NON-GORENSTEIN ISOLATED SINGULARITIES OF GRADED
COUNTABLE COHEN-MACAULAY TYPE

arXiv:1307.6206v1 [] 23 Jul 2013

BRANDEN STONE

Abstract. In this paper we show a partial answer the a question of C. Huneke
and G. Leuschke (2003): Let R be a standard graded Cohen-Macaulay ring of
graded countable Cohen-Macaulay representation type, and assume that R has
an isolated singularity. Is R then necessarily of graded finite Cohen-Macaulay
representation type? In particular, this question has an affirmative answer for
standard graded non-Gorenstein rings as well as for standard graded Gorenstein rings of minimal multiplicity. Along the way, we obtain a partial classification of graded Cohen-Macaulay rings of graded countable Cohen-Macaulay
type.

1. Motivation and Introduction
For motivational purposes, we first consider local Cohen-Macaulay rings. Throughout, a local Cohen-Macaulay ring is said to have finite Cohen-Macaulay type (respectively, countable Cohen-Macaulay type) if it has only finitely (respectively, countably)
many isomorphism classes of indecomposable maximal Cohen-Macaulay modules.
In [2], M. Auslander showed that local rings of finite Cohen-Macaulay type must
have an isolated singularity. It was later shown in [4] that this was true for standard
graded rings as well. Concerning countable Cohen-Macaulay type, it was shown in
the excellent case by C. Huneke and G. Leuschke that a local ring of countable
Cohen-Macaulay type does not necessarily have an isolated singularity, but the
singular locus has dimension at most one [12](the graded version of this is found
in [19]). Given this result, it is natural to try and find a local Cohen-Macaulay
ring of countable (and infinite) Cohen-Macaulay type with an isolated singularity.
Examining the known examples countable type rings with an isolated singularity,
C. Huneke and G. Leuschke found they were actually finite type! This led to the
following question:
Question 1.1 ([12]). Let R be a complete local Cohen-Macaulay ring of countable
Cohen-Macaulay representation type, and assume that R has an isolated singularity.
Is R then necessarily of finite Cohen-Macaulay representation type?
In this paper, we are mainly concerned with the graded version of this question
(see Question 1.2). In Corollary 6.1, we are able to show a positive answer for
standard graded non-Gorenstein rings as well as for graded Gorenstein rings of
minimal multiplicity. The main strategy is to classify the rings of graded countable
Cohen-Macaulay type and compare this list with a the classification of graded finite
Cohen-Macaulay type in [9].
The author was partially funded by the NSF grant, Kansas Partnership for Graduate Fellows
in K-12 Education (DGE-0742523).
1

2

B. STONE

The sections of this paper break the problem up into cases based on the dimension
of the ring, with Section 5 discussing the issues surrounding Gorenstein rings. The
results are summarized in Section 6.
1.1. Preliminaries and known results. We sayL
that a ring R is standard graded
if, as an abelian group, it has a decomposition R = i>0 Ri such that Ri Rj ⊆ Ri+j
for all i, j > 0, R = R0 [R1 ], and R0 is a field. Further, we will always assume that
a standard graded ring is Noetherian. Unless otherwise stated, we will denote
(R, m, k) byP
the standard graded ring with m being the irrelevant maximal ideal,
∞
that is m = i=1 Ri , and k := R0 = R/m being an uncountable algebraically closed
field of characteristic zero. We let e(R) be the multiplicity of the maximal ideal m.
When R is a d-dimensional Cohen-Macaulay ring, if e(R) = dimk (m/m2 )−dim R+1
then R is said to have minimal multiplicity. In the case when k is infinite, the
following are equivalent:
(i) R has minimal multiplicity;
(ii) there exists a regular sequence x1 , . . . , xd such that m2 = (x1 , . . . , xd )m;
(iii) the h-vector of R is of the form (1, n).
Here we define the h-vector of R to be the sequence (dimk (R/(x1 , . . . , xd ))i )i>0
where x1 , . . . , xd is a homogeneous regular sequence of degree one.
A standard graded Cohen-Macaulay ring (R, m, k) has graded finite Cohen-Macaulay
type (respectively, graded countable Cohen-Macaulay type) if it has only finitely (respectively, countably) many indecomposable, maximal Cohen-Macaulay modules
up to a shift in degree.
The graded version of Question 1.1 is as follows.
Question 1.2. Let R be a standard graded Cohen-Macaulay ring of graded countable Cohen-Macaulay representation type, and assume that R has an isolated singularity. Is R then necessarily of graded finite Cohen-Macaulay representation type?
Remark 1.3. It is known that if the completion of a standard graded ring R with
respect to the maximal ideal has finite (respectively countable) type, then R must
have graded finite (respectively countable) type (see [4, Prop 8 and 9] or [20, Corollary 2.5]). Hence any affirmative answer to Question 1.1 can be passed to a positive
answer of Question 1.2.
With this in mind, Question 1.2 has a positive answer if the ring is a hypersurface. In particular, it was shown in [14, 6] that if R is a complete hypersurface
containing an algebraically closed field k (of characteristic different from 2), then
R is of finite Cohen-Macaulay type if and only if R is the local ring of a simple
hypersurface singularity in the sense of [1]. For example, if we let k = C, then
R is one of the complete ADE singularities over C. That is, R is isomorphic to
kJx, y, z2 , . . . , zr K/(f ), where f is one of the following polynomials:
(An ) : xn+1 + y 2 + z22 + · · · + zr2 , n > 1;

