Z 5 ) 2 , Y p , Z p

split as k[Y, Z](m) ⊕ k[Y, Z](n) for suitable m, n ∈ Z.
The question now is how to compute the degree shifts m and n. If the quotients were
of the form k[Y, Z]/((Y + Z)d1 , Y d2 , Z d3 ), we could make use of Han’s δ-function,
which computes the difference of the degree shifts in this situation. It has a unique
continuous extension to [0, ∞)3 satisfying δ(pt) = p · δ(t) for char(k) = p > 0,
t ∈ [0, ∞)3 and can be computed with Han’s Theorem 4.5. This is useful for us
since one can reduce the non-standard-graded situation to a standard-graded case
as we will see in a moment.
Definition 4.4. Let
be the odd lattice.


Lodd := u ∈ N3 |u1 + u2 + u3 is odd

Theorem 4.5 (Han). Let t = (t1 , t2 , t3 ) ∈ [0, ∞)3 . If the ti do not satisfy the strict
triangle inequality (w.l.o.g. t1 ≥ t2 + t3 ), we have δ(t) = t1 − t2 − t3 . If the ti satisfy
the strict triangle inequality and there are s ∈ Z and u ∈ Lodd with kps t − uk1 < 1,
then there is such a pair (s, u) with minimal s and with this pair (s, u) we get
1
δ(t) = s · (1 − kps t − uk1 ) .
p
Otherwise, we have δ(t) = 0.
The proof and more properties can be found in [Han91] as well as in [Mon06].
To make use of this result in a non-standard-graded setting, lets fix some notations.
Setup 4.6. Let k be a field of characteristic p > 0, let R := k[U, V ] equipped with
the positive grading deg(U ) = α, deg(V ) = β. Fix a, b ∈ N≥1 such that a · α = b · β.
For a1 , a2 , a3 ∈ N≥1 the R-module
SyzR ((U a + V b )a1 , U a2 , V a3 )

splits by Hilbert’s Syzygy Theorem as R(m) ⊕ R(n) for some m, n ∈ Z. Let
τ (a1 , a2 , a3 ) := |m − n|.

This function τ has a unique continuous extension to [0, ∞)3 , again with the property τ (pt) = p · τ (t). See [Bri13, Chapter 2] for the details.

THE HILBERT-KUNZ FUNCTIONS OF TWO-DIMENSIONAL RINGS OF TYPE ADE

21

Lemma 4.7. With the notations from above and t ∈ [0, ∞)3 we have


t2 t3
.
τ (t1 , t2 , t3 ) = a · α · δ t1 , ,
a b

Proof. For t ∈ N3 both sides multiply by p when t is replaced by pt. Since rationals
nb
of the form na
pj resp. pj are dense in [0, ∞), it suffices to prove the statement for
t = (t1 , at2 , bt3 ) with ti ∈ N. Let S be the standard-graded polynomial ring k[X, Y ].
Consider the map φ : S → R, X 7→ U a , Y 7→ V b , which is homogeneous of degree
a · α. Denote by ψ the induced map on the projective spectra. We obtain
ψ ∗ (SyzProj(S) ((X + Y )t1 , X t2 , Y t3 ))

∼
=

// SyzProj(R) ((U a + V b )t1 , U at2 , V bt3 )

∼
=


ψ ∗ (OProj(S) (l) ⊕ OProj(S) (l + δ))

∼
=

// OProj(R) (aαl) ⊕ OProj(S) (aα(l + δ))

for some l ∈ Z and δ = δ(t1 , t2 , t3 ). Now the claim follows, since we have the
isomorphism
Syz
((U a + V b )t1 , U at2 , V bt3 ) ∼
= OProj(R) (m) ⊕ OProj(R) (m + τ )
Proj(R)

for some m ∈ Z and τ = τ (t1 , at2 , bt3 ).



This enables to prove the following.
Lemma 4.8.
e

e

e

SyzR (X p , Y p , Z p )



31 e


Syz
(X,
Y,
Z)
−
(p
−
1)
if pe mod 30 ∈ {±1, ±11},
R

2
∼


=

31

 SyzR (X, Y, Z 2 ) − (pe − 1) + 3
if pe mod 30 ∈ {±7, ±13}.
2

Proof. Going back to Equation (4.1), we obtain by Lemma 4.7 the equality
 e

p ∓ 1 pe pe
|m − n| = 30 · δ
.
, ,
2
3 5
First we assume e = 1 and p = 60 · l + 1 for some l ∈ N. We want to compute




1
1
p−1 p p
= δ 30 · l, 10 · l + , 6 · l +
.
, ,
δ
2
3 5
3
5
It is easy to see that the s from Han’s Theorem
 has to be non-negative. Since the
taxi-cap distance of 30 · l, 10 · l + 13 , 6 · l + 51 to the odd lattice is given by

 