(Dn ) : xn−1 + xy 2 + z22 + · · · + zr2 , n > 4;
(E6 ) : x4 + y 3 + z22 + · · · + zr2 ;
(E7 ) : x3 y + y 3 + z22 + · · · + zr2 ;
(E8 ) : x5 + y 3 + z22 + · · · + zr2 .

NON-GORENSTEIN, GRADED COUNTABLE COHEN-MACAULAY TYPE

3

It was further shown in [6] that a complete hypersurface singularity over an algebraically closed uncountable field k has (infinite) countable Cohen-Macaulay type
if and only if it is isomorphic to one of the following:
(1.1)
(1.2)

(A∞ ) : kJx, y, z2 , . . . , zr K/(y 2 + z22 + · · · + zr2 );

(D∞ ) : kJx, y, z2 , . . . , zr K/(xy 2 + z22 + · · · + zr2 ).

As both (A∞ ) and (D∞ ) are non-isolated singularities, if a hypersurface has an
isolated singularity and is countable type then it must have finite type.
In 2011, R. Karr and R. Wiegand [13, Theorem 1.4] showed the one dimensional
case as well. This was under the assumption that the integral closure of the ring R
is finitely generated as an R-module. We recover these results in Section 3.
2. Zero Dimensional Rings
It is well known that a zero dimensional equicharacteristic local ring R is a
hypersurface if and only if R is of finite Cohen-Macaulay type [11, Satz 1.5]. We
show the graded countable analog to this statement in Proposition 2.1.
Proposition 2.1. Let (R, m, k) be a 0-dimensional standard graded Cohen-Macaulay
ring. Further assume that k is an uncountable field. Then the following are equivalent:
(1) R is of graded finite Cohen-Macaulay type;
(2) R is of graded countable Cohen-Macaulay type;
(3) R is a hypersurface ring.
Proof. The implication (1) implies (2) is straight forward. To show (2) implies
(3), assume that R is not a hypersurface. Thus there must be two linear forms
a, b ∈ m \ m2 that are basis elements of m/m2 . By [20, Lemma 4.1], there are
uncountably many distinct homogeneous ideals {Iα }α∈k in R. In this context, we
have that Iα = (a + αb)R. Consider the graded indecomposable maximal CohenMacaulay modules {R/Iα }α∈k . As each of these modules have different annihilators,
we know they are not isomorphic. A contradiction as we assumed that R was of
graded countable type.
To prove that (3) implies (1), we consider the m-adic completion of R and then
apply [11, Satz 1.5] to see that the completion is of finite Cohen-Macaulay type.
By Remark 1.3, we know that R also has graded finite Cohen-Macaulay type. 
We thus have a complete classification of graded countable Cohen-Macaulay
type for zero dimensional standard graded rings. Further, Proposition 2.1 gives a
positive answer to Question 1.2 for zero dimensional standard graded rings with
uncountable residue field.
3. One Dimensional Rings
In the one-dimensional case, Question 1.2 has a positive answer as shown by R.
Karr and R. Wiegand [13, Theorem 1.4]. In this section, we examine the DrozdRoı̆ter conditions and give a classification of one dimensional reduced standard
graded rings of graded countable Cohen-Macaulay type. Thus retrieving the result
of R. Karr and R. Wiegand.

4

B. STONE

3.1. Finite Type and the Drozd-Roı̆ter conditions. As detailed by N. Cimen,
R. Wiegand, and S. Wiegand [7], if (R, m, k) is a one dimensional, Noetherian,
reduced, local Cohen-Macaulay ring such that the integral closure of R, say S,
is finitely generated as an R-module, then we know precisely when R has finite
Cohen-Macaulay type. This happens when the following conditions occur:
DR1 S is generated by 3 elements as an R-module;
DR2 the intersection of the maximal R-submodules of S/R is cyclic as an Rmodule.
These conditions are called the Drozd-Roı̆ter conditions. It was further shown in [7,
Proposition 1.12] that the Drozd-Roı̆ter conditions are equivalent to the following:
(dr1)

dimk (S/mS) 6 3;


R + mS
dimk
6 1.
R + m2 S

(dr2)

In order to have a different grasp of what it means to satisfy the Drozd-Roı̆ter
conditions, we state another set of equivalent conditions. This result is stated in the
next proposition where λ represents length as an R-module, and ∗ is the integral
closure of ideals.
Proposition 3.1. Let (R, m, k) be a one-dimensional, Noetherian, reduced, local
Cohen-Macaulay ring with finite integral closure and uncountable residue field k.
Let x be a minimal reduction of the maximal ideal m. The Drozd-Roı̆ter conditions
are equivalent to the following:
(3.1)

e(R) 6 3;

(3.2)

λ(m2 /xm) 6 1.

Proof. To show that (3.1) holds, we will show that e(R) = dimk (S/mS) where S
is the integral closure of R. As x is a reduction of m, we know that xS is also a
reduction of mS. But this holds if and only if mS ⊆ xS. As principal ideals are
integrally closed in S, we have that
xS ⊆ mS ⊆ xS = xS,
and hence xS = mS. By assumption, S is finitely generated as an R-module. Therefore we have that the map S → S defined by multiplication by x is an injection.
Hence, we have the following commutative diagram in Figure 1.
0

0

K

0


/R

·x


/R


/ R/xR

/0

0


/S

·x


/S


/ S/xS

/0


C0


C


C0

Figure 1. A commutative diagram where K, C 0 , and C are the
respective kernel and cokernels.

NON-GORENSTEIN, GRADED COUNTABLE COHEN-MACAULAY TYPE

5

Consider the exact sequence in the right side of Figure 1,
(3.3)

0

/K

/ R/xR

/ S/xS

/C

/ 0.

Notice by the Snake Lemma applied to Figure 1 that λ(K) = λ(C); here λ represents length as R-modules. As C and K have the same length, (3.3) shows that
R/xR and S/xS also have the same length. Since x is a minimal reduction of the
maximal ideal, we know that e(R) = λ(R/xR). Therefore,
e(R) = λ(R/xR) = λ(S/xS) = λ(S/mS) = dimk (S/mS).
In order to show (3.2), first notice that
(3.4)

R + mS
mS
mS
xS
' 2
' 2
' 2
.
R + m2 S
m S + (R ∩ mS)
m S+m
x S+m

For simplicity, we define B as follows,




R + mS
xS
B := dimk
=λ
.
R + m2 S
x2 S + m
We now consider the short exact sequence
0

2
/ m S+m
m2 S

/ mS
m2 S
/

mS
m2 S + m

/ 0.

Rewriting the two terms on the left gives us
m2 S + m
m
' 2
' m/m2 ;
m2 S
m S∩m
S
S
mS
'
=
.
m2 S
mS
xS

(3.5)
(3.6)

Combining (3.5) and (3.6) with the above short exact sequence yields
 


S
(3.7)
λ
= λ m/m2 + B.
xS
On the other hand, consider the following short exact sequence
(3.8)

0

2
/ m + xR
xR

/ R
xR
/

R
m2

+ xR

/ 0,

along with the isomorphisms
(3.9)

m2 + xR
m2
m2
'
=
.
xR
xm
xR ∩ m2

Note that the equality in (3.9) can be justified as follows. Since xS = mS, we know
that m2 = m2 S ∩ R = x2 S ∩ R. If y ∈ xR ∩ m2 , then y = xr ∈ m2 = x2 S ∩ R. This
forces r ∈ xS ∩ R = m. Hence y ∈ xm. Equality follows as xm ⊆ xR ∩ m.
Computing length in (3.8) gives us
!




R
m2
R
λ
=λ
+λ
.
xR
xm
m2 + xR

6

B. STONE

We can repeat the above steps with the following short exact sequence and
isomorphisms:
0

2
/ m + xR
m2

m2 + xR

/ R
m2

R
m2

+ xR

/ 0;

xR
R
' .
mxR
m
∩ xR
Once again, if we compute the length, we have that




R
R
=1+λ
(3.10)
λ
m2
m2 + xR




R
S
Combining (3.7) and (3.10) with the fact that λ xR
= λ xS
and λ m2 =
m


λ R2 − 1, we have
m
!




m2
R
m
λ
+λ
=λ
+B
xm
m2 + xR
m2


R
=λ
−1+B
m2


R
− 1 + B.
=1+λ
m2 + xR
!
m2
Simplifying we see that λ
= B. We now have
xm
m2

λ

'

xR

/

m2

'

e(R) = dimk (S/mS);
!


m2
R + mS
= dimk
.
xm
R + m2 S

Applying (dr1) and (dr2) yields the desired result.



Given Proposition 3.1, we can construct a couple of examples.
Example 3.2. Consider the ring R = kJt3 , t7 K. This is a one-dimensional domain
3 7
with e(R) = 3. The element t3 is a minimal
 of the maximal ideal (t , t )R.
 reduction
If we compute the length, we see that λ m2 /t3 m = 2. Hence, by Proposition 3.1,
we have that R is not of finite type.

Example 3.3. Let R = kJx, yK/(x3 y − xy 3 ). This ring is one-dimensional and
reduced. If we compute the multiplicity, we find that e(R) = 4. Thus, we immediately have from Proposition 3.1
 that R is not of finite type. Computing the length
none-the-less, we find that λ m2 /(x + 2y)m = 1.
3.2. Graded Countable Type. In [20, Theorem 5.3], one dimensional standard
graded rings of graded countable type were shown to either be of minimal multiplicity or a hypersurface. Turning to the case of minimal multiplicity, we find the
following.