30 · l
30 · l
10 · l + 1  − 10 · l + 1 = 2 + 1 = 13 ,
3
3 5
15
6·l
6 · l + 51
1

p p
2
the other possibilities for p
we obtain δ p−1
2 , 3 , 5 = 15 and |m − n| = 4. Treating

p p
modulo 60 and computing the values of δ p−1
,
,
one
obtains
2
3 5
=

HSyz (X p ,Y p ,Z p ) (t)
 R p−1
 t31· 2 · HSyz (X,Y,Z) (t)
R

t

31· p−1
2 −3

if p mod 30 ∈ {±1, ±11},

· HSyzR (X,Y,Z 2 ) (t) if p mod 30 ∈ {±7, ±13},

22

DANIEL BRINKMANN

This already shows
e

e

e

e
SyzR (X p , Y p , Z p ) ∼
= SyzR (X, Y, Z)(− 31
2 (p − 1))

if p mod 30 ∈ {±1, ±11}. Computing the Hilbert-series of SyzR (X p , Y p , Z 2p ) one
finds

 t37· p−1
2
· HSyzR (X,Y,Z 2 ) (t)
if p mod 30 ∈ {±1, ±11},
p
p
2p
HSyzR (X ,Y ,Z ) (t) =
p−1
 t37· 2 +3 · H
SyzR (X,Y,Z) (t) if p mod 30 ∈ {±7, ±13}.


The previous lemma shows that we have a (0, 1)-Frobenius periodicity (on the
punctured spectrum) if p mod 30 ∈ {±1, ±11} and a (0, 2)-Frobenius periodicity if
p mod 30 ∈ {±7, ±13}.
Similarly, one obtains Frobenius periodicities of Syz(X, Y, Z) on the punctured
spectrum for all ADE-rings.
5. The Hilbert-Kunz functions of the ADE-rings
In this section we put everything together to compute the Hlbert-Kunz functions
of the ADE-rings.
Theorem 5.1. The Hilbert-Kunz function of R = k[X, Y, Z]/(X 2 + Y 3 + Z 5 ) is
given by

2 · p2e
if e ≥ 1 and p ∈ {2, 3, 5},







191
1

if p mod 30 ∈ {±7, ±13} and e is odd,
p2e −
2−
e 7−→
120
120





1
119

 2−
p2e −
otherwise.
120
120
Proof. By Lemma 4.8 we know the isomorphism classes of the syzygy modules
e
e
e
SyzR (X p , Y p , Z p ). This allows us to use Lemma 1.10 and we obtain
30 · Q(15, 10, 6) · p2e − 30 · Q(15, 10, 6) + 30 · 120
30 · 120
239 · p2e − 239 + 120
=
120


1
119
,
= 2−
p2e −
120
120

HK(R, pe ) =

if pe mod 30 ∈ {±1, ±11} and

30 · Q(15, 10, 6) · p2e − 30 · Q(15, 10, 12) + 30 · 240
30 · 120
239 · p2e − 431 + 240
=
120


1
191
= 2−
p2e −
,
24
120

HK(R, pe ) =

if pe mod 30 ∈ {±7, ±13}.

THE HILBERT-KUNZ FUNCTIONS OF TWO-DIMENSIONAL RINGS OF TYPE ADE

23

If p ∈ {2, 3, 5} we see that already SyzR (X p , Y p , Z p ) splits, since in each case X p ,
Y p or Z p is a polynomial in the other two variables. For example if p = 2, we have
SyzR (X 2 , Y 2 , Z 2 ) = SyzR (Y 3 + Z 5 , Y 2 , Z 2 )
∼
= R(−30) ⊕ R(−32),

e
e
e
hence SyzR (X 2 , Y 2 , Z 2 ) ∼
= R(−15 · 2e ) ⊕ R(−16 · 2e ). Using Lemma 1.11 one
e
2e
obtains HKR (R, p ) = 2 · p for e ≥ 1.


Similarly, one obtains the Hilbert-Kunz functions of the rings of type E6 and E7 .
In the case of an E6 singularity we already know from Lemma 3.4 that



 e
e
e
13
SyzR X p , Y p , Z p ∼
= SyzR (X, Y, Z) − (pe − 1)
2

holds for all e ≥ 0 and all p ≥ 5 (since there is no other possibility). Using Lemmata
1.10 and 1.11 one obtains the following Hilbert-Kunz function.
Theorem 5.2. The Hilbert-Kunz function of R = k[X, Y, Z]/(X 2 + Y 3 + Z 4 ) is
given by

 2 · p2e
if e ≥ 1 and p ∈ {2, 3},



e 7−→
23
1

p2e −
otherwise.
 2−
24
24

If R is of type E7 and p ≥ 5, we have similarly to the E8 case the two possibilities


19
p
p
p ∼
SyzR (X , Y , Z ) = SyzR (X, Y, Z) − · (p − 1) or
2


19
p
p
p ∼
2
SyzR (X , Y , Z ) = SyzR (X, Y, Z ) − · (p − 1) + 2 .
2

Computing the Hilbert-series, one sees that the first case appears if p mod 24 ∈
{±1, ±7} yielding a (0, 1)-Frobenius periodicity of Syz(X, Y, Z) on the punctured
spectrum. If p mod 24 ∈ {±5, ±11} the second case holds and one obtains a (0, 2)Frobenius periodicity on the punctured spectrum.
Theorem 5.3. The Hilbert-Kunz function of E7 is given by