NON-GORENSTEIN, GRADED COUNTABLE COHEN-MACAULAY TYPE

7

Theorem 3.4. Let (R, m, k) be a standard graded one dimensional Cohen-Macaulay
ring with uncountable residue field k. If the h-vector of R is (1, n) with n > 3, then
R is not of graded countable Cohen-Macaulay type.
Proof. Let x be a minimal homogeneous reduction of the maximal ideal m, and let
x, u, v, w, c4 , · · · , cn ∈ m be elements of a minimal k-basis of m/m2 . By assumption
n > 3, so we are guaranteed at least four elements in the basis of m/m2 . Without
losing any generality, we assume that w is the fourth basis element. Assume that
there is a graded isomorphism
Iα = (x, u + αv) ' (x, u + βv) = Iβ
where α, β are elements of k. As dim(R) = 1, these ideals are graded indecomposable maximal Cohen-Macaulay modules. Since this isomorphism is graded of
degree 0, we have that
x 7→ δ1 x + δ2 (u + βv)
u + αv 7→ δ3 x + δ4 (u + βv)
where det(δi ) is a unit and δi are elements of k. We have that
(3.11)

0 = δ3 x2 + δ4 x(u + βv) − δ1 x(u + αv) − δ2 (u + αv)(u + βv).

Notice that (u + αv)(u + βv) is an element of m2 . As R is of minimal multiplicity,
we have that xm = m2 . Hence we can view elements of m2 as elements of xm. In
particular we view u2 , uv, v 2 in the following way
 
x
u
 
 2 

v
u
x(a10 x + a11 u + a12 v + a13 w + a15 c4 + · · · + a1n cn )
 

uv  = x(a20 x + a21 u + a22 v + a23 w + a25 c4 + · · · + a2n cn ) = x · A · 
w


v2
x(a30 x + a31 u + a32 v + a33 w + a35 c4 + · · · + a3n cn )
 c4 
 .. 
.
cn
where the matrix A = (aij ), 1 6 i 6 3, 0 6 j 6 n. Since u2 , uv, v 2 are homogeneous
elements, the grading forces the entries of A to be elements of k. Further, if we let

t
Φ = 1 α + β αβ and b = x u v w c4 · · · cn ,
then we can use matrix notation to write
(u + αv)(u + βv) = u2 + (α + β)uv + αβv 2 = x · Φ · A · b.
We can now cancel the x in equation (3.11) and rewrite it as

t
δ3 − ΦA0 δ2
 δ4 − δ1 − ΦA1 δ2 


βδ4 − αδ1 − ΦA2 δ2 




−ΦA3 δ2
(3.12)
0=
 ·b


−ΦA
δ
4 2




..


.
−ΦAn δ2

8

B. STONE

where Ai are the columns of the matrix A. All of the elements in the coefficient
matrix of (3.12) are elements of k and hence equal zero as x, u, v, w, c4 , . . . , cn form
a k-basis.
At this point we focus on δ2 . If δ2 6= 0, then the fact that ΦA3 δ2 = 0 implies
that
(3.13)

a13 + (α + β)a23 + αβa33 = 0

in the field k. As the aij are independent of our choice of α and β, Equation (3.13)
shows that every α, β such that Iα ' Iβ is a solution to
f (X, Y ) = a13 + (X + Y )a23 + XY a33 ∈ k[X, Y ].
This forces f (X, Y ) to be identically zero, a contradiction. Hence there are uncountably many Iα that are not isomorphic.
If we let δ2 = 0, then Equation (3.13) becomes the relation
(3.14)

δ3 x + (δ4 − δ1 )u + (βδ4 − αδ1 )v = 0.

As x, u, v are independent over k, we have that the coefficients are zero. In particular
δ4 − δ1 = 0. Since δ1 δ4 − δ2 δ3 6= 0, we know that δ1 = δ4 6= 0. Thus the fact that
βδ4 − αδ1 = 0 implies that α = β. Hence there are uncountably many nonisomrophic ideals Iα .

Given the above results, we are now ready to characterize one dimension standard
graded rings of graded countable Cohen-Macaulay type.
Corollary 3.5. Let (R, m, k) be a one-dimensional standard graded Cohen-Macaulay
ring with uncountable residue field k. If R is of graded countable Cohen-Macaulay
type, then R is either of minimal multiplicity with h-vector (1, 2), or is isomorphic
to one of the following hypersurfaces:
(1) k[x];
(2) k[x, y]/(xy);
(3) k[x, y]/(xy(x + y));
(4) k[x, y]/(xy 2 );
(5) k[x, y]/(y 2 ).
Proof. A direct application of [20, Theorem 5.3] and Theorem 3.4 show that R is
either a hypersurface ring, or has minimal multiplicity with h-vector (1, 2).
Concerning the hypersurfaces, items (1)-(3) have graded finite Cohen-Macaulay
type as can be seen from [6] or [9]. The hypersurfaces (4) and (5) are not graded
finite Cohen-Macaulay type, but their completions are the one dimensional (A∞ )
and (D∞ ) hypersurface singularities shown in (1.1) and (1.2) on page 3. It was
shown by R. Buchweitz, G. Greuel, and F. Schreyer in [6] that these are the only
hypersurfaces that are countable but not finite Cohen-Macaulay type. Hence by
Remark 1.3, the rings (4) and (5) are of graded countable Cohen-Macaulay type. 
Corollary 3.6. Let (R, m, k) be a one-dimensional standard graded Cohen-Macaulay
ring with uncountable residue field k. If R is of graded countable Cohen-Macaulay
type then e(R) 6 3.
Proof. As shown in [20, Corollary 4.5], the possible h-vectors are (1), (1, n), or
(1, n, 1) for some integer n. Combining this with Corollary 3.5, we know that the
possible h-vectors of R are the following:
(1), (1, 1), (1, 2), (1, 1, 1).