2 · p2e
if e ≥ 1 and p ∈ {2, 3},







71
1

p2e −
if p mod 24 ∈ {±5, ±11} and e is odd,
2−
e 7−→
48
48






47
1

 2−
p2e −
otherwise.
48
48

It remains to treat the family of Dn singularities. In this case we failed to compute
e
e
e
the Hilbert-series of SyzR (X p , Y p , Z p ) in general. If one fixes a concrete value
for n, it is possible to compute the Hilbert-series and everything works as in the E
situations. In the general case we change the approach to identify the isomorphism
e
e
e
class of SyzR (X p , Y p , Z p ).
Let R := k[X, Y, Z]/(X 2 + Y n+1 + Y Z 2 ) with deg(X) = n + 1, deg(Y ) = 2,
deg(Z) = n and char(k) = p ∤ 4n. Then R has an Dn+2 singularity and injects into
a two-dimensional ring with an A2n−1 singularity. The inclusion is given by
φ : R −→ k[U, V, W ]/(U 2n − V W )

24

DANIEL BRINKMANN

and the pull-back behavior of the maximal Cohen-Macaulay modules can be read
off Table 3.2. We will again use curly letters for punctured spectra and modules
over them. By Lemma 3.4 we have
∼ M2m
(5.1)
Fe∗ (Syz (m)) = Fe∗ (M2 ) =
R

for all e ∈ N and m = m(q) ∈ {1, . . . , n2 }. The parameter m = m(q) can be
computed by considering the pull-back via φ: M2 pulls back to N1 ⊕ N2n−1 on
A2n−1 . The e-th Frobenius pull-back of this module is Nr ⊕ N2n−r with r ≡ pe
mod 2n (and 1 ≤ r ≤ 2n − 1). Note that r is odd by the assumption p ∤ 4n. If
r ≤ n − 1, only Mr+1 pulls back to Nr ⊕ N2n−r via φ. If r ≥ n, only M2n−r−1
pulls back to Nr ⊕ N2n−r via φ. But Mr+1 and M2n−r−1 are isomorphic on R as
one sees by computing the first syzygy modules from the matrix factorizations by
using different columns (cf. [Bri13, Section 3.5]). We find

 e
e
e
SyzR U p , V p , W p ∼
= SyzR (U, V (r+1)/2 , W ),

where r ≡ pe mod 2n with 1 ≤ r ≤ 2n − 1. This isomorphism induces a global
isomorphism
 e

e
e
SyzR U p , V p , W p ∼
= SyzR (U, V (r+1)/2 , W )(m),

for some m ∈ Z. The Hilbert-Kunz function can be computed via Lemma 1.10.
Note that we can define all appearing syzygy modules in the case p|4n as well. Since
the computations made to obtain Table 3.2 are correct in all odd characteristics, the
above computation of the Hilbert-Kunz function remains correct in the case p|4n
with p odd, if we would know that our list of maximal Cohen-Macaulay modules is
complete.
By a theorem of Herzog (cf. [LW12, Theorem 6.3]) we only need that R is a direct
summand of the polynomial ring k[u, v], on which the group Dn acts, to have an
one to one correspondence between indecomposable, maximal Cohen-Macaulay Rmodules and direct summands of k[u, v] as an R-module.
Leuschke and Wiegand point out in [LW12, Remark 6.22] that R is a direct summand of k[u, v] and that the direct summands of k[u, v] as an R-module are the
same as in the non-modular case (meaning that the algebraic definition is the same,
although the modules in the modular case might not come from representations of
the group Dn ). Therefore, the formula above is a full description of the HilbertKunz function for odd characteristics.
To compute the Hilbert-Kunz function for p = 2, we can use Lemma 1.11. All in
all, one obtains
Theorem 5.4. The Hilbert-Kunz function of Dn+2 is given by (with r ≡ pe mod
2n for 0 ≤ r ≤ 2n − 1)

2e

if e ≥ 1, p = 2,
2· p


2
e 7−→
1
r
r+1

+
otherwise.
p2e −
 2−
4n
2
4n

Note that our explicit knowledge of the Hilbert-Kunz functions enables us to compute the F -signature functions by using [HL02, Theorem 11]. If R is of type D or
E, one has
Fsign (R, pe ) = 2p2e − HK(R, pe ).

THE HILBERT-KUNZ FUNCTIONS OF TWO-DIMENSIONAL RINGS OF TYPE ADE

25

References
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[BD08]

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[HM93]
[Kle86]
[Knö87]
[KST07]
[Kun76]
[LW12]
[Mon83]
[Mon06]
[Tri05]
[Wit02]

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DANIEL BRINKMANN

[WY00] K.-i. Watanabe and K.-i. Yoshida. Hilbert-Kunz multiplicity and an inequality between
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Institut für Mathematik, Universität Osnabrück
E-mail address: dabrinkm@uos.de