NON-GORENSTEIN, GRADED COUNTABLE COHEN-MACAULAY TYPE

Hence we have that e(R) 6 3.

9



An obvious improvement to Corollary 3.5 would be to classify the rings whose
h-vector is (1, 2). Currently, there are two known examples of one-dimensional
standard graded Cohen-Macaulay rings of graded countable Cohen-Macaulay type
having h-vector (1, 2):
k[x, y, z]/(xy, yz, z 2 );


x y z
k[x, y, z]/det2
.
y z x

(3.15)
(3.16)

The completion of (3.15) is the local endomorphism ring of (x, y)R where R =
kJx, yK/(xy 2 ), the one-dimensional (D∞ ) hypersurface. This endomorphism ring
is known to be of countable Cohen-Macaulay type (see [15, Theorem 1.5] or [16,
Example 14.23]) and thus by Remark 1.3, (3.15) is of graded countable CohenMacaulay type. It is worth noting that (3.15) is not reduced.
As for (3.16), this is the only one dimensional ring of finite type and h-vector
(1, 2) found in the list of graded finite Cohen-Macaulay type rings [9]. Given this
result, if we assume our ring to have an isolated singularity, then we have a positive
answer to Question 1.2.
Corollary 3.7. Let (R, m, k) be a one-dimensional, reduced, standard graded CohenMacaulay ring with uncountable residue field k. Further assume that R has a finite
integral closure. If R is of graded countable Cohen-Macaulay type, then R is of
graded finite type and isomorphic to one of the following:
(1) k[x];
(2) k[x, y]/(xy);
(3) k[x, y]/(xy(x +y));

x y z
(4) k[x, y, z]/det2
.
y z x
Proof. By Remark 1.3, we can pass to the
Since local rings of minimal

 completion.
2
multiplicity with h-vector (1, 2) have λ m /xm 6 1, we can apply Proposition
3.1 and Corollary 3.5 to obtain the desired result.

4. Non-Gorenstein Rings of Dimension at least Two
We are unable to classify standard graded rings of graded countable type with
dimension larger than one. However, if we assume an isolated singularity, we have
a positive answer to Question 1.2 in the non-Gorenstein case.
4.1. Non-Gorenstein Rings of Dimension Two. As shown in [9], two dimensional standard graded Cohen-Macaulay rings with an isolated singularity must be
a domain and have minimal multiplicity. In [8], it can be seen that standard graded
Cohen-Macaulay domains of minimal multiplicity have the following classification:
(i) quadratic hypersurfaces (not necessarily nondegenerate);
(ii) cones over the Veronese embedding P2 ,→ P5 . I.e., rings isomorphic to


x0 x1 x2
.
k[x0 , x1 , . . . , xn ] det2 x1 x3 x4  ,
x2 x4 x5
where for n > 5;

10

B. STONE

(iii) rational normal scrolls.
The ring defined in (i) is Gorenstein and is discussed in Section 5. The ring in (ii)
is of graded finite (hence countable) Cohen-Macaulay type when n = 5 as can be
seen in [9] or [21, Example 17.6.1]. For n > 6, the dimension of the singular locus is
larger than 1, hence they are not of graded countable type. However when n = 6,
it is unclear if the ring is graded countable type or not. Notice that this case does
not concern Question 1.2 as the ring does not have an isolated singularity, but does
play a role in a general classification. Further, the dimension of each ring in (ii) is
at least three. As such, it is left to consider the rational normal scrolls defined as
follows.
Definition 4.1. Let 0 6 a0 6 a1 6 · · · 6 ak be given integers and let {xij | 0 6
j 6 ai , 0 6 i 6 k} be a set of variables over a field k. Then, take the ideal I of the
polynomial ring S = k[xij | 0 6 j 6 ai , 0 6 i 6 k] generated by all the 2 × 2-minors
of the matrix:

 0
x0 x01 · · · x0a0 −1 | · · · | xk0 xk1 · · · xkak −1
.
| · · · | xk1 xk2 · · ·
xkak
x01 x02 · · ·
x0a0
Define the graded ring R to be the quotient S/I with deg(xij ) = 1 for all i, j, and
call R the scroll of type (a0 , a1 , . . . , ak ).
Notice that the polynomial ring k[x0 , x1 , . . . , xN ] is a rational normal scroll of
Pk
type (0, · · · , 0, 1), where N = i=0 ai +k+1. Further, a scroll is of type (0, · · · , 0, 1)
if and only if it is regular (for basic properties of scrolls, see [8, 17]). In general,
it is known that scrolls are integrally closed Cohen-Macaulay domain of dimension
k + 2.
Proposition 4.2. Let (R, m, k) be a standard graded Cohen-Macaulay ring that is
not Gorenstein and dim(R) = 2. Further assume that R has an isolated singularity
and that k is an uncountable field. If R is of graded countable Cohen-Macaulay
type, then R is of graded finite type and is isomorphic to


.
x0 . . . xn−1
k[x0 , x1 , . . . , xn ] det2
,
x1 . . .
xn
where n > 2.
Proof. It was shown in [9] that two dimensional, non-Gorenstein rings with an
isolated singularity have minimal multiplicity.
By the above discussion, we know that R must be isomorphic to a two dimensional scroll. The only non-Gorenstein scrolls of dimension two are of type (m)
where m > 2. These are the rings listed in the statement of Proposition 4.2. It
is known that these rings are of graded finite Cohen-Macaulay type (see [9] or [3,
Theorem 2.3]).

Proposition 4.2 gives a partial (positive) answer to Question 1.2. Ideally though,
we would like to remove the isolated singularity condition from the hypothesis of
Proposition 4.2. Doing so would show that graded countable Cohen-Macaulay type
implies graded finite Cohen-Macaulay type for non-Gorenstein two dimensional
rings.

NON-GORENSTEIN, GRADED COUNTABLE COHEN-MACAULAY TYPE

11

4.2. Non-Gorenstein Rings of Dimension at least Three. Let (R, m, k) be a
standard graded Cohen-Macaulay ring of graded countable Cohen-Macaulay type,
that is not Gorenstein and dim R > 3. By [20, Proposition 4.6], we know that R
must be a domain and have minimal multiplicity. Thus we only have to consider
the three classes of rings listed in Section 4.1 in order to obtain the following.
Proposition 4.3. Let (R, m, k) be a standard graded Cohen-Macaulay ring with
uncountable residue field k and an isolated singularity. Further assume that R is
not Gorenstein and dim R > 3. If R is of graded countable Cohen-Macaulay type,
then R is of graded finite type and is isomorphic one of the following rings:


x1 x2 x4
(1) k[x1 , . . . , x5 ]/det2
;
x2 x3 x5
(2) k[x1 , . . . , x6 ]/det2 (sym 3 × 3).
Proof. Consider the classes of rings listed in Section 4.1. We can ignore the rings
in (i) as they are Gorenstein. The rings in (ii) only have an isolated singularity
when n = 5. This ring is known to be of graded finite Cohen-Macaulay type [9].
As such, we list it above.
Concerning the scrolls in (iii), M. Auslander and I. Reiten classify the scrolls of
finite type in [5]. In particular, they show that if k > 1 and R is not of type (1, 1)
or (1, 2), then R has |k| many indecomposable graded Cohen-Macaulay modules,
up to shifts. As it is, the graded scrolls of type (1, 1) and (1, 2) are the only graded
scrolls of dimension at least 3 that have graded countable Cohen-Macaulay type.
Notice that a scroll of type (1, 1) is Gorenstein and hence we are left with the scroll
of type (1, 2).

Proposition 4.3 gives another case where Question 1.2 has a positive answer.
Concerning the general classification, in order to remove the isolated singularity
condition from the hypothesis, one only needs to determine the graded CohenMacaulay type of


x0 x1 x2
.
(4.1)
k[x0 , x1 , . . . , x6 ] det2 x1 x3 x4  .
x2 x4 x5
Knowing this information would give a complete classification for non-Gorenstein
rings of dimension at least three.
5. Gorenstein Rings
Not much is known about Gorenstein rings of graded countable Cohen-Macaulay
type. However, it is well known that standard graded Gorenstein rings of graded
finite Cohen-Macaulay type are hypersurfaces [11, Satz 1.2]. This fact is heavily
exploited in the classification of standard graded Cohen-Macaulay rings of graded
finite Cohen-Macaulay type [9]. The countable analog of this fact is still unknown.
Conjecture 5.1. A Gorenstein ring of countable Cohen-Macaulay type is a hypersurface.
Using the concept of super-stretched, Conjecture 5.1 was shown to be true for
standard graded rings of dimension at most one [20], but the conjecture remains
open for higher dimensions. By the results of [14, 6] on hypersurfaces, a positive result to Conjecture 5.1 would completely classify Gorenstein rings of graded
countable Cohen-Macaulay type.

12

B. STONE

If we restrict to rings of minimal multiplicity, then we are only dealing with
hypersurfaces, as h-vectors of Gorenstein rings are symmetric. As such, by [14, 6]
we have a complete characterization. In particular, consider a standard graded
Gorenstein ring (R, m, k) of minimal multiplicity, i.e. the h-vector is (1, 1). Hence
if R is of graded countable Cohen-Macaulay type, then R is isomorphic to one of
the following hypersurfaces:
(A1 ) : k[x1 , . . . , xn ]/(x21 + · · · + x2n ), n > 1;
(A∞ ) : k[x1 , . . . , xn ]/(x22 + · · · + x2n ), n > 1.
Remark 5.2. If we further assume that a standard graded Gorenstein ring of minimal multiplicity has an isolated singularity, then graded countable Cohen-Macaulay
type implies graded finite Cohen-Macaulay type. To see this, notice that only (A1 )
has a zero dimensional singular locus.
It is worth noting that standard graded rings of dimension at least 2 and with
graded finite Cohen-Macaulay type have minimal multiplicity. This can be seen by
examining the list graded rings of graded finite Cohen-Macaulay type in [9]. With
these results, we ask the following question.
Question 5.3. If (R, m, k) is a standard graded Gorenstein ring of graded countable
Cohen-Macaulay type and dim(R) > 2, is R necessarily of minimal multiplicity?
A positive answer to Question 5.3 would ultimately force the ring to be a hypersurface and affirm Conjecture 5.1. Further it would show that Question 1.2 has an
affirmative answer for Gorenstein rings of graded countable Cohen-Macaulay type.
Remark 5.4. Since Conjecture 5.1 is still open, the natural place to look are rings of
complete intersection. According to [20], we know that standard graded complete
intersection with graded countable Cohen-Macaulay type are either a hypersurface
or defined by two quadrics. With this fact, one could use tools from representation
theory [18] and the analysis of the h-vector, to show that the ring is indeed a
hypersurface no matter what the dimension. As these techniques are outside the
scope of the paper, we leave it here as a remark.
6. Summary of Results
In Corollary 6.1 we summarize the previous results in relation to Question 1.2. As
noted earlier, proving Conjecture 5.1 would give a complete answer to the question.
In the rest of the section, we consider what can be shown when the ring in question
does not necessarily have an isolated singularity. In particular, we state a partial
classification of standard graded rings of graded countable Cohen-Macaulay type.
Corollary 6.1. Let (R, m, k) be a standard graded ring of graded countable CohenMacaulay type. Further assume that R has an isolated singularity and that k is an
uncountable algebraically closed field of characteristic zero. Then R is of graded
finite type if one of the following hold:
(1) dim(R) 6 1;
(2) R is not Gorenstein;
(3) R is of minimal multiplicity.
Proof. If the dimension of R is at most 1, then we can directly apply Proposition
2.1 and Corollary 3.7. For non-Gorenstein rings of dimension at least 2, we can

NON-GORENSTEIN, GRADED COUNTABLE COHEN-MACAULAY TYPE

13

apply Propositions 4.2 and 4.3. If R is Gorenstein of minimal multiplicity and
dim(R) > 2, then we have graded finite Cohen-Macaulay type from Remark 5.2
and the discussion preceding it.

6.1. Partial classification in the general case. Throughout this paper, we have
obtained a partial classification of standard graded rings of graded countable CohenMacaulay type. For uniformity’s sake, we state the partial classification in the same
format as the classification of graded finite Cohen-Macaulay type found in [9]. As
such, notice that rings listed in the arbitrary dimension section are omitted from
the other cases. In particular, the A∞ hypersurface is only listed in the arbitrary
dimension section. For more details on rings of countable Cohen-Macaulay type,
see [16].
Arbitrary Dimension: The following rings are shown to be graded countable type
in [9, 14, 6]. This list is not known to be complete.
k[x1 , . . . , xn ], n > 1;
k[x1 , . . . , xn ]/(x21 + · · · + x2n ), n > 1;
k[x1 , . . . , xn ]/(x22 + · · · + x2n ), n > 1.
Dimension Zero: As seen in Proposition 2.1, graded countable type is the same
as graded finite type. By [9], we have the following complete list.
k[x]/(xm ), m > 1.
Dimension One: We do not have a complete list of dimension one graded countable type rings. However, by Corollary 3.7 and the discussion before it, we know
these rings must have h-vector (1, 2) or be isomorphic to
k[x, y]/(xy);
k[x, y]/(xy(x + y));
k[x, y]/(xy 2 );

k[x, y, z]/det2

x
y

y
z


z
;
x

k[x, y, z]/(xy, yz, z 2 ).
Dimension Two: According to Proposition 4.2, the non-Gorenstein graded countable type rings with an isolated singularity are known. Besides the ring given in
the Arbitrary Dimension section, we are not aware of any other graded countable
type ring with a non-isolated singularity. Thus we state what was already recorded
in [9].


.
x1 . . .
xn
k[x1 , . . . , xn+1 ] det2
, n > 2.
x2 . . . xn+1
Dimension Three: If we restrict the proof of Proposition 4.3 to three dimensional
rings, we can omit the isolated singularity condition. As such, we know the rings
below are the only non-Gorenstein graded countable type rings of dimension three.
As the Gorenstein case is open, it is not know if this is a complete list or not.


.
x1 x2 x4
k[x1 , . . . , x5 ] det2
;
x2 x3 x5
k[x1 , . . . , x6 ]/det2 (sym 3 × 3).

14

B. STONE

Remark 6.2. According to the remarks after Proposition 4.3, for non-Gorenstein
rings of dimension at least four, the only possible ring is (4.1).
7. Acknowledgements
The author would like to thank the anonymous referee for valuable feedback and
suggestions regarding the content of this paper. For instance, the referee pointed
out the existence of the ring (3.15) as well as some discrepancies in the initial
version of Section 4. Further, the author is especially thankful to Craig Huneke
for several useful conversations. Additionally, results in this note were inspired by
many Macaulay2 [10] computations.
References
Arnol0 d.

[1] V. I.
Critical points of smooth functions. In Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, pages 19–39. Canad. Math. Congress, Montreal, Que., 1975.
[2] Maurice Auslander. Isolated singularities and existence of almost split sequences. In Representation theory, II (Ottawa, Ont., 1984), volume 1178 of Lecture Notes in Math., pages
194–242. Springer, Berlin, 1986.
[3] Maurice Auslander and Idun Reiten. Almost split sequences for Z-graded rings. In Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), volume 1273 of
Lecture Notes in Math., pages 232–243. Springer, Berlin, 1987.
[4] Maurice Auslander and Idun Reiten. Cohen-Macaulay modules for graded Cohen-Macaulay
rings and their completions. In Commutative algebra (Berkeley, CA, 1987), volume 15 of
Math. Sci. Res. Inst. Publ., pages 21–31. Springer, New York, 1989.
[5] Maurice Auslander and Idun Reiten. The Cohen-Macaulay type of Cohen-Macaulay rings.
Adv. in Math., 73(1):1–23, 1989.
[6] R.-O. Buchweitz, G.-M. Greuel, and F.-O. Schreyer. Cohen-Macaulay modules on hypersurface singularities. II. Invent. Math., 88(1):165–182, 1987.
[7] Nuri Cimen, Roger Wiegand, and Sylvia Wiegand. One-dimensional rings of finite representation type. In Abelian groups and modules (Padova, 1994), volume 343 of Math. Appl., pages
95–121. Kluwer Acad. Publ., Dordrecht, 1995.
[8] David Eisenbud and Joe Harris. On varieties of minimal degree (a centennial account). In
Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), volume 46 of Proc. Sympos.
Pure Math., pages 3–13. Amer. Math. Soc., Providence, RI, 1987.
[9] David Eisenbud and Jürgen Herzog. The classification of homogeneous Cohen-Macaulay rings
of finite representation type. Math. Ann., 280(2):347–352, 1988.
[10] Daniel R. Grayson and Michael E. Stillman. Macaulay2, a software system for research in
algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/.
[11] Jürgen Herzog. Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren
Cohen-Macaulay-Moduln. Math. Ann., 233(1):21–34, 1978.
[12] Craig Huneke and Graham J. Leuschke. Local rings of countable Cohen-Macaulay type. Proc.
Amer. Math. Soc., 131(10):3003–3007 (electronic), 2003.
[13] Ryan Karr and Roger Wiegand. Direct-sum behavior of modules over one-dimensional
rings. In Commutative algebra—Noetherian and non-Noetherian perspectives, pages 251–275.
Springer, New York, 2011.
[14] Horst Knörrer. Cohen-Macaulay modules on hypersurface singularities. I. Invent. Math.,
88(1):153–164, 1987.
[15] Graham J. Leuschke and Roger Wiegand. Local rings of bounded Cohen-Macaulay type.
Algebr. Represent. Theory, 8(2):225–238, 2005.
[16] Graham J. Leuschke and Roger Wiegand. Cohen-Macaulay representations, volume 181 of
Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI,
2012.
[17] Rosa M. Miró-Roig. The representation type of rational normal scrolls. Rend. Circ. Mat.
Palermo (2), 62(1):153–164, 2013.

NON-GORENSTEIN, GRADED COUNTABLE COHEN-MACAULAY TYPE

15

[18] Greg Stevenson. Subcategories of singularity categories via tensor actions. arXiv.org,
, May 2012. arXiv:1105.4698.
[19] Branden Stone. Super-stretched and graded countable Cohen-Macaulay type. 2012. Thesis
(Ph.D.)–University of Kansas.
[20] Branden Stone. Super-stretched and graded countable Cohen-Macaulay type. arXiv.org,
, Jan 2013. arXiv:1301.3593.
[21] Yuji Yoshino. Cohen-Macaulay modules over Cohen-Macaulay rings, volume 146 of London
Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1990.
Branden Stone, Mathematics Program, Bard College/Bard Prison Initiative, P.O.
Box 5000, Annandale-on-Hudson, NY 12504
E-mail address: bstone@bard.edu

